Reaction Engineering
Edited by
Th. Meyer, J. Keurentjes Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

Reaction Engineering
Edited by
Th. Meyer, J. Keurentjes Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

E. S. Wilks (Ed.)Industrial Polymers Handbook Products, Processes, Applications ISBN 3-527-30260-3 H.-G. Elias (Ed.)Macromolecules
Vols. 1–4
ISBN 3-527-31172-6, 3-527-31173-4, 3-527-31174-2,3-527-31175-0 M. F. Kemmere, Th. Meyer (Eds.)Supercritical Carbon Dioxide in Polymer Reaction Engineering ISBN 3-527-31092-4 M. Xanthos (Ed.)Functional Fillers for Plastics ISBN 3-527-31054-1 R. C. Advincula, W. J. Brittain, K. C. Caster, J. Ru̇he (Eds.)Polymer Brushes ISBN 3-527-31033-9 H.-G. Elias (Ed.)An Introduction to Plastics Second, Completely Revised Edition ISBN 3-527-29602-6

Handbook of Polymer Reaction Engineering Thierry Meyer, Jos Keurentjes

Editors 9 All books published by Wiley-VCH are carefully produced. Nevertheless, authors,Dr. Thierry Meyer editors, and publisher do not warrant the Swiss Federal Institute of Technology information contained in these books,Institute of Process Science including this book, to be free of errors.EPFL, ISP-GPM Readers are advised to keep in mind that 1015 Lausanne statements, data, illustrations, procedural Switzerland details or other items may inadvertently be Prof. Jos T. F. Keurentjes Process Development Group Library of Congress Card No.: Applied for Eindhoven University of Technology P.O. Box 513 British Library Cataloging-in-Publication 5600 MB Eindhoven Data: A catalogue record for this book is The Netherlands available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publica-tion in the Deutsche Nationalbibliografie;detailed bibliographic data is available in the Internet at hhttp://dnb.ddb.dei.8 2005 WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form –nor transmitted or translated into machine language without written permission from the publishers. Registered names,trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.Printed in the Federal Republic of Germany Printed on acid-free paper Composition Asco Typesetters, Hong Kong Printing betz-druck gmbh, Darmstadt Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN-13 978-3-527-31014-2 ISBN-10 3-527-31014-2

V
A principal difference between science and engineering is intent. Science is used to bring understanding and order to a specific object of study – to build a body of knowledge with truth and observable laws. Engineering is more applied and prac-tical, focused on using and exploiting scientific understanding and scientific prin-ciples to make products to benefit mankind. A polymer reaction engineer seeks the applied and practical as the title implies, but the path to success is most often through polymer science. This truth is steeped in history – there are many exam-ples of polymeric products commercialized without adequate understanding of the chemistry and physics of the underlying polymerization. Polymer reaction engi-neers, faced with detriments in process safety, product quality or product cost, be-come the driving force behind many polymer science developments. As such, poly-mer reaction engineering is more a collaboration of polymer science and reaction engineering. A collaboration where polymer reaction engineers develop a firm un-derstanding of the many aspects of polymer chemistry and physics to successfully apply chemical engineering principles to new product developments. Only through the integration of science and engineering are such products realized.This handbook is a testimony to this melding of polymer science and chemical engineering that defines polymer reaction engineering. Thierry Meyer and Jos Keurentjes have compiled a strong list of contributors with an even balance from academia and industry. The text offers a comprehensive view of polymer reaction engineering. The text starts with an overview describing the important integration of science and engineering in polymer reaction engineering and ends with recent and potential breakthrough developments in polymer processing. The middle chapters are divided into three sections. The first section is devoted to the science and chemistry of the major types of polymerization. Included are step and chain growth polymerizations with chapters devoted specifically to several different chain growth methods. The central section of the middle chapters is dedicated to poly-mer reaction engineering tools and methods. The very important topics of safety and process control are detailed and help frame the conditions through which suc-cessful scale-ups are achieved. The last section of the middle chapters is focused on the physics and physical nature of formed polymers including their physical and mechanical structure. In these chapters, many of the processes that modify poly-

mers through man-made and natural change are discussed. The details of polymer end use are also presented.This tome represents the first published handbook on polymer reaction engi-neering and should be well received in academia and industry. Polymer reaction engineering became recognized as a separate engineering discipline in the 1970’s.It is well past due that such a handbook be published. The broad scope and depth of coverage should make it an important reference for years to come.Michael C. Grady, ScD Senior Engineering Associate
DuPont
Philadelphia, Pennsylvania

VII
Volume 1
Foreword V
Preface XXIX List of Contributors XXXI 1 Polymer Reaction Engineering, an Integrated Approach 1 Th. Meyer and J. T. F. Keurentjes
1.1 Polymer Materials 1
1.2 A Short History of Polymer Reaction Engineering 4
1.3 The Position of Polymer Reaction Engineering 5
1.4 Toward Integrated Polymer Reaction Engineering 7
1.5 The Disciplines in Polymer Reaction Engineering 9
1.5.1 Polymerization Mechanisms 11
1.5.2 Fundamental and Engineering Sciences 12
1.6 The Future: Product-inspired Polymer Reaction Engineering 14
1.7 Concluding Remarks 15
References 152 Polymer Thermodynamics 17 Theodoor W. de Loos
2.1 Introduction 17
2.2 Thermodynamics and Phase Behavior of Polymer Solutions 18
2.2.1 Thermodynamic Principles of Phase Equilibria 18
2.2.2 Fugacity and Activity 18
2.2.3 Equilibrium Conditions 20
2.2.4 Low-pressure Vapor–Liquid Equilibria 21
2.2.5 Flory–Huggins Theory and Liquid–Liquid Equilibria 21
2.2.6 High-pressure Liquid–Liquid and Vapor–Liquid Equilibria 25
2.3 Activity Coefficient Models 29
2.3.1 Flory–Huggins Theory 30

2.3.2 Hansen Solubility Parameters 32
2.3.3 Correlation of Solvent Activities 34
2.3.4 Group Contribution Models 35
2.4 Equation of State Models 39
2.4.1 The Sanchez–Lacombe Lattice Fluid Theory 40
2.4.2 Statistical Associating-fluid Theory 44
2.4.2.1 SAFT and PC-SAFT Hard Chain Term 44
2.4.2.2 SAFT Dispersion Term 45
2.4.2.3 The PC-SAFT Dispersion Term 46
2.4.2.4 SAFT and PC-SAFT Applications 47
2.4.2.5 Extension to Copolymers 48
2.5 Conclusions 50
Notation 52 References 543 Polycondensation 57 Mário Rui P. F. N. Costa and Rolf Bachmann
3.1 Basic Concepts 57
3.1.1 A Brief History 57
3.1.2 Molecular Weight Growth and Carothers’ Equation 59
3.1.3 The Principle of Equal Reactivity and the Prediction of the Evolution of
Functional Group Concentrations 62
3.1.4 Effect of Reaction Media on Equilibrium and Rate Parameters 62
3.1.5 Polycondensation Reactions with Substitution Effects 64
3.1.6 Exchange Reactions 66
3.1.7 Ring-forming Reactions 67
3.1.8 Modeling of Polymerization Schemes 68
3.2 Mass Transfer Issues in Polycondensations 69
3.2.1 Removal of Volatile By-products 69
3.2.2 Solid-state Polycondensation 80
3.2.3 Interfacial Polycondensation 82
3.3 Polycondensation Processes in Detail 85
3.3.1 Polyesters 85
3.3.1.1 Structure and Production Processes 85
3.3.1.2 Acid-catalyzed Esterification and Alcoholysis 86
3.3.1.3 Catalysis by Metallic Compounds 87
3.3.1.4 Side Reactions in Aromatic Polyester Production 89
3.3.1.5 Side Reactions in the Formation of Unsaturated Polyesters 90
3.3.1.6 Modeling of Processes of Aromatic Polyester Production 91
3.3.1.7 Modeling of Processes for Unsaturated Polyester Production 92
3.3.2 Polycarbonates 93
3.3.2.1 General Introduction 93
3.3.2.2 Interfacial Polycondensation 94
3.3.2.3 Melt Transesterification 96
3.3.3 Polyamides 98

3.3.3.1 Introduction 98
3.3.3.2 Kinetic Modeling 98
3.3.3.3 Nonoxidative Thermal Degradation Reactions 100
3.3.3.4 Process Modeling 101
3.3.4 Polymerizations with Formaldehyde: Amino Resins (Urea and
Melamine) and Phenolics 102
3.3.4.1 Formaldehyde Solutions in Water 102
3.3.4.2 Amino Resins 102
3.3.4.3 Phenolic Resins 107
3.3.5 Epoxy Resins 108
3.3.6 Polyurethanes and Polyureas 109
3.4 Modeling of Complex Polycondensation Reactions 113
3.4.1 Overview 113
3.4.2 Description of Reactions in Polycondensations of Several Monomers
with Substitution Effects 113
3.4.3 Equilibrium Polycondensations with Several Monomers 117
3.4.4 Kinetic Modeling of Irreversible Polycondensations 129
3.4.5 Kinetic Modeling of Linear Reversible Polycondensations 133
Notation 136 References 1444 Free-radical Polymerization: Homogeneous 153 Robin A. Hutchinson
4.1 FRP Properties and Applications 153
4.2 Chain Initiation 154
4.3 Polymerization Mechanisms and Kinetics 156
4.3.1 Homopolymerization 157
4.3.1.1 Basic Mechanisms 157
4.3.1.2 Kinetic Coefficients 161
4.3.1.3 Additional Mechanisms 169
4.3.2 Copolymerization 179
4.3.2.1 Basic Mechanisms 179
4.3.2.2 Kinetic Coefficients 183
4.3.2.3 Additional Mechanisms 187
4.3.3 Diffusion-controlled Reactions 190
4.4 Polymer Reaction Engineering 193
4.4.1 Types of Industrial Reactors 195
4.4.1.1 Batch Processes 195
4.4.1.2 Semi-batch Processes 196
4.4.1.3 Continuous Processes 196
4.4.2 Mathematical Modeling of FRP Kinetics 197
4.4.2.1 Method of Moments 198
4.4.2.2 Modeling of Distributions 201
4.4.3 Reactor Modeling 203
4.4.3.1 Batch Polymerization 204

4.4.3.2 Continuous Polymerization 204
4.4.3.3 Complex Flowsheets 205
4.4.3.4 Computational Fluid Dynamics (CFD) 205
4.4.3.5 Model-based Control 206
4.5 Summary 206
Notation 206 References 2095 Free-radical Polymerization: Suspension 213 B. W. Brooks
5.1 Key Features of Suspension Polymerization 213
5.1.1 Basic Ideas 213
5.1.2 Essential Process Components 214
5.1.3 Polymerization Kinetics 214
5.1.4 Fluid–Fluid Dispersions and Reactor Type 215
5.1.5 Uses of Products from Suspension Polymerization 216
5.2 Stability and Size Control of Drops 216
5.2.1 Stabilizer Types 217
5.2.2 Drop Breakage Mechanisms 218
5.2.3 Drop Coalescence 222
5.2.4 Drop Size Distributions 223
5.2.5 Drop Mixing 224
5.3 Events at High Monomer Conversion 228
5.3.1 Breakage of Highly Viscous Drops 229
5.3.2 Polymerization Kinetics in Viscous Drops 229
5.3.3 Consequences of Polymer Precipitation Inside Drops 230
5.4 Reaction Engineering for Suspension Polymerization 234
5.4.1 Dispersion Maintenance and Reactor Choice 234
5.4.2 Agitation and Heat Transfer in Suspensions 235
5.4.3 Scaleup Limitations with Suspension Polymerization 237
5.4.4 Reactor Safety with Suspension Processes 238
5.4.5 Component Addition during Polymerization 238
5.5 ‘‘Inverse’’ Suspension Polymerization 239
5.5.1 Initiator Types 239
5.5.2 Drop Mixing with Redox Initiators 240
5.6 Future Developments 240
5.6.1 Developing Startup Procedures for Batch and Semi-batch Reactors 240
5.6.2 Maintaining Turbulence Uniformity in Batch Reactors 242
5.6.3 Developing Viable Continuous-flow Processes 242
5.6.4 Quantitative Allowance for the Effects of Changes in the Properties of
the Continuous Phase 242
5.6.5 Further Study of the Role of Secondary Suspending Agents 243
5.6.6 Further Characterization of Stabilizers from Inorganic Powders 243
Notation 243 References 244

6 Emulsion Polymerization 249 José C. de la Cal, José R. Leiza, Jose M. Asua, Alessandro Buttè, Guiseppe Storti,and Massimo Morbidelli
6.1 Introduction 249
6.2 Features of Emulsion Polymerization 250
6.2.1 Description of the Process 250
6.2.2 Radical Compartmentalization 254
6.2.3 Advantages of Emulsion Polymerization 254
6.3 Alternative Polymerization Techniques 256
6.4 Kinetics of Emulsion Polymerization 258
6.4.1 Monomer Partitioning 259
6.4.2 Average Number of Radicals per Particle 260
6.4.3 Number of Polymer Particles 264
6.4.3.1 Heterogeneous Nucleation 264
6.4.3.2 Homogeneous Nucleation 266
6.4.3.3 Simultaneous Heterogeneous and Homogeneous Nucleation 267
6.4.3.4 Coagulative Nucleation 267
6.5 Molecular Weights 267
6.5.1 Linear Polymers 268
6.5.1.1 Zero–One System (Smith–Ewart Cases 1 and 2) 268
6.5.1.2 Pseudo Bulk System (Smith–Ewart Case 3) 270
6.5.2 Nonlinear Polymers: Branching, Crosslinking, and Gel Formation
6.6 Particle Morphology 273
6.7 Living Polymerization in Emulsion 275
6.7.1 Chemistry of LRP 275
6.7.1.1 Nitroxide-mediated Polymerization (NMP) 277
6.7.1.2 Atom-transfer Radical Polymerization (ATRP) 277
6.7.1.3 Degenerative Transfer (DT) 278
6.7.1.4 Reversible Addition–Fragmentation Transfer (RAFT) Polymerization
6.7.2 Polymerization of LRP in Homogeneous Systems 280
6.7.3 Kinetics of LRP in Heterogeneous Systems 282
6.7.4 Application of LRP in Heterogeneous Systems 284
6.7.4.1 Ab-initio Emulsion Polymerization 284
6.7.4.2 Miniemulsion Polymerization 285
6.8 Emulsion Polymerization Reactors 286
6.8.1 Reactor Types and Processes 286
6.8.1.1 Stirred-tank Reactors 286
6.8.1.2 Tubular Reactors 287
6.8.2 Reactor Equipment 288
6.8.2.1 Mixing 289
6.8.2.2 Heat Transfer 290
6.9 Reaction Engineering 290
6.9.1 Mass Balances 291

6.9.2 Heat Balance 292
6.9.3 Polymer Particle Population Balance (Particle Size Distribution)
6.9.4 Scaleup 295
6.10 On-line Monitoring in Emulsion Polymerization Reactors 296
6.10.1 On-line Sensor Selection 297
6.10.1.1 Latex Gas Chromatography 298
6.10.1.2 Head-space Gas Chromatography 298
6.10.1.3 Densimetry 298
6.10.1.4 Ultrasound 299
6.10.1.5 Spectroscopic Techniques 299
6.10.1.6 Reaction Calorimetry 302
6.11 Control of Emulsion Polymerization Reactors 305
Notation 312 References 3177 Ionic Polymerization 323 Klaus-Dieter Hungenberg
7.1 Introduction 323
7.2 Anionic Polymerization 325
7.2.1 Anionic Polymerization of Hydrocarbon Monomers – Living
Polymerization 326
7.2.1.1 Association Behavior/Kinetics 326
7.2.1.2 Molecular Weight Distribution of Living Polymers 331
7.2.1.3 Side Reactions 336
7.2.1.4 Copolymerization 338
7.2.1.5 Tailor-made Polymers by Living Polymerization – Optimization 341
7.2.1.6 Industrial Aspects – Production of Living Polymers 343
7.2.2 Anionic Polymerization of Vinyl Monomers Containing Heteroatoms
7.2.3 Anionic Polymerization of Monomers Containing Hetero Double Bonds
7.2.4 Anionic Polymerization via Ring Opening 346
7.3 Cationic Polymerization 351
7.3.1 Cationic Polymerization of Vinyl Monomers 351
7.3.2 Cationic Ring-opening Polymerization 353
7.4 Conclusion 356
Notation 357 References 3598 Coordination Polymerization 365 João B. P. Soares and Leonardo C. Simon
8.1 Polyolefin Properties and Applications 365
8.1.1 Introduction 365
8.1.2 Types of Polyolefins and Their Properties 366

Contents XIII
8.1.3 The Importance of Proper Microstructural Determination and Control
in Polyolefins 369
8.2 Catalysts for Olefin Polymerization 372
8.2.1 Ziegler–Natta, Phillips, and Vanadium Catalysts 378
8.2.2 Metallocene Catalysts 379
8.2.3 Late Transition Metal Catalysts 381
8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts
8.3.1 Comparison between Coordination and Free-radical Polymerization
8.3.2 Polymerization Kinetics with Single-site Catalysts 383
8.3.2.1 Homopolymerization 383
8.3.2.2 Copolymerization 388
8.3.3 Polymerization Kinetics with Multiple-site Catalysts 392
8.3.4 Long-chain Branch Formation 395
8.4 Single Particle Models – Mass- and Heat-transfer Resistances 399
8.5 Macroscopic Reactor Modeling – Population Balances and the Method
of Moments 408
8.5.1 Homopolymerization 408
8.5.2 Copolymerization 413
8.6 Types of Industrial Reactors 416
8.6.1 Gas-phase Reactors 420
8.6.2 Slurry Reactors 422
8.6.3 Solution Reactors 423
8.6.4 Multizone Reactors 425
Notation 425 References 4289 Mathematical Methods 431 P. D. Iedema and N. H. Kolhapure
9.1 Introduction 431
9.2 Discrete Galerkin h–p Finite Element Method 432
9.3 Method of Moments 435
9.3.1 Introduction 435
9.3.2 Linear Polymerization 435
9.3.3 Nonlinear Polymerization 438
9.4 Comparison of Galerkin-FEM and Method of Moments 440
9.5 Classes Approach 444
9.5.1 Introduction 444
9.5.2 Computing the CLD of Poly(vinyl acetate) for a Maximum of One TDB
per Chain 444
9.5.3 Multiradicals in Radical Polymerization 446
9.6 Pseudo-distribution Approach 449
9.6.1 Introduction 449
9.6.2 CLD/DBD for Mixed-metallocene Polymerization of Ethylene 451

9.6.2.1 Formulation of Pseudo-distribution Problem 451
9.6.2.2 Construction of the Full 2D Distribution 456
9.6.3 CLD/Number of Terminal Double Bonds (TDB) Distribution for
Poly(vinyl acetate) – More than one TDB per Chain 458
9.6.3.1 General Case 458
9.6.3.2 TDB Pseudo-distribution Approach for a Maximum of one TDB per
Chain 466
9.6.3.3 TDB Pseudo-distribution Approach for More than one TDB per Chain
9.6.4 Radical Polymerization of Ethylene to Low-density Polyethylene (LDPE)
9.6.4.1 Introduction 469
9.6.5 Radical Copolymerization 473
9.6.5.1 Introduction 473
9.6.5.2 Balance Equations 474
9.7 Probability Generating Functions 480
9.7.1 Introduction 480
9.7.2 Probability Generating Functions in a Transformation Method 480
9.7.3 Probability Generating Functions and Cascade Theory 481
9.8 Monte Carlo Simulations 485
9.8.1 Introduction 485
9.8.2 Weight-fraction Sampling of Primary Polymers: Batch Reactor,
Transfer to Polymer 486
9.8.3 Example 490
9.8.4 CSTR with Transfer to Polymer 491
9.8.5 Comparison of Galerkin-FEM Classes Model and CSTR with Transfer
to Polymer 492
9.8.6 Batch Reactor, Terminal Double Bond Incorporation 493
9.8.7 CSTR, Terminal Double Bond Incorporation 497
9.8.8 Incorporation of Recombination Termination 498
9.8.9 Incorporation of Random Scission, Linear Chains, Batch Reactor 498
9.8.10 Combined Scission/Branching 501
9.8.11 Scission in a CSTR 501
9.9 Prediction of Branched Architectures by Conditional Monte Carlo
Sampling 502
9.9.1 Introduction 502
9.9.2 Branched Architectures from Radical Polymerization in a CSTR 503
9.9.3 Branched Architectures from Polymerization of Olefins with Single
and Mixed Branch-forming Metallocene Catalysts in a CSTR 505
9.9.3.1 Introduction 505
9.9.3.2 Single-catalyst System 505
9.9.3.3 Synthesis of Topology 505
9.9.3.4 Mixed-catalyst System 508
9.9.4 Mathematical Methods for Characterization of Branched Architectures
9.9.4.1 Graph Theoretical Connectivity Matrices 510

9.9.4.2 Characterization of Architectures by Radius of Gyration 511
9.9.4.3 Characterization of Architectures by Seniorities and Priorities 512
9.10 Computational Fluid Dynamics for Polymerization Reactors 517
9.10.1 Introduction 517
9.10.1.1 Modeling Challenges 517
9.10.2 Development and Optimization of Modern Polymerization Reactors
9.10.2.1 Benefits of CFD 519
9.10.2.2 Limitations of CFD 519
9.10.3 Integration of CFD with Polymerization Kinetics 520
9.10.3.1 Classification and Complexity of CFD Models 521
9.10.3.2 Treatment of Polymerization Kinetics 522
9.10.3.3 Illustration of Homogeneous Reactor Model Formulation 522
9.10.4 Target Applications 523
9.10.4.1 Illustrative Case Studies 523
9.10.5 Concluding Remarks 528
Acknowledgments 530 References 53010 Scaleup of Polymerization Processes 533 E. Bruce Nauman
10.1 Historic and Economic Perspective 533
10.2 The Limits of Scale 533
10.3 Scaleup Goals 534
10.4 General Approaches 535
10.5 Scaleup Factors 537
10.6 Stirred-tank Reactors 537
10.7 Design Considerations for Stirred Tanks 541
10.8 Multiphase Stirred Tanks 542
10.9 Stirred Tanks in Series 542
10.10 Tubular Reactors 543
10.11 Static Mixers 545
10.12 Design Considerations for Tubular Reactors 546
10.13 Extruder and Extruder-like Reactors 549
10.14 Casting Systems 549
10.15 Concluding Remarks 550
Notation 550 References 551
Volume 2
11 Safety of Polymerization Processes 553 Francis Stoessel
11.1 Introduction 553
11.2 Principles of Chemical Reactor Safety Applied to Polymerization 554
11.2.1 Cooling Failure Scenario 554

11.2.2 Criticality Classes Applied to Polymerization Reactors 557
11.2.2.1 Description of the Criticality Classes 558
11.2.3 Heat Balance of Reactors 559
11.2.3.1 Heat Production 559
11.2.3.2 Heat Exchange 560
11.2.3.3 Heat Accumulation 561
11.2.3.4 Convective Heat Transport due to Feed 561
11.2.3.5 Stirrer 561
11.2.3.6 Heat Losses 562
11.2.3.7 Simplified Expression of the Heat Balance 562
11.2.4 Dynamic Control of Reactors 562
11.2.5 Thermal Stability of Polymerization Reaction Masses 563
11.3 Specific Safety Aspects of Polymerization Reactions 564
11.3.1 Kinetic Aspects 564
11.3.2 Thermochemical Aspects 565
11.3.3 Factors Leading to Changing Heat Release Rates 568
11.4 Cooling of Polymerization Reactors 570
11.4.1 Indirect Cooling: Heat Exchange Across the Reactor Wall 570
11.4.2 Hot Cooling: Cooling by Evaporation 574
11.4.3 Importance of the Viscosity 578
11.5 Chemical Engineering for the Safety of Polymerization Processes
11.5.1 Batch Processes 579
11.5.2 Semi-batch Processes 580
11.5.2.1 Initiation 581
11.5.2.2 Feed 582
11.5.2.3 Final Stage 583
11.5.2.4 Practical Aspects 583
11.5.3 Continuous Processes 584
11.5.3.1 Concentration Stability 584
11.5.3.2 Particle Number Stability 584
11.5.4 Design Measures for Safety 585
11.5.4.1 Process Design 586
11.5.4.2 Reactor Design 586
11.5.4.3 Control of Feed 587
11.5.4.4 Emergency Cooling 587
11.5.4.5 Inhibition 588
11.5.4.6 Quenching 588
11.5.4.7 Dumping 588
11.5.4.8 Controlled Depressurization 588
11.5.4.9 Pressure Relief 588
11.5.4.10 Time Factor 589
11.6 Conclusion 589
References 590 Notation 591

Contents XVII 12 Measurement and Control of Polymerization Reactors 595 John R. Richards and John P. Congalidis
12.1 Introduction 595
12.1.1 Definitions 595
12.1.2 Measurement Error 597
12.2 Measurement Techniques 598
12.2.1 Temperature 599
12.2.1.1 Resistance Thermometers 599
12.2.1.2 Thermocouples 600
12.2.1.3 Expansion Thermometers 601
12.2.1.4 Radiation Pyrometers 601
12.2.2 Pressure Measurement 602
12.2.3 Weight 604
12.2.4 Liquid Level 605
12.2.5 Flow 608
12.2.6 Densitometry, Dilatometery, and Gravimetry 617
12.2.7 Viscosity 619
12.2.8 Composition 620
12.2.9 Surface Tension 622
12.2.10 Molecular Weight Distribution (MWD) 622
12.2.11 Particle Size Distribution (PSD) 623
12.3 Sensor Signal Processing 625
12.3.1 Sensors and Transmitters 625
12.3.2 Converters 626
12.3.3 Indicators 626
12.3.4 Filtering Techniques 627
12.4 Regulatory Control Engineering 627
12.4.1 General 627
12.4.2 Process Dynamics 630
12.4.2.1 First-order System 631
12.4.2.2 Second-order System 632
12.4.2.3 High-order and Dead Time Systems 636
12.4.2.4 First-order Plus Dead Time System 636
12.4.2.5 Integrating System 638
12.4.2.6 Integrator plus Dead Time System 639
12.4.3 Controllers 639
12.4.3.1 Proportional Control 640
12.4.3.2 Integral Control 641
12.4.3.3 Derivative Control 641
12.4.3.4 PI, PD, and PID Control 642
12.4.3.5 Digital Controllers 642
12.4.3.6 Controller Tuning 644
12.4.3.7 On–Off Controllers 646
12.4.3.8 Self-operated Regulators 647
12.4.4 Valve Position Controllers 650

12.4.5 Single-loop Controllers 650
12.4.6 Digital Control Systems 650
12.4.7 Actuators 652
12.5 Advanced Control Engineering 656
12.5.1 Feedforward Control 659
12.5.1.1 Steady-state Model Feedforward Control 660
12.5.1.2 Ratio Control 660
12.5.2 Cascade Control 661
12.5.3 Feedforward–Feedback Control 663
12.5.4 State Estimation Techniques 666
12.5.5 Model Predictive Control 668
12.5.6 Batch and Semi-batch Control 669
12.5.6.1 Operation and Variability 669
12.5.6.2 Statistical Process Control 671
12.5.7 Future Trends 671
Notation 672 References 67513 Polymer Properties through Structure 679 Uday Shankar Agarwal
13.1 Thermal Properties of Polymers 679
13.1.1 Crystalline and Amorphous Polymers 680
13.1.2 Influence of Polymer Structure on Crystallizability of Polymers 682
13.1.3 The Glass Transition Temperature 683
13.1.4 Influence of Polymer Structure on Tg of Polymers 684
13.1.5 The Crystallization Temperature and the Melting Point 686
13.1.6 Tuning Polymer Crystallization for Properties 686
13.1.7 Morphology of Crystalline Polymers 688
13.1.8 Tailoring Polymer Properties through Modification, Additives, and
Reinforcement 690
13.1.8.1 New Morphologies through Block Copolymers 691
13.1.8.2 Polymeric Nanocomposites 692
13.2 Polymer Conformation and Related Properties 692
13.2.1 The Chain Conformation 692
13.2.2 Solubility of Polymers 694
13.2.3 Dilute Solution Zero-shear Viscosity 695
13.2.3.1 Polymers as Dumbbells 696
13.2.3.2 Polymers as Chains of Beads and Springs 697
13.2.4 Viscosity of Concentrated Solutions and Melts 698
13.2.5 Nonlinear Polymers 699
13.2.6 Rigid Rod-like Polymers 701
13.3 Polymer Rheology 702
13.3.1 The Viscous Response: Shear Thinning 702
13.3.2 Normal Stresses during Shear Flow 703
13.3.3 Extensional Thickening 705

13.3.4 The Elastic Response 706
13.3.4.1 Ideal Elastic Response 706
13.3.4.2 Rubberlike Elasticity 706
13.3.5 The Viscoelastic Response 707
13.3.5.1 Linear Viscoelasticity in Dynamic Oscillatory Flow 709
13.3.6 Influence of Polymer Branching Architecture in Bulk Polymers 711
13.3.7 Polymers as Rheology Modifiers 712
13.3.8 Rheological Control with Block Copolymers 714
13.3.9 Polymer-like Structures through Noncovalent Associations 715
13.4 Summary 715
Notation 716 References 71814 Polymer Mechanical Properties 721 Christopher J. G. Plummer
14.1 Introduction 721
14.1.1 Long-chain Molecules 721
14.1.2 Simple Statistical Descriptions of Long-chain Molecules 722
14.2 Elasticity 724
14.2.1 Deformation of an Elastic Solid 724
14.2.2 Thermodynamics of Rubber Elasticity 725
14.2.3 Statistical Mechanical Approach to Rubber Elasticity 727
14.3 Viscoelasticity 729
14.3.1 Linear Viscoelasticity 729
14.3.2 Time–Temperature Superposition 734
14.3.3 Molecular Models for Polymer Dynamics 736
14.3.4 Nonlinear Viscoelasticity 740
14.4 Yield and Fracture 741
14.4.1 Yield in Polymers 741
14.4.2 Models for Yield 744
14.4.3 Semicrystalline Polymers 746
14.4.4 Crazing and Fracture 748
14.5 Conclusion 752
References 75515 Polymer Degradation and Stabilization 757 Tuan Quoc Nguyen
15.1 Introduction 757
15.2 General Features of Polymer Degradation 759
15.2.1 Degradative Reactions 759
15.2.1.1 Initiation 760
15.2.1.2 Propagation 760
15.2.1.3 Chain Branching 761
15.2.1.4 Termination 762
15.2.2 Some Nonradical Degradation Mechanisms 763

15.2.3 Physical Factors 763
15.2.3.1 Glass Transition Temperature 764
15.2.3.2 Polymer Morphology 766
15.3 Degradation Detection Methods 767
15.3.1 Mechanical Tests 768
15.3.2 Gel Permeation Chromatography 771
15.3.3 Fourier Transform Infrared Spectroscopy 773
15.3.4 Magnetic Resonance Spectroscopy 775
15.3.4.1 Nuclear Magnetic Resonance (NMR) 775
15.3.4.2 Electron Spin Resonance (ESR) 776
15.3.5 Oxygen Uptake 776
15.3.6 Chemiluminescence 778
15.4 Thermal Degradation 778
15.4.1 Thermal Stability 779
15.4.2 Polymer Structure and Thermal Stability 779
15.4.3 Computer Simulation 780
15.4.4 Thermal Oxidative Degradation of Polypropylene 782
15.4.4.1 Initiation 782
15.4.4.2 Propagation 784
15.4.4.3 Chain Branching 785
15.4.4.4 Termination 786
15.4.4.5 Secondary Reactions 786
15.4.4.6 Formation of Volatile Compounds 788
15.4.5 Homogeneous versus Heterogeneous Kinetics 789
15.4.6 Applications of Thermal Degradation 790
15.4.6.1 Analytical Pyrolysis 790
15.4.6.2 Introduction of New Chemical Functionalities 791
15.4.6.3 Chemical Modification of Polymer Structure 791
15.4.6.4 Metal Injection Molding (MIM) 792
15.4.6.5 Recycling 792
15.5 Photodegradation 793
15.5.1 Absorption of UV Radiation by Polymers 793
15.5.2 The Solar Spectrum 796
15.5.3 Photo-oxidation Profile 796
15.5.4 Influence of Wavelength: the Activation and Action Spectrum 799
15.5.5 Photodegradation Mechanisms 802
15.5.5.1 Photoinitiation 802
15.5.5.2 The Norrish Photoprocesses 803
15.5.5.3 Photo-Fries Rearrangement 803
15.6 Radiolytic Degradation 805
15.6.1 Interaction of High-energy Radiation with Matter 805
15.6.2 Radiation Chemistry 807
15.6.3 Radiolysis Stabilization 810
15.6.4 Applications 811
15.6.4.1 Radiation Sterilization 812

15.6.4.2 Controlled Degradation and Crosslinking 812
15.7 Mechanochemical Degradation 813
15.7.1 Initiation by Mechanical Stresses 813
15.7.1.1 Effect of Tensile Stress on Chemical Reactivity 813
15.7.1.2 Breaking Strength of a Covalent Bond 814
15.7.1.3 Rate of Stress-activated Chain Scission 815
15.7.2 Extrusion Degradation 816
15.7.3 Applications 817
15.8 Control and Prevention of Aging of Plastic Materials 818
15.8.1 Antioxidants 818
15.8.1.1 Radical Antioxidants 818
15.8.1.2 Hindered Amine Stabilizers (HAS) 819
15.8.1.3 Peroxide Decomposers 821
15.8.2 Photostabilizers 822
15.8.3 PVC Heat Stabilizers 823
15.8.4 Other Classes of Stabilizers 824
15.8.4.1 Metal Deactivators 824
15.8.4.2 Antiozonants 824
15.9 Lifetime Prediction 824
15.10 Conclusions 826
Notation 827 References 83016 Thermosets 833 Rolf A. T. M. van Benthem, Lars J. Evers, Jo Mattheij, Ad Hofland, Leendert J.Molhoek, Ad J. de Koning, Johan F. G. A. Jansen, and Martin van Duin
16.1 Introduction 833
16.1.1 Thermoset Materials 833
16.1.2 Networks 834
16.1.3 Advantages 835
16.1.4 Curing Resins 835
16.1.5 Functionality 835
16.1.6 Formulation 836
16.1.7 Production 837
16.1.8 General Areas of Application 837
16.2 Phenolic Resins 838
16.2.1 Introduction 838
16.2.2 Chemistry 838
16.2.2.1 Resols 840
16.2.2.2 Novolacs 840
16.2.2.3 Epoxy-novolacs 841
16.2.2.4 Discoloration 841
16.2.3 Production 842
16.2.4 Properties and Applications 842
16.3 Amino Resins 843

16.3.1 Introduction 843
16.3.2 Chemistry 843
16.3.2.1 Polymerization Chemistry 845
16.3.3 Production 848
16.3.4 Properties and Applications 849
16.4 Epoxy Resins 849
16.4.1 Introduction 849
16.4.2 Chemistry 850
16.4.2.1 Cure 851
16.4.3 Production 853
16.4.3.1 Standard Liquid 853
16.4.3.2 Taffy Process 854
16.4.3.3 Advancement Process 854
16.4.4 Properties and Applications 855
16.5 Alkyd Resins 855
16.5.1 Introduction 855
16.5.2 Chemistry 856
16.5.2.1 The Alkyd Constant 858
16.5.2.2 Autoxidative Drying 858
16.5.3 Production 859
16.5.4 Properties and Applications 861
16.5.4.1 Short Oil Alkyds 861
16.5.4.2 Long Oil Alkyds 861
16.5.4.3 Medium Oil Alkyds 861
16.5.5 Alkyd Emulsions 861
16.5.5.1 The Inversion Process 862
16.6 Saturated Polyester Resins 862
16.6.1 Introduction 862
16.6.2 Chemistry 863
16.6.3 Production 865
16.6.3.1 Monitoring the Reaction 865
16.6.4 Properties and Applications 866
16.6.5 Powder Coatings 866
16.6.5.1 Application 867
16.6.5.2 Crosslinking 868
16.6.5.3 Advantages 869
16.7 Unsaturated Polyester Resins and Composites 869
16.7.1 Introduction 869
16.7.2 Chemistry 869
16.7.2.1 Crosslinking 871
16.7.2.2 Styrene Emission 871
16.7.2.3 Vinyl Ester Resins 873
16.7.3 Production 874
16.7.4 Reinforcement 875
16.7.5 Fillers 878

Contents XXIII
16.7.6 Processing 879
16.7.6.1 Hand Lay-up and Spray-up 882
16.7.6.2 Continuous Lamination 882
16.7.6.3 Filament Winding 882
16.7.6.4 Centrifugal Casting 882
16.7.6.5 Pultrusion 883
16.7.6.6 Cold-press Molding 883
16.7.6.7 Resin Infusion 883
16.7.6.8 Resin-transfer Molding 883
16.7.6.9 Hot-press Molding 883
16.7.6.10 Casting, Encapsulation, and Coating (Non-reinforced Applications)
16.7.7 Design Considerations: Mechanical Properties of Composites 886
16.8 Acrylate Resins and UV Curing 889
16.8.1 Introduction 889
16.8.2 Chemistry 890
16.8.3 Production 891
16.8.3.1 Epoxy Acrylates 891
16.8.3.2 Polyester Acrylates 891
16.8.3.3 Urethane Acrylates 892
16.8.3.4 Inside-out 893
16.8.3.5 Outside-in 894
16.8.3.6 Comparing Inside-out with Outside-in 894
16.8.3.7 Stabilization 894
16.8.3.8 Dilution 895
16.8.3.9 Safety 895
16.8.4 Properties 895
16.8.5 Introduction to UV Curing 896
16.8.5.1 General Introduction to UV-initiated Radical Polymerization 896
16.8.5.2 Photoinitiators for Radical Polymerization 897
16.8.5.3 Resin 897
16.8.5.4 Reactive Diluent 898
16.8.5.5 Curing Process 899
16.8.5.6 Cationic Curing 900
16.8.5.7 Base-mediated Curing 901
16.9 Rubber 901
16.9.1 Introduction 901
16.9.1.1 Types of Rubber 902
16.9.2 Polymerization 903
16.9.3 Crosslinking 904
16.9.3.1 Sulfur Vulcanization 904
16.9.3.2 Peroxide Curing 905
16.9.3.3 Processing 906
16.9.4 Properties and Applications 907
16.9.4.1 Advantages and Disadvantages 907

16.9.4.2 Thermoplastic Vulcanizates 907
Notation 908 References 90917 Fibers 911 J. A. Juijn
17.1 Introduction 911
17.1.1 A Fiber World 911
17.1.2 Scope of this Chapter 912
17.2 Fiber Terminology 912
17.2.1 Definitions: Fibers, Filaments, Spinning 912
17.2.2 Synthetic Yarns 914
17.2.3 Titer: Tex and Denier 914
17.2.4 Tenacity and Modulus: g denierC1, N texC1 , or GPa 915
17.2.5 Yarn Appearance 916
17.2.6 Textile, Carpet, and Industrial Yarns 917
17.2.7 Physical Structure 918
17.3 Fiber Polymers: Choice of Spinning Process 920
17.3.1 Polymer Requirements 920
17.3.2 Selection of Spinning Process 920
17.3.3 Spinnability 922
17.4 Melt Spinning 923
17.4.1 Extrusion 923
17.4.2 Polymer Lines and Spin-box 924
17.4.3 Spinning Pumps 925
17.4.4 Spinning Assembly 926
17.4.4.1 Filtration 926
17.4.4.2 Spinning Plate 926
17.4.5 Quenching 928
17.4.6 Finish 929
17.4.7 Spinning Speed 931
17.4.8 Winding 931
17.4.9 Drawing 931
17.4.10 Relaxation and Stabilization 934
17.4.11 Process Integration 934
17.4.12 Rheology 934
17.4.12.1 Shear Viscosity 934
17.4.12.2 Elasticity 936
17.4.12.3 Elongational Viscosity 936
17.4.13 Process Calculations 936
17.4.13.1 Mass Flow 937
17.4.13.2 Volume Flow 937
17.4.13.3 Extrusion Speed and Elongation in the Spin-line 937
17.4.13.4 Pressure Drop over the Spinning Holes 938

17.4.14 Polyester (Poly(ethylene terephthalate), PET) 938
17.4.14.1 PET Polymer 938
17.4.14.2 Spinning of PET 939
17.4.14.3 PET Staple Fiber 939
17.4.14.4 PET Textile Filament Yarns 940
17.4.14.5 PET Industrial Yarns 940
17.4.15 Polyamide (PA6 and PA66) 941
17.4.15.1 PA Polymer 941
17.4.15.2 PA Spinning 941
17.4.15.3 PA Staple Fiber 942
17.4.15.4 PA Textile Filament Yarns 942
17.4.15.5 PA Industrial Yarns 942
17.4.16 Polypropylene (PP) 943
17.4.16.1 PP Polymer 943
17.4.16.2 PP Spinning 943
17.4.16.3 PP Staple Fiber 943
17.4.16.4 PP Split Fiber 943
17.4.16.5 PP Filament Yarns 944
17.5 Solution Spinning 944
17.5.1 Preparation of Spinning Dope 944
17.5.2 Dry Spinning 944
17.5.2.1 Cellulose Acetate 945
17.5.2.2 Acrylics 946
17.5.2.3 Poly(vinyl alcohol) 946
17.5.3 Wet Spinning 946
17.5.3.1 Viscose Rayon 948
17.5.3.2 Acrylics 951
17.5.3.3 Poly(vinyl alcohol) 952
17.6 Comparison of Melt and Solution Spinning 953
17.7 High-modulus, High-strength Fibers 956
17.7.1 Air-gap Spinning 956
17.7.1.1 Aramids 956
17.7.1.2 Other Liquid-crystalline Polymers 960
17.7.2 Gel Spinning 961
17.7.2.1 Theory 961
17.7.2.2 Gel Spinning of Polyethylene 962
17.7.2.3 Other Gel-spun or Superdrawn Fibers 964
17.7.3 Carbon Fiber 965
17.7.3.1 Carbon Fiber from PAN 965
17.7.3.2 Carbon Fiber from Pitch 966
17.7.3.3 Applications of Carbon Fibers 966
17.7.4 Other Advanced Fibers 966
Notation 967 Acknowledgments 969 References 969

18 Removal of Monomers and VOCs from Polymers 971 Marı́a J. Barandiaran and José M. Asua
18.1 Introduction 971
18.2 Polymer Melts and Solutions 972
18.2.1 Devolatilization 973
18.2.1.1 Fundamentals 973
18.2.1.2 Implementation of Devolatilization 975
18.2.1.3 Equipment 975
18.3 Polyolefins 979
18.4 Waterborne Dispersions 979
18.4.1 Post-polymerization 980
18.4.1.1 Modeling Post-polymerization 981
18.4.2 Devolatilization 981
18.4.2.1 Modeling 982
18.4.2.2 Rate-limiting Steps 985
18.4.2.3 Devolatilization under Equilibrium Conditions 986
18.4.2.4 Equipment 986
18.4.3 Combined Processes 988
18.4.4 Alternative Processes 989
18.5 Summary 989
Notation 990 References 99119 Nano- and Microstructuring of Polymers 995 Christiane de Witz, Carlos Sánchez, Cees Bastiaansen, and Dirk J. Broer
19.1 Introduction 995
19.1.1 Patterning Techniques 996
19.1.2 Photoembossing 998
19.2 Materials and their Photoresponsive Behavior 999
19.3 Single-exposure Photoembossing 1001
19.4 Dual-exposure Photoembossing 1007
19.5 Complex Surface Structures from Interfering UV Laser Beams 1007
19.6 Surface Structure Development under Fluids 1010
19.7 Conclusion 1012
Acknowledgments 1012 Notation 1013 References 101320 Chemical Analysis for Polymer Engineers 1015 Peter Schoenmakers and Petra Aarnoutse
20.1 Introduction 1015
20.2 Process Analysis 1017
20.2.1 Near-infrared Spectroscopy 1017
20.2.2 In-situ Raman Spectroscopy 1018
20.2.3 At-line Conversion Measurements 1020

Contents XXVII
20.3 Polymer Analysis 1022
20.3.1 Basic Laboratory Measurements 1022
20.3.1.1 Conversion 1022
20.3.2 Detailed Molecular Analysis 1023
20.3.2.1 FTIR Spectroscopy 1023
20.3.2.2 NMR Spectroscopy 1024
20.3.2.3 Mass Spectrometry 1025
20.3.3 Polymer Distributions 1030
20.3.3.1 Molecular Weight Distributions 1030
20.3.3.2 Functionality-type Distributions 1034
20.3.3.3 Chemical Composition Distributions (CCDs) 1037
20.3.3.4 Degree of Branching Distributions 1040
20.3.3.5 Complex Polymers (Multiple Distributions) 1041
Notation 1044 References 104521 Recent Developments in Polymer Processes 1047 Maartje Kemmere
21.1 Introduction 1047
21.2 Polymer Processes in Supercritical Carbon Dioxide 1048
21.2.1 Interactions of Carbon Dioxide with Polymers and Monomers 1050
21.2.1.1 Solubility in Carbon Dioxide 1051
21.2.1.2 Sorption and Swelling of Polymers 1052
21.2.1.3 Phase Behavior of Monomer–Polymer–Carbon Dioxide Systems 1054
21.2.2 Polymerization Processes in Supercritical Carbon Dioxide 1055
21.2.3 Polymer Processing in Supercritical Carbon Dioxide 1058
21.2.3.1 Extraction 1060
21.2.3.2 Impregnation and Dyeing 1061
21.3 Ultrasound-induced Radical Polymerization 1062
21.3.1 Ultrasound and Cavitation in Liquids 1063
21.3.2 Radical Formation by Cavitation 1065
21.3.3 Cavitation-induced Polymerization 1067
21.3.3.1 Bulk Polymerization 1067
21.3.3.2 Precipitation Polymerization 1069
21.3.3.3 Emulsion Polymerization 1070
21.3.4 Cavitation-induced Polymer Scission 1072
21.3.5 Synthesis of Block Copolymers 1073
21.4 Concluding Remarks and Outlook for the Future 1074
Acknowledgments 1076 Notation 1076 References 1077
Index 1083

Preface
Freshly started as chairman and secretary of the Working Party on Polymer Reac-tion Engineering it never crossed our mind to edit a book on this subject. This changed when Wiley-VCH asked if the working party would be able to provide a translation of the Handbuch der Technischen Polymerchemie, written in 1993 by Adolf Echte. We decided to do so, but not exactly. Very rapidly we were convinced that we needed a completely new book, covering the field of polymer reaction engi-neering in a modern, broad and multidisciplinary approach. Many of the working party members directly agreed to participate, others needed somewhat stronger persuasion techniques, and for some chapters we hired authors from other in-stitutions. In June 2003 we had completed the list of contributors, coming from Europe, Canada and the USA. Now, roughly one year later, the new handbook is there.The quality an edited book like this very much depends on the quality of the in-dividual contributions. It has been a great pleasure for us to see that all authors have taken their writing jobs very seriously. With these contributions, we are sure that this book represents the state of the art in polymer reaction engineering. It is intended to attract equally readers that are new in the field as well as readers that may be considered expert in some of the topics but want to broaden their knowledge. We are convinced that the multidisciplinary and synergetic approach presented in this book may act as an eye-opener for research and development ac-tivities going on in strongly related areas. We hope the reader will take advantage of this approach, where the references given in the various chapters may be a start-ing point for further reading.Reading books, you often read the preface as well. We have seen numerous ex-amples from which the frustration is quite obvious. Of course things may not al-ways work out the way you plan, that has also been the case for this book. Maybe we were just lucky, but we have greatly enjoyed doing this. Editing this book has also been a starting point for the editors to become friends, including Swiss cheese fondue and Dutch ‘‘Friese nagelkaastaart’’ in a friendly home setting. From that perspective also Francine and Maartje have had their part both of the workload but also of the fun of all this.Finally, we would like to thank Karin Sora and Renate Doetzer from Wiley-VCH

for their help with the editing process. They really know to find the balance be-tween waiting and pushing in order not to diverge too far from the schedule.Lausanne & Eindhoven, fall 2004 Thierry Meyer & Jos Keurentjes

List of Contributors P. Aarnoutse Prof. M. J. Barandiaran Polymer-Analysis Group The University of the Basque Country Department of Chemical Engineering Institute for Polymer Materials(ITS) (POLYMAT)Faculty of Science, University of Manuel Lardizabal, 3 Amsterdam 20018 Donostia-San Sebastián Nieuwe Achtergracht 166 Spain 1018 WV Amsterdam The Netherlands Dr. C. W. M. Bastiaansen Eindhoven University of Technology Dr. U. S. Agarwal Den Dolech 2 Polymer Technology Group 5600 MB Eindhoven Department of Chemical Engineering The Netherlands and Chemistry Eindhoven University of Technology Prof. D. J. Broer P.O. Box 513 Philips Research Laboratories 5600 MB Eindhoven Prof. Holstlaan 4 The Netherlands 5656 AA Eindhoven The Netherlands Prof. J. M. Asua
and
The University of the Basque Country Institute for Polymer Materials Eindhoven University of Technology(POLYMAT) Den Dolech 2 Paseo Manuel Lardizabal 3 5600 MB Eindhoven 20018 Donostia-San Sebastián The Netherlands
Spain
and
Dr. R. Bachmann Dutch Polymer Institute (DPI)Bayer AG P.O. Box 902 ZT-TE-SVT 5600 AX Eindhoven 51368 Leverkusen The Netherlands
Germany

Polymer Technology Prof. J. C. de la Cal Department of Chemical Engineering The University of the Basque Country and Chemistry Institute for Polymer Materials Eindhoven University of Technology (POLYMAT)P.O. Box 513 Paseo Manuel Lardizabal, 35600 MB Eindhoven 20018 Donostia-San Sebastián The Netherlands Spain Prof. B. W. Brooks Dr. A. J. de Koning Loughborough University DSM Research Department of Chemical Oude Postbaan 1 Engineering 6167 RG Geleen Loughborough The Netherlands Leicestershire, LE11 3TU United Kingdom Dr. T. W. de Loos Delft University of Technology Dr. A. Buttè Faculty of Applied Sciences Swiss Federal Institute of Department Chemical Technology Technology Julianalaan 136 Zurich, ETHZ 2628 BL Delft Institut für Chemie- und The Netherlands Bioingenieurwissenschafften Gruppe Morbidelli C. de Witz ETH Hoenggerberg/HCI F135 Philips Research Laboratories 8093 Zurich Prof. Holstlaan 4 Switzerland 5656 AA Eindhoven The Netherlands Dr. J. P. Congalidis E.I. du Pont de Nemours and Dr. L. J. Evers Company DSM Melamine DuPont Central Research and Oude Postbaan 1 Development 6167 RG Geleen Experimental Station The Netherlands Wilmington, DE 19880 USA Dr. A. Hofland DSM Coating Resins Dr. M. R. P. F. N. Costa Ceintuurbaan 5 Faculty of Engineering 8022 AW Zwolle University of Porto The Netherlands Rua Roberto Frias, s/n 4200-465 Porto Dr. K.-D. Hungenberg Portugal BASF AG Polymer Technology, B167056 Ludwigshafen
Germany

Prof. R. A. Hutchinson N. H. Kolhapure Department of Chemical Engineering DuPont Engineering Research and Queen’s University Technology Dupuis Hall, 19 Division St. 1007 N. Market St.Kingston, Ontario K7M 2G9 Wilmington, DE 19898-0001
Canada USA
Dr. P. D. Iedema Prof. J. R. Leiza Department of Chemical Engineering The University of the Basque Country University of Amsterdam Institute for Polymer Materials Nieuwe Achtergracht 166 (POLYMAT)1018 WV Amsterdam Paseo Manuel Lardizabal 3 The Netherlands 20018 Donostia-San Sebastián
Spain
Dr. J. F. G. A. Jansen DSM Research J. Mattheij Oude Postbaan 1 DSM Melamine 6167 RG Geleen Oude Postbaan 1 The Netherlands 6167 RG Geleen The Netherlands Dr. J. A. Juijn Research Institute Dr. T. Meyer Department QRI Swiss Federal Institute of Technology P.O. Box 9600 Institute of Process Science 6800 TC Arnheim EPFL, ISP-GPM The Netherlands 1015 Lausanne Switzerland Dr. M. F. Kemmere Process Development Group L. J. Molhoek Department of Chemical Engineering DSM Coating Resins and Chemistry Ceintuurbaan 5 Eindhoven University of Technology 8022 AW Zwolle P.O. Box 513 The Netherlands 5600 MB Eindhoven The Netherlands Prof. M. Morbidelli Swiss Federal Institute of Technology Prof. J. T. F. Keurentjes Zurich, ETHZ Process Development Group Institut für Chemie- und Eindhoven University of Technology Bioingenieurwissenschaften P.O. Box 513 Gruppe Morbidelli 5600 MB Eindhoven ETH Hoenggerberg/HCI F135 The Netherlands 8093 Zurich Switzerland

Prof. E. B. Nauman Prof. P. J. Schoenmakers The Isermann Department of Chemical Polymer-Analysis Group and Biological Engineering Department of Chemical Engineering Rensselaer Polytechnic Institute (ITS)Troy, NY 12180 Faculty of Science, University of USA Amsterdam Nieuwe Achtergracht 166 Dr. Q. T. Nguyen 1018 WV Amsterdam Laboratory of Polymers (LP) The Netherlands Ecole Polytechnique Fédérale de Lausanne Prof. L. C. Simon 1015 Lausanne Department of Chemical Engineering Switzerland University of Waterloo 200 University Avenue West J. C. Plummer Waterloo, Ontario N2L 3G1 Laboratory of Composite and Polymer Canada Technology (LTC)Ecole Polytechnique Fédérale Prof. J. B. P. Soares de Lausanne Department of Chemical Engineering 1015 Lausanne University of Waterloo Switzerland 200 University Avenue West Waterloo, Ontario N2L 3G1 Dr. J. R. Richards Canada E. I. du pont de Nemours and Company DuPont Engineering and Research Prof. F. Stoessel Technology Swiss Institute for the Promotion of Experimental Station Safety and Security Wilmington, DE 19880 Chemical Process Safety Consulting USA Klybeckstrasse 141
WKL-32.322
Dr. C. Sánchez 4002 Basel Eindhoven University of Technology Switzerland Den Dolech 25600 MB Eindhoven Prof. G. Storti The Netherlands Swiss Federal Institute of Technology Zurich, ETHZ
and
Institut für Chemie- und Dutch Polymer Institute (DPI) Bioingenieurwissenschaften P.O. Box 902 Gruppe Morbidelli 5600 AX Eindhoven ETH Hoenggerberg/HCI F125 The Netherlands 8093 Zurich Switzerland

Prof. R. A. T .M. van Benthem Dr. M. van Duin Coating Technology DSM Research Department of Chemical Engineering Oude Postbaan 1 and Chemistry 6167 RG Geleen Eindhoven University of Technology The Netherlands P.O. Box 5135600 MB Eindhoven The Netherlands

Polymer Reaction Engineering, an Integrated Th. Meyer and J. T. F. Keurentjes
1.1
Polymer Materials Synthetic polymers can be denoted as the materials of the 20th century. Since World War II the production volume of polymers has increased by a factor of 50 to a current value of more than 120 million tonnes annually (Figure 1.1). The con-sumption per capita has also increased over the years to a worldwide average of ap-proximately 20 kg per annum in the year 2000. In terms of volumetric output, the production of polymers exceeds that of iron and steel. The enormous growth of synthetic polymers is due tot the fact that they are lightweight materials, act as in-sulators for electricity and heat, cover a wide range of properties from soft packag-ing materials to fibers stronger than steel, and allow for relatively easy processing.Annual Production 120
106 to/an
Consumption,
kg/hab 80
World Population , 20
109 people
1940 1950 1960 1970 1980 1990 2000 2010
Year
Fig. 1.1. Polymer production and the evolution of the population since 1940 [1].Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

2 1 Polymer Reaction Engineering, an Integrated Approach Tab. 1.1. Applications and 2002 Western European markets for the major thermoplastics [1].Thermoplastic Market Applications LDPE 7935 pallet and agricultural film, bags, toys, coatings,containers, pipes PP 7803 film, battery cases, microwave-proof containers, crates,automotive parts, electrical components PVC 5792 window frames, pipes, flooring, wallpaper, bottles, clingfilm, toys, guttering, cable insulation, credit cards,medical products HDPE 5269 containers, toys, housewares, industrial wrappings andfilms, pipes PET 3234 bottles, textile fibers, film food packaging PS/EPS 3279 electrical appliances, thermal insulation, tape cassettes,cups and plates, toys PA 1399 film for food packaging (oil, cheese, ‘‘boil-in-bag’’), high-temperature engineering applications, textile fibers ABS/SAN 788 general appliance moldings PMMA 317 transparent all-weather sheet, electrical insulators,bathroom units, automotive parts Moreover, parts with complex shapes can be made at low cost and at high speed by shaping polymers or monomers in the liquid state.The polymer market can be divided into thermoplastics and thermosets. The ma-jor thermoplastics include high-density polyethylene (HDPE), low-density polyeth-ylene (LDPE), polyethylene terephthalate (PET), polypropylene (PP), polystyrene(PS and EPS), poly(vinyl chloride) (PVC), polyamide (PA), poly(methyl methacry-late) (PMMA) and styrene copolymers (ABS, SAN). The most important applica-tions of thermoplastics are summarized in Table 1.1. The total Western European demand for thermoplastics was 37.4 million tonnes in 2002, a growth of about 9%as compared to 2001 [1]. Thermoplastics are used not only in the manufacture of many typical plastics applications such as packaging and automotive parts, but also in non-plastic applications such as textile fibers and coatings. These non-plastic ap-plications account for about 14% of all thermoplastics consumed.The major thermosets include epoxy resins, phenolics, and polyurethanes (PU),for which the major applications are summarized in Table 1.2. It has to be noted,Tab. 1.2. Applications and 2002 Western European markets for the major thermosets [1].Thermoset Market [10 3 tonnes] Applications PU 3089 coatings, finishes, cushions, mattresses, vehicle seats Phenolics 912 general appliance moldings, adhesives, appliances,automotive parts, electrical components Epoxy resins 420 adhesives, automotive components, E&E components,sports equipment, boats

1.1 Polymer Materials 3
Domestic
22.3%
Automotive
7.0%
Large industry
5.2%
Agriculture Electrical and 2.5%electronics
7.3%
Building and Packaging construction 38.1%
17.6%
Fig. 1.2. Plastic consumption in 2002 by industry sectors in Western Europe [1].however, that about one-third of the market for thermosets is for relatively small-scale specialty products. The total Western European market for thermosets was
10.4 million tonnes in 2002, about 1% below the 2001 value.
The major application areas of polymers can be defined as follows (Figure 1.2).Automotive industry Motorists want high-performing cars combined with reliabil-ity, safety, comfort, competitive pricing, fuel efficiency, and, increasingly, reassur-ance about the impact on the environment. Lightweight polymeric materials are increasingly used in this sector (Daimler Benz’s Smart is a nice example), also con-tributing to a 10% reduction in passenger fuel consumption across Europe.Building and construction Polymeric materials are used in the building and con-struction sector, for example for insulation, piping, and window frames. In 2002 this sector accounted for 17.6% of the total polymer consumption.Electrical and electronic industry Many applications in this field arise from newly designed polymeric materials, for example for polymeric solar cells and holo-graphic films. It is interesting to note that, while the number of applications in this field is increasing, the weight of the polymers used per unit is decreasing.Packaging The packaging sector remains the largest consumer of synthetic poly-mers, approximately 38% of the total market. This is mainly due to the fact that these materials are lightweight, flexible, and easy to process, and are therefore increasingly being substituted for other materials. Although polymer packaging ranks first in terms of units sold, it is only third if judged on weight.Agriculture As agricultural applications account for about 2.5% of the total of syn-thetic polymers consumed in Europe, they play only a marginal role. Irrigation and

4 1 Polymer Reaction Engineering, an Integrated Approach drainage systems provide effective solutions to crop growing, and polymeric films and greenhouses can increase horticultural production substantially. The use of so-called ‘‘super absorbers’’ for increased irrigation efficiency in arid areas can be con-sidered an important emerging market.A Short History of Polymer Reaction Engineering In Table 1.3 a comprehensive overview of the major developments in the polymer industry is given. In the 19th century, polymers produced by Nature, such as cellu-lose, Hevea brasiliensis latex (natural rubber), and starch, were processed to manu-facture useful products. This practice was often based on experimental discoveries.As an example, in 1839 Goodyear discovered by mistake the sulfur vulcanization of natural rubber, making it possible for Ford to develop the automotive market. In those times no polymers were produced synthetically.Early in the 20th century (1920), the first empirical description of macromole-cules was developed by Staudinger [2]. At the same time, new methods were devel-oped to determine the specific characteristics of these materials. In the 1930s many research groups (for examples see refs. 3–7) developed models for the chain length distribution in batch reactors resulting from different polymer chemistries, a meth-odology that was further developed in the 1940s leading to more complex and com-prehensive models, some of which are still being used today.Tab. 1.3. The history of polymers in brief.19th century natural polymer and derivatives (vulcanized rubber, celluloid)1920 concept of macromolecules postulated by Staudinger 1930–1940 first systematic synthesis of polymers synthesis of polyamides (nylon) by Carothers at DuPont discovery of polyethylene at ICI (Fawcett and Gibson)1940–1950 synthetic rubbers and synthetic fibers 1950–1960 stereospecific polymerizations by Ziegler and Natta, the birth of polypropylene discovery of polymer single crystals (Keller, Fischer, Till)development of polycarbonate 1960–1970 discovery of PPO at GE by Hay and commercialization of PPO/PS blends (Noryl2 )1970–1980 liquid-crystalline polymers 1980–1990 superstrong fibers (Aramid2 , polyethylene)functional polymers (conductive, light-emitting)1990–2000 metallocene-based catalysts; novel polyolefins hybrid systems(polymer/ceramic, polymer/metals)2000– Nature-inspired catalysts synthesis of polymers by bacteria and plants

1.3 The Position of Polymer Reaction Engineering 5
Around 1940, partly inspired by World War II, a more systematic search for new synthetic polymer materials as a replacement for scarce natural materials led to the development of nylon (DuPont) and polyethylene (ICI) [8, 9]. This was followed by the development of synthetic rubbers and synthetic fibers. In the same period,Denbigh [10] was one of the first to introduce chemical reaction engineering con-cepts into polymer science by considering polymerization reactions at both the chemical and at the process levels. Processes were classified as homocontinuous and heterocontinuous, depending on the mixing level. This pioneering approach also acted as a catalyst for the further development of polymer reaction engineer-The development of catalysts based on transition metals by Ziegler and Natta[11] allowed the development of stereospecific propylene polymerization processes and ethylene polymerization in the 1950s. Several process schemes were developed at that time, of which some are still in use. The major problem in process develop-ment has been to deal with the heat of polymerization, an issue that was solved, for example, by using an inert solvent as a heat sink or by flashing of monomer fol-lowed by condensation outside the reactor. In the same period, polycarbonate and(somewhat later) poly(propylene oxide) (PPO) were developed. The main character-istic of the polymers developed so far was that they were bulk materials, to be pro-duced in extremely large quantities.In the 1970s, a paradigm shift occurred when polymers with more specific prop-erties started to be produced. This included various liquid crystalline polymers leading, for example, to the production of superstrong fibers such as Aramid2 /Kevlar 2 [12]. The development of functional polymers for the conduction of light and electricity and optical switches also started then [13]. In the near future this will probably lead to highly effective and flexible polymer solar cells [14].In the 1990s, metallocene catalysts were developed for polyolefin production that surpassed the Ziegler–Natta catalysts in terms of selectivity and reactivity [15, 16].Additionally, various hybrid materials were combining properties of both the poly-mer (lightweight, flexible) and a solid material, which could be metal (conductive)or ceramic (insulating), leading to materials with specific properties applicable, for example, as protective coatings [17].Current developments include the mimicking of nature (enzymes) for the syn-thesis of quite complex polymers like natural silk. Also, bacteria and plants are be-ing modified to produce polymers of interest [18]. However, this can be expected to require polymer reaction engineering developments that are as yet difficult to foresee.
1.3
The Position of Polymer Reaction Engineering Traditional chemical reaction engineering has its basis in the application of scien-tific principles (from disciplines such as chemistry, physics, biology, and mathe-matics) and engineering knowledge (transfer of heat, mass, and momentum) to

6 1 Polymer Reaction Engineering, an Integrated Approach Energy Integrated heat recovery Less energy demanding processes Raw material Coal, Oil Natural gas Renewable feedstock Integrated and Safety Plant operation inherent safety Market Capacity Quality control Flexible production Pollution Recycling Water Air Green solvents control Clean processes Process Optimization Intensification Scientific From empiricism to strategy Multidisciplinarity methodology 1980 1990 2000 2010 Fig. 1.3. Changing priorities in industrial chemical engineering research.the solution of problems of practical, industrial, and societal importance. Since the 1970s, a changing focus in chemical reaction engineering can be observed, which is summarized in Figure 1.3.To deal with more stringent requirements in terms of energy consumption re-quires a shift from heat loss minimization toward novel intensified process con-cepts that intrinsically require less energy. Safety should now be considered as an intrinsic plant property rather than a responsive action, and the plant needs to beflexible to be able to respond quickly to changes in the market. Last but not least,new concepts will be required to provide a basis for sustainable future develop-ments, that is, the use of renewable resources and processes based on ‘‘green’’solvents. As a result of this changing focus, a shift toward a multidisciplinary approach can be observed.For PRE this implies the combination of several disciplines such as polymer chemistry, thermodynamics, characterization, modeling, safety, mechanics, phys-ics, and process technology. PRE problems are often of a multi-scale and multi-functional nature to achieve a multi-objective goal. One particular feature of PRE is that the scope ranges from the micro scale on a molecular level up to the macro scale of complete industrial systems. PRE plays a crucial role in the transfer of in-formation across the boundaries of different scale regions and to provide a compre-hensive and coherent basis for the description of these processes [19].As depicted in Figure 1.4, there is a direct link between time and size scale, from which it is obvious that the micro and macro scales are not related to the same time scale [20]. As an example, molecular dynamics calculations are addressing a time scale in the order of femto- to nanoseconds, whereas process system integra-tion evolves on the scale of years. Engineers have traditionally been working at the meso scale, which is represented by the middle portion of Figure 1.4, using phe-

1.4 Toward Integrated Polymer Reaction Engineering 7
Time scale
Years System integration Environmental, Global
modeling
Engineering design Days Process models
Continuum
Models
Heterogeneous
Minutes
Phenomenological
Models
Microstructure Milliseconds Molecular level Elementary reactions Molecular modeling Nanoseconds Chemical equilibrium Atomic level Picoseconds Fundamental Quantum techniques Quantum chemistry Femtoseconds 1Å 10Å 100Å 1µm 1mm 1m 1km
Size scale
Fig. 1.4. Activities in PRE with their corresponding time and size scales.nomenological and continuum models. Today these limits are pushed in two direc-tions, both toward a more fundamental understanding and at the same time to-ward a more global scale. In the past, the ‘‘micro-region’’ has traditionally been the domain of physicists and chemists, whereas the ‘‘macro-region’’ has been thefield, rather, of process or plant engineers. Today, it becomes obvious that only us-ing a multidisciplinary, parallel, and synergetic approach can lead to successful de-velopments. Polymer reaction engineering will play an essential role as the core and the coordinator of this complex process.
1.4
Toward Integrated Polymer Reaction Engineering As will be obvious from the foregoing discussion, PRE is composed of many disci-plines all linked together. These disciplines can be either mature or emergent, but they have a common gateway (see Figure 1.5). Although there is not necessarily a direct connection between them, there exists a common core in which the different disciplines make their own specific contribution to a general objective.The frontiers in PRE are determined by what we know, understand, and are able to quantify, and these frontiers are moving with growing knowledge, competences,and experience. Efforts to push these limits will induce innovative developments leading to emerging technologies and products, and will also strengthen the multi-disciplinary approach. In general terms, PRE can be defined as the science that

8 1 Polymer Reaction Engineering, an Integrated Approach Materials sciences Thermodynamics Novel processes Inherent safety Materials Application Environment Recycling, Disposal Modeling and Polymer Reaction simulation Engineering Process integration (PRE)optimization New products Polymer chemistry Post-reaction Reaction kinetics processes Nano-, Micro-Novel processes Bio-Measurement and control Fig. 1.5. The expanding sphere of polymer reaction engineering.brings molecules to an end-use product. We can either consider it like a black box(Figure 1.6) or we can try to define the interconnected disciplines that compose this black box (Figure 1.7). Provided the required product properties can be met,we expect that sustainability is the common denominator for all the disciplines in-volved in this process.The process of transforming raw materials into valuable end-use products is not a one-way procedure but rather an iterative process in which we try to optimize all the parameters involved. The selection of the proper chemistry and technology should include an evaluation of environmental, safety, and economic parameters.Moreover, questions regarding the possible use of renewable resources and mini-mizing the energy requirement will have to be answered. Defining PRE in this manner appears to be very close to the procedure of life cycle analysis (LCA) [21].Polymer Reaction Raw materials End use product Engineering Fig. 1.6. PRE as a black box process.

1.5 The Disciplines in Polymer Reaction Engineering 9
Materials sciences Thermodynamics Novel processes Raw materials Inherent safety Materials Application Products
Energies
Environment Profit Needs Recycling, Disposal Modeling and simulation Satisfaction
Laws
Integrated PRE Process integration Knowledge
Economy
optimization New products Polymer chemistry Post-reaction Reaction kinetics processes Nano-, Micro-Novel processes Bio-Measurement and control
Renewable
Fig. 1.7. The integrated approach for sustainable PRE.Life cycle analysis is a tool assisting decision making in the engineering process.LCA includes the information on the history of the materials used, and the dif-ferent process and raw material alternatives, as well as the final product require-ments. LCA is an instrument driven by environmental considerations against a background of technical and economic specifications, and involves the so-called 3-P concept (people, planet, and profit). The LCA-based PRE methodology (Figure
1.8) [22] leads to an optimization of all the parameters involved and a reduction of
the costs. This seems to be contradictory at first sight, but integrating all the as-pects often leads to cost reductions. In our view, the use of this approach will lead to a ‘‘sustainable integrated PRE’’.
1.5
The Disciplines in Polymer Reaction Engineering The different disciplines involved in PRE can be represented using the academia–industry dichotomy (Figure 1.9). The interests of the two types of players are not identical: the differences are similar to the differences in their mission statements.Nevertheless, we can observe that a great overlap is present in the middle zone,

10 1 Polymer Reaction Engineering, an Integrated Approach
management
Specification Balances recycling raw material- technical - energy- economic - material- ecologic - emission- safety product use synthesis - waste
- sewage
processing
Evaluation leads to closed loop assessment of costs Fig. 1.8. Life cycle analysis of parts, methods, products, and systems.where interests, tools, and knowledge are similar, thus providing a strong basis for partnership.As stated above, PRE is composed of a large number of disciplines, which are described in more detail in the following chapters of this handbook. These disci-plines are interconnected by a synergetic and multidisciplinary approach, and com-
Academia
Fundamental Novel Thermodynamics kinetics technologies Molecular Measurement modeling and control Reactor design Environment Polymer physics Polymer chemistry Safety
Quality
assurance
Applied Market modeling Process modeling economics
Industry
Fig. 1.9. Overlap of industrial and academic disciplines.

1.5 The Disciplines in Polymer Reaction Engineering 11
Fig. 1.10. Product-driven PRE, based on an orthogonal relationship between science and engineering.mercial products are the final achievement resulting from this methodology. This could be expressed by an orthogonal representation (Figure 1.10) where polymer sciences are linked with engineering sciences. Every type of polymerization will have its own specific features, models, and engineering aspects involved. From Fig-ure 1.10 it will be obvious that only teamwork, bringing together several fields of expertise, can lead to the final objective.
1.5.1
Polymerization Mechanisms Polymerization reactions can be classified depending on the reaction mechanism involved and can be either step-growth or chain-growth. These mechanisms differ basically with the time scale of the process. In step-growth polymerization (like polycondensation), the polymer chain growth proceeds slowly from monomer to dimer, trimer, and so on, until the final polymer size is formed at high monomer conversions. Both the chain lifetime and the polymerization time are often in the order of hours. In chain-growth polymerization (like ionic or free-radical polymer-ization), macromolecules grow to full size in a much shorter time (seconds being the order of magnitude) than required for high monomer conversion. High molec-ular weights are already obtained at low monomer conversion, which is in great contrast to step-growth polymerizations. Also, unlike step-growth polymerization,chain-growth polymerization requires the presence of an active center.Condensation polymers are the result of a condensation reaction between mono-mers, with or without the formation of a condensation by-product (Chapter 3). Ex-amples of polymers produced by condensation are polyamide[6.6], (Nylon 6,6) the result of the intermolecular condensation of hexamethylenediamine and adipic acid, and polyamide[6], (Nylon 6) which is the product of intramolecular condensa-tion of a-caprolactam. This type of reaction is generally sensitive to thermodynamic equilibrium and requires the removal of the by-product, which is often volatile.

12 1 Polymer Reaction Engineering, an Integrated Approach The polymers produced by condensation reactions can be either linear or non-linear, depending on the number of functional groups per monomer. The polymer-ization process can be performed in bulk (liquid or solid state) or as an interfacial Free-radical polymerization (FRP) can be performed homogeneously (in bulk, so-lution, or suspension; Chapters 4 and 5) or heterogeneously (emulsion, precipita-tion; Chapter 6). The active site is always a radical that can be unstable (classical FRP) or stabilized as in pseudo-living FRP. Radicals can be formed by the homo-lytic bond rupture of initiators (molecules sensitive to homolytic cleavage, such as peroxides, photosensitive molecules, or bisazo compounds) or by complex mecha-nisms creating radicals from monomer units using thermal or high-energy sources, such as X-rays, g-irradiation, or UV. This type of polymerization usually comprises several steps: initiation, propagation, various transfer mechanisms,and termination.In ionic polymerizations a cation or anion is the active site (Chapter 7). A heter-olytic process leads to charged parts of molecules that can induce the polymeriza-tion by nucleophilic or electrophilic processes. These reactions generally evolve at low temperatures (even as low as 120 C) due to the high reactivity of ions. Also,they are very sensitive to impurities present in the monomer or solvent. These re-actions are not always terminated, so lead to living polymerization. This process is often used to build tailor-made copolymers.Coordination polymerizations require a transition metal catalyst (Chapter 8). Poly-olefins are often produced by this kind of reaction where the catalyst (Ziegler–Natta, for example) acts as the active site but also as the steric regulator, which makes it possible to build polymers with a defined tacticity. Nowadays a great re-search effort is devoted to the synthesis of new transition metal-based catalysts,such as metallocenes, to produce new products.
1.5.2
Fundamental and Engineering Sciences Apart from the various polymerization mechanisms involved, a large number of other disciplines will have to be involved, according to the matrix depicted in Figure 1.10.Thermodynamics is essential to understand the physicochemical properties of the individual reactants, solvents, and products involved (Chapter 2). Also, it pro-vides information on the interaction between the various components present in the reaction mixture, from which phenomena such as phase behavior and parti-tioning can be derived. This information is usually accessible by using the appro-priate equation of state for a given system studied. A close collaboration between chemical physicists, chemists, and chemical engineers is required to take full ad-vantage of this fundamental knowledge.Polymer solutions (solid, bulk, solution, complex media) have to be characterized by several specific analytical tools (Chapter 20). Techniques such as NMR, ESR,

1.5 The Disciplines in Polymer Reaction Engineering 13
electron microscopy, chromatography, electrophoresis, viscometry, calorimetry, and laser diffraction are widely used to determine polymer properties, often in combi-nation. The main characteristics being analyzed are the chain length distribution,degree of branching, composition, tacticity, morphology, particle size, and chemical and mechanical properties. Polymer mechanics (Chapter 14) usually concerns thefinal product rather than the polymerization reaction. Nevertheless, as polymers are usually judged on their end-use properties (Chapter 13), the final product needs specific and often customer-based analysis. This is described more specifi-cally for two application areas, namely the use of thermosets for coating applica-tions (Chapter 16) and the production of polymeric fibers (Chapter 17).Measurement and control are indispensable to achievement of a robust and safe process (Chapter 12). Since the early 1990s, a tremendous effort has been observed in the development of new in-line analytical techniques, including spectroscopy(UV, IR, Raman, laser, and so on), ultrasonic sensing, chromatography, and diffrac-tion or electrical methods. New control schemes appear where the reaction is per-formed just below the constraint limits, independently of the reaction kinetics. All these techniques tend to lead to safer and more robust processes while increasing productivity and product quality at the same time.Safety cannot be treated as a separate discipline as it is already integrated from the early chemistry and process development (Chapter 11). Safety deals with a wide variety of technological aspects with respect to the environment (water, air, soil,and living species). However, economic aspects are usually taken into consider-ation also. Modern process development intrinsically includes safety and environ-mental aspects in all stages of the development.Modeling is probably the tool of excellence for engineers (Chapter 9). It is used to simulate the reaction and the process system in order to shorten the time for development. It is based on models that can be physical or chemical, semi-empirical or empirical, descriptive or more fundamental. To describe the develop-ment of the molecular weight distribution upon reaction, moment methods or equations based on population balance are often used.Scaleup is a widely used term to define the methodology that allows scaling up of a process from small to larger scale (Chapter 10). Often the scaleup process be-gins with a scaledown approach in order to have reliable and representative equip-ment already at the laboratory scale. Scaleup is always dependent on the system studied and requires a proper understanding of the performance of process equip-ment involved at different scales. In polymer reaction engineering, heat transfer and mixing can be considered as two major issues in this perspective. Modern computing techniques such as computational fluid dynamics and process simula-tion become more and more important in the optimization of process parameters and the equipment hardware.Volatile organic compound (VOC) content in the final product is related to prod-uct properties and legislation (FDA approval in the USA, for example). All the pro-cesses aiming to lower the residual VOC content in the product are denoted as ‘‘re-movable’’ (Chapter 18). These processes can differ from each other, depending on

14 1 Polymer Reaction Engineering, an Integrated Approach the techniques involved. Devolatilization, post-process reaction, and extraction are some of the methodologies employed for this purpose.Stability and degradation of polymers (Chapter 15) become relevant especially during post processing or moulding processes. Temperature, oxidation and me-chanic stresses are the main contributors to product degradation.Currently, there is a strong emphasis on the synthesis of novel functional poly-mers shaped on a nano scale (Chapter 19) and the development of sustainable production processes (Chapter 21). The latter includes process intensification as a methodology, the use of ‘‘green’’ solvents, and the use of renewable resources.Many of the new processes under development are focusing on one or more of these topics, for which the use of supercritical fluids is currently being imple-mented on an industrial scale.
1.6
The Future: Product-inspired Polymer Reaction Engineering Innovation times in industry have shown a steady decrease since the 1970s. Classic thinking is that process development becomes increasingly important as industry matures [23]. This is due to the fact that in an early phase of the lifetime of an in-dustry, when product concepts are still being created, the rate of product innova-tion exceeds the rate of process innovation. This period continues until a dominant design has emerged and opportunities for radical product innovation decrease. In this phase, the shift is toward process innovations to reduce cost price.The half-time of product innovation (time-to-market) in the early 1970s was about ten years. Currently, two years is often considered long. This acceleration of innovation time is the result of competitive pressure in the market. As a rule of thumb, the first company to enter the market with a new product can get up to 60% of market share, so there is a high reward for being first.As has been discussed above, chemical engineering has been the basis for poly-mer reaction engineering in the past. In recent discussions, however, it has been emphasized that a need exists to refocus chemical engineering toward product-driven process engineering [24, 25]. The thinking about a process should then start with the customer or consumer: which of the two depends on the structure of the supply chain. The wishes of the consumer and consumer-perceived product prop-erties have to be translated into physical and chemical product properties. In this way, the main physical attributes of a product are determined, including an idea about the microstructure. Next, a functional analysis is performed to determine the lowest number of transformations needed to create the product; this is fol-lowed by a morphological analysis [26]. Finally, a conceptual process design exer-cise is performed to generate possible process routes to achieve the desired product properties. This sequence of events is the core of product-inspired polymer reac-tion engineering. A key characteristic of this approach is the fact that it avoids the classical ‘‘unit operation trap’’, because it does not fix the mindset to consider only traditional reactor design and separation process steps to build a process.

1.7
Concluding Remarks In the foregoing we have presented a general framework for sustainable polymer reaction engineering. Its most important characteristic lies in the concerted multi-disciplinary approach, rather than focusing on individual competencies. Given the volume of polymer production, it will be of major importance that environmental and safety issues become an integral part of the development process. In combina-tion with tools such as life cycle analysis and product-inspired PRE, this will allow the development of sustainable new polymer processes.1 Association of Plastics Manufacturers Kivits, L. J. IJzendoorn, M. T.in Europe, Annual report, 2002. Rispens, J. C. Hummelen, R. A. J.2 H. Staudinger, Chem. Ber., 1920, 53, Janssen, Adv. Funct. Mater., 2002, 12,
1073. 665.
3 W. Kuhn, Berichte de Deutschen Chem. 15 W. Kaminsky, H. Sinn (Eds.) Transi-Gesellsch., 1930, 63, 1503. tion metals and organometallics as 4 W. H. Chalmers, J. Am. Chem. Soc., catalysts for olefin polymerisation,1934, 56, 912. Springer Verlag, Berlin, 1987.5 H. Dostal, H. Mark, Trans. Faraday 16 J. M. Benedikt, B. L. Goodall,Soc., 1936, 32, 54. Metallocene-catalyzed polymers, B.F.6 G. V. Schulz, Z. Physik. Chem., 1935, Goodrich, Brecksville, 1998.B30, 379. 17 J. Sinke, Appl. Comp. Mater., 2003, 10,7 P. J. Flory, J. Am. Chem. Soc., 1936, 293.58, 1877. 18 E. R. Howells, Chem. Ind., 1982, 508.8 W. H. Carothers, US Patent 2 130 19 A. Penlidis, Can. J. Chem. Eng., 1994,948, 1937. 72, 385.9 E. W. Fawcett, R. O. Gibson, J. 20 A. Sapre, J. R. Katzer, Ind. Eng.Chem. Soc., 1934, 386. Chem. Res., 1995, 34, 2202.10 K. G. Denbigh, Trans. Faraday Soc., 21 A. Azapagic, Chem. Eng. J., 1999, 73,1947, 43, 648. 1.11 K. Ziegler, E. Holzkamp, H. Breil, 22 P. Eyerer, J. Polym. Eng., 1996, 15,H. Martin, Angew. Chem., 1955, 67, 197.
541. 23 W. J. Abernathy, J. M. Utterback,
12 P. W. Morgan, S. L. Kwolek, Technol. Rev., 1978, 80, 40.Macromolecules, 1975, 8, 104. 24 E. L. Cussler, G. D. Moggeridge,13 H. Sasabe, T. Wada, Polymers for Chemical product design, Cambridge electronic applications, in: Compre- University Press, Cambridge, 2001.hensive Polymer Science, vol. 7, S. L. 25 E. L. Cussler, J. Wei, AIChE J., 2003,Aggarwal (Ed.), Pergamon Press, 49, 1072.Oxford, 1989. 26 C. J. King, AIChE Monograph Ser.,14 J. K. J. van Duren, J. Loos, F. 1974, 70, 1.Morrissey, C. M. Leewis, K. P. H.

Theodoor W. de Loos
2.1
Introduction The phase behavior of polymer solutions plays an important role in polymer pro-duction and processing. Many polymers are produced by solution polymerization.Solvent choice, solvent recovery and the removal of traces of solvent from the poly-mer product are important factors in these processes. An example is the produc-tion of linear low-density polyethylene (LLDPE), which is a copolymer of ethylene and a 1-alkene. Hydrocarbons are used as solvents in this process. The reactor con-ditions are limited at high temperature by the onset of a liquid–liquid phase split,characterized by a lower critical solution temperature, and at low temperature by crystallization of LLDPE. The pressure must be high enough to keep the ethylene in solution. Another well-known example is the production of low-density polyethy-lene (LDPE). In this process ethylene is compressed together with an initiator and the LDPE is formed by radical polymerization. The reactor pressure chosen must be high enough to dissolve the polymer in its monomer. In practice reactor pres-sures are higher than 200 MPa. Since the conversion of ethylene to LDPE is incom-plete, LDPE has to be separated from unreacted ethylene, which is recycled to the reactor. To save energy this is done by pressure reduction in two steps, which in-volve a high-pressure vapor–liquid flash.From a thermodynamic point of view polymer solutions are complicated solu-tions. A polymer is not a single component but a multicomponent mixture charac-terized by a molecular weight distribution or by average molecular weights, such as the number-average molecular weight Mn or the weight-average molecular weight Mw . In the case of a copolymer different types of copolymers are possible, for ex-ample random copolymers and block copolymers, and the comonomer content may vary. Because of their asymmetric nature the entropy of mixing of polymer solutions is much lower than in the case of a mixture of two low molecular weight compounds; also, the pure solvent and the polymer have rather different free
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

volumes. Because of this, polymer–solvent systems often show a liquid–liquid phase split.In this chapter it is not possible to give an in-depth treatment of the thermody-namics of polymer solutions. For further reading, see Refs. 1–6.
2.2
Thermodynamics and Phase Behavior of Polymer Solutions
2.2.1
Thermodynamic Principles of Phase Equilibria The equilibrium conditions for phase equilibria can be derived in the simplest way using the Gibbs energy G. According to the second law of thermodynamics, the total Gibbs energy of a closed system at constant temperature and pressure is a minimum at equilibrium. If this condition is combined with the condition that the total number of moles of component i is constant in a closed system [Eq. (1),where nai is number of moles of component i in phase a], the equilibrium condi-tions given by Eq. (2) can easily be derived for a system of p phases and N compo-nents [7].
X
nai ¼ constant ð1Þmai ¼ mib ¼


¼ mpi for i ¼ 1; 2; . . . ; N ð2ÞThe chemical potential of component i in phase a is defined by Eq. (3), where g is the molar Gibbs energy.
0
X 1
q nai g a
B C
mai ¼ ð3Þ
qnai
P; T; nj0i
2.2.2
Fugacity and Activity In the thermodynamic treatment of phase equilibria, auxiliary thermodynamic functions such as the fugacity coefficient and the activity coefficient are often used. These functions are C closely related to the Gibbs energy. The fugacity of com-ponent i in a mixture, fi , is defined by Eq. (4a) together with (4b).dmi 1 RT ln fi at constant T ð4aÞ
fi
lim ¼ 1 ð4bÞ
P!0 Pi

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 19
According to this definition, fi is equal C to the partial pressure Pi in the case of an ideal gas. The fugacity coefficient fi is defined by Eq. (5) and is a measure of the deviation from ideal gas behavior.
C fi
fi ¼ ð5Þ
The fugacity coefficient can be calculated from an equation of state by Eq. (6) or (7)
C ð P
RT

RT ln fi ¼ Vi  dP ð6Þ
0 P
0X 1
ð "
 # n i RT
C V
qP RT @ A
RT ln fi ¼   þ RT ln i ð7Þy qn i V; T; nj0i V PV According to Eq. (4), the equilibrium relation in Eq. (2) can be replaced by Eq. (8).
C C C
fi a ¼ fi b ¼


¼ fi p for i ¼ 1; 2; . . . ; N ð8ÞThe activity a i is defined as the ratio of fi and the fugacity of component i in the standard state fi 0 at the same P and T [Eq. (9)].fi ðP; T; xÞ
ai 1 0 ð9Þ
fi ðP; T; x 0 ÞAn ideal solution is defined by Eq. (10).a iid 1 x i ð10ÞThe activity coefficient of component i, gi [Eq. (11)], is a measure of the deviation from ideal solution behavior, so the fugacity of a nonideal liquid solution can be written as Eq. (12).
gi 1 ð11Þ
a iid
fi ¼ x i gi fi 0 ð12ÞThe activity coefficient gi can be calculated from a model for the molar excess Gibbs energy g E, Eq. (13).
0
X 1
B q n i g C RT ln gi ¼ ð13Þqn i P; T; nj0i

In this approach the standard state fugacity fi 0 of a liquid component is usually the fugacity of the pure liquid component, and is closely related to the vapor pressure Pisat of that component. On the vapor-pressure curve of a pure component, we have conditions according to Eqs. (14).fi L ðPisat ; TÞ ¼ fi V ðPisat ; TÞ ¼ fVi ðPisat ; TÞPisat ð14ÞFrom Eq. (4a) we obtain Eq. (15), where viL is the molar volume of pure liquid i.ðP ! ðP !
dmiL viL
fi L ðP; TÞ ¼ fi L ðPisat ; TÞ exp ¼ fi L ðPisat ; TÞ exp dP ð15ÞPisat RT Pisat RT Combining Eqs. (14) and (15), we get Eq. (16).ðP !
viL
fi ðP; TÞ ¼ fVi ðPisat ; TÞPisat exp
dP ð16Þ
Pi sat RT
At low pressure the fugacity coefficient and the exponential term are close to 1, so Eq. (17) holds.fi L A Pisat ð17Þ
2.2.3
Equilibrium Conditions For low-pressure vapor–liquid equilibria (VLE) the equilibrium condition of Eq.
(18) is usually written as Eq. (19), in which fi 0 is calculated with Eq. (16) or (17)
and ji with Eq. (6) or (7); x i and yi are the liquid-phase and vapor-phase mole frac-tion of component i, respectively.
C C
fi L ¼ fi V ð18Þ
0 C
x i gi fi ¼ yi ji P ð19ÞThe truncated virial equation or the ideal gas equation is often used as the equa-tion of state. In the latter case all fugacity coefficients are 1, so the simplest form of Eq. (19) is Eq. (20).x i gi Pisat ¼ yi P ð20ÞIn the case of low-pressure liquid–liquid equilibria (LLE) the equilibrium condition is given by Eq. (21).
C C
fi 0 ¼ fi 00 or ðx i gi fi 0 Þ 0 ¼ ðx i gi fi 0 Þ 00 ð21Þ

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 21
The two liquid phases are indicated by prime ( 0 ) and double prime ( 00 ). If for both liquid phases the standard fugacity is chosen as the pure liquid component, Eq.ðx i gi Þ 0 ¼ ðx i gi Þ 00 ð22ÞFor the calculation of high-pressure vapor–liquid equilibria or liquid–liquid equi-libria an equation of state is always used for both phases and the equilibrium con-dition used is given by Eq. (23).C C C C C C fi a ¼ fi b or ðx i ji PÞa ¼ ðx i ji PÞb or ðx i ji Þa ¼ ðx i ji Þb ð23Þ
2.2.4
Low-pressure Vapor–Liquid Equilibria Since polymers have no vapor pressure and as a consequence the vapor phase does not contain polymer, the equilibrium conditions for low-pressure vapor–liquid equilibria of polymer solutions as given by Eq. (20) are only applicable to the solvent s as in Eq. (24), or in a case where the weight fraction of polymer wp is used as a composition variable as in Eq. (25), where Ws is the weight fraction based activity coefficient of the solvent.P ¼ xs gs Pssat ð24ÞP ¼ ð1  wp ÞWs Pssat ð25ÞThe relation between a mole fraction based activity coefficient gi and a weight frac-tion based activity coefficient Wi of component i is given by Eq. (26), where wi and Mi are the weight fraction and molecular weight of component i, respectively. gs and Ws can be obtained from a correlation of experimental data using a suitable model or from a predictive model (see Section 2.3).
gi wi XN
wj
¼ ¼ Mi ð26Þ
Wi x i j¼1
M j
2.2.5
Flory–Huggins Theory and Liquid–Liquid Equilibria The Flory–Huggins theory of polymer solutions [1] is based on a rigid lattice model in which a polymer molecule is assumed to consist of r segments of the size of a solvent molecule. The Flory–Huggins expression for the Gibbs energy of mixing Dmix G for Ns moles of solvent and Np moles of polymer is given by Eq. (27).
Dmix G
¼ Ns ln js þ Np ln jp þ js jp ðNs þ rNp Þw ð27Þ
RT

The first two terms on the right-hand side of Eq. (27) represent the so-called com-binatorial entropy of mixing, and the third term is an approximation for the en-thalpy of mixing. In practice w, the Flory–Huggins interaction parameter, is used as an adjustable, temperature-dependent parameter. js and jp are the segment fractions of solvent and polymer, respectively, defined by Eq. (28).
Ns rNp
js ¼ and jp ¼ ð28ÞNs þ rNp Ns þ rNp According to the original Flory–Huggins theory w is given by Eq. (29), in which ess ; epp , and esp are the interaction energies between two solvent molecules, two polymer segments, and a solvent molecule and a polymer segment, respectively.ð2esp  ess  epp Þ
wz ð29Þ
Often r is approximated by the ratio of the molar volumes of pure liquid polymer and pure solvent. The segment fractions of solvent and polymer are then equal to the volume fractions of solvent and polymer, fs and fp . Equation (27) is derived us-ing many assumptions and approximations (for a discussion, see Ref. 8), but on the basis of this rather simple expression many features of the phase behavior of polymer solutions can be explained. The expressions for the mole fraction based activity coefficients of solvent and polymer are Eqs. (30) and (31), respectively.
 

jp 1
ln gs ¼ ln ð1  jp Þ þ þ 1  jp þ jp2 w ð30Þ
r r
ln gp ¼ ln½ð1  jp Þr þ jp   ðr  1Þð1  jp Þ þ rð1  jp Þ 2 w ð31ÞAs can be seen from Eq. (30), the activity coefficient of the solvent is strongly de-pendent on r for low values of r, but at high values of r gs becomes practically inde-pendent of r. This implies that the equilibrium pressure of low-pressure VLE for polymer–solvent systems is hardly dependent on the molecular weight of the In the original formulation of the Flory–Huggins theory Dmix G increases with decreasing temperature, which leads to a liquid–liquid phase split with an upper critical solution temperature. Per mole of lattice sites, the entropy of mixing of a polymer–solvent system is much less than for a solvent–solvent system; because of this, polymer–solvent systems show a stronger tendency to demix than solvent–solvent systems. This is illustrated in Figure 2.1, which shows schematically the influence of r on the location and shape of the liquid–liquid two-phase region.The figure shows that with increasing r, the two-phase region increases in size and becomes more asymmetric. The critical point, the temperature maximum of

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 23
0.40
0.50
r=1000
0.60
r=100
χ
0.70
0.80
0.90
r=10
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60
φp
Fig. 2.1. Liquid–liquid equilibria for polymer–solvent systems at different polymer chain lengths r. Calculated from the Flory–Huggins theory.the two-phase region, shifts with increasing r to lower values of w, higher tempera-ture, and lower values of jp . At r ¼ y the critical point is found at a limiting value of w ¼ 12 , T ¼ y, and jp ¼ 0. This follows directly from the critical point conditions[Eq. (32), where G is the Gibbs energy per mole of lattice sites].
qG q2G
¼ ¼0 ð32Þ
qjp qjp2
P; T
This leads to Eqs. (33) and (34) for the Flory–Huggins expression [Eq. (27)].


crit 1 1 2
w ¼ 1 þ pffiffi ð33Þ
2 r
jpcrit ¼ pffiffi ð34Þ
1þ r

The theta temperature y is the highest possible upper critical solution temperature(UCST) within the framework of the Flory–Huggins theory.Many polymer–solvent systems, at temperatures above the UCST, show a second region with liquid–liquid immiscibilty (Figure 2.2), which is characterized by the occurrence of a lower critical solution temperature (LCST) [9]. This phenomenon can not be explained on the basis of the original Flory–Huggins theory. Delmas and Patterson [10] calculated binary critical curves of polymer–solvent systems us-ing the Flory equation of state [11, 12]. According to these authors the reason for the occurrence of this LCST is the large difference in thermal expansion of pure liquid polymer and pure solvent, which leads to an increasing difference in free volume with increasing temperature and an LCST-type liquid–liquid phase split at temperatures below the critical point of the solvent. If the molecular weight of the polymer is increased, the UCST and LCST approach each other. In some cases, this can lead to the phase diagram of Figure 2.3, in which the liquid–liquid region has the shape of an hourglass, as in the polystyrene–acetone system (Figure 2.4) [13].However, another possibility is that the UCST and LCST never meet and that each liquid–liquid region shows a (different) y temperature for r ¼ y. This type of be-havior is found for polystyrene–methylcyclohexane [14]; see Figure 2.5.Another type of LCST is connected to specific interactions like hydrogen bond-ing. This type of LCST is found at temperatures below the UCST. The resulting phase diagram is presented in Figure 2.6, which shows a closed-loop region of liquid–liquid immiscibility. This type of phase diagram is not predicted by the original Flory–Huggins theory, either.
L1+L2
L1+L2
Solvent φp Polymer Fig. 2.2.Liquid–liquid equilibria showing a low-temperature two-phase region with a UCST and a high-temperature two-phase region with a LCST.

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 25
L L
Solvent φp Polymer Fig. 2.3. Hourglass-shaped two-phase liquid–liquid equilibria.
2.2.6
High-pressure Liquid–Liquid and Vapor–Liquid Equilibria Figure 2.7 shows schematically the phase behavior of a binary polymer solution at constant composition in a P; T diagram. In the case of curve a, the mixture at low temperature shows a liquid–liquid region characterized by UCST behavior and at high temperature a liquid–liquid region characterized by LCST behavior. Both two-phase regions are bounded at low pressures by a three-phase liquid–liquid–vapor curve, which separates the liquid–liquid region from a vapor–liquid region. The curve that separates a liquid–liquid region from the one-phase liquid region is often called a cloud-point curve. The liquid–liquid–vapor curve is found in many cases very close to the vapor-pressure curve of the pure solvent because the poly-mer concentration in the vapor phase and in the solvent-rich liquid phase is low.Curve b shows the case where the low-temperature liquid–liquid region and the high-temperature liquid–liquid region are merged into a single liquid–liquid re-gion. In this case the cloud-point curve shows a pressure minimum. At pressures below this minimum, hourglass-shaped two-phase regions are found. At higher pressures the phase behavior is as represented by Figure 2.2. An example of this behavior is shown by the polystyrene–acetone system [15] (see Figure 2.8). In the case of curve c the cloud point curve is found at much higher pressure and no longer shows a minimum.In going from case a to case c, the mutual solubility of polymer and solvent de-creases. Stronger polymer–polymer or solvent–solvent interactions lead to a lower mutual solubility, while stronger polymer–solvent interactions lead to a higher mu-tual solubility [15]. As discussed in Section 2.2.5, increasing molecular weight of

Fig. 2.4.Liquid–liquid equilibria in the polystyrene–acetone system at the indicated polystyrene molecular weights(in g mol1 ). Reproduced with permission from Ref. 13.the polymer also leads to a decrease in mutual solubility for the same polymer–solvent system. An example of this effect is given by Zeman and Patterson [16] for the polystyrene–acetone system.A similar effect is observed when the chain length of the solvent is increased.Ehrlich and Kurpen [17] determined cloud-point curves of 5 wt.% LDPE in ethane,propane, butane, and pentane. These results are reproduced in Figure 2.9, which shows that with increasing molecular weight of the solvent the cloud-point pres-sure increases and dP=dT of the cloud-point curves change sign from positive to negative. Since ethylene is a worse solvent than ethane, the cloud point pres-sures for the system ethylene þ LDPE [18, 19] are higher than for the system

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 27
Fig. 2.5. Liquid–liquid equilibria in the polystyrene–methylcyclohexane system at the indicated molecular weights of polystyrene (in g mol1 ). Reproduced with permission from ethane þ LDPE. For this type of systems it is not possible to make a clear dis-tinction between vapor–liquid equilibria and liquid–liquid equilibria. The poly-mer-rich phase can be considered to be a liquid phase, but the solvent-rich phase is liquidlike at low temperature, but vaporlike at high temperature. De Loos et al.[20] measured cloud-point curves for LLDPE systems plus hexane, plus heptane,and plus octane (see Figure 2.10). In this case also, the polymer is more soluble in the higher molecular weight solvent, which is shown by the shift of the LCST-type cloud-point curve to higher temperatures. An increased degree of branching of the polymer leads to a better mutual solubility [21].

L1+L2
Solvent
φp Polymer
Fig. 2.6. Closed-loop liquid–liquid equilibria.
P c
Fig. 2.7.P; T phase diagram for a constant cloud pressure; (c) system in which the low-composition polymer–solvent system: (a) and high-temperature regions of demixing have system with separate low- and high- merged, without a minimum cloud pressure.temperature regions of demixing; (b) system in The full curves are cloud-point curves; the which the low- and high temperature regions broken curves are liquid–liquid vapor curves.of demixing have merged, showing a minimum

Fig. 2.9. Cloud-point isopleths of LDPE in various n-alkane solvent at 5 wt.% polymer. The short dash–long dash curve is the solidification boundary of LDPE. Reproduced with permission from Ref. 17.in the free volume of polymer and solvent can be accounted for by adding a free volume contribution [4]. Further, Eq. (27) can be extended to account for the poly-dispersity of the polymer.The system-dependent parameters in these models can be adjusted to experi-mental data or predicted from a group contribution approach [4, 26].
2.3.1
Flory–Huggins Theory For a solution of a polydisperse polymer with m polymer components in one sol-vent, the Flory–Huggins expression for the Gibbs energy of mixing per mole of lat-

2.3 Activity Coefficient Models 31
Fig. 2.10. Cloud-point curves of poly(E-co-1-octene)–n-alkane systems at P ¼ 3 MPa. Mn ¼ 33 kg mol1 , Mw ¼ 124 kg mol1 ,Mz ¼ 420 kg mol1 . The systems show LCST phase behavior.Reproduced with permission from Ref. 20.tice sites can be written as Eq. (35). The sum is only over the polymer components,ji is the segment fraction of polymer component i, and jp ¼ ji ¼ ð1  js Þ is the
i¼1
overall polymer segment fraction.
Dmix G Xm
¼ js ln js þ ji ri1 ln ji þ js jp g sp ð35Þ
RT i¼1
g sp ¼ f ðT; jp Þ ð36ÞTo avoid confusion with the polymer–solvent interaction parameter, the symbol g sp[Eq. (36)] is used instead of w, which is independent of concentration. Equations(37a)–(37c) are examples of expressions used for the temperature and composition dependence of g sp [27–29].g sp ¼ a þ þ cT þ djp þ ejp2 ð37aÞ
bþ
g sp ¼ a þ T ð37bÞ
1  djp

aþ þ c ln T g sp ¼ T ð37cÞ
1  djp
We can derive Eq. (38) for the solvent, and Eq. (39) for polymer component i,where rn is number-average chain length of the polymer, which is defined by Eq.
(40) and which is proportional to the number-average molecular weight of the

 !
1 qg sp 2
lnðxs gs Þ ¼ ln js þ 1  j þ g sp  js j ð38Þ
rn p qjp p


ri jp qg sp 2 lnðx i gi Þ ¼ ln ji þ 1   ri js þ ri g sp  jp j ð39Þ
rn qjs i
X
n i ri
rn ¼ i¼1 ð40Þ
Xm
i¼1
Using Eqs. (37) and (38), vapor–liquid and liquid–liquid equilibria can be calcu-lated as discussed above. Reference 5 gives details of liquid–liquid equilibrium cal-culations in polydisperse polymer–solvent systems. In Figure 2.11 the experimen-tally determined phase behavior of three PEG–water systems is compared with the calculated phase behavior of these systems using Eq. (37c) to represent g sp as a function of temperature and polymer segment fraction. The parameters were fitted to the data [29].The description of the influence of pressure on polymer–solvent phase behavior is also possible using a pressure-dependent g sp [27, 30–32]. However, this approach is purely empirical.
2.3.2
Hansen Solubility Parameters According to the regular solution theory of Hildebrand the w-parameter can be ap-proximated by Eq. (41) [8], where vs is the molar volume of the solvent and ds and dp are the solubility parameters of solvent and polymer, respectively. Since these solubility parameters are pure component parameters, Eq. (41) combined with Eq.
(27) results in a predictive model. However, since many simplifications are in-
volved, the results of this model can be considered as only a rough estimate. Fol-lowing the slogan ‘‘like dissolves like’’, a good solvent for a polymer is a solvent for which ds and dp have similar values.
vs
w¼ ðds  dp Þ 2 ð41Þ

2.3 Activity Coefficient Models 33
Fig. 2.11. Closed-loop liquid–liquid equilibria in the PEG–water system. Symbols: experimental data, M ¼ 3:35 kg mol1
(0), M ¼ 8 kg/mol (5), M ¼ 15 kg/mol (4); curves fitted using
Eq. (37c). Reproduced with permission from Ref. 29.Hansen suggested refining the solubility parameter theory by the introduction of contributions from dispersive interactions (d), polar interactions ( p) and hydrogen bond formation (hb), as in Eq. (42) [33].w¼ ½ðds; d  dp; d Þ 2 þ ðds; p  dp; p Þ 2 þ ðds; hb  dp; hb Þ 2  ð42Þ
RT
Recently, Lindvig et al. [34, 35] showed that Eq. (42) systematically overestimates the infinite dilution activity coefficient of the solvent and proposed an alternative expression, Eq. (43).w¼a ½ðds; d  dp; d Þ 2 þ 0:25ðds; p  dp; p Þ 2 þ 0:25ðds; hb  dp; hb Þ 2  ð43Þ
RT

These authors showed that for a number of polymer–solvent system with a ¼ 0:6 this method performs similarly to group contribution methods using volume frac-tions to represent the segment fractions in the Flory–Huggins model. Values of solubility parameters are tabulated by Barton [36].
2.3.3
Correlation of Solvent Activities The combination of Eq. (35) with one of the Eqs. (36) or similar empirical equa-tions is not very successful for describing the phase behavior of polymer–solvent systems with strong interactive species and of systems which only differ in mo-lecular weight. Since the mid-1990s activity coefficient models have been proposed based on a combination of the Flory–Huggins type of expression for the combina-torial entropy of mixing and segment-based local composition models to account for the contribution from energetic interactions (the residual contribution to the activity coefficient).In 1993 Chen [37] proposed a correlative model that used a combination of the Flory–Huggins expression for the combinatorial entropy of mixing and the non-random two-liquid (NRTL) theory [38]. The same approach was followed by Wu et al. [39]. These authors used Freed’s expression [41, 42] for the combinatorial entropy of mixing, which is more accurate than the Flory–Huggins expression, in combination with the NRTL theory. The nonrandom factor model of Haghtalab and Vera [42] was modified by Zafarini-Moattar and Sadeghi [43] for polymer solu-tions. In 2004, Pedrosa et al. [44] suggested use of the UNIQUAC theory [45] in-stead of the NRTL theory and tested various combinations of combinatorial con-tributions and residual contributions using a database of 70 low-pressure VLE systems. These authors have concluded that the combination of a segment-based NRTL or UNIQUAC residual term with a good combinatorial term is able to pro-duce the best correlations of experimental VLE data and can also be used to predict the influence of the molecular weight of the polymer on polymer–solvent VLE.As an example we will give here the expression for the activity coefficient of the solvent for a model which combines the p free-volume combinatorial term with a segment-based Wu-NRTL residual term [44]. The p free-volume combinatorial term combines combinatorial contributions and free-volume contributions [45].The activity coefficient of the solvent is given by Eq. (44).ln gs ¼ ln gscombfv þ ln gsres ð44ÞThe combinatorial/free-volume term is given by Eq. (45).
fsfv f fv
ln gscombfv ¼ ln þ1 s ð45Þ
xs xs
The free-volume fraction fsfv is calculated from Eq. (46).

2.3 Activity Coefficient Models 35
xj vi
The free volume of a component is defined by Eq. (47), where vi is the molar volume of liquid component i, vi; vdW is the hard-core volume or van der Waals volume of this component and can be calculated using the Bondi tables [46], and p is a correction factor, which is calculated from Eq. (48) in the p free-volume model [45].vi ¼ ðvi  vi; vdW Þ p ð47Þ
vs
p¼1 ð48Þ
vp
The residual term is given by Eq. (49) together with Eqs. (50), where asp and a ps are adjustable interaction parameters, a is the NRTL-nonrandomness parameter,which was fixed by Pedrosa et al. at 0.4.tps Gps tsp Gsp2 ln gsres ¼ qs X p2 þ ð49ÞðXs þ X p Gps Þ 2 ðX p þ Xs Gsp Þ 2


a ij
tij ¼ exp and Gij ¼ expðatij Þ ð50Þ
RT
X i is the effective mole fraction of segments of species i and qi is the effective seg-ment number of species i, which are given by Eqs. (51) and (52); rs ¼ 1 and rp ¼ r.
x i qi
Xi ¼ X ð51Þ
xj q j


qi ¼ ri 1  2a 1  ð52Þ
2.3.4
Group Contribution Models Thermodynamic properties can be predicted from group contribution methods.In these models molecules are divided in functional groups. Group contribution models for activity coefficients consider the interactions between functional groups rather than between molecules. Since the number of functional groups is much lower than the number of possible molecules composed of these groups, only a limited number of group interaction parameters have to be known to describe a large number of systems. These group interactions are obtained from regression

of experimental data. This makes the group contribution methods purely predic-tive. However, since details of molecular structure are not considered, group contri-bution methods for activity coefficients are in general less accurate than correlative models.Oishi and Prausnitz [47] proposed writing the solvent activity coefficient for polymer–solvent systems as the sum of three terms [Eq. (53)].ln gs ¼ ln gscomb þ ln gsfv þ ln gsres ð53ÞThe combinatorial contribution gscomb accounts for differences in size and shape
fv
of the molecules. The free-volume contribution gs accounts for changes in free volume due to mixing, caused by the large difference between the free volumes of pure solvent and polymer. For ordinary liquid mixtures in far from critical condi-tions, this term is usually negligible. The residual contribution gsres accounts for energy interactions. In the approach of Oishi and Prausnitz, the combinatorial con-tribution is represented by the Staverman–Guggenheim expression, a modification of the Flory–Huggins equation, also used in the UNIFAC group contribution model [48]; for the residual contribution also, the corresponding expression of the UNIFAC model is used. The free-volume contribution is calculated from the Flory equation of state [49]. This group contribution model and those of Chen et al. [50]and Danner and High [51] are discussed in Ref. 4. Here we will discuss the en-tropic free volume model of Elbro et al. [52] and Kontogeorgis et al. [53], and the group contribution Flory (GC–Flory) model of Bogdanic et al. [54, 55], which is a modification of the model of Chen et al. [50].In the entropic-free volume model, the activity coefficient of the solvent is given by Eqs. (44)–(48) with p ¼ 1 [52]. The residual contribution is represented by the residual contribution of the UNIFAC model with temperature-dependent interac-tion parameters [53]. The liquid molar volumes needed for the calculation of the free volume of a component can be taken from experiment or calculated from the Tait equation [4] or by the group contribution method of Elbro et al. [56]. This model is relatively easy to use.In the GC–Flory model, the solvent activity coefficient is calculated from Eq.
(53).
The combinatorial term is calculated by means of the original Flory–Huggins ex-pression, Eq. (54).
fi f
ln gicomb ¼ ln þ1 i ð54Þ
xi xi
The free-volume and residual terms are calculated from a modification of the orig-inal Flory equation of state, Eq. (55), where ~v is the reduced volume, defined by Eq.
(56).
RT v~1/3 þ C E attr
P¼  ð55Þ
v v~1/3  1 v

2.3 Activity Coefficient Models 37
v
The molar hard-core v  is calculated from the pure component hard-core molar volumes vi using a linear mixing rule [Eq. (57)]. The same type of mixing rule is used for the number of external degrees of freedom parameter C [Eq. (58)].
X
v ¼ x i vi ð57Þ
X
C¼ x i Ci ð58ÞThe pure component hard-core volumes and C parameters are calculated from the group contribution expressions in Eqs. (59) and (60),
X
vi ¼ 21:9662 uðiÞ
m Rm ð59Þ
m
X 
 X
1 1 R
Ci ¼ uðiÞ
n CT0 ; n þ CT; n  þ X n Cn0 ð60Þ
T T0 n Rm
m
ðiÞ
where un is the number of groups of type n in molecule i and T0 is a reference temperature. R n is the normalized van der Waals volume of group n as used in the UNIFAC method. The attraction term E attr is related to the UNIFAC model by Eq. (61), z is the lattice coordination number, chosen to be equal to 10, and qi ,the surface area of molecule i, is given by Eq. (62)
2 X 3
yj expðDeji /RTÞDeji
X1 6
6 j 7
E attr ¼ zqi x i 6
4eii þ
X 7
2 y expðDe /RTÞ 5
k ki
k

1/3 
v~  1 X
X ðiÞ  3R ln xi un CT; n ð61Þ
v~ i n
X
qi ¼ vðiÞ
n Qn ð62Þ
Q n is the normalized van der Waals surface of group n, as in UNIFAC. The inter-action energy parameters eji and Deji are given by Eqs. (63), in which eij0 is calcu-lated from a group contribution expression [Eq. (64), together with Eq. (65)].
eij0
eij ¼ and Deij ¼ eij  eii ð63Þ

X X
eji0 ¼ yðiÞm yðn jÞ enm ð64Þ
m n
enm ¼ ðenn emm Þ 1/2 þ Denm ð65ÞIn these expressions the volume fraction fi of molecule i, the segment fraction yi of molecule i, and the segment fraction yðiÞn of n in molecule i are defined by Eqs.
(66)–(68).
xiv 
fi ¼ X i  ð66Þ
xj vj
x i qi
yi ¼ X ð67Þ
xj q j
ðiÞ
u Q
yðiÞ
n ¼
Xn n ð68Þ
uðiÞ
m Qm
Note that indices m and n refer to groups m and n, and i and j to molecules i and j.The resulting expressions for the free-volume contribution and the residual con-tribution in the activity coefficient are Eqs. (69) and (70).
1/3
fv v~i  1 v~i ln gi ¼ 3ð1 þ Ci Þ ln  Ci ln ð69Þv~1/3  1 v~
1 1 X
ln gires ¼ zqi 4 ½eii ð~ vi Þ þ 1  ln vÞ  eii ð~ yj expðDeji /RTÞ
2 RT j
X yj expðDeji /RTÞ
 X 5 ð70Þ
j yk expðDe ki /RTÞ
k
To apply this method, the liquid molar volumes of the mixture and of the pure components need to be known. At a given pressure and temperature these values can be calculated from the equation of state. However, since eji according to Eq.
(63) is volume-dependent, this involves an iterative procedure similar to that
described by Danner and High [4] for the method of Chen et al. [50]. Figure
2.12 shows the experimental solvent activity of the system poly(propylene oxide)–
benzene at 347.85 K compared to the correlation by UNIQUAC and predictions by the GC–Flory model. The result of the correlation is almost perfect. The predicted solvent activities by the GC–Flory model are also very close to the experimental values. In Figure 2.13 a comparison is shown of experimental solvent activity

log Ω∞cal
log Ω∞exp
Fig. 2.13. Comparison of calculated activity coefficients at infinite dilution using the GC–Flory model with experimental values for many polymer–solvent systems. Reproduced with permission from Ref. 54.and co-workers and can be used to model the phase behavior of mixtures of small and large molecules, including polymers, over a wide range of pressure and tem-perature [57–60]. Here, we will discuss only the two equations of state methods:the relatively simple Sanchez–Lacombe (S–L) lattice fluid model [61, 62], and the statistical associating-fluid theory (SAFT) [63, 64], which has now become one of the standard equations of state for polymer solutions.
2.4.1
The Sanchez–Lacombe Lattice Fluid Theory Like the Flory–Huggins model, the Sanchez–Lacombe lattice fluid theory is based on the assumption that segments of solvent molecules and polymer molecules oc-cupy the lattice sites of a rigid lattice, but vacant lattice sites are also allowed. The number of vacant lattice sites, and as a consequence the total number of lattice sites, are pressure-dependent, and in this way compressibility is introduced.

2.4 Equation of State Models 41
The resulting equation of state for a pure component is given by Eq. (71).


p~v~ 1 1 1
¼  1 þ v~ ln 1   ð71ÞT r v~ v~T~The reduced volume v~, the reduced pressure p~, and the reduced temperature T~ are defined by Eqs. (72a)–(72c).
v n0 þ rn
v~ ¼ ¼ ð72aÞ
v rn
P Pv 
p~ ¼ 
¼  ð72bÞ
P e
kT
T~ ¼  ð72cÞThe parameters with an asterisk ( ) are the so-called characteristic parameters. In practice r, the number of segments per molecule, e  , the interaction energy param-eter, and v  , the molar volume of a lattice site, are used as the independent pure-component parameters; n is the number of moles of the component and n0 is the number of moles of vacant lattice sites. If v is the volume per mole of segments,the total volume of the system is given by V ¼ nrv. For high molecular weights r is large, so it can be concluded from Eq. (71) that the density of polymer melts is not very dependent on molecular weight and that the PVT behavior of polymer melts follows the corresponding states principle (see Figure 2.14) [62].For mixtures the same equation of state is used, but the characteristic parame-ters r; e  , and v  are composition-dependent. Neau [65] gives an overview of differ-ent mixing rules proposed in the literature. Often-used mixing rules are given in Eqs. (73)–(75). In Eq. (75) k ij is an adjustable binary interaction parameter which equals zero for i ¼ j and which can fitted to binary experimental data.
X
r¼ x i ri ð73Þ
X
v ¼ ji vi ð74ÞXX qffiffiffiffiffiffiffiffie ¼ ji jj eii ejj ð1  k ij Þ ð75ÞThe segment fraction ji of component i is given by Eq. (76).
x i ri
ji ¼ X ð76Þ
x i ri

Fig. 2.14. Corresponding states behavior of various liquid–polymer PVT data according to the Sanchez–Lacombe model.Symbols represent experimental data; curves are calculated from Eq. (71). Reproduced with permission from Ref. 62.Neau [65] also showed that the expressions for the chemical potential used in earlier literature to calculate phase equilibria are thermodynamically inconsistent.According to Neau, the correct expression for the fugacity coefficient for the SL model is Eq. (77).

2.4 Equation of State Models 43

 !"
 #

1 1 z1 nr qv ln j^i ¼ ln z þ ri 2  ln 1  þ X v~T~ v~ xj r j v  qn i nj 0n i
"
 #
1 nr qe 
 ð77Þ
v~T~ e  qn i nj 0n i The compressibility factor z is given by Eq. (78).
Pv p~v~
z¼ ¼ r ð78Þ
RT T~
In Figure 2.15 [66] experimental isothermal cloud-point curves of the linear low density polyethylene þ hexane system are compared with the results of a fit of these data using the Sanchez–Lacombe equation of state. The pure component parameters of hexane were calculated from the critical point of hexane and its acentric factor [67]. The pure component parameters of the polymer were obtained from a simultaneous fit of PVT data and the data presented in Figure 2.15. The equations solved were those described by Koak and Heidemann [68]. The binary interaction parameter was linearly dependent on temperature. The polymer was
P / MPa
450 K
470 K
490 K
2 Critical Point Cloud Point
1 Spinodal
Critical Point
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Weight fraction LLDPE Fig. 2.15.Isothermal cloud-point curves of LLDPE þ n-hexane.Symbols: experiments; curves: modified Sanchez–Lacombe fit.[66].

represented by 36 pseudo-components. The SL theory is very well able to describe the experimental phase behavior.
2.4.2
Statistical Associating-fluid Theory The statistical associating-fluid theory (SAFT) developed by Chapman et al. [63, 64]is based on the thermodynamic perturbation theory of Wertheim [69]. Since itfirst appeared, many different versions of SAFT have been published. The different SAFT versions and their application have been reviewed by Muller and Gubbins[70].For polymer solutions the SAFT version of Huang and Radosz [71, 72] is the most widely used. In 2000, a promising new version of SAFT for polymer solutions called PC-SAFT (perturbed chain-SAFT), was proposed by Gross and Sadowski[73]. Here we will restrict ourselves to these two SAFT versions.The basics of both equations of state are equal, and can be written as separate contributions to the molar Helmholtz energy a. The molar residual Helmholtz en-ergy a res, which is the difference between the molar Helmholtz energy of the sys-tem and the molar Heltmholtz energy of the same system in the ideal gas state in the same conditions of temperature, pressure, and composition, is calculated as the sum of the contributions of a hard chain term a hc , a dispersion term a disp , and an association term a assoc [Eq. (79)].a res ¼ a  a ig ¼ a hc þ a disp þ a assoc ð79ÞFrom this expression the pressure and chemical potential can be derived [Eqs.
(80) and (81), respectively] using standard thermodynamic relationships
(A ¼ n i a).


qA
p¼ ð80Þ
qV T; n


qA
mi ¼ ð81Þ
qn i V; T; nj0i For polymer solutions the association term is normally not used. The two SAFT versions discussed here do not explicitly account for polarity and differ only in the way the dispersion contributions are calculated.
2.4.2.1 SAFT and PC-SAFT Hard Chain Term
In the molecular picture behind SAFT a chain consists of mi hard-sphere seg-ments. These hard-sphere segments are bonded by covalent bonds. The hard-sphere term of both SAFT versions is the sum of two contributions: a hard-sphere contribution and a term due the connectivity of these hard-sphere segments, as

2.4 Equation of State Models 45
given by Eq. (82), where m is the average chain length of the molecules in the mix-ture [Eq. (83)] [64].a hc a hs a chain
¼m þ ð82Þ
RT RT RT
X
m¼ x i mi ð83ÞThe hard-sphere contribution a hs is represented by the Boublik–Mansoori hard-sphere equation of state for mixtures of hard spheres, Eq. (84) [74, 75].
"
3  #
a hs 6 z23 þ 3z1 z2 z3  3z1 z2 z32 z2¼ þ 2  z0 lnð1  z3 Þ ð84ÞRT pr z3 ð1  z3 Þ 2 z3 In this equation zk is given by Eq. (85), where r is the number density, dii is the temperature-dependent hard-sphere diameter obtained from Eq. (86); mi , the hard-sphere diameter si, and the energy parameter ei are pure component pa-rameters.
p X
zk ¼ r x i mi ðdii Þ k ð85Þ


3ei
dii ðTÞ ¼ si 1  0:12 exp ð86Þ
kT
The chain term a chain is given by Eq. (87), where gðdii Þ hs is the so-called hard-sphere radial distribution function at close contact [Eq. (88)] [74, 75].
a chain X
¼ x i ð1  mi Þ ln½gðdii Þ hs  ð87Þ


1 didj 3z2 d i d j 2 3z22 gijhs ¼ þ þ ð88Þð1  z3 Þ d i þ d j ð1  z3 Þ d i þ d j ð1  z3 Þ 3
2.4.2.2 SAFT Dispersion Term
In the Huang and Radosz version of SAFT [71, 72] the Chen–Kreglewski disper-sion term is used. This term is obtained from a fit to the physical property data of argon and is given by Eq. (89) [76], where t is a constant equal to 0.74048. The con-stants Dij are given by Chen and Kreglewski [76].
 i   j
a disp X 4 X 9
u z3
¼ Dij ð89Þ
RT i¼1 j¼1
kT t

For mixtures, the van der Waals one-fluid mixing rules or the volume fraction mix-ing rules can be used.The van der Waals mixing rules are given by Eq. (90), where the combining rules are Eqs. (91) and (92), in which k ij is an adjustable binary interaction parameter.
XX  
uij 3
x i xj d
u i j
kT ij
¼ XX ð90Þ
kT x i xj dij3
di þ dj
dij ¼ ð91Þ
pffiffiffiffiffiffiffiffiffi
uij ¼ ð1  k ij Þ uii ujj ð92ÞThe volume fraction mixing rules are given by Eq. (93), with volume fractions de-fined by Eq. (94).
u XX uij
¼ fi fj ð93Þ
kT i j
kT
x i mi d 3i fi ¼ X ð94Þxj m j d 3j The combing rule for uij is again given by Eq. (92).
2.4.2.3 The PC-SAFT Dispersion Term
In SAFT the dispersion term represents the interactions between individual seg-ments, while in PC-SAFT the dispersion term represents the interactions of chains of segments. The expression for a disp derived by Gross and Sadowski is Eq. (95),where the terms on the right-hand side are defined by Eqs. (96) and (97).a disp a1 a2
¼ þ ð95Þ
RT RT RT

 X 6
a1 u
¼ 2prm 2 s3 a i ðmÞh i ð96Þ
RT kT i¼0
" #1
a2 8h  2h 2 20h  27h 2 þ 12h 3  2h 4¼ prm 1 þ m þ ð1  mÞRT ð1  hÞ 4 ð1  hÞ 2 ð2  hÞ 2


2 u 2 3X 6
m s bi ðmÞh i ð97Þ
kT i¼0

2.4 Equation of State Models 47
The reduced fluid density h is defined by Eq. (98), in which NAv is Avogadro’s number.
pNAv
h¼ rmd 3 ð98ÞThe parameters a i and bi are dependent on m, as Eqs. (99) and (100) state.m1 ðm  1Þðm  2Þa i ðmÞ ¼ a 0i þ a 1i þ a 2i ð99Þ
m m2
m1 ðm  1Þðm  2Þbi ðmÞ ¼ b0i þ b1i þ b2i ð100Þ
m m2
The constants a 0i –b2i are fitted to the thermophysical properties of n-alkanes and are given by Gross and Sadowski [73].
2.4.2.4 SAFT and PC-SAFT Applications
Both SAFT and PC-SAFT contain pure component parameters: the energy parame-ter e or u, the hard-sphere diameter s, or the hard-sphere volume v 00 , and the num-ber of segments m per molecule. For small (solvent) molecules these parameters are obtained from a fit of vapor pressure data and saturated liquid volume data.Since they do not have a vapor pressure, this fit is not possible for polymers, and the pure component polymer parameters are obtained from a fit to PVT data of the molten polymer or from a fit to PVT data and binary phase equilibrium data. For the description of a mixture one needs one binary interaction parameter k ij per binary, which has to be fitted to phase equilibrium data. If necessary, k ij can be made temperature-dependent. In general, phase equilibria are very sensitive to the k ij value.In Figure 2.16 [77], experimental results from a system of ethylene þ HDPE are compared with modeling results using SAFT and Sanchez–Lacombe models. In both cases k ij is taken to be linearly dependent on temperature. Due to the polydis-persity of the polymer, the cloud-point curves show a dip in which the critical point is located. If in the modeling the polymer is assumed to be monodisperse, this behavior cannot be reproduced. The figure shows that both SAFT and Sanchez–Lacombe models give a reasonable description of the experimental phase behavior,although at high and low polymer concentrations the deviations become larger.The same system was modeled by Tumakaka et al. [78] using SAFT and PC-SAFT. The results are presented in Figure 2.17, which clearly shows that in this case PC-SAFT gives a better result than SAFT. In the same paper these authors present PC-SAFT modeling results for LDPE þ solvent systems at constant poly-mer concentration. The pure LDPE parameters were fitted to the experimental data of ethane þ LDPE. These parameters were subsequently used to describe the LDPE þ ethane, þ propene, þ propane, þ butane, and þ 1-butene systems, using a

Fig. 2.16. Isothermal cloudpoint curves of the HDPE þ ethylene system. Mn ¼ 43 kg mol1 , Mw ¼ 118 kg mol1 , Mz ¼ 231 kg mol1 . Symbols: experimental data;curves: modeling results: (a) SAFT model; (b) Sanchez–Lacombe model. Reproduced with permission from Ref. 77.temperature-independent k ij . The results are very good (see Figure 2.18), but it should be kept in mind that the modeling is restricted to one polymer concen-tration.
2.4.2.5 Extension to Copolymers
For the modeling of the phase behavior of copolymer–solvent systems, the copoly-mer can be treated as a homopolymer with effective pure component parameters.Examples of this approach are given by McHugh and co-workers [79, 80]. The dis-advantage of this approach is that the pure component polymer parameters depend on the type and composition of the copolymer. Pure component polymer parame-ters are obtained from binary polymer–solvent phase equilibrium data. With these parameters it is possible to model the phase behavior of the same polymer with another solvent.A better approach is the copolymer SAFT approach of Radosz and co-workers[81–83], in which the copolymer parameters are estimated on the basis of the mo-lecular weight and structure only. For an AB-type copolymer there are three binary interaction parameters, the interaction parameters between A segments and seg-ments of the solvent molecule, the interaction parameter between B segments

2.4 Equation of State Models 49
Fig. 2.17. Isothermal cloud-point curves of the HDPE þ ethylene system. Mn ¼ 43 kg mol1 , Mw ¼ 118 kg mol1 , Mz ¼ 231 kg mol1 . Symbols: experimental data;curves: modeling results. Reproduced with permission from Ref. 78.and segments of the solvent molecule, and the interaction parameter between A and B segments. The first two binary interaction parameters can be obtained from the phase behavior of the two homopolymer systems, while the third has to befitted to some copolymer–solvent data. Once these parameters are known, predic-tions can be made for copolymer–solvent systems with the same type of copolymer but with a different copolymer composition. The same approach was followed by Gross et al. [84] for the PC-SAFT model. The result is known as copolymer PC-SAFT. The two parameters that characterize the polymer structure are the fraction of type A segments in the polymer molecule and the bonding fraction which gives the fraction of bonds between segment types A and B. The original literature gives details.Figures 2.19 and 2.20 show experimental cloud-point curves and PC-SAFT mod-eling of the ethylene þ poly(E-co-EA) and ethane þ poly(E-co-BA) systems, respec-tively [85], at a polymer concentration of 5 wt.%. The model correctly predicts the change in the location of the cloud point with changing comonomer concentra-tion in the polymer. Especially interesting is the ethylene þ poly(E-co-EA) system,in that the curve of cloud-point pressure as a function of the EA concentration in the polymer shows a minimum. This behavior is correctly described by the model.

Fig. 2.18. Constant composition cloud-point curves of LDPE–solvent systems at 5 wt.% polymer. Symbols: experimental data; curves: modeling results using PC-SAFT with one temperature-independent k ij for each system. Reproduced with permission from Ref. 78.
2.5
Conclusions Polymer–solvent systems behave in many respects in the same way as systems of low molecular weight components. Differences between polymer–solvent systems and low molecular weight systems are mainly caused by the fact that polymers have no vapor pressure, that polymers are composed of many components of different molecular weights, and that there is a large difference between the free volumes of the solvent and of the polymer.The thermodynamic models for polymer–solvent systems are less advanced than for systems of low molecular weight compounds. In general low-pressure vapor–liquid equilibria can be described very well with a variety of models. Once the ad-justable parameters in these models are fitted to experimental data, reliable predic-tions can be made for other conditions, for example at a different temperature or for a system with the same solvent and the same type of polymer with a different

Fig. 2.19. Constant composition cloud-point the range 113–157 kg mol1 . Symbols:curves for poly(E-co-EA)–ethylene systems with experimental data; curves: PC-SAFT different repeat-unit compositions at 5 wt.% calculations. Reproduced with permission from polymer. The copolymer molecular weight is in Ref. 85.molecular weight. Vapor–liquid-equilibria can be predicted with different group contribution models. Most recent models give reliable predictions of comparable accuracy for different polymer–solvent systems. A disadvantage of this type of model is that the parameters for the groups of interest should be available.In general the thermodynamic modeling of low-pressure liquid–liquid equilibria is more difficult than for vapor–liquid equilibria. This also holds for polymer–solvent systems. Reliable prediction methods are not available. The correlation of liquid–liquid data using the extended Flory–Huggins type models [27–29] gives reasonably good results. Again, once the adjustable parameters in these models are known, predictions can be made for other conditions.High-pressure fluid-phase equilibria can only be modeled using equations of state. However, the equation of state models contain adjustable binary interaction parameters that have to be fitted to data. Small variations in these parameters in general have a large influence on the predicted phase equilibria. The most promis-ing models for high-pressure phase equilibria of polymer solutions are the SAFT and the PC-SAFT ones.

Fig. 2.20. Constant composition cloud-point the range 155–283 kg mol1 . Symbols:curves for poly(E-co-BA)–ethylene systems with experimental data; curves: PC-SAFT different repeat-unit compositions at 5 wt.% calculations. Reproduced with permission from polymer. The copolymer molecular weight is in Ref. 85.
Notation
Symbols
a activity, molar Helmholtz energy A Helmholtz energy C number of external degrees of freedom d temperature-dependent hard-sphere diameter DC constant
f fugacity
g molar Gibbs energy; interaction parameter; radial distribution
function
G Gibbs energy Gij NRTL parameter G Gibbs energy per mole of lattice sites k Boltzmann’s constant k ij binary interaction parameter m number of polymer components; number of segments M molecular weight n number of moles

N number of components p correction factor
P pressure
q effective segment number; surface area Q normalized van der Waals surface r number of segments in a molecule rn number-average chain length R gas constant Rm ; Rn normalized van der Waals volume T absolute temperature u energy parameter v molar volume
V volume
w weight fraction x; y mole fraction X segment fraction z lattice coordination number; compressibility factor
Greek
a NRTL nonrandomness parameter a; b; p phase g activity coefficient (mole fraction basis)d solubility parameter; density e energy parameter y theta temperature; segment fraction jC segment fraction; volume fraction f fugacity coefficient m chemical potential W activity coefficient (weight fraction basis)s temperature-independent hard-sphere diameter t NRTL interaction parameter; constant in SAFT z density-related variable [Eq. (85)]h reduced density u number of groups w interaction parameter
Subscripts
d dispersion hb hydrogen bonding i; j component
mix mixing
p polymer; polar
s solvent
vdW van der Waals

Superscripts 0 standard state

hard core, indicates characteristic parameter
@ reduced
assoc association attr attraction chain chain formation contribution comb combinatorial crit critical disp dispersion
E excess
fv free volume hc hard chain hs hard sphere id ideal mixture ig ideal gas
L liquid
res residual sat saturated V gas; vapor
Acronyms
LCST lower critical solution temperature LDPE low-density polyethylene LLDPE linear low-density polyethylene LLE liquid–liquid equilibria NRTL nonrandom two-liquid PCHT perturbed hard-chain theory poly(E-co-BA) poly(ethylene-co-BA)poly(E-co-EA) poly(ethylene-co-EA)SAFT statistical associating-fluid theory S–L Sanchez–Lacombe lattice fluid model UCST upper critical solution temperature VLE vapor–liquid equilibria
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53 G. M. Kontogeorgis, A. Freden- Phys, 1971, 54, 1523.slund, D. Tassios, Ind. Eng. Chem. 76 S. S. Chen, A. Kreglewski, Ber.Res. 1993, 32, 362. Bunsenges Phys. Chem. 1977, 81, 1048.54 G. Bogdanic, A. Fredenslund, Ind. 77 N. Koak, R. M. Visser, Th. W. de Eng. Chem. Res. 1994, 33, 1331. Loos, Fluid Phase Equilib. 1999, 158–55 G. Bogdanic, A. Fredenslund, Ind. 160, 835.Eng. Chem. Res. 1995, 34, 324. 78 F. Tumakaka, J. Gross, G. Sadowski,56 H. S. Elbro, A. Fredenslund, P. Fluid Phase Equilib. 2002, 194–197,Rasmussen, Ind. Eng. Chem. Res. 541.1991, 30, 2576. 79 B. M. Hasch, M. A. McHugh, J.57 S. Beret, J. M. Prausnitz, Polym. Sci. B 1995, 33, 715.Macromolecules 1975, 6, 878. 80 S. H. Lee, B. M. Hasch, M. A.58 S. Beret, J. M. Prausnitz, AIChE J. McHugh, Fluid Phase Equilib. 1996,1975, 21, 1123. 117, 61.59 M. D. Donohue, J. M. Praunitz, 81 M. Banaszak, C. K. Chen, M.AIChE J. 1978, 24, 849. Radosz, Macromolecules, 1996, 29,60 D. D. Liu, J. M. Pruasnitz, Ind. Eng. 6481.Chem. Process Des. Dev. 1980, 19, 205. 82 C. Pan, M. Radosz, Ind. Eng. Chem.61 I. C. Sanchez, R. H. Lacombe, J. Res., 1998, 37, 3169.Phys. Chem. 1976, 80, 2352. 83 C. Pan, B. Folie, M. Radosz, Ind.62 I. C. Sanchez, R. H. Lacombe, Eng. Chem. Res., 1999, 38, 2842.Macromolecules 1978, 11, 1145. 84 J. Gross, O. Spuhl, F. Tumakaka, G.63 W. G. Chapman, K. E. Gubbins, G. Sadowski, Ind. Eng. Chem. Res., 2003,Jackson, M. Radosz, Fluid Phase 42, 1266.Equilib. 1989, 52, 31. 85 F. Becker, M. Buback, H. Latz, G.64 W. G. Chapman, K. E. Gubbins, G. Sadowski, F. Tumakaka, Fluid Phase Jackson, M. Radosz, Ind. Eng, Chem. Equilib. 2004, 215, 263.Res. 1990, 29, 1709.

Mário Rui P. F. N. Costa and Rolf Bachmann
3.1
Basic Concepts
3.1.1
A Brief History The concept of producing larger molecules starting from smaller ones containing suitable groups (such as carboxyls and hydroxyls), by reaction to create a bond be-tween inert chemical moieties by splitting off a smaller molecule (a condensation),can be traced back to the middle of the nineteenth century [1]. It was discovered when preparing oligomers of ethylene glycol, but, in spite of the good impact the work had at that time, it fell into oblivion a little later. Phenol–formaldehyde (in the form of resins and varnishes) was the first commercial product (in 1907) con-sisting of an entirely synthetic polymer [2]; its formation was also an example of polycondensation; but this early development was purely empirical and contrib-uted very little to scientific progress.Staudinger’s concept of the existence of macromolecules came two decades later[3]. It paved the way to the systematic study aimed at producing synthetic fibers carried out by Carothers at DuPont in the late 1920s and early 1930s [4], from which stems the still-used concepts of addition and condensation polymerizations.In his famous book, Flory [5] justly criticized this terminology, since closely related polymerization mechanisms involving reactions of groups with active hy-drogen atoms with epoxides and isocyanates do not lead to the splitting of a by-product; he proposed a classification into ‘‘step growth’’ and ‘‘chain growth poly-merization’’. These expressions became widely used, even if they are quite vague;much better would be ‘‘random’’ and ‘‘sequential’’ polymerization [6].A survey of important examples of polycondensation reactions can be found in Table 3.1.
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

Tab. 3.1. A survey of technically important polycondensation reactions.Functional groups or monomers Connecting group By-product aCOOH aOH aCOOa H2 O Carboxylic acid hydroxyl ester aCOOR aOH aCOOa RaOH Ester hydroxyl ester aCOOH aNH2 aCONHa H2 O Carboxylic acid primary amine amide ClCOCl/aCOCl aOH O HCl Phosgene/chloroformate hydroxyl C
O O
carbonate
Na2 S/aSNa aCl aSa NaCl Di-sodium sulfide chloro end group (as in sulfide monomer/sodium sulfide ClC6 H4 Cl)
end group
NaOa aCl aOa NaCl Sodium alkoxide/phenoxide, chloro end group, e.g., ether
e.g., O
CH3 Cl S Cl NaO C ONa O
CH3
aNbCbO aOH O none Isocyanate hydroxyl C
O NH
urethane
aNbCbO aNH2 O none Isocyanate primary amine C
NH NH
urea
CH CH2 aNHa OH none O secondary amine N CH2 CH 1,2-epoxide b-hydroxy-t-amine aCH2 OH (NH2 )2 CO/NH2 CONHa O H2 O Methylol urea/N-alkylurea C
CH2NH NH
N,N 0 -dialkylurea AraCH2 OH Ar 0 aCH2 OH AraCH2 aAr 0 HCHO Aromatic methylol aromatic methylol In some polycondensations, new functional groups are created by reactions of the monomers. Thus, methylol end groups are formed by the initial reactions of formaldehyde with other monomers, and their condensation reactions shown in the last two rows of Table 3.1 occur in later stages of formaldehyde/urea and formaldehyde/phenol polymerizations (see Section 3.3.4).

3.1.2
Molecular Weight Growth and Carothers’ Equation Carothers was mainly interested in polyesterifications, and later in polyamidations,of compounds with the structures AXA and BYB, or sometimes AXB, in which X and Y are inert chemical moieties whereas A and B are the active carboxyl and hy-droxyl or amine groups, giving rise to a connecting ester or amide group Z and splitting off a by-product (water) W [Eq. (1)].AþB!ZþW ð1ÞIf reacting groups A and B do not belong to the same molecule, and thus do not create a ring, this reaction leads to a decrease of one polymer molecule (compris-ing the starting monomers).A count of the polymer molecules and their mass in the system can easily be carried out with the help of p, the conversion of reference groups A. Starting with 1 mol of monomer AXB, the remaining moles of polymer are 1
p. Polymer mass per mole of AXB is equal to its molecular weight MAXB , minus the mass lost in by-product formation, p  MW , yielding Eq. (2), allowing the number-average molecu-lar weight (mass of polymer/moles of polymer) to be predicted.
MAXB
pMW
Mn ¼ ð2Þ
1
p
With two monomers, on introduction of the initial stoichiometric ratio r ¼ ½B0 /½A0 between mole concentrations of groups A and B, a similar reasoning leads to Eq. (3).MAXA þ rMBYB
2pMW
Mn ¼ ð3Þ
1 þ r
2p
High molecular weights (say, above 20 000, needed for use as fibers) are possible only by reaching high conversions when monomer molecular weights are a few hundreds. With two different monomers, stoichiometric ratios very close to unity are also needed. Well-known relationships for number-average degree of polymer-ization xn are obtained from Eqs. (2) and (3) by replacing the molecular weights of monomers by 1 and the molecular weight of by-product by 0.The effect of ring formation on Mn of open chain molecules can be taken into account by introducing pc , the conversion of end groups yielding rings, in Eqs.
(2) and (3). For instance, when there are two monomers AXA and BYB, Eq. (4) is
obtained.MAXA ð1
pc Þ þ ðr
pc ÞMBYB
2ð p
pc ÞMW
Mn ¼ ð4Þ
1 þ r
2ð p
pc Þ

Fig. 3.1. Average degrees of polymerization and sol fraction in XA f polycondensations without intramolecular reaction, versus conversion p of A groups with monomer functionalities f ¼ 2 and f ¼ 3, for equilibrium or batch reaction starting from monomer.Prediction of pc and the average molecular weight of rings is not a trivial task.Carothers has stated that the fraction of rings in bulk polycondensations is negligi-ble, except when five- or six-membered rings (or higher in the case of rings con-taining Si, P, and similar heavier elements lower in the periodic table) can result.Recent experimental results show this concept is not valid in the case of irrevers-ible polycondensations (see Section 3.1.7).Monomers with three or more functional groups lead to the formation of branched polymers. A network of macroscopic dimensions (a gel) appears for high enough conversions and monomer branching. In Figure 3.1 are presented classical predictions (see Section 3.4.3), of average degrees of polymerization and weight fraction of finite molecules (sol fraction) for the single monomer polycon-densations of XA3 and XA2 . Notice that, in contrast with linear polycondensations,which require high conversions for obtaining high values of x n , low-value maxima(4 if f ¼ 3) at gel point are observed with these nonlinear polycondensations.Close to full conversion, the sol is nearly pure unconverted monomer.Stockmayer [7] has obtained the distribution for the fractions of the initial mono-mer units present in each finite polymer molecules (Eq. (5), where Px is the set of isomers with x repeating units X).f ð fx
xÞ!x½Px /½X  ¼ xp x
1 ð1
pÞ xð f
2Þþ2 ð5Þx!ð fx
2x þ 2Þ!

3.1 Basic Concepts 61
Fig. 3.2. Equilibrium distributions of monomer units x ½Px =½X for XA f polycondensations without intramolecular reaction versus conversion p of A groups with monomer functionalities f ¼ 2 and 3.If the functionality f ¼ 2, the even more notorious Schulz–Flory [8] distribution is obtained [Eq. (6)].x½Px /½X  ¼ xð1
pÞ 2 p x
1 ð6ÞThis latter distribution has a relative maximum for x G x n, in contrast to the gen-eral Stockmayer distribution, which decreases monotonically (see Figure 3.2).More complex chemical systems lead to molecular weight distributions that are qualitatively similar: linear polycondensations have most often geometrical or nearly geometrical distributions with polydispersities M w /M n close to 2, whereas nonlinear polycondensations lead to extremely broad distributions near the gel point, but always dominated in number and even in weight by the lower molecular weight species.This chapter concentrates on processes for making either high molecular weight,mainly linear polymers, or else low molecular weight, branched oligomers which will be later used to form networks. In any case, consideration of initial stoichiom-etry, and possible change of functionality due to side reactions, is essential to pre-dict average molecular weights, and to avoid premature gelation in the processes described here.It is also possible to prepare low molecular weight mixtures of ring molecules for further reactive processing (using, for instance, anion polymerization), starting

with monomers [8] or even with recycled polymer [9], by dilution with inert sol-vents and a suitable catalyst (see also Section 3.1.7). The discussion of these other-wise interesting processes falls outside the scope of this chapter.
3.1.3
The Principle of Equal Reactivity and the Prediction of the Evolution of Functional Group Concentrations Carothers’ successful synthesis of high molecular weight, aliphatic polyesters could only be carried out thanks to his care in reducing the effect of the hydrolysis reac-tion through use of high vacuum. Further research carried out by his former col-laborator P. J. Flory in order to assess the effect of molecular weight on chemical kinetics would eventually lead to the establishment of the principle of equal reac-tivity. A parallel research, which also has contributed to the establishment of the same concept and to the prediction of equilibrium chain length distribution(CLD), has been led by G. V. Schulz [10] in Germany.Assuming there is no mutual interference of end groups in the same molecule(which does often happen with groups attached to aromatic rings), the specific rate of consumption, RA , of end groups A by an irreversible reaction reaction with groups B can be written as Eq. (7).
RA ¼ k½A½B ð7ÞEquation (7) states that a single apparent second-order rate constant k should de-scribe all intermolecular reactions of the involved groups A and B for a certain con-centration of catalyst(s) (which may be A or B themselves) and solvent, regardless of the polymer molecule to which they belong. A change of composition of the re-action medium will often cause k also to change.If the reaction is reversible, a more general expression [Eq. (8)] for RA should be used.
RA ¼ k½A½B
k Z ½Z ð8ÞIn the case where, as often happens, a by-product W exists, the apparent first-order reverse rate constant k Z is related, through Eq. (9), to apparent equilibrium ratio K and by-product concentration ½W .k Z ¼ k½W /K ð9Þ
3.1.4
Effect of Reaction Media on Equilibrium and Rate Parameters Polycondensations must be carried out in bulk or with a concentration of solvent as low as possible (viscosity being the limiting factor) in order to minimize the for-mation of rings. If the end groups have polarities and/or hydrogen acceptor or do-

3.1 Basic Concepts 63
nor affinities very different from the bonds, this is likely to lead to changes in the value of apparent constant k, which will be observed as a dependence on conver-sion or stoichiometric ratio.Known examples are the apparent increase in reactivity with conversion in ester-ifications and the strong effect of water concentration on apparent equilibrium and rate constants in amidation systems (see Sections 3.3.1 and 3.3.3 respectively).Should some thermodynamic model of the chemical system be available, knowl-edge of activity coefficients g of chemical groups A; B; Z and by-product W would allow the apparent equilibrium ratio K to be related to the thermodynamic K 0 (a true constant, a function only of temperature) through Eq. (10).gA gB ½Z½W K ¼ K0 ¼ ð10Þg Z gW ½A½BThis kind of computation can be done using the UNIFAC contribution group method [11], but its empirical character and lack of available parameters at high temperatures limit drastically its usefulness.Solvent or bulk media effects can be modeled by taking advantage of group con-tribution methods [12]. Multiple kinetic experiments with different media compo-sitions are needed in order to compare the free energy of interaction involving the hypothetical transition state group (see Ref. 13 for an example). No such studies have ever been tried with polymerization reactions.Progress in computational quantum mechanics should ultimately make the practical use of these correction methods more widespread.In nonlinear polycondensations, it is possible to observe a limiting conversion,the topological limit [14] when unreacted groups in network are too far apart to meet. Rate constants will fall before reaching the topological limit if glass transi-tion temperature Tg is attained (the glass effect). This situation occurs with many thermosetting systems of practical interest, epoxies being the most studied, as dis-cussed in the reviews by Mita and Horie [15] and Dušek [14].A new theory for describing this phenomenon, leaving aside the long-used ‘‘free volume’’ concept, was recently put forward by Corezzi et al. [16]. It is based upon the Adam–Gibbs [17] entropy theory of glass transition. Formation of more cova-lent bonds by polymerization decreases the number of available statistical configu-rations of the chemical system, thus decreasing configurational entropy, suggest-ing Eq. (11) relating structural relaxation time tr to number-average degree of polymerization x n .tr ¼ t0 exp½b g ðTÞx n 
b g0 ð11Þ
bg ¼
T
T0
Function b g ðTÞ is empirical and could be different from that in Eq. (11) [16]. There is for the moment no published use of this theory to fit apparent kinetic constants.

A possible suggestion is the following (untested) correlation with yet another two parameters, k 0 and Cg , based on the Rabinowitch equation for diffusional effects on chemical reactions [Eq. (12)].1/k ¼ 1/k 0 þ Cg tr ð12ÞIn Dušek’s terminology, diffusion control of chemical reactions may be specific,when the bond formation rate depends on the mobilities of individual reacting molecules or substructures, or else nonspecific, when the reaction rate depends only on the average properties of the system.Experimental evidence points toward prevalence of nonspecific rate control: for instance, gel conversion in epoxide resin cure is the same regardless of glass effect[18].Lacking a better alternative, an empirical dependence of rate parameters on con-version, for a given starting composition, must be tried. Examples of this approach will be discussed along with their respective chemical systems (bulk polyester and polyamide formation). It is seen that the glass effect could be taken into account using a similar procedure.
3.1.5
Polycondensation Reactions with Substitution Effects When the reaction of a functional group changes the reactivity of its neighbors, a substitution effect exists.A first-shell substitution effect (FSSE) is the simplest kind of departure from ideal behavior. It is the situation where reactivity is affected only by the reaction of the functional groups attached to the same monomer unit. FSSEs are encountered in many polycondensations, as reactivities of functional groups in monomers are often different from the reactivities of end groups in polymers because of mutual steric, resonance, or electrostatic interactions.For instance, a glycol HOaXaOH shows no substitution effect in an esterifica-tion reaction if the reactivity of the hydroxyls in aCOOaXaOH is the same: a single kind of group aOH needs to be considered and the description of the reaction scheme is much simpler.A second-shell substitution effect (SSSE) occurs when the reactivity of a func-tional group is affected by the reaction of all the groups attached not only to the same monomer unit, but also to the units linked to that one. This behavior has been recognized in urea/formaldehyde formation (see Section 3.3.4.2).Linear polycondensations AXA þ BYB with FSSEs in both root units X and Y are examples found in some important processes, such as the esterification of tereph-thalic acid and ethylene glycol leading to poly(ethylene terephthalate) (PET). Four different rate constants ki are needed to describe forward reactions, and possibly an equal number of apparent rate constants (kiZ , i ¼ 1 . . . 4) are required to describe reverse reactions:

3.1 Basic Concepts 65
k1 k2
AXA þ BYB T AXZY B þ W AXA þ BYZ T AXZY Z þ W
k1Z k2Z
ð13Þ
k3 k4
AXZ þ BYB T BYZX Z þ W AXZ þ BYZ T ZXZY Z þ W
k3Z k4Z
Recall that apparent first-order rate constants of the reverse reactions kiZ are related to the apparent second-order rate constants of the forward reactions ki through the apparent equilibrium ratios Ki and the concentrations of by-product:
ki
Ki ¼ ð14Þ
kiZ ½W 
If there is no by-product, as happens with isocyanate þ hydroxyl reactions, ½W  ¼ 1(dimensionless) should be used in Eq. (14).This example illustrates several difficulties encountered in modeling reversible polycondensation reactions with substitution effects. A major problem is the possi-ble change in the nature of bonds because another bond connecting a different unit has been destroyed. There is no closed set of rate equations in terms of the concentrations of functional groups ½AXA; ½BYB; ½ZD ; ½ZA ; ½ZB , and ½ZP ,even with the help of the two stoichiometric restrictions ½X  ¼ ½AXA þ ½ZD  þ½ZA  þ ½ZP  and ½Y  ¼ ½BYB þ ½ZD  þ ½ZB  þ ½ZP . Introduction of more complex chemical entities does not alleviate the problem.For instance, sequence ZB XZB (a particular kind of monad, a repeating unit X with its pendent chemical groups) is formed by destruction of either of the sequen-ces ZB XZP YZP or ZB XZP YZA (which are dyads, sets of two connected repeating units) by reverse reactions at the rightmost Z group as written above. In order to predict the concentrations of monads, one needs the concentrations of dyads; this demands knowledge of the concentrations of triads, and so on. This hierarchy of equations can not be broken without obtaining the whole chain length distribu-tion; for further details see Section 3.4.5.In fact, the above kinetic model considers SSSEs for the reverse reaction.An FSSE model taking into account reverse reaction must consider equality of rate constants of reverse reactions k1Z ¼ k2Z ¼ k3Z ¼ k4Z ¼ k Z . Since ½ZA  ¼2½AXZA YZA XA þ ½AXZA YZP  and ½ZB  ¼ 2½BYZB XZB YB þ ½BYZB XZP , rates of formation of groups can be written as shown in Eqs. (15).RAXA ¼
4k1 ½AXA½BYB
2k2 ½AXAð½ZD  þ ½ZB Þ þ k Z ð½ZD  þ ½ZA ÞRBYB ¼
4k1 ½AXA½BYB
2k3 ½BYBð½ZD  þ ½ZA Þ þ k Z ð½ZD  þ ½ZB ÞRZD ¼ 4k1 ½AXA½BYB
½ZD ½2k2 ½AXA þ 2k3 ½BYB
ð15Þ
þ k4 ð2½ZD  þ ½ZA  þ ½ZB Þ
k Z ð½ZD 
½ZA 
½ZB ÞRZA ¼ 2k2 ½AXAð½ZD  þ ½ZB Þ þ 2k4 ½ZD 2
½ZA ð2k3 ½BYB þ k4 ½ZB Þ
2k Z ½ZA RZB ¼ 2k3 ½BYBð½ZD  þ ½ZA Þ þ 2k4 ½ZD 2
½ZB ð2k2 ½AXA þ k4 ½ZA Þ
2k Z ½ZB 

In conclusion, descriptions of substitution effects not only require knowledge of more kinetic parameters, but also may lead to added mathematical difficulties,even for just the prediction of basic characteristics of the chemical system, such as the concentrations of functional groups.
3.1.6
Exchange Reactions Exchange reactions without formation of by-products, such as the acidolysis and aminolysis reactions present in polyamidations, do not increase number-average molecular weight, but can be an important cause of relaxation of chain length dis-tributions toward the equilibrium. Kotliar has presented a general review of these reactions [19].Direct bond exchanges in the complete absence of by-product (such as amide–amide exchange) are at best very slow (if they indeed exist) and, as these reactions can be safely neglected, only exchange reactions involving an end group and a bond should be of any importance. The exchange of bonds Z1 and Z2 (necessarily formed with a common by-product W), can be described through Eqs. (16).
k1 k2
XA þ YC T XZ1 Y þ W XB þ YC T XZ2 Y þ W
k1Z k2Z
k12
ð16Þ
XA þ X 0 Z2 Y T XZ1 Y þ X 0 B
k21
These reactions modify the counts of functional groups of similar chemical nature(for example, distinguishable kinds of amides/amines/carboxylic acids). If there is only a single kind of bond, there is no net creation or destruction of func-tional groups or bonds, but a reshuffling of pieces of the reacting molecules takes place.In order not to violate the condition of chemical equilibrium, rate constants of forward and reverse exchange reactions can be related to equilibrium constants of the condensation reactions in which the same functional groups are involved.In this example, the equilibrium ratio of the exchange reaction must be in accord with Eq. (17).
k12 K1
¼ ð17Þ
k21 K2
This operation will reduce the number of kinetic parameters in the kinetic scheme.Also, several of the various equilibrium constants can also be related through a rea-soning due to Gordon and Scantlebury [20] for the XA f polycondensation, which consists of checking the formation of the same final products through successive equilibria but with different intermediate stages.

3.1 Basic Concepts 67
3.1.7
Ring-forming Reactions Prediction of the entropy of cyclization for the gaussian chain molecular model by Jacobson and Stockmayer [21] could relate rate and equilibrium constants of intramolecular ring-forming reactions to their analogous intermolecular counter-parts. Their theory has later been refined [22, 23] as it provides a good test for statistical–mechanical treatments of chain molecules.Measured ring concentrations for thermodynamically controlled (reversible) poly-condensations in bulk are usually only a few percent, so this is not a major factor in those systems.For irreversible polycondensations, recent studies by Kricheldorf [24, 25] using MALDI mass spectroscopy have in several circumstances detected quite an appre-ciable concentration of ring molecules. Unfortunately, dependence of ionization and thus of instrumental response factor of polymer molecules on the nature of the end groups [26] prevents a quantitative exploration of those findings. But it can be concluded that extension of kinetic modeling in order to take into account the presence of rings is more important than was previously acknowledged.The rationale for this unexpected concentration of rings is the irreversible char-acter of their formation. Unless there is a slight excess of some of the end groups,presence of both kinds of end groups in the same molecule will inevitably lead to competition of ring formation with respect to polycondensation. The limiting fac-tor for molecular weight increase becomes ring formation, not closeness to stoichi-ometry of the reagents.In what follows, Cn is a ring with n bonds and L n is a linear molecule containing n
1 bonds. The formation of that ring moiety can occur by intramolecular reac-tion of functional groups, or by a ‘‘back-biting’’ reaction, analogous to the exchange reactions discussed in the previous section, according to the schemes represented in Eqs. (18) and (19).
k c ðnÞ
L n
!
Cn þ W ð18Þ
k cZ
k aE
L m þ Cn

!

L mþn ð19Þ
k cE ðnÞ
Except for small rings, rate constants k cZ , representing breakage of bonds in the ring, and k aE , describing the addition of rings to the end of chains through the ex-change reactions, should be identical to their open-chain counterparts, k Z and k E respectively. The reverse constants depend on the size of the ring. According to the Jacobson–Stockmayer theory, valid for gaussian chains (and therefore only for rings with at least some tens of bonds, and bulk media or with low solvent concen-tration) k c and k cE should be related to the corresponding parameters for intermo-lecular relations through Eqs. (20)–(21).kjc ðnÞ ¼ kj k c ðnÞ ð20Þ

kijcE ðnÞ ¼ kijE k c ðnÞ ð21Þk c ðnÞ ¼ ð3/2pnb 2 Þ 3/2 ð22Þ
NAL
In Eq. (22), the small contributions of the distances of the ends of functional groups b were assimilated to the length of a bond, and NAL is the Avogadro–Löschmidt number.If there is only one kind of bond, an equilibrium constant of cyclization K c ðnÞcan be defined from the equilibrium in Eq. (23).½Cn ½L m K c ðnÞ ¼ ¼ k c ðnÞ/n ð23Þ
½L nþm 
3.1.8
Modeling of Polymerization Schemes Methods for predicting molecular weight distributions at chemical equilibrium and for irreversible polycondensations are presented with some detail below; see Sec-tions 3.4.3 and 3.4.4.Systems at chemical equilibrium are amenable to description by the domain of calculus of probabilities known as the theory of branching processes [27]. Batch re-actions starting from monomers can sometimes be described by the same solu-tions, increasing the practical importance of such an approach. Exchange reactions also drive molecular weight distributions toward equilibrium. Many reactions of great technological interest, such as melt polyesterifications, can be tackled using this approach.Applicability of statistical–probabilistic methods far from chemical equilibrium cannot be guaranteed, and the kinetic approaches described in Sections 3.4.4 and
3.4.5 are a better choice.
Use of statistical–probabilistic methods for describing polycondensations started with Flory, who used them to compute the equilibrium chain length distribution of linear systems and later was able to predict gelation conditions for multifunctional monomers. Stockmayer [7] could extend this method to the computation of chain length distributions and average molecular weights of nonlinear polymers.Gordon [20, 28] recognized that nonlinear polymers in the absence of intramo-lecular reaction can be described by a Galton–Watson branching process. Relatively complex chemical systems, presenting substitution effects, became tractable, and even some properties of polymer networks relevant to rubber elasticity theory [29]and average radius of gyration [30] as well as other polymer properties [31–33]could be computed.Alternative approaches to Gordon’s branching theory have later been developed,mainly the so-called recursive method [34, 35], which is in a number of ways simpler to understand and use, but lacks the power of the older theory in many situations.

3.2 Mass Transfer Issues in Polycondensations 69
Reversible polycondensations can be tackled using the concept of molecular frag-ments introduced in Section 3.1.5. It is possible to establish closed sets of rate equations for those fragments in many important cases (the main difficulty being the presence of higher-order substitution effects). For a more detailed discussion,together with a short analysis of the much simpler case of linear polycondensa-tions with a single kind of bond (a single monomer AXB or two monomers with different groups AXA þ BYB), see Section 3.4.5.It is noteworthy that CLD in those chemical systems (and probably in similar al-ternating polyesterifications and polyamidations) is nearly always fairly close to equilibrium, the main discrepancies occurring as far as the concentrations of thefirst linear oligomers are concerned. So, a method which predicts those concentra-tions correctly is what is really needed in practice. The main factor controlling the CLD should be the presence of multifunctional impurities, which should be care-fully tracked by the kinetic model.Prediction of the whole CLD should be possible only with Monte Carlo methods,with their usual drawbacks [36]. Nevertheless, polycondensations are easier to sim-ulate by Monte Carlo than other polymerizations, since all reactions have more or less the same time scale.These simulations are invaluable for investigating the effect of space correlations between reacting groups and their effect on ring formation and elastic properties[37]. Prediction of molecular size distribution (which hopefully can be determined by size exclusion chromatography) is also a useful result [38] which is difficult to obtain otherwise.
3.2
Mass Transfer Issues in Polycondensations This section deals with situations where mass transfer effects of by-products (devo-latilization, solid-state polymerization) and monomers (interfacial polymerization)become so important as to cause a spatial change in polymer molecular weight distributions.
3.2.1
Removal of Volatile By-products Devolatilization in irreversible polycondensations is carried out in the later stages of the process and is similar to other polymerizations. For reversible polycondensa-tions, however, it differs from the devolatilization of other polymers in a number of ways. Reaction and removal of by-product are intimately connected. Monomers may be removed in substantial quantities. Removal of volatiles takes place during the whole course of the reaction.

In the first stages of the reaction, reversible polycondensates readily form boiling liquids and care has to be taken to avoid removal of monomers, foaming, or prod-uct entrainment.Equilibrium constants are of the order of unity for polyesters and polycarbon-ates, and of the order of hundreds for polyamides. In order to obtain end-group conversions above 0.99, this implies a concentration of by-product ½W  below 10
4 times the concentration of bonds in the former two systems. The high temperature of the processes increases the vapor pressure of the by-products and their equilib-rium solubility decreases, but even so partial pressures of some millibars for poly-ester and polycarbonate processes are required; they may be obtained by using a combination of vacuum and inert stripping gas. These low partial pressures and consequently nearly infinite dilution often justify the application of Henry’s law to describe the vapor–liquid equilibrium according to Eq. (24), in which PW is the equilibrium vapor pressure of by-product W, HW is its Henry constant for the poly-mer melt (in the usual range of 10–1000 bar), wW is mass fraction in the melt, PW is the vapor pressure of pure W, and WW is its weight fraction activity coefficient.PW ¼ HW wW ¼ PW WW wW ð24ÞThe second part of Eq. (24) is useful below the critical temperature of the by-product, when some thermodynamic model is available. For instance, given the densities of melt and by-product respectively as rP ; rW , Flory–Huggins theory leads to Eq. (25).
rP
WW ¼ expð1 þ wÞ ð25Þ
rW
Similar expressions hold for the volatile monomers, which at high dilutions can be treated independently. At low conversions, the infinite dilution is not applicable and the multicomponent Flory–Huggins equation [Eq. (26)], for example, should be used instead. Here the fYi are the volume (or segment) fractions of the various components, i ¼ 1 up to NY being the volatile components and the polymer being component NY þ 1; the yYi are their mole fractions, VYi the molar volumes, and the wij the binary interaction parameters.
rP B X
N Y þ1 f
Yj X
N Y þ1
f NX X
Y þ1 NY þ1 C W Yi ¼ expB@
VYi þ fYi wij
Yi yYi fYk wjk C
A ð26Þ
rYi j¼1
V Y j j¼1
yY i j¼1 k¼ jþ1
j0i
At high conversions, mass transfer resistance to by-product removal is a key factor,not because of low diffusivities D of the by-products (in the range 10
9 to 10
11 m 2 s
1 ), and consequent mass transfer coefficients, but because of low interfacial areas.Because of the huge increase in viscosity of the melt when the reacting mixture changes from the initial mixture of monomers/oligomers to high polymer, a possi-

3.2 Mass Transfer Issues in Polycondensations 71
ble solution is the use of different continuous stirred reactors (CSTRs) in series,with different kinds of stirrers. Other processes consist of simple unstirred bubble In either case, these reacting mixtures at the beginning of the processes resem-ble boiling liquids. Stirring helps the formation of bubbles by cavitation and breaks existing bubbles, increasing interfacial area. Inert gas injection will also help (but loss of volatile monomers becomes more difficult to counterbalance). These mech-anisms become inefficient for high enough molecular weight with the consequent higher viscosity (above 200 Pa s), and special film-forming equipment must be used.Equipment for devolatilization of residual monomers and solvents without chemical reaction is used in most polymerization processes, vented extruders be-ing a common choice. They are also used in the final stages of reversible polycon-densations [39], as well as other specially developed devices that we discuss next.The basic design of these latter systems consists in a partially filled horizontal cylindrical vessel in which the reaction mixture moves axially with the help of a screw or a rotor with blades. The impeller periodically extracts part of the bulk liquid, leaves it exposed as a thin film to the gas phase for a short time t f , and re-mixes it again with the main stream as it flows toward the outlet.According to the mechanism of creation of film, three classes of devices can be distinguished [40–42]: Wiped-film reactors (WFRs) [43], where the impeller blades throw the polymer melt against the inner surface of the vessel (Figure 3.3), subjecting it to a high shear rate (in the range 10 3 –10 4 s
1 ) in the gaps between their tips and the wall. This high shear is favorable to conveying pseudoplastic, high-viscosity poly-mers, but can conversely bring about problems for chains which break under shear, such as polybutylene terephthalate. Vented extruders work using a similar principle in their central section (where deeper screws lead to partial filling of the channel) and their reactor models can be considered a variant of this class. Rotating disk contactors (RDCs) (Figure 3.4), in which the disks periodically dip into the pool of reacting mixture. The polymer film they carry is exposed to the gas phase before being mixed again with the bulk liquid. Falling-strand or falling-film evaporators.Fig. 3.3. Diagrammatic representation of a wiped-film reactor.

Fig. 3.4. Diagrammatic representation of a rotating disk contactor.The currently used description of homogeneous diffusion of volatile by-products in polymer media during reversible polycondensations is due to Secor [44]. It consid-ers polymer molecules immobile. The flux of small molecules has a negligible con-vective contribution; only the diffusional flux with respect to the polymer needs be considered, and the microscopic mass balance of a generic volatile component(usually, but not always, a by-product) Yi and a group A i belonging to the polymer may be written, neglecting density variations, as in Eq. (27).
q½Yi 
¼ DYi ‘ 2 ½Yi  þ RYi
qt
ð27Þ
q½A i 
¼ RA i
qt
As the resistance to mass transfer in the gas phase may be neglected, the diffusiveflux of evaporation N_ Yi , in moles per unit area and unit time, will be obtained with the help of a mass transfer coefficient kfYi . There are two equivalent alternative def-initions [Eq. (28)], one using a driving force in terms of mole concentrations, the other in terms of activities (asterisks mark a concentration or activity value at the interface, considered to lie at y ¼ 0).
q½Yi 
N_ Yi ¼
DYi ¼
kfYi ð½Yi 
½Yi   Þ ¼
kfYðaYw
aYi Þ ð28Þ
qy jy¼0 i
Combining this expression with the microscopic mass balances in the polymerfilm, it is possible to predict mass transfer coefficients for simple geometries and to take into account possible coupling with chemical reactions. For instance, if the polymer film is immobile, has a constant depth L, and mass transfer occurs along the y direction with a plane geometry, time-averaged mass transfer coefficient of

3.2 Mass Transfer Issues in Polycondensations 73
volatile species kfi after an exposure time t f would be computed by solving Eq. (27)with initial and boundary conditions as given in Eqs. (29).½Yi jt¼0 ¼ ½Yi 0 ; ½A i jt¼0 ¼ ½A i 0½Yi jy¼0 ¼ ½Yi   ; ¼0 ð29Þ
qy jy¼L
ð tf
DYi dt
0 qy jy¼0
kfYi t f ¼
½Yi 0
½Yi  In view of the kinds of gas–liquid contact described above, an obvious model is provided by the penetration theory (infinite depth L), where a portion of fluid with uniform concentration profile is suddenly exposed to the gas phase, and is replaced by fresh fluid after an exposure time t f . For an infinite film depth L and negligible chemical reaction, this yields the well-known result [Eq. (30)] for the time-averaged mass transfer coefficient.
sffiffiffiffiffiffiffi
DYi
kfYi ¼ 2 ð30Þ
pt f
Pell and Davis [45] were the first to actually measure a diffusion coefficient for a volatile by-product of a polycondensation, using PET formation in films of varying depth (although obtaining values of D much higher than those nowadays ac-cepted). An early example of discussion of coupling of diffusion/chemical reactions in these systems may be found in Ref. 46.The first model associating the axial transport along the reactor (direction z) with the cross-flow transfer of volatile by product (direction y) (see Figure 3.5) is due to Amon and Denson [47] (Ault and Mellichamp [46] considered that all the polymer was in the film) and was developed for WFRs. It assumes plug flow in the pool,which implies a negligible hold-up of the liquid in the film. A time-averaged mass Volatile by-product
x
Low M JT
polymer inlet Polymer outlet z=0 z z=L y Fig. 3.5. Simple model for a WFR.

transfer coefficient is obtained at each axial position by solving Eqs. (27) and (29),and the mass balance of the pool is written as Eqs. (31), where u z is the axial su-perficial velocity (volumetric flow rate of polymer Q divided by cross-section area of the pool) and a v is the film area per unit volume.
d½Yi 
uz ¼ RYi
kfYi a v ð½Yi 
½Yi   Þ
dz
d½A i 
uz ¼ RA i ð31Þ
dz
½Yi jz¼0 ¼ ½Yi 0 Since the main mass transfer area is the film on the barrel wall, the exposure time would be calculated [47] through Eq. (32), where d T is the inner barrel diameter, L x is the film perimeter (L x G pd T , if the nip is small) and n_ is the screw rotational speed in rotations per unit time.
Lx
tf ¼ ð32Þ
pd T n_
A better model [48] takes into account the movement of the film along the wall with velocity u x ¼ pd T n_ between coordinates x ¼ 0 and x ¼ L x and therefore adds a convection term, leading to Eqs. (33), where the ½A i f and ½Yi f are the concentra-tions of the species in the film. There is no need to consider cylindrical geometry for the film, since its thickness is low.q½Yi f q½Yi f q 2 ½Yi f q½A i f q½A i fþ ux ¼ DYi þ RYi þ ux ¼ RA i qt qx qy 2 qt qx
ðL
q½Yi  q½Yi  uxþ uz ¼ R Yi þ ½Yi f ðt; yÞ dy
u z ½Yi 
qt qz L 0
ðL
q½A i  q½A i  uxþ uz ¼ RA i þ ½A i f ðt; yÞ dy
u z ½A i  ð33Þ
qt qz L 0
½Yi jz¼0 ¼ ½Yi 0 ðtÞ ½A i jz¼0 ¼ ½A i 0 ðtÞ½Yi f jx ¼0 ðt; y; zÞ ¼ ½Yi ðt; zÞ ½A i f jx ¼0 ðt; y; zÞ ¼ ½A i ðt; zÞ
q½Yi f
¼0 ½Yi f ðt; 0Þ ¼ ½Yi  
qy jy¼L
With some modifications, a model of single-screw vented extruders can also be de-veloped. We will present here a slightly extended version of the treatment by Rob-erts [49] and Biesenberger and Sebastian [50]. Fundamental studies on fluid me-chanics and mass transfer without reaction are reported in Refs. 51 and 52.

3.2 Mass Transfer Issues in Polycondensations 75
Film
Fig. 3.6. Scheme of flow and mass transfer in single-screw vented extruders.A scheme of flow and mass transfer in single-screw vented extruders, also illus-trating some key geometrical parameters, is shown in Figure 3.6.As polymer flows inside the channel along a trajectory in a helix, with a total length LB /sin y, in which y is the angle of the screw, the coordinate z will be taken along this helicoidal path. Dimensions of the channel will be LW (width) by H(depth), with a fraction fL filled with liquid. As the movement of screw pushes the polymer pool, a fraction passes through the space between the barrel and the screw, forming the desired evaporating film. If the fluid is Newtonian, the film width is h/2, where h is the clearance. The average transverse velocity of the screw dragging the film, vT , is given by Eq. (34).vT ¼ pd T n_ sin y ð34ÞThe film which is wiped from the channel re-enters at a distance d upstream of its departure point given by Eq. (35) [50] and the time of exposure of the film t f is given by Eq. (36).d ¼ pd T fL cos y ð35Þt f ¼ ð1
fL Þ/n_ ð36ÞThe concentrations ½A i f ðzÞ and ½Yi f ðzÞ respectively for nonvolatile and volatile components in the film, back-mixed at axial position z, are obtained by solving Eq. (27) for t ¼ t f with initial conditions described by Eqs. (37) (a more exact model would consider convection as in Eq. (33)):½Yi jt¼0 ¼ ½Yi ðz þ dÞ; ½A i jt¼0 ¼ ½A i ðz þ dÞ ð37Þ

Assuming plug flow in the channel (a trivial change would be to add an axial dis-persion coefficient), the mass balances in the channel taking into account also the devolatilization from the pool (mass transfer coefficients kfpYi ) thus becomes those given in Eqs. (38).d½Yi  vT h kfpYi uz ¼ RYi þ ð½Yi f
½Yi Þ
ð½Yi 
½Yi   Þdz fL LW H fL LW d½A i  vT h uz ¼ RA i þ ð½A i f
½A i Þ ð38Þ
dz fL LW H
½Yi jz¼0 ¼ ½Yi 0 ½A i jz¼0 ¼ ½A i 0 The presence of the delay d may be circumvented by using a Taylor expansion about z [50] in order to obtain a system of second-order ODE. Also according to Refs. 49 and 50, the exposure time of the pool may be estimated through Eq.
(39).
H
t fP ¼ ð39Þpd T n_ sin y Another integration of Eq. (27) for t ¼ t fP with a trivial modification of Eq. (30) will provide an estimation of the mass transfer coefficients of the pool kfpYi .Twin-screw extruders have the advantage of being self-cleaning and can work with extremely high viscosity, above 10 6 Pa s [53], making them a good choice for polyamide and polyurethane final stages of reaction, thanks to the possibility of using reduced space times. Their detailed modeling is more difficult than with single-screw devices, and few fundamental studies [54] have been carried out.Rates of mass transfer have been predicted for a co-rotating twin screw using pen-etration theory and experiments have been done with a transparent device (for observing whether bubbles were present or not in the devolatilization zone). Ob-served results of kf a v were proportional to the speed of rotation to the power of
0.5, as penetration theory predicts, but were three times lower than theoretical pre-
dictions, which has been attributed to the very low liquid hold-up and consequent lack of coverage.In contrast to WFRs and vented extruders, use of staged models for describing rotating disk contactors is a natural choice [55], since the J compartments with liq-uid hold-ups Vm j can be approximated as CSTRs. Taking into account the possibil-ity of back-flow, the overall volume flow rate leaving the jth CSTR, Q j , will be di-vided into a fraction b j going back to CSTR j
1 and 1
b j going to tank j þ 1,except for the first CSTR, in which b 1 ¼ 0. Also, Q Jþ1 ¼ 0, and the gas phase will be considered well mixed with uniform temperature (see Figure 3.7). Thus, mass balances at a steady state of volatile and nonvolatile components in the jth com-partment ( j ¼ 1, J) may be written as in Eqs. (40).

3.2 Mass Transfer Issues in Polycondensations 77
Fig. 3.7. Staged model of an RDC.Q j
1 ð1
b j
1 Þ½A i j
1 þ Q jþ1 b jþ1 ½A i jþ1 ¼ Q j ½A i j
RA i Vm j Q j
1 ð1
b j
1 Þ½Yi j
1 þ Q jþ1 b jþ1 ½Yi jþ1 ð40Þ¼ Q j ½Yi j
RYi Vm j þ kfYij avj Vm j ð½Yi j
½Yi   ÞMass transfer coefficients can be computed using penetration theory as described Murakami et al. [56] obtained correlations for hold-up and mixing time in RDCs.Fractional dead space (between 0 and 18% for their experimental data) can be pre-dicted from the mixing time. This dead space should obviously be kept to a mini-mum, because polymer staying there will degrade due to secondary reactions to possibly discolored or gelled material, and product quality may be seriously harmed. Local film thickness and hold-up have been correlated to physical proper-ties (surface tension, viscosity) and geometrical parameters [57]. Residence time distribution has been shown to become narrower when viscosity grows [58].Interfacial area [58] can be correlated with relative filling level H/d T and the number of disk rings per unit length ( J/LT ) [Eq. (41)].J 1:72
1:87H/d T
av ¼ ð41Þ
LT 0:085 þ 0:955H/d T An experimental study on mass transfer in disk-ring contactors of diameter dR using low-viscosity acrylamide solutions as a model fluid [60] has led to the corre-lation of Eq. (42), but this correlation should only be used as a first approximation[40], in view of the complexities introduced by polymer viscoelasticity.
 2 0:5
kf dR d n_
Sh ¼ ¼ 1:59 R ð42Þ
D D
Falling-film or falling-strand devolatilizators receive increasing attention as they re-quire no heavy and expensive machinery: Polymer is simply pumped through small slits or holes. Efficient surface renewal is achieved by shear thinning during the fall [59]; strands and films become very thin and may be completely depleted of

volatiles. Depletion or possible tearing determine the optimal height of strands/films for a given viscosity and initial film diameter. Reaction considerably increases the efficiency of these devices (by a factor of up to ten) as it replenishes the volatile by-product. Guides to the strands/films, such as wires, thin rods, or grids, will fur-ther enhance the role of the reaction while reducing the effect of shear thinning and tearing.In all these devices, an additional component of interfacial area is provided by the gas bubbles, which can result from sparging with inert gas (widely used for devolatilization without chemical reaction) or from boiling. Observed rates of mass transfer are often several times higher than predicted [50] and this discrep-ancy has been linked to the presence of bubbles.The Laplace–Kelvin equation predicts that an isolated gas bubble should redis-solve if its size is below a critical threshold, and conversely it should grow if its ra-dius r b exceeds the critical value given by Eq. (43) [50], where s is the surface ten-sion, Pme is the local pressure over the bubble (it may be simply the hydrostatic pressure, but in other circumstances it may be controlled by medium elasticity[61]), and Pb is the pressure inside the bubble, equal to the sum of the partial vapor pressures due to inert gases and volatile components if physical equilibrium and gas-phase ideality hold. So, it is possible that no bubbling occurs if the pressure is high enough or the content of volatiles is too low, but this is often not the case.
2s
rbc ¼ ð43Þ
Pb
Pme
Bubble nucleation in polymer media is usually heterogeneous [62]. It starts when shear stress manages to detach the small bubbles which are stuck in cracks and crevices of vessel walls, internals, or suspended dust. Once the source of heteroge-neous nucleation is spent, only high supersaturation will restart boiling. Of course,in many practical situations there are already small air bubbles in the polymer, and they will start foaming too. Favelukis et al. [63] have confirmed this view, and have developed a theory for bubble growth which may explain the higher mass transfer rates obtained with vented extruders and similar devices for high rotation speeds.A predictive theory for bubble nucleation was developed in this same research [64].Gestring and Mewes [65] have studied polymer degassing both with and without bubbling using a transparent drum with a rotating blade (similar to the screw of a vented extruder). Measured values of mass transfer rates without bubbling are three times lower than predicted by penetration theory, because neither pool norfilm is well stirred – which could explain the failure of predictions in Ref. 54. Rates of mass transfer in the presence of foaming were about 40 times higher than in the bubble-free regime.Trace devolatilization with the help of a stripper agent has a greatly enhanced ef-ficiency. An important recent result is that, when it forms, foam grows until a limiting volume is reached, regardless of initial volatile content and presence of stripper gas [66], and thus the devolatilization section in vented extruders should

3.2 Mass Transfer Issues in Polycondensations 79
have enough room for that expansion, and residence time should also be sufficient to allow the final density to be reached.A patent [67] for enhancing mass transfer in this class of reactors proposes the introduction of inert gas into the polymer in order to force the formation of bubbles (the forced gas sweeping process). An experimental and theoretical model has also been presented [68] and will be briefly summarized here.The reactor was a rotating disk contactor for making bisphenol A (BPA) polycar-bonate. The reactor model uses a staged approach, and the crux is the prediction of the mass transfer rate of by-product (phenol). The two relationships expressed in Eqs. (44) for the volume of a gas bubble Vb as a function of the gas flow rate Q g[69] and of its rising velocity u b in a laminar regime [70] were the key data.
 1/4  
4p 15mQ g 3/4
Vb ¼
3 2rg
  
2gd b Q g d b 1/2 ub ¼ 1þ ð44Þ
Cd u b Vb
Cd ¼ þ1
Re
In these equations m; r, respectively, are the viscosity and density of the liquid, g is the acceleration of gravity, d b is the bubble diameter, and Cd is the drag coefficient.The mass transfer coefficient was predicted using penetration theory, and the expo-sure time was computed [Eq. (45)] as the rising time of a bubble in the melt (at height hR above the gas injection point).
hR
tf ¼ ð45Þ
ub
The interfacial area per unit volume was obtained from the number of bubbles Nb and the melt volume Vm [Eqs. (46)].
Nb
a v ¼ pdb2
Vm
ð46Þ
Q g tf
Nb ¼
Vb
The observed bubble frequency agreed with the theoretical predictions, as also did the profiles of x n versus reaction time.It is interesting to finish this complex section with such a case study, suggesting that for some problems ‘‘classical’’ Chemical Engineering of the 1960s can still help in present-day industrial and scientific problems.

The design and operation of most polycondensation reactors for devolatilization of volatile by-products (namely vented extruders) is clearly a very difficult problem because of the complexity of the flows (with or without foaming). Further progress is likely to require a heavy use of computational fluid dynamics, as simplified models seem to have arrived at their limits.
3.2.2
Solid-state Polycondensation Current industrial processes for the production of high molecular weight, linear,aromatic polyesters and polyamides, for use as plastics and fibers, use solid-state polycondensation (SSP) for their last stages. This is the kind of process that will be treated in this section: the polycondensation of semi-crystalline, low molecular weight polymers to high molecular weight ones, occurring below the melting tem-perature of high polymer, but well above the glass transition temperature. We will disregard polycondensation of crystalline monomers, which is also covered in the review by Pilati [71].Because of the need to provide enough interfacial area to allow the removal of volatile by-products, the polymer has to be in a powdery form. One of the several optimization problems of these processes is to specify an economical starting par-ticle size.A practical difficulty is the possible tendency of the particles to stick, which will make the process unfeasible. It is counteracted by starting with polymers with suf-ficiently high crystallinity, and by adding glass beads. Another problem may be the sublimation of oligomers, as in nylon-6, which may clog the bed. A precrystalliza-tion step for PET, to make it attain at least 40% crystallinity before SSP starts, is present in industrial processes since early 1970’s.The main reason for using SSP is the achievement of molecular weights higher than would be possible in melt polycondensation, owing to the selectivity gain of polycondensation with respect to degradation reactions. Therefore, in a batch SSP,a maximum in molecular weight versus time is expected, and this maximum will occur at shorter times and will lead to lower molecular weights as the temperature is increased.The key assumptions made in order to interpret SSP are due to Gostoli, Pilati et al. [72, 73] (see Figure 3.8): All chain end groups belong to the amorphous regions. No reactions occur in crystalline regions. Chemical reactions follow the same kinetics as in melt, taking into account the change in the volume of the reaction media affecting functional group concentrations. Equilibrium CLD holds locally.The overall polydispersity of polymer will be greater than the equilibrium value (2

3.2 Mass Transfer Issues in Polycondensations 81
Fig. 3.8. Phase separation in solid-state polycondensation;A, B are the end groups, W is volatile by-product.for linear polycondensations) because of the radial profile of Mn in the diffusion-In the same way as in heterogeneous catalysis, effective diffusion coefficients DYie (for fluxes with respect to the total geometric area) are decreased relative to the values in the melt DYi because of the obstruction due to the crystalline phase at a volume fraction fcr and because of the tortuosity factor tD (which depends on the structure of the solid, values of 1.5 to 3 being common); the relationship is given in Eq. (47).
1
fcr
DYie ¼ DYi ð47Þ
tD
Notice that volume and mass fractions wcr of the crystalline phase are different be-cause of the slight difference in density with respect to the amorphous phase.An unavoidable complication is thus the description of the build-up of the crys-talline phase, which affects mass transfer and chemical reactions by increasing functional group concentrations in the amorphous phase. The rate of crystalliza-tion is often described by the Avrami equation [Eq. (48)].wcr ¼ 1
expð
kcr t ncr Þ ð48ÞThe exponent ncr is a function of nucleation and growth type, which is not constant for the entire course of crystallization. Mallon and Ray [74] have put forward in-

stead of Avrami equation an empirical rate law in terms of residual amorphous phase volume fraction.The microscopic mass balance of polymer functional groups and volatile compo-nents given in Eqs. (27) has to be modified in order to take into account the vari-able reaction volume due to polymer crystallization. Here we do not follow the notation in Ref. 74; rather, we use concentrations and rates of reaction per unit volume of amorphous phase instead of concentrations per unit volume of particle½A i p ¼ ½A i /ð1
fcr Þ, and so on, in order to use the same kinetic rate laws [Eqs.
(49)] as in the melt.
 
q½ð1
fcr Þ½Yi  1
fcr¼ ‘ DYi ‘½Yi  þ ð1
fcr ÞRYi
qt tD
ð49Þ
q½ð1
fcr Þ½A i ¼ ð1
fcr ÞRA i
qt
Examples of the use of this approach with PET and nylon-6,6, including a success-ful comparison with available experimental data, can be found in Ref. 74.Industrial-scale SSP is carried out in moving packed bed, fluidized bed, and stirred bed reactors; Mallon and Ray have published a brief discussion of idealized models of these reactors [75]. Fluidized beds have a serious drawback because of the high consumption of gas needed to keep the bed in a fluidized state, and resi-dence time distribution is unfavorable to high conversions. Stirred beds in series are a possible solution, depending on economic details.A model for SSP of nylon-6,6 in a moving bed reactor, considering its complex geometry and its start-up and shutdown operation [76], can serve as a guide for dealing with more complex real-life situations.
3.2.3
Interfacial Polycondensation Typical chemical systems are fast reactions between two difunctional monomers,AXA þ BYB. The first monomer (diamine, bisphenolate) is dissolved in a water so-lution (in alkaline media in both cases), and the other monomer, with low water solubility (acid chloride, phosgene), is usually dissolved in an organic solvent. Ei-ther the neutral form of AXA is in an appreciable amount (in the case of amines),or a phase transfer catalyst is needed (as in polycarbonate synthesis), since ionized forms will not dissolve in the organic phase. A decrease in the pH is often used to quench interfacial polyamidation.The chain extension of water dispersions of isocyanates with water-dissolved amines, in order to make polyurea dispersions, shares some similarities with the former (amine þ acid chloride) systems.Another common feature among these chemical systems is the presence of a parasite reaction consuming end groups B by reaction with water.

3.2 Mass Transfer Issues in Polycondensations 83
Addition of a monofunctional chain stopper to the organic phase is advisable in order to control the final M n .In most cases (namely polyamides and polyureas), the polymer is insoluble in monomer BYB or in its solvent, and precipitates as soon as it forms, often yielding a shell through which AXA has to diffuse. Polycarbonates are an exception, as they are completely soluble in the methylene chloride solvent chosen for their produc-tion. Also, the organic phase is continuous in this latter case, which is a rather ex-ceptional situation for interfacial polycondensations.Also with the exception of polycarbonates, interfacial polycondensation is mostly used in the production of specialty products, such as membranes [80–82] and mi-crocapsules [83–87].Early important contributions on interfacial polycondensations are described by Morgan [77] as well as in Refs. 78 and 79. These studies show that the reaction occurs in a layer close to the interface, on the organic side; the adjective ‘‘interfa-cial’’ is thus rather misleading. Reaction is very fast and mass transfer resistance is an important factor.Models have for a long time concentrated on describing the velocity of con-sumption of water-soluble monomer and consequent rate of film growth [83–87].Although a possible framework for describing the simpler case of polycarbo-nate formation has been presented by Mills [88], the first attempt at predicting molecular weight distributions for more typical interfacial polycondensation sys-tems is due to Karoda, Kulkarni et al. [89, 90], based on experimental data by Johnson [91]. This work will be the basis of the analysis next presented in this section.Assuming an apparent second-order rate constant in the organic phase k of the order 10 2 to 10 4 m 3 /kmol s [77] and a concentration of functional groups B in the bulk organic phase ½Bb ¼ 1 kmol m
3 , the characteristic reaction time for con-sumption of monomer AXA is of the order of 10
4 to 10
2 s. In the absence of polymer, the diffusion coefficient of AXA in organic solvent þ BYB, assuming a molecular weight of up to a few hundreds, should be of the order of 10
10 to 10
11 m 2 s
1 , and this yields a characteristic width of reaction zone LR ¼ 10
8 to 10
7 m. So, polymerization occurs in a thin shell beneath the film of precipitated polymer. There will be no functional groups B at the water interface and a steep gradient of concentration of monomer BYB will be rapidly established inside the organic phase.Instead of solving microscopic mass balances for the concentration profiles, Kar-ode et al. [89, 90] use average concentrations in the reaction zone and consider its thickness LR constant (see Figure 3.9).Furthermore, the rate of movement of the interface between the organic phase and the precipitated polymer film is considered to be slow, so that a pseudo-steady approximation for the diffusion of AXA through the polymer film and for diffusion of BYB inside the organic phase should be valid. Mass balances of monomers in the reaction zone and in bulk aqueous and organic phases resulting from these as-sumptions are given in Eqs. (50).

[AXA]blk
[BYB]blk
[BYB]blk
He [AXA]blk
[AXA]
[BYB]
Hi [AXA]
[BYB]
LP LR x
Bulk Swollen Bulk aqueous polymer Reaction zone organic phase film phase Fig. 3.9. Model for interfacial polycondensation.d½AXA He ½AXAblk
Hi ½AXALR ¼ LR RAXA þ DAXA
dt Lp
d½BYB
LR ¼ LR RBYB þ kf BYB ð½BYBblk
½BYBÞ
dt
ð50Þ
d½AXAblk He ½AXAblk
Hi ½AXAVaq ¼
av; aq DAXA
dt Lp
d½BYBblk
Vorg ¼
av; org kf BYB ð½BYBblk
½BYBÞ
dt
There are two partition coefficients He and Hi for monomer AXA: He is the ratio of bulk concentration in the aqueous phase ½AXAblk to concentra-tion in the outer interface of polymer film (thickness L p ). Hi is the ratio of monomer concentration in the polymer film to concentration in the organic phase.Vaq and Vorg are the volumes of aqueous and organic phases and av; aq ; av; org are their interfacial areas per unit volume (trivially related). No mass transfer resis-tance is assumed to exist outside the polymer film (although it can be easily in-cluded), but it is considered inside the organic phase, with the help of a mass transfer coefficient kf BYB (which can be obtained by penetration theory). Introduc-

3.3 Polycondensation Processes in Detail 85
ing the rates of reaction of polymer species (see Section 3.4.4), their mass balances can be written as Eq. (51).
d½Pn 
¼ RPnIJ
k nuc; n ð½PnIJ 
½PnIJ sat Þ I; J ¼ A; B; C ð51Þ
dt
Polymer precipitation is taken into account through the model of Kamide et al. [92]with a phenomenological rate of nucleation k nuc; n (nil for n ¼ 1, taken as indepen-dent of molecular weight for n > 1) [93]. The ‘‘saturation’’ concentrations of poly-mer species are taken as the values of their concentrations in the lower branch of the spinodal curve for the liquid–liquid equilibrium with organic solvent. The ear-lier paper by Karode et al. [89] considered only spinodal decomposition.A coherent film is predicted to form when the sum of projected areas for all phase-separated polymer nuclei (assumed spherical) is equal to the interfacial area. Film thickness is predicted through the overall mass balance of precipitated polymer.Besides thermodynamic data on the liquid–liquid equilibria of polymer/solvent and hydrophilic monomer polymer/solvent, this model needs the rate of nuclea-tion k nuc , considered as an adjustable parameter, it also tries to fit the diffusion co-efficient of monomer in the polymer film DAXA , and, as it postulates a constant width of the reaction zone, it has also to fit the kinetic parameters.In spite of its limitations in predictive power (a natural consequence of the com-plexity of the phenomena involved), this approach has given an important new in-sight on these processes. Film permeation properties should depend on the mode of phase separation, nucleation giving better crystallinity. Molecular weight de-pends on the competition between reaction and precipitation of polymer: a more powerful solvent should lead to higher molecular weight. The existence of a sharp maximum of average molecular weight as a function of the concentration of mono-mer in the organic phase when the two monomer fluxes toward the reaction zone are balanced, remarked upon by P. W. Morgan, could at last be explained by this model – 40 years later.
3.3
Polycondensation Processes in Detail
3.3.1
Polyesters
3.3.1.1Structure and Production Processes
Two kinds of polyesters will be discussed in this section: Linear, crystalline, high molecular weight (M n between 15 000 and 100 000), used as plastics and fibers, the most important being poly(ethylene terephthalate)(PET) and poly(butylene terephthalate) (PBT);

 Unsaturated low molecular weight (Mn between 1000 and 10 000), often branched, used as macromonomers for synthesis of thermosets (polyester res-ins), or thermosetting materials by themselves (alkyd resins). They are prepared from several monomers, namely phthalic and maleic anhydrides, adipic acid, iso-phthalic acid, natural fatty acids or triglycerides, and a great variety of multifunc-tional alcohols. In a few special cases, they may be saturated and/or linear for use as macromonomers in the production of polyurethanes or other polymers.
3.3.1.2 Acid-catalyzed Esterification and Alcoholysis
The two main reactions in these processes are esterification and alcoholysis, which share similar mechanisms, with a hypothetical tetrahedral intermediate.Reactions between anhydrides and alcohols are much faster than carboxyl þalcohol reaction, and full conversion of anhydrides (unless they are in excess) will occur in less than one minute while being melted and blended with the rest of the mixture. This stage is slightly exothermal and care is needed to avoid sudden boiling of the reacting mixture. Kinetic parameters of this reaction (apart from selectivities with respect to hydroxyls) are usually not needed.Acidolysis reactions have a different mechanism, involving mixed anhydrides[94]. Their rate is comparable to the esterification reactions only above 250 C.Esterification also occurs in high-temperature alcoholysis of aromatic esters, as carboxyl end groups are formed by side reactions, and should also be considered in the kinetic modeling of these processes.In spite of being the first reaction ever studied [95], esterification has been under investigation ever since, and much knowledge has accumulated, even if some points are still less clear. The basic kinetic model for polyesterification was estab-lished by Flory and is summarized in his classic book [5]. Esterification was shown to be acid-catalyzed, it is first-order with respect to hydroxyls and, with respect to carboxyls, its order is either one in the presence of foreign strong protic acids, or two in their absence [Eq. (52)].
R COOH ¼ kscat ½OH½COOH 2 ðno foreign acidÞ
ð52Þ

R COOH ¼ kfcat ½OH½COOH½Cat-H ðstrong foreign acid Cat-HÞHowever, these rate laws can only be observed at low concentrations of hydroxyl and carboxyl groups; otherwise a higher dielectric constant, association through hydrogen bonds, and generic nonidealities will introduce changes in the apparent values of the kinetic constants in Eq. (52). Experimental verification of these rate laws is more delicate than it seems at first sight (the effect of reverse hydrolysis re-action must be adequately taken into account or eliminated by the experimental set-up) and has been repeated by Hamann et al. [96], but proposal of other kinetics has continued. In their extensive review, Fradet and Maréchal [97] have found that the overall order of esterification in the chemical literature is claimed to vary from zero to six!It is nevertheless useful to have some relationship, even empirical, that could ex-tend the validity of Eq. (52) to the whole range of concentrations of functional

3.3 Polycondensation Processes in Detail 87
groups, and such a relationship (Eq. (53) where p is carboxyl conversion) has been proposed by Chen and Wu [98, 99].k ¼ kA expðapÞ ð53ÞIn Eq. (53) the empirical parameter a is a function of the initial stoichiometric ratio r. It was theoretically linked to the dependence of the dissociation equilibrium con-stant of the carboxylic acid on the dielectric constant of the medium and to the de-pendence of the latter on the carboxylic acid concentration through conversion p. A good fit of experimental data has been obtained with the parameter a varying in the range 0.2 to 1.2 both for the self-catalyzed and the foreign acid-catalyzed esterifica-tions of adipic acid with different diols.In their modeling of unsaturated polyester resin formation, Beigzadeh et al.[100] and Zetterlund et al. [101] have also found the Chen–Wu relationship more useful than the rather cumbersome empirical models of Paatero et al. [102] or Leh-tonen et al. [103]. Zetterlund et al. [101] have also presented interesting experimen-tal data on the simultaneous self- and cross-catalysis by two carboxyl groups,namely those formed by the addition of maleic and phthalic anhydride to 1,2–propanediol; they show there is an anti-synergistic effect: that is, the total rate reac-tion of the two carboxyl groups is lower than the sum of the rates of reaction of the individual acids with the same concentration and at the same temperature.
3.3.1.3 Catalysis by Metallic Compounds
Self-catalyzed esterification is often too slow to be of practical use, especially be-cause hydroxyl-terminated polymers are either sought or are a consequence of the process (aromatic polyesterifications are carried out with a large excess of hydroxyls at the beginning of the process, because of the low solubility of the diacid), and strong protic acids are not advisable, as they would catalyze polymer hydrolysis if allowed to remain with the polymer. Even volatile catalysts such as methanesul-fonic acid are avoided. Therefore, metallic salts are currently used as catalysts,both for esterification and alcoholysis. Strong bases, such as lithium hydroxide,can also be used, but for alcoholysis only (as in polycarbonate formation).Metals in metallic complexes can catalyze esterification and alcoholysis through two distinct mechanisms [117]: Metals of groups II–III (such as Zn, Mn, Ce, Pb), usually introduced as carboxy-lates, complex the oxygen in carbonyl esters preferentially. Metals of groups III–VI (namely Ti, Sb, Ge, Bi), usually introduced as alkoxides,dialkyltin oxides R2 SnO and carboxylates such as dibutyltin dilaurate, coordinate with the acylic oxygen of esters.Their activity with respect to esterification and alcoholysis has been compared by Habib and Málek [105, 106] and Chung [107], who found volcano-shaped relation-ships with different optima of activity of the several metals in terms of metal elec-tronegativity according to Tanaka [108] for the glycolysis of dimethyl terephthalate

(DMT) and for the polycondensation of bis(2-hydroxyethyl) terephthalate (BHET).A much more careful analysis [117] has considered the catalyst concentrations and the differences between the catalyzed reactions.There are striking differences [109, 110] between the two groups of metals as regards sensitivity to inhibition by carboxyls (which poison the first group) or by hydroxyls (which poison the second group). Titanium has the best balance of prop-erties, because it is little inhibited by carboxyls and hydroxyls (which poison Sb)and also efficiently catalyzes esterification. It is also active at surprisingly low con-centrations [117].The choice of catalyst also depends on secondary reactions (Ti causes yellowing),toxicity (a problem for Sb) and price (expensive Ge will nevertheless yield a white polymer).The metal responsible for the catalysis is usually involved in several complexes.An FTIR/NMR study of alcohol exchange with titanates in bulk or concentrated solutions in apolar solvents [111] has shown that these compounds are present in stable dimers, trimers, and other associations. These complexes exchange alkoxide groups very rapidly, even at temperatures below ambient.Kinetic models have to take into account both the existence of these polynuclear complexes and the poisoning phenomena, and are more complicated than for acid-catalyzed reactions. Alcoholysis of DMT, BHET, and other aromatic and aliphatic esters is now becoming better understood. A key step was to recognize that in sys-tems such as DMT þ ethylene glycol (EG) there are two reversible alcoholysis reac-tions, one consisting in the attack of methyl esters either by EG or the hydroxyethyl end groups, the other in the attack of the hydroxyethyl ester groups which produ-ces oligomers detected by HPLC [Eq. (54)].
kEG
HOCH2 CH2 OH þ XCOOCH3


!


XCOOCH2 CH2 OH þ CH3 OH
kEG =KEG
ð54Þ
kHE
XCOOCH2 CH2 OH þ XCOOCH3



!



XCOOCH2 CH2 OOCX þ CH3 OH
kHE =KHE
Besnoin and Choi [112] were the first to actually use experimentally measured oligomer concentrations to validate this kinetic scheme for Zn catalyst, as was soon confirmed and perfected by others [113–117].These reactions are first order with respect to the esters and hydroxyls, but the order with respect to the catalyst becomes zero at catalyst concentrations over a few millimoles per gram (no power rate law [114]). Moreover, the ratio kEG /kHE much depends on the catalyst (it may vary from 1 to 5) and even on the catalyst concentration. Mixtures of divalent metal catalysts can have considerable synergis-tic effects [117], which cannot be explained unless polynuclear complexes partici-pate in the reaction.It is noteworthy that, in the similar system dimethyl 2,6-naphthalenedicarboxy-late þ 1,3-propanediol, the reactivity of hydroxyl groups in monomer and in hy-droxypropyl chain ends is the same [118].There are also studies for DMT þ 1,4-butanediol with Ti and divalent metal cata-lysts, for which an order of one with respect to Ti and the hydroxyl and ester

3.3 Polycondensation Processes in Detail 89
groups has been reported [119]. The analysis of rate data is more difficult because of the relatively high importance of the secondary reaction leading to THF forma-Models for Ti-catalyzed esterification are still more complex. Titanates are hydro-lyzed by water and form oligomeric a(RO, R 0 O)aTiOa structures (unless the hy-droxyl excess is large). These structures are more active than the monomeric tita-nate [120]. Too much water will lead to a drop in activity [110], probably due to formation of insoluble products. Order one was found with respect to acid, hy-droxyl, and Ti (at a concentration of a few parts per million) [121], but there is a slight inhibition effect by the ester groups.The increase in activity brought about by vestiges of water has also been ob-served both for Ti and Zr (this latter is even more active) in the model reaction of octadecanoic acid þ octadecanol [122]. No simple first-order rate law with respect to hydroxyls and carboxyls was found in that research.Dialkyltin catalysts (such as dibutyltin dilaurate) have catalytic properties for es-terification and alcoholysis similar to Ti and Zr [123]. The SnaC bond is fairly sta-ble, but the upper acceptable temperature limit is around 220 C. On the other hand, thanks to the added flexibility provided by the organic group and the possi-bility of oligomerization, it is possible to prepare catalysts that are quite insensitive to deactivation by vestiges of humidity [124]. Nowadays these catalysts are often used in alkyd resin production.
3.3.1.4 Side Reactions in Aromatic Polyester Production
The important side products formed in nonthermooxidative secondary reactions in PET formation are acetaldehyde, vinyl end groups, and diethylene glycol (DEG)units, together with carboxyl end groups. DEG units decrease the melting point,crystallinity and thermal stability. Acetaldehyde, even at parts-per-million level, is detectable by its flavor in drink bottles. Vinyl esters and acetaldehyde lead to chro-mophoric products. Carboxyl end groups promote hydrolytic and thermal instabil-ity. Additionally, these side reactions lead to a decrease in molecular weight, which is particularly undesirable for product use in markets such as tire cord and soft drink bottles.Side reactions can be minimized to a certain extent by choice of operating condi-tions and catalysis. Knowledge of this chemistry has obvious economic advantages,and many details (for example, the influence of catalysts and stabilizers) cannot be found in the open literature.Thermal scission of ethylene diester groups is a likely source of carboxyl and vinyl end groups [125] (Scheme 3.1):
O O
O H OH
C C
C CH O C + CH O O CH2 O CH2 Scheme 3.1. Vinyl end group formation from PET.

This is a first-order reaction, with no effect of catalysts or additives.Further alcoholysis of vinyl esters yields acetaldehyde [Eq. (a)].XCOOCHbCH2 þ HOCH2 CH2 OOCY ! XCOOCH2 CH2 OOCY þ CH3 CHO
ðaÞ
DEG units are observed to form mainly in the first stages of the process. The pres-ence of a hydroxyl ester group is indispensable, as it can be shown experimentally that DEG forms in negligible amounts if ethylene glycol is heated alone in the ab-sence of acid catalysts [126]. Reimschuessel [127] has suggested the attack of ethyl-ene glycol or a hydroxyethyl end group as a possible source of the DEG moieties(see Scheme 3.2). The analogous intramolecular etherification is the source of side product dioxane [125] (see Scheme 3.3).
O C
C CH2 O + H O CH2 O CH2 CH2 OX
O C
C + CH2 O
OH CH2
O CH2
CH2 OX
Scheme 3.2. Formation of DEG moieties in PET.O O CH2 CH2
C C + O O
OCH2CH2 OH CH2 CH2
OH O
CH2CH2
Scheme 3.3. Dioxane formation in PET.Poly(butylene terephthalate) is also subject to analogous side reactions [128],the formation of 1-butenyl end groups (Scheme 3.4) followed by splitting-off of 1,3-butadiene or tetrahydrofuran (Scheme 3.5). This latter reaction is not affected by the metallic catalyst, but is catalyzed by the carboxyl end groups, making pro-duction of PBT by direct esterification of terephthalic acid and 1,4-butanediol dis-advantageous.
3.3.1.5 Side Reactions in the Formation of Unsaturated Polyesters
A recent review on the chemistry of unsaturated polyesters has been published by Malik et al. [129]. Besides cis–trans isomerization [130], the other important side reaction is Ordelt reaction [131–133] (see Scheme 3.6) which increases branching and consumes double bonds. Both reactions are reversible and acid-catalyzed.

3.3 Polycondensation Processes in Detail 91
H CH2 CH2 C
C CH CH2 O
O CH2
OH C
CH2 O
C + H2C CH CH2 H2C CH CH CH2 + HO Scheme 3.4. Vinyl end group formation from PBT.
OH CH2 CH2
C CH2 CH2
C + CH2 CH2 O CH2 CH2 O O
HO
Scheme 3.5. THF splitting-off from PBT.R OH + HC CH HC CH2 R'OOC COOR'' R'OOC COOR''Scheme 3.6. Ordelt reaction.
3.3.1.6 Modeling of Processes of Aromatic Polyester Production
Continuous processes are currently used in the manufacture of PET. Several mod-els have been developed, with the aim of contributing to a better design and opera-tion. A brief discussion of their main assumptions and predictive capacities fol-lows. Since polycondensation in solid-state and in film-forming devices has been analyzed previously, only the specific aspects of the initial process stages still needs to be covered.Earlier models for continuous processes based on DMT [134, 135] study the in-fluence of variables, such as the initial stoichiometric ratio, reactor temperatures,and average residence times for CSTRs in series connected with distillation col-umns, on process performance. They show it is possible to optimize conversion and minimize formation of side products. Even if kinetics and physical equilibria are now much better known, their qualitative conclusions should still hold.PBT production, for which no published process models have been found,should be described by a similar approach.Yamada et al. [136] and more recently Kang et al. [137] have presented models of the direct esterification process of terephthalic acid (TPA) with ethylene glycol (Fig-ure 3.10). As TPA has a low solubility because of its high melting point, the first reactor in the train (‘‘esterification’’ reactor) is operated at a higher pressure and

Fig. 3.10. Simple model for WFR used for the direct esterification process of terephthalic acid (TPA) with ethylene glycol.temperature than the ‘‘pre-polycondensation’’ reactor in order to counterbalance this lack of solubility. The calculations show it is also possible to optimize conver-sion and minimize side reactions by a choice of average reaction times, tempera-tures, and pressures. There are, however, no actual plant data to validate these con-clusions. The kinetic model uses a ‘‘fragment’’ approach similar to what was recommended in previous sections, although without taking into account the influ-ence of alcoholysis and acidolysis on monomer concentrations. Also, no thermo-dynamic model has been used for predicting the solubility of TPA, only an inter-polation between values in pure ethylene glycol and different oligomers.An integrated view of recent processes for PET has presented by Yao and Ray[138]. Most of the DEG units are shown to be produced essentially in the esterifi-cation and pre-polycondensation reactors, with very little change afterwards. Mini-mizing vinyl ester formation, thus improving molecular weight and product qual-ity, is achieved by decreasing residence time in the ‘‘finishing’’ wiped-film reactor,but increasing the residence time in the solid-state polymerization reactor, which is operated at a lower temperature under a stream of inert gas.
3.3.1.7 Modeling of Processes for Unsaturated Polyester Production
Nava gives a brief description of the industrial processes [139]. Polyester resins should ideally be produced with a certain predefined viscosity in their solution in acrylic/vinyl monomers and with a known, reproducible, distribution of double bonds. Trans double bonds are much more reactive in free-radical polymerization,and the amounts of each one should be known. Carboxyl end groups are in some processes further converted to metal carboxylates (typically, by adding MgO) in or-der to thicken the solution, so it is also important to control their concentration.Therefore, it is worth developing models for these processes, and a few recent studies have appeared in this area [101, 104], but for now they only aim at predict-ing the concentrations of functional groups, which is not a minor task in view of the large number of parameters needed. Modeling of the full process requires con-sideration of the losses of volatile monomers (lack of reliable vapor–liquid equilib-ria data is a problem), and the aforementioned problems of taking into account the low solubility of isophthalic and terephthalic acids are also pending.

3.3 Polycondensation Processes in Detail 93
Description of the branched structure of the resins is less problematic than in the case of alkyds, because the conversion of double bonds by Ordelt reaction is in the range 10 to 20%, so the molecules are almost comb-like and the approxima-tions used by Yang and Pascault [140] should be reasonable.Modeling of alkyd resin production is a rather formidable task because of the high number of distinguishable chemical groups, the branched structure of the polymer, a nonnegligible amount of intramolecular reaction, and side reactions of the double bonds in fatty acids. The usual problems found in previously discussed polyesterifications, namely lack of data for liquid–vapor and liquid–solid equilibria and associated mass transfers, are also present.Industrial processes [141–143] use batch stirred reactors, connected to partial condensers in order to recover volatile glycols. Azeotropic distillation with xylene(for instance) is often used. Monomers may be added in several steps to over-come solubility problems. Vacuum and inert gas sparging is also used in different stages. Catalysts (mostly alkyltin salts) are used to convert most of the carboxyl groups, as specifications often require less than 1 mg g
1 KOH acid value. Long reaction times are sometimes avoided through different techniques for eliminating residual carboxyls.The same plant usually produces different varieties of resins, according, for in-stance, to fatty acid content. New compositions are often sought in order to im-prove end use properties or to compensate for fluctuations of raw material prices.Therefore, it is often necessary to look for initial amounts of monomers which sat-isfy stoichiometric constraints (such as mole ratio and fatty acid content), will not lead to gelation, and meanwhile keep an acceptably high M n (for instance).An early review of the foundations of the macromolecular chemistry of alkyds has been presented by Kienle [143] and simple methods for predicting gelation conversion have been reviewed by Jonason [144]. These predictions are not very ac-curate, but good data on intramolecular reactions and differences of reactivity of functional groups will be needed if better control of physico-chemical properties is sought.
3.3.2
Polycarbonates
3.3.2.1 General Introduction
Polycarbonates are polyesters of carbonic acid formed by reaction of diols (aro-matic, aliphatic or a mixture of both) with a derivative of carbonic acid. The first preparations of polycarbonates were reported by Einhorn in 1898 [155], by reaction of phosgene with resorcinol or hydroquinone in a pyridine solution. Bischoff and van Hedenström in 1902 [156] obtained the same aromatic polycarbonates via transesterification with diphenyl carbonate (DPC). Thus the main routes to poly-carbonates were established early, but the properties of the products seemed unin-teresting. Around 1930 aliphatic polycarbonates were studied by Carothers and van Natta [157]. These carbonates have low melting points and thermal resistance and are not commercially interesting as stand-alone thermoplastics. Low molecular

weight, aliphatic polycarbonates with hydroxy end groups, however, are widely used as a diol component for the synthesis of polyurethanes and polyurethane–urea elastomers.Following the work of Whinfield and Dickson [158], who in 1941 succeeded in preparing high molecular weight, high melting polyesters, the chemistry of polycarbonates was re-examined. This led to the preparation of a linear, high molecular weight polycarbonate derived from bisphenol A [BPA, or 2,2-di(4-hydroxyphenyl)propane] by Schnell at Bayer [159, 160] and shortly afterwards by Fox at General Electric [161]. BPA polycarbonate (BPA-PC) proved to be an out-standing engineering thermoplastic that differs from the other polyesters in that it is noncrystalline with a high glass transition temperature (about 150 C) and re-tains its high transparency and toughness after molding. It has thermal stability up to over 300 C as well as excellent mechanical, optical, and electrical properties, in-herent fire resistance, and food compatibility. Improvements of several polymer properties such as heat resistance, melt flow, and birefringence were achieved with different (co)monomers [163, 164]. However, BPA-PC remains the commer-cially most important polycarbonate.The original processes – phosgenation in pyridine solution and melt transesterification – were soon replaced by interfacial polycondensation with phos-gene, which proceeds at low temperatures and allows the easy production of high molecular weight polymer. It still remains the predominant production process al-though interest in the simpler, non–phosgene-based and environmentally more at-tractive transesterification process revived in the 1990s; problems with the earlier melt carbonates, in particular the poor resin color, could be overcome. Polycarbon-ate demand has enjoyed steady growth and total production capacity in 2003 was about 2.5 million tons per year. Approximately 12% is produced by transesterifica-tion and the percentage is expected to increase. The presentation here focuses on the engineering aspects of BPA homopolymer production. Detailed reviews of the synthesis and application of polycarbonates are given in Refs. 164–171.
3.3.2.2 Interfacial Polycondensation
In the interfacial process, BPA and phosgene react at the boundary of two immis-cible liquids, an aqueous alkaline BPA solution and an organic phase containing phosgene. The overall reaction is shown in Scheme 3.7.
CH3 CH3
-2n NaCl
n NaO C ONa + n COCl2 O C O
CH3 C
CH3 n
Scheme 3.7. Overall stoichiometry of bisphenol A polycarbonate formation.The synthesis proceeds in two steps: phosgenation of BPA forming oligomeric carbonates with phenolic and chloroformate end groups; and polycondensation of the oligomers (see Scheme 3.8). For the phosgenation, BPA is first dissolved in an

3.3 Polycondensation Processes in Detail 95
* ... C Cl + * ... Na * ... C ... * + NaCl
m n m n
m m
* ... C Cl + 4 NaOH * ... Na + NaCl + Na2CO3 + H2O
m m
Scheme 3.8. Formation of polycarbonate by interfacial polycondensation.aqueous alkaline solution as sodium bisphenolate, and phosgene is dissolved in a solvent of chlorinated hydrocarbons (such as dichloromethane and monochloro-benzene) which also dissolves polycarbonate. The reaction is started by dispersing the two phases. Alternatively, liquid–gas phosgene (boiling point 4 C) is fed into a slurry of BPA in the presence of an organic solvent. At high BPA concentrations a fourth solid phase may be present [172]. Part of the phosgene is hydrolyzed with NaOH to NaCl and Na2 CO3 and an excess of phosgene (10–20 mol%) is required to compensate for hydrolysis and provide an excess of chloroformate end groups for the following reaction step. Reaction temperatures are between 20 and 50 C and the pH is kept between 9 and 13 by the addition of NaOH.In the polycondensation step, a monofunctional phenol (such as 3–5 mol% phe-nol, p-tert-butylphenol, p-cumylphenol) is added as a chain terminator to control the molecular weight of the final polycarbonate. Reaction partners are now end groups (chloroformate and phenolic aOH; see above) and reaction rates decrease.The final polycondensation stages are catalyzed by tertiary amines. The amines re-act with the chloroformate end groups to form intermediate quaternary acylium salts which then react with phenolate to form carbonate and OH
ions, hydrolyz-ing the chloroformate end groups, or to form a urethane in a side reaction [164].Detailed mechanistic studies of the catalyst reaction were performed by Aquino et al. [173] and Kosky et al. [174].Numerous variations of the interfacial process have been published. The reac-tions can be carried out in batch in stirred tank reactors or continuously in series of CSTRs and tubular reactors. Intensive mixing with dispersion and redispersion is required throughout the reaction stages. After the reaction is complete, the brine phase is separated and the polymer solution washed to remove residual amine and base. Several processes for devolatilization are in use, including solventless precip-itation, steam precipitation, spray drying, falling-strand devolatilization, and vac-uum extrusion in devolatilizing extruders.Phosgenation is generally mass transfer-limited [175, 176] and its rate depends on mixing as well as on pH and the volume ratio of the organic and aqueous phases. Although the polycondensation reaction of end groups is slower, both rates still show the same dependencies due to the interfacial nature of the reaction. The following phenomena contribute to the reaction process:

 Dispersion of the two phases. Rates always depend on mixing. Effective kinetic rate constants can be formulated as a function of energy dissipation or interfacial area. The type of emulsion. Both types of emulsions, oil in water (o/w) as well as water in oil (w/o), can be found. The partition of phenols between the phases and its pH dependence [177]. Silva and Kosky [178] studied the reaction of hydrolysis taking into account the differ-ent phases and the partitioning of BPA between them. Monofunctional phenols with better solubility in the organic phase show a better efficiency as chain termi-nators. Mass transfer to and across the boundaries. Intrinsic (kinetic) reaction rates.Due to the low reaction temperature and the use of chain terminator, the molecu-lar weight distribution in interfacial synthesis is kinetically controlled and may be far from thermodynamic equilibrium. In two parametric studies Mills [179] and Munjal [180] have tried to model the full molecular weight distribution of polycar-bonate. Varying the ratio of mass transfer/kinetic rates, they show how mass trans-fer limitations can lead to a higher polydispersity or a higher oligomer content.
3.3.2.3Melt Transesterification
The melt process is based on the transesterification of diphenylcarbonate (DPC)and BPA (see Scheme 3.9).n HO OH + n OCO O OC + 2n OH Scheme 3.9. Formation of bisphenol A polycarbonate by transesterification.The melt process requires no solvent and polycarbonate is produced directly without the need for cleaning, drying, and devolatilization. The only by-product,phenol, is removed and can be recycled to the production of DPC or BPA. The pro-cess, however, requires high temperatures (190–320 C) so that the polymer re-mains molten and the viscosity can still be handled. With improved quality of the starting materials and improved high-viscosity reactors these temperatures no lon-ger affect polycarbonate quality and most grades can now be produced via the melt process. The transesterification is a reversible reaction with an equilibrium con-stant close to unity, so that the by-product phenol has to be removed for the reac-tion to proceed. Pressures below 0.1 hPa may have to be applied for the final stage.

3.3 Polycondensation Processes in Detail 97
Small amounts of basic catalyst (< 0.01 mol% of alkyl-ammonium, or phosphonium-salts) are mixed together with the DPC and BPA and the reaction is started at the lower temperature. Conversion is driven by the removal of phenol and the pressure is successively reduced while the temperature is increased. Sim-ple reactors such as CSTRs, tubular reactors or falling-film evaporators can be used for the first stages. Viscosity increases dramatically from 5 mPa s up to 1000 Pa s as conversion increases and special reactors are needed for dealing with the high viscosity at high conversions [149–151].The melt process depends on the reaction rates, the thermodynamic phase equi-librium, and mass transfer between the phases. Detailed mechanistic studies of the Li-catalyzed melt process have been published by Choi et al. [152], and of the solu-bility of DPC and phenol in polycarbonate by Webb [153]. The liquid–gas equilib-rium has to take into account at least two components: phenol and DPC.For a semi-batch operation for the first stages, optimal variations of pressure and temperature can be calculated based on the above relationships plus the assump-tion of phase equilibrium, or on a simple relationship for the mass transfer of each volatile component Yi (Eq. (55), with the mass transfer rates per unit volume Ji of component Yi , mass transfer coefficient of component i kfi , interface area per unit volume a v , and equilibrium concentration ½Yi   at the interface).Ji ¼ kfi a v ð½Yi 
½Yi   Þ ð55ÞMass transfer coefficients may be obtained by fitting to process data. Including DPC loss, production capacity (reaction time), and unwanted side products in a cost function, the optimization leads to a balancing of mass transfer and reaction rate. This means that an optimal process is neither entirely mass-transfer nor ki-netically controlled. To avoid side reactions that impair product quality, the lowest temperature that kinetics and mass transfer allow is chosen. The results of the semi-batch optimization can be transferred to the design of a staged, continuous process [154].The balancing of reaction versus mass transfer rate can similarly be applied for the last stages. The relative DPC loss compared to phenol removal is quite high for these stages. A high fraction of phenol end groups compared to OH end groups is desirable for product quality. Therefore, excess DPC has to be used at the begin-ning of the reaction. Unfortunately, a high fraction of phenol end groups reduces the concentration of one reaction partner and hence reaction rates [146].Mass transfer and hence the speed with which phenol can be removed are re-duced with increasing viscosity. Reactors are needed that provide a high rate of surface renewal. These may be wiped-film evaporators, single- or twin-screw ex-truders, reactors of the rotating disk type or falling-strand evaporators [145, 149,150]. Whereas the first three reactor types create surface actively, in falling-strand evaporators the melt is simply pumped through small holes. High surface renewal rates are achieved by small holes, but also by shear thinning of the falling strands[59]. The effect of shear thinning films can also be achieved in rotating disk reac-tors with disks that contain holes. Apart from surface renewal, mass transfer is de-

termined by diffusion. Measurement of diffusion coefficients is difficult because of the parallel reaction. An alternative is the determination of diffusion constants us-ing molecular dynamics simulations [148].
3.3.3
Polyamides
3.3.3.1 Introduction
Since their discovery by Carothers [181], aliphatic polyamides such as nylon-6,6 and nylon-6 are important textile fibers and plastics. Similar polyamides produced by melt reaction of aliphatic diacids þ diamines, or by hydrolytic polymerization of lactams, have some interest as engineering plastics, and will also be discussed in this section.Aliphatic, low molecular weight, branched polyamides of dimer fatty acids with di- and triethylenediamine are among the most widely used curing agents for epoxide resins. The chemistry of their formation is similar to that observed for the aforementioned crystalline linear polyamides.Aromatic polyamides are specialty products [182], used for high-performance fi-bers and composites, which are produced by solution or interfacial processes from acid chlorides and amines.
3.3.3.2 Kinetic Modeling
The only published kinetic data on nylon-6,6 reaction are due to Ogata [183, 184],and there is in general a great scarcity of information on the amide þ acid reaction.Much more data are available on nylon-6 formation, such as in Refs. 185–187.Because of the obvious identity in average chemical group composition of both polymers, it should be possible to reconcile both kinetic laws [188]. However,nylon-6,6 data have been obtained at high water concentrations and relatively low temperatures, and data on nylon-6 are in the opposite situation. Degradation reactions of nylon-6,6 through adipic acid chain ends (see Section 3.3.3.3) compli-cate the interpretation of kinetic data at higher temperatures. Schaffer, McAuley et al. [189] have recently obtained experimental data on the nylon-6,12 system,which does not present these problems, and these kinetics are now much better understood.Apparent equilibrium and kinetic constants in these systems are seen to depend on water concentration. It obviously changes the activity coefficients of functional groups, but no thermodynamic model has ever been used to completely describe the mixture. The activity of water can be directly measured from knowledge of its vapor pressure, and it has been claimed that a correlation based upon a Flory–Huggins model can predict it [190], but no model exists for taking into account the group interactions. Some interesting ideas can be found in Ref. 191, but no actual thermodynamic model has been developed, concentrations of hypothetical species having been used throughout that paper.The crux of the treatment by Schaffer et al. [189] lies in the empirical correlation

3.3 Polycondensation Processes in Detail 99
expressed by Eq. (56), where bA and mA are assumed to be constant parameters;activity coefficients are based on mole fractions.¼ bA þ mA ½H2 O ð56Þ
gCONH
As the overall composition of the system may be described by two variables, such as [H2 O] and [COOH], a dependence on [COOH] might be added in Eq. (56).However, this is not needed, as no effect of the mole ratio [COOH]/[NH2 ] on the apparent equilibrium constant has ever been detected. Insertion of the above rela-tionship into the mass action law provides a relationship between apparent equilib-rium constant K a and true thermodynamic constant K 0 [Eq. (57)].K a ¼ K 0 gH2 O ðbA þ mA ½H2 OÞ ð57ÞIt is seen that parameter bA is absorbed by the unknown thermodynamic constant,and only the ratio g A ¼ mA /bA can actually be found from experimental data. The correlations found for the activity coefficient of water are given in Eqs. (58).
 
gH2 O ¼ exp 9:624
ðNylon 6,12Þ
  ð58Þ
gH2 O ¼ exp 6:390
ðNylon 6,6ÞChoosing a reference temperature T0 ¼ 549 K, Eq. (57) is rewritten as Eq. (59).
  
1 þ g A ½H2 O DH 1 1 K a ¼ K a0 exp

gH2 O /gH2 O ðT0 Þ R T T0
  ð59Þ
bA DH DS
K a0 ¼ exp
þgH2 O ðT0 Þ RT0 R Parameters describing equilibrium (DH; K a0 , and g A ) have been fitted simulta-neously with a mass transfer time constant for water k m, appropriate to their exper-imental set-up, and activation energy Ec and pre-exponential factors kc0 or ku0 de-scribing forward reaction through a third- or second-order rate law:
    
Ec 1 1 ½H2 O½CONHR CONH ¼ kc0 exp

½COOH ½NH2 ½COOH

R T T0 Ka
   
Ec 1 1 ½H2 O½CONHR CONH ¼ ku0 exp

½NH2 ½COOH
ð60Þ
R T T0 Ka

Tab. 3.2. Equilibrium and rate parameters for nylon formation.Parameter Units Estimate 95% confidence interval DH kcal mol
1
1.82 G1.42 K a0 – 63.1 G6.2 gA g mmol
1 2:03  10
2 G0:45  10
2 Ec kcal mol
1 22.6 G7.7 kc0 g 2 mmol
2 h
1 3:19  10
4 G0:71  10
4 km h
1 24.3 G15.4 The optimum values of parameters for the third-order model are shown in Table
3.2. Their approximate correlation matrix can be found in the same reference. It
shows that parameters describing equilibrium are highly correlated with g A , and the pre-exponential factor is highly correlated with k m . Further, there is little change in parameters for the second-order model, which yields ku0 ¼ 2:64  10 7 mg mol
1 h
1 and has a similar sum of weighed squared residuals. Thus, it is not yet possible to determine the reaction order without performing experiments with excess of diamine or diacid, which have not yet been reported at the time of writ-ing. The authors state the parameters thus obtained for nylon-6,12 should hold for the other aliphatic polyamides, just correcting the equilibrium constant, which is lower by a factor of 0.4 to 0.5 in nylon-6,12 relative to nylon-6,6.There are several studies concluding that the apparent order of reaction changes from two to three as conversion grows. The reason might be a nonideality effect similar to the one observed in esterifications.Miller [192] has carried out an experimental study on aminolysis and acidolysis reactions using carefully dried model compounds. As in esterifications, acidolysis is slower and is explained by a mechanism involving the formation of intermediate anhydrides. Its activation energy is 27 kcal mol
1 , whereas aminolysis has the much smaller activation energy of 13 kcal mol
1 . The rate of aminolysis was shown to be first order in carboxylic acid.These results are valuable not only for dealing with block polymers, but also in kinetic modeling, particularly with cyclic lactams: a narrow CLD will not occur be-cause of the reorganization brought about by aminolysis.Hydrolysis of caprolactam is autocatalytic. Mallon and Ray [191] have suggested that its initial rate is determined by the presence of impurities.The same authors have also remarked that the rate constant of addition of capro-lactam to amine end groups is about what would be expected for an aminolysis reaction.It should also be possible to predict the concentrations of higher cyclic oligomers, but the only usable data concern the equilibrium concentrations, and experimental confirmation has not yet been possible [191].
3.3.3.3 Nonoxidative Thermal Degradation Reactions
The main degradation reaction of nylon-6 is decarboxylation through interaction of a carboxyl end group and caprolactam or an amide in the polymer chain (see

3.3 Polycondensation Processes in Detail 101
Scheme 3.10) [193, 194]. A slow deamination reaction has also been shown to occur.
X X
HO C N C N X N
k1 O CH2
OH CH2 k2 C
CH2 O C
O C H2C CH2- H2 O H2C CH2 - CO
CH2 2
H2C H2C CH2
CH2 CH2
CH2 CH2
Scheme 3.10. Nonoxidative thermal degradation of nylon-6.Nylon-6,6 also degrades in the absence of oxygen, to a much greater extent than nylon-6, and it eventually gels, as explained by the simplified mechanism in Scheme 3.11 [195, 196].XNHCO(CH2)4COOH - H2O
NH Y k1
O O C k2 O
C CH2 CH2 C + NH2 Y X NH CH2 CH2 XNH
- CO2
X k3 X
N N
k4
+ 2 NH2 Z Z Z + 2 NH3 Scheme 3.11. Nonoxidative thermal degradation of nylon-6,6.Cyclopentanone units are created, with a decrease in molecular weight, either from carboxyl end groups or by intrachain reaction. Losses of CO2 and NH3 lead to an imbalance of end groups, a nuisance for dyeing, and also to crosslinking.
3.3.3.4 Process Modeling
Recent progress in kinetic and reactor modeling make it possible to assist the reac-tor design and process operations with unprecedented exactness. A recent analysis of the nylon-6,6 process [138] looked for improvements based on expansion of the solid-state polymerization, but the gain was minor compared to what happened with PET. The reason is the much reduced sensitivity of polyamides to by-product removal as compared to polyesters, since the equilibrium constant of the former is much greater.Since the early 1970s, many researchers have modeled nylon-6 processes [197],as reviewed by Kumar and Gupta [198]. There is an interesting optimization prob-lem inherent to this process, which consists in adding just as much water as is needed to start polymerization, and to get rid of it in the later stages. Because of

the autocatalytic nature of the caprolactam hydrolysis, back-mixing increases con-version.A simple and widely used reactor is the VK column (simplified continuous col-umn), essentially a vertical tube at atmospheric pressure [193, 199], stirred by the boiling action of water leaving the reactor mixture in the top zone.Extensive pilot-plant tests carried by Jacobs and Schweigman [199], and further data from industrial plant, have lead to a simple model of the VK column, consist-ing in one or two CSTRs in series, followed by a plug-flow reactor.Other designs [200] have improved performance by preventing water evapora-tion at the top of the column through the use of above-atmospheric pressure.The bottom one-third of the columns has a homogenizing function, not only physical but also chemical, through the aminolysis reaction [201].Vacuum stripping with a low residence time is used to eliminate most of the large amount of caprolactam which remains because of the chemical equilibrium of the back-biting reaction. An alternative is a hot-water extraction step, which will also extract higher cyclic oligomers.
3.3.4
Polymerizations with Formaldehyde: Amino Resins (Urea and Melamine) and
Phenolics
3.3.4.1 Formaldehyde Solutions in Water
Formaldehyde is a gas at room temperature. It may be sold as a low molecular weight, solid polymer (paraformaldehyde), and more conveniently as 37% or 55%water solutions, which usually contain some methanol. Under such conditions,nearly all the formaldehyde is transformed into methanediol and higher oligomers(see Scheme 3.12), usually end-capped by methanol, in order to reduce the average molecular weight and prevent precipitation of paraformaldehyde.HCHO + H2O HOCH2OH(CH2O)xOH + HOCH2OH (CH2O)x+1OH + H2O HCHO + CH3OH HOCH2OCH3(CH2O)xOH+ HOCH2OCH3 (CH2O)x+1OCH3 + CH3OH Scheme 3.12. Formaldehyde/water/methanol equilibria.These reactions are not very fast at room temperature: characteristic reaction times are of the order of minutes.The various equilibrium constants have been measured using NMR [202] and a model describing the vapor–liquid equilibrium in that system has been developed.
3.3.4.2 Amino Resins
Reaction between a water solution of formaldehyde with urea, melamine, and sim-ilar molecules (such as acrylamide) leads to hydroxymethylation of the nitrogens

3.3 Polycondensation Processes in Detail 103
[Eq. (b)] and further condensations produce the so-called amino resins [Eq. (c)], of which 80% are based on urea [203], the rest being nearly all produced from mela-XaNH2 þ HCHO ! XaNHaCH2 OH ðbÞXaNHaCH2 OH þ YaNH2 ! XaNHaCH2 Y þ H2 O ðcÞ
NH2
H2N N
1 NH2
Scheme 3.13. Melamine.Since melamine is made from urea and ammonia, it is more expensive. Mela-mine resins are therefore chosen when one can get an appreciable benefit from their better hydrolytic or thermal resistance.Urea–formaldehyde (UF) resins are mainly used as adhesives for wood. Lami-nated sheets (tables and counter tops) are a major application for melamine resins,which stay in the outer decorative surface. Molding compounds, their first big ap-plication, is still a major market, taking advantage of their extreme hardness and heat resistance. Coatings, textile finishing, paper additives, leather tanning and foundry binders, for which methanol- or butanol-etherified resins are usually em-ployed, are important markets discussed in Ref. 203.A major problem with the use of UF resins is their formaldehyde emission due to hydrolysis. Formation of melamine resins is much less reversible and therefore food contact with them is allowed.Besides earlier classic data on the kinetics of reactions between urea, formalde-hyde, and UF oligomers by de Jong and de Jonge [204–206], only experiments by Price et al. [207] at higher temperatures in a sealed reactor are of immediate use to establish a kinetic model of the chemical system. Kumar and Sood [208] have pro-posed an FSSE model for the early stage of this polycondensation. A modified ver-sion of that model introduces the groups presented in Scheme 3.14, where theirfive urea monads U0 . . . U4 have been kept but three formaldehyde monads have been used instead of two. Formation of tetrasubstituted urea is known to be negli-gible.Both de Jong and de Jonge, and Price et al., have considered that hydrolysis reactions are unimolecular. Kumar and Sood [208] have considered it could be bi-molecular, which seems to be reasonable. Available experimental data could not de-cide for any of the alternatives, since the water concentration was always the same;this matter needs to be solved, because higher initial concentrations of formalde-

O O O
U0 = C U1 = C U2 = C H2N NH2 H2N NH- -HN NH-
O O
U3= U4 = C
C -HN
H2N N N
F0 = C F1 = -CH2OH F2 = -CH2-
H H
Scheme 3.14. Monads in the FSSE model of urea/formaldehyde polycondensation.hyde are often used nowadays. Chemical transformations according to this new model are shown in Scheme 3.15.
k1 k6
→ →
U 0 + F0 U 1 + F1 + W U 0 + F1 U 1 + F2 + W
←
k
 ←

k

h1 h2
k k
→2
→7
U 1 + F0 U + F1 + W U 1 + F1 U + F2 + W
←
k
 2 ←

k
 2
h1 h2
k k
→
→
U 1 + F0 U 3 + F1 + W U 1 + F1 U 3 + F2 + W
←
k
 ←

k

h1 h2
k4 k
→ →9
U 2 + F0 U 4 + F1 + W U 2 + F1 U + F2 + W
←
k
 ←

k
 4
h1 h2
k5 k10
→ →
U 3 + F0 U 4 + F1 + W U 3 + F1 U 4 + F2 + W
←
k
 ←

k

h1 h2
Scheme 3.15. Kinetic scheme of FSSE model of urea/formaldehyde polycondensation.A possible simplification consists in distinguishing only rate constants of for-ward reactions according to the number of hydrogens involved, and therefore k1 ¼ 2k2 ¼ 4k5 and k3 ¼ k4 , k6 ¼ 2k7 ¼ 4k10 and k8 ¼ k9 , reducing the number of unknown parameters.Since the only information in the experimental data of Price et al. [207] is form-aldehyde concentration versus time, it is not possible to estimate rate constants k6 to k10 from them. Although only at low temperatures, rate constants k1 ; k3 , and kh1(this one measured from the rate of formation/hydrolysis of both monomethylol urea and dimethylolurea) are nevertheless available from other sources, such as Ref. 209. A crucial check of the FSSE hypothesis is the equality of the first-order hydrolysis constants of methylol groups in mono- and dimethylolurea: the rate con-stants per mole of the chemical substances should be double for dimethylolurea.The value of this ratio is 1.68, standard deviation 0.42, for 15 values reported by Landqvist with different buffers (pH 6, 7, 9.2, 10) at temperatures 20, 30 and 40 C.

3.3 Polycondensation Processes in Detail 105
However, that ratio is 6.9 according to de Jong and de Jonge; the activation energy of the hydrolysis is the same (20 kcal mol
1 ), though, according to both research The equilibrium constants of the first and second hydroxymethylations of urea are, respectively, 990 and 253 at 35 C, and there is a decrease of about 3 in the for-ward rate constants of the successive substitutions. Interestingly, the rate con-stants for the reaction of methylenediurea with formaldehyde or monomethylo-lurea are identical, respectively, to those observed for urea þ formaldehyde and urea þ monomethylolurea [210, 211].So, there seems to be enough evidence to take FSSEs into account for urea, but unfortunately it seems there might be no such thing as a single ‘‘aCH2 OH’’ group,and SSSEs should be needed for fully describing this chemistry.For simplicity, we will keep using the above model in the discussion.Rate constants are known to depend on pH (although not much between pH 4 and 9); there is catalysis by OH
and Hþ , so these constants should be written as in Eq. (61) [204].ki ¼ ki0 þ kiOH bOH
c þ kiH bHþ c ð61ÞRate constants k6 to k10 in this scheme can be estimated from data of reactions in acid media between urea, mono- and dimethylolurea [206]. In the same work, re-action between methylols was found negligible (the temperature was at most 50 C). For these reactions, the terms ki0 ; kiOH can be neglected; the reactions are very slow at pH > 4.More recent work has concentrated on analysis supported by quantitative 13 C NMR [212–216] and size exclusion chromatography [217] has completed this view of the chemistry. At high temperatures and alkaline pH, methylol groups form methylol ether bridges and uron rings 2 (Scheme 3.16).
O O O
2 C C C + H2O HOCH2N N N NH-CH2OCH2N N
O O
C C
N N N N
HOCH2 CH2 CH2 CH2 + H2O
O O
H
Scheme 3.16. Reactions at alkaline pH in urea/formaldehyde polycondensation.The preparation and possible industrial uses of uron UF resins may be found in Ref. 218. NMR shows that these intra- or intermolecular ether bonds are destroyed with liberation of formaldehyde at acid pH.

Another question, that should not be overlooked, is the incomplete solubility of the polymer in water. Except at low conversions, the reaction medium resembles a colloidal dispersion [219]. Gelation may be physical, before or instead of being chemical [220].Amino resins are nearly always made in batch processes, consisting of thermo-stated reaction kettles connected to a condenser. The current method of synthesis has three stages [221]: synthesis of methylolated oligomers at pH 8 to 8.5; acid condensation at pH 4 to 5; addition of urea to decrease the final stoichiometric ratio to 1 to 1.3.The goal is to reduce formaldehyde emissions [222], while keeping the properties of products (such as wood panels) at an acceptable level. Since the formaldehyde/urea molar ratio had to be decreased, the process has also become more difficult to control.Modeling of the process has increased its potential importance in this context,but the difficulties of putting it into practice are considerable, because of the daunt-ing complexity of the chemistry. Notice that it should be integrated with the mod-eling of the cure stage (a complex combination of heat, mass, and mechanical modeling, in the case of wood panel manufacture). Cure is performed with ammo-nium chloride as catalyst, which also acts as a formaldehyde scavenger. Its chemis-try is not fully understood, because of the much higher temperatures than in resin synthesis, which lead possibly to ladder structures.Melamine resins have a similar chemistry, the main difference being the re-duced importance of hydrolysis reactions. They are prepared using two stages, al-kaline addition of formaldehyde, and acid polycondensation.The initial stage of the melamine/formaldehyde reaction has been studied by Nastke et al. [223], who succeeded in providing evidence not only of hydroxymethy-lation, but also of the formation of methylene and methylene ether bridges using polarography. The functionality of melamine is six, with a negative substitution ef-fect of about 40% [224, 225].As with UF, formaldehyde addition is acid- and base-catalyzed (sensitive to pH),and equilibrium constants are 100 to 200. Methylene bridges are also formed only at acid pH. Direct analysis of methylene ether bridges has also been performed by NMR [226].A simplified model (no reaction reversibility) of melamine–formaldehyde forma-tion based on Tomita’s kinetic scheme [225] has been presented [227] and after-wards extended to reaction in a CSTR [228], also considering reaction reversibility.There is no experimental validation, but it is noteworthy for the use of a program to calculate the CLD of a nonlinear reversible polycondensation in order to over-come the astronomical number of reaction possibilities in the rates of formation of individual oligomers.

3.3 Polycondensation Processes in Detail 107
3.3.4.3Phenolic Resins
Two subclasses have to be distinguished [229]: resols, which are highly branched, low molecular weight (150–1500) polymers with a formaldehyde/phenol stoichiometric ratio between 1.2 to 3, formed at al-
kaline pH;
 novolacs, made at acid pH, with a formaldehyde/phenol stoichiometric ratio be-tween 0.5 and 0.8, which have a different and much less branched structure than resols. They are low molecular weight (500–5000) thermoplastics, further cross-linked with hexamethylenetetramine.Phenolic resins are mainly used as wood adhesives, laminates, molded parts, insu-lating varnishes, abrasives, and rigid foams.Novolacs can be made using either strong acid catalysts (sulfuric acid is pre-ferred) or at pH 4 to 7 using carboxylates of divalent metals (such as Zn, Mn,Mg). These catalysts complex phenol and methanediol and lead to formation of o-methylolphenol in a first step. Subsequent addition steps may also be ortho-directed (Scheme 3.17), or more randomly distributed (in the case of Zn).H M2+ OH OH
OH
O OH
- M2+ - M2+ CH2 CH2 CH2OH ...OH - H2 O - H2 O
+ OH
Scheme 3.17. Formation of ortho-directed methylene bridges.Novolacs prepared with acid catalysts have a more random structure. The branching density is low because nonterminal rings are less reactive. This is caused by molecular coiling, which is especially important in high ortho-novolacs.Phenol groups tend to associate through hydrogen bonds and change the mole-cular conformation (Scheme 3.18). Nonterminal units are often assumed to stay
CH2 CH2
O H H O
H H CH2
O H
O H O
CH2 CH2
Scheme 3.18. Hypothetical intramolecular hydrogen bonds in novolac resins.

preferentially inside the molecular coils and decrease their reactivity because of this.Mathematical models describing formation of novolacs both in batch and contin-uous reactors have been developed by Frontini et al. [230] and Kumar et al. [231].They consider the existence of at most one methylol group per molecule, and lump together all isomers with the same unit counts. A Monte Carlo method [232] can also be used in order to obtain a more detailed description at molecular level.The initial addition of formaldehyde to phenol in alkaline media could for thefirst time be successfully described, thanks to Zavitsas and collaborators [233]. Re-sol formation in further reactions is a complex process, owing to the several differ-ent aromatic reaction sites and substitution effects [234]; a total of 19 can be distin-guished [235, 236]. Concentrations of fragments have been computed, assuming irreversible reactions. Number-average and weight-average molecular weights have been predicted using the ‘‘recursive’’ approach. Experimental determination of the necessary structural and kinetic parameters is a huge task, which requires exten-sive use of 13 C NMR and synthesis of model compounds. The research informa-tion [237, 238] allows modeling of these reacting systems to be elaborated with better chemical support. Fairly good agreement of model and experimentally mea-sured functional group concentrations in resol formation at various stoichiometric ratios has been claimed [239] and it is expected that a trustworthy quantitative de-scription of these systems may eventually be achieved. Validation of these predic-tions is plagued by experimental difficulties, and has mainly been carried out through measurement of individual oligomer and functional group concentrations.Reaction is exothermal (DH ¼
80 kJ mol
1 ). Heat of reaction is removed using water reflux, sometimes with a small amount of inert solvent (aromatics are inert only if they carry deactivating groups), and relatively small batch reactors ( 2 to 10 m 3 ) are usually preferred.Resol reactors have been the subject of studies, such as Ref. 240, concerning op-eration in accident situations. In novolac production, formaldehyde can be fed con-tinuously in order to increase safety.
3.3.5
Epoxy Resins The three-membered cyclic ether group oxirane, a 1,2-epoxide, or an epoxy group reacts with substances containing an active hydrogen group, such as amines, phe-nols, and carboxylic acids, or can be polymerized with anion or cation initiators,therefore yielding a great variety of potentially useful polymers. Nearly all of them are thermosetting, and used as coatings and adhesives. Their cure processes,which should be considered in close connection with the method of producing the final material, will not be discussed in this section. Instead, a brief review of processes to make the macromonomers containing epoxy groups, the so-called epoxy resins, will be presented.The most widely used epoxy resins are formed through the reaction between epichlorohydrin (ECH; 3 in Scheme 3.19) and bisphenol A.

3.3 Polycondensation Processes in Detail 109
+ -NaCl
ClCH2CH CH2 HO C OH CH2 CHCH2O C OCH2CHCH2O C OCH2CH CH2 Scheme 3.19. Formation of epoxy resin by reaction of bisphenol A with epichlorohydrin.The so-called ‘‘taffy process’’ consists in the two-phase reaction of an alkaline so-lution of bisphenol A with ECH in stoichiometric excess [241]. The main reaction as described above is accompanied by side reactions, such as hydrolysis and alco-holysis of chlorine and epoxides in ECH. These reactions create molecules with functionality one or even zero, and must of course be minimized.The kinetics has been studied by Enikolopyan et al. [242] and Gao [243], among others. Branching formation occurs to a low extent and can usually be neglected;the reaction can be described as a linear irreversible polycondensation AXA þ BYC,with A ¼ aOH, B ¼ aCl, and C ¼ epoxide (ECH and oligomers have different reactivities).The similar epoxidation of novolacs with ECH has been additionally studied by Oyanguren and Williams [244]. In this system, intramolecular ring formation has been measured.The so-called ‘‘advancement process’’ consists in the melt reaction of bisphenol A (or a similar monomer) with a bifunctional epoxy resin in the presence of a cat-alyst, with the goal of producing a higher molecular weight, bifunctional, epoxy resin. This process leads to branching, due to the reaction of the pendent hydroxyl group with epoxide, and eventually gelation occurs. Its kinetics has recently been studied by Smith and Ishida [276]. The activation energy of the branching reaction was found to be higher (20 as compared to 18 kcal mol
1 ) than that of chain exten-sion, and both constants have been determined both for the catalyzed and noncata-lyzed reaction.
3.3.6
Polyurethanes and Polyureas Urethane polymers were discovered by Baeyer in 1937 [246, 247], using the addi-tion of alcohols to isocyanates leading to carbamates (or urethanes), 4 in Scheme
3.20.
C O R'
R NCO + HO R' R N Scheme 3.20. Addition of alcohols to isocyanates.

The analogous, much faster, reaction with primary amines produces N-substituted ureas 5 (Scheme 3.21).
C NH R'
R NCO + H2N R' R N
H 5
Scheme 3.21. Addition of primary amines to isocyanates.Reaction with water produces an amine and carbon dioxide through an unstable carbamic acid intermediate (Scheme 3.22).
C O H H
R NCO + H2O R N R N + CO2
H H
Scheme 3.22. Reaction of water with isocyanates.As the amine reacts again with isocyanate, this reaction will lead to branching, as will consecutive reactions with carbamates, leading to allophanates 6 (Scheme
3.23) and with ureas, leading to biurets 7 (Scheme 3.24).
O O
C O R' C O R'R NCO + R N R N
H C N R 6
H
Scheme 3.23. Reaction of urethanes with isocyanates leading to allophanates.
O O
C NH R' C NH R'R NCO + R N R N
H C N R 7
H
Scheme 3.24. Reaction of ureas with isocyanates leading to biurets.Trimerization of isocyanates, leading to the thermostable isocyanurate ring 8,occurs with basic catalysts (multifunctional amines, carboxylates, alkoxides, and so on) through allophanate intermediates [252–254] (Scheme 3.25).Isocyanate groups also react to form uretdiones 9. This is an equilibrium reac-tion, which is mainly important at high isocyanate concentration (Scheme 3.26).Most urethane polymers are thermosets [247, 248]. They are mainly used in the production of foams, taking advantage of the reaction of water and of the pre-cipitation of insoluble ureas, which stabilize the foam even before the system

3.3 Polycondensation Processes in Detail 111
C O C H R R
R' N N
R N + R NCO R N O + R' OH
N R
C N H C N C
O O O N O
R R R
Scheme 3.25. Formation of isocyanurates.2 R NCO R N N R Scheme 3.26. Formation of uretdiones.gels chemically. Amine- or hydroxyethyl-capped branched polyols of molecular weight of the order of a few thousands are preferred. They are most often star-shaped poly(oxypropylene), in combination with shorter polyols such as glycerine,additives such as surfactants and demolding agents, blowing agents, and water.Rigid foams use more branched polyols than do flexible foams.Another important use is the fabrication of plastics by RIM (reaction injection molding) (see Figure 3.11), which exploits the very fast reaction which can be Fig. 3.11. Scheme for an RIM machine with a jet impingement mix-head (on a very exaggerated scale), in its recirculation and injection modes.

achieved either with amines or with hydroxyls, in the presence of catalysts such as dibutyltin carboxylates, tertiary amines, or a combination of the two [250]. The ex-cellent book by Macosko [249] extensively covers this technology, which is particu-larly well adapted to the fabrication of big, flat, molded parts, and also to complex elastomeric objects (viscosity is very low during mold filling). The whole produc-tion cycle can take less than one minute from injection to demolding.As with the other thermosets in this chapter, these processes will not be dis-cussed in this section. We will nevertheless give a few hints of the reaction engi-neering of the production of polymers and macromonomers based on this chemis-try; other relevant uses are adhesives, binders, coatings, thermoplastic elastomers,and fibers.The catalysis of isocyanate reactions has been extensively studied because of its critical importance in many of these processes. Noncatalyzed (or rather, self-catalyzed) reactions may sometimes be fast enough in practice: isocyanate reac-tions with amines are so fast that only recent studies using stopped-flow methods could lead to useful data [255, 256], metallic or tertiary amine catalysts being inef-fective in this case.As often happens with polymerization reactions, simple rate laws can seldom describe the whole course of reaction because of catalysis or inhibition by the urethane groups formed or by the initial reagents. Self-association of metallic cata-lysts, or their loose complexation by products or reagents, also prevents correlation of rates of reaction by simple proportionality or even power-law relations. Catalysis of isocyanate reaction with hydroxyls [250, 251] is by far the best understood.It might be thought that modeling of polyurethane processes would be relatively straightforward, given that reactions are mostly irreversible and the methods de-scribed in Section 3.4.4 should deal with them without difficulties.In reality, allophanate and biuret formation is reversible at temperatures above 130 C [252]. Formation of isocyanurates causes a reorganization of the CLD akin to what happens in reversible polycondensations because of exchange reactions.Many polyurethanes are block polymers prepared with a diisocyanate, a short diol such as 1,4-butanediol or 1,6-hexanediol, or a diamine (the chain extender),and a diol with molecular weight between 500 and 4000, based on a polyether,polyester, polycarbonate, poly(butadiene) or other. Most often, the preparation is performed in two steps: firstly, reaction of the longer polyol with the isocyanate,then with the chain extender in the second stage.An important and desirable feature of polyurethanes and polyureas is the phase separation of the small isocyanate/chain extender blocks, which is possible pro-vided the thermodynamics is favorable: this means a high enough concentration and chain length of ‘‘hard’’ blocks. These ‘‘hard’’ blocks act as physical crosslinks at a temperature lower than their melting point, and thermoplastic elastomers (in-cluding elastic fibers) can therefore be obtained.From the point of view of the prediction of structure, this brings about a complex problem, which shares some characteristics with solid-state polycondensation, and has been tackled through the use of Monte Carlo methods [257, 258, 259].Reactors for these processes range from simple batch or continuous stirred tank

3.4 Modeling of Complex Polycondensation Reactions 113
reactors for very low molecular weight or solvent-based processes, to tubular reac-tors with static mixers or extruders. Owing to the strong exothermicity of the main reaction, cooling has to be used in order to prevent side reactions from becoming Many processes at low temperature and in homogeneous phase can nevertheless be analyzed through the methods described in Section 3.4.4, as they now stand.
3.4
Modeling of Complex Polycondensation Reactions
3.4.1
Overview
Rate equations allowing the prediction of concentrations of reactive groups, includ-ing more complex molecular fragments (monads, dyads and so on) from mass bal-ance equations are established in Section 3.4.2 for irreversible reactions or systems with at most FSSEs if reversible reactions exist. Stoichiometric coefficients are in-troduced in order to obtain a fairly general formalism, later exploited in Section
3.4.4.
Knowledge of the distributions of numbers of bonds connecting repeating units makes it possible to describe molecular structure at chemical equilibrium. In Sec-tion 3.4.3, the main results concerning molecular weights and network properties of chemical systems verifying FSSEs are presented, using the theory of branching processes. The presence of rings is also allowed. The results of Section 3.4.2 are useful in order to predict average numbers of bonds for each monad, needed for predicting average molecular weights.Irreversible polycondensations can be tackled quite easily using a general kinetic approach developed in Section 3.4.4, allowing prediction of average molecular weights before or after gelation, and even molecular weight distributions (lumping together isomers with the same numbers of groups).Prediction of molecular weight distributions for reversible, linear, alternating polycondensation is discussed in Section 3.4.5. Mathematical difficulties grow con-siderably in the presence of SSSEs. There seems to be no alternative to Monte Carlo methods for dealing with reversible nonlinear polycondensations or even lin-ear polycondensations where more than two kinds of bonds are present.
3.4.2
Description of Reactions in Polycondensations of Several Monomers with Substitution Effects The goal of this section is to present a general nomenclature of chemical entities and reactions which provides a concise form for writing the rate equations and mass balances of chemical species. The existence of substitution effects seriously complicates the task, because the same molecular entity has to be labeled in a dif-

ferent way, not because it has intrinsically changed, but because its neighbors have reacted.We introduce the following nomenclature for the species (monomer units or functional groups) and vectors h; g; e storing their indices:Xh i monomer units, h i A 1 . . . NX A gi
; A giþ functional groups which react forming bond ZiR ; indices gi
; giþ A
1 . . . NA
We i by-products, e i A 1 . . . NW Z iR bonds, i A 1 . . . NR
þ
Ze
i ; Zeþi ‘‘half-bonds’’, e
i ; ei A 1 . . . NZ The NA functional (or end) groups A i will be distinguished (even if chemically sim-ilar) according to the monomer unit Xh i where they are attached. h is yet another vector of indices, with size NA .NR reactions involving pairs of functional groups create connections between monomer units, possibly (with the well-known exceptions of epoxides and isocya-nate reactions) also forming by-products. A total number NW of by-products Wi will be considered for the sake of generality. By-product We i is formed by the reac-tion between functional groups A gi
and A giþ creating the bond ZiR .Vectors e; gþ and g
have sizes NR . This definition allows for more than one pos-sible kind of bond between two given monomer units, as happens for instance if a carboxylic acid reacts with glycerol, which possesses distinguishable primary and secondary hydroxyls.In some of these NR reactions, such as in the case of self-condensations of sila-nols in silicone formation and of methylols in formaldehyde polymerizations, an end group may react with itself. The number of such reactions in this subset will be defined as NRs .For each bond, it is useful to define a positive sense in the direction of the unit with higher or equal index, which will be coincident for the NRs reactions consid-ered above. So, there is a total number of NZ ¼ 2NR
NRs kinds of directed bonds Z i , incident on monomer units Xzi ; the set of monomer units and bonds are the vertices and edges of a directed graph (or digraph). Vector z, of size NZ , therefore contains the indices of the repeating units to which each directed bond points.There is a one-to-one correspondence between directed bonds and half-bonds hanging from the repeating units at each side.The directed bonds associated with ZiR will be named Ze
i and Zeþi , using yet an-other pair of vectors of indices, eþ and e
, of sizes NZ . Hence, the vectors z and h,defining respectively their incident and adjacent monomer units, will be related through hgiþ ¼ zeþi and hgi
¼ ze
i .In order to avoid multiple levels of indexing in equations, the notation given by Eqs. (62) (loosely inspired by indirect addressing of computer assembly languages)will be used hereafter.A giþ 1 A½iþ Zeþi 1 Z½iþ We i 1 W½i Xh i 1 X½iA Xzi 1 X½iZ ð62Þ

3.4 Modeling of Complex Polycondensation Reactions 115
For the NR reactions which create new bonds from functional groups and also (very often) a by-product, we will introduce apparent second-order rate constants of the forward reaction (ki , i ¼ 1; NR ), as well as apparent first-order rate constants of the backward reactions (kiZ ), related to the former through the equilibrium ratios Ki and the concentration of the corresponding by-product, if it exists, according to
½W½i 
kiZ ¼ ki ð63Þ
Ki
This may look rather artificial, but it helps in situations such as the thermal de-composition of urethanes or ureas, which are first-order reactions. By convention,in such cases we will introduce a nil by-product W0 (assuming that the indices are counted starting from one) with a constant unit concentration.If the groups react independently, by definition there is no substitution effect. The condensation reaction creating a bond (the same as two half-bonds) can be written as Eq. (64).
ki
A½i
 þ A½iþ T Z½i
 þ Z½iþ þ W½i ð64Þ
kiZ
To account for first-shell substitution effects (FSSEs), a more general expression[Eq. (65)] can be written, using stoichiometric coefficients n:
X
NA X
NW X
NZ
ðninA
þ ninAþ ÞAn þ nW in Wn þ ðninZ
þ ninZþ ÞZn ¼ 0 ð65Þn¼1 n¼1 n¼1 ninA
; ninZ
correspond, respectively, to functional groups An and oriented bonds Zn coming out of the monomer unit in which stood A½i
 , and a similar convention is used for ninAþ ; ninZþ.
Aþ Zþ
No FSSE means that, for every reaction i, nigA
Z

¼ n þ ¼
1, nie
¼ n þ ¼ 1, and,
igi iei
moreover, all other stoichiometric coefficients are nil. Likewise, it is useful to intro-duce the stoichiometric coefficients of by-products, nW in .This trick only works with FSSEs. Higher-order substitution effects (see Sections
3.1.5 and 3.4.5) are much more difficult to describe.
In the absence of FSSEs, the exchange of bonds ZiR and ZjR , necessarily forming the same by-product (e i ¼ ej ), can be described through Eqs. (66).
kijE

X
A½i
 þ X þ
Z½ j
 Z½ jþ X þþ
!
þ

X Z½i
 Z½iþ X þ A½ j
 X
þþ
kjiE

ð66Þ
kijEþ

þ
þþ
!
þ
þþX A½iþ þ X Z½ j
 Z½ jþ X
X Z½i
 Z½iþ X þ A½ jþ X
kjiEþ
In the presence of FSSEs, they would be written with an algebraic notation as in Eqs. (67).

X
NA X
NZ
AE

AE
þ
AE
þþ ZE

ZE
þ
ZE
þþðnijn þ nijn þ njin ÞAn þ ðnijn þ nijn þ nijn ÞZn ¼ 0
n¼1 n¼1
ð67Þ
X
NA X
NZ
AEþ
AEþþ
AEþþþ ZEþ
ZEþþ
ZEþþþðnijn þ nijn þ nijn ÞAn þ ðnijn þ nijn
nijn ÞZn ¼ 0
n¼1 n¼1
AE

ZE


Stoichiometric coefficients nijn ; nijn are the changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X
where either AE
þþ ZE
þþ AEþþþ ZEþþþA½i
 or A½iþ were attached, and nijn ; nijn ; nijn ; nijn are the changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X þþ where stood the living group A½ j
 or A½ jþ. The other stoichiometric coef-AE
þ
ZE
þ
AEþþ
ZEþþ
ficients nijn ; nijn ; nijn ; nijn are the changes in the numbers of groups attached to root unit X þ
which gets connected to the unit where stood the attack-ing group, and they are nil if there are no substitution effects.The above reactions modify the counts of functional groups of similar chemical nature (for example, distinguishable kinds of amides/amines/carboxylic acids). If there is only a single kind of bond, there is no net creation or destruction of func-tional groups or bonds, as shown by the cancellation of the stoichiometric coeffi-cients in Eq. (67), but a reshuffling of pieces of the reacting molecules takes place.Rates of production by chemical reaction of the various groups are obtained us-ing Eqs. (68)–(70).
X
NR
R An ¼ ðninA
þ ninAþ Þðki ½A½i
 ½A½iþ 
kiZ ½Z½iþ Þ
i¼1
X
NR X
NR
þ de ij ½kijE
½A½i
 ½Z½ jþ ðnijn
EA
EA
þ

þ nijn EA
þþ
þ nijn Þ
i¼1 j¼iþ1
þ kjiE
½A½ j
 ½Z½iþ ðnjin
EA
EA
þ

þ njin EA
þþþ njin Þ ð68Þ
X
NR
R Zn ¼ ðninZ
þ ninZþ Þðki ½A½i
 ½A½iþ 
kiZ ½Z½iþ Þ
i¼1
NR X
X NR
þ de ij ½kijE
½A½i
 ½Z½ jþ ðnijn
ZA
ZA
þ

þ nijn ZA
þþ
þ nijn Þ
i¼1 j¼iþ1
þ kjiE
½A½ j
 ½Z½iþ ðnjin
ZA
ZA
þ

þ njin ZA
þþþ njin Þ ð69Þ
X
NR
RWn ¼ k i nW Z in ð½A½i
 ½A½iþ 
ki ½Z½iþ Þ ð70Þ
i¼1
The reaction volume changes mostly because of by-product removal, and little be-cause of density changes. The relative rate of change of reaction volume RV can be

3.4 Modeling of Complex Polycondensation Reactions 117
estimated as the sum of products of the molar volume of by-products by their rate of elimination by phase change.The mass balances of the functional groups in a batch reactor can thus be writ-ten as Eqs. (71).
d½An 
¼ RAn
RV ½An 
dt
ð71Þ
d½Zn 
¼ RZn
RV ½Zn 
dt
A convenient way of computing the concentrations of groups at chemical equilib-rium consists in integrating the system of ODE [Eq. (71)] until close to steady state.
3.4.3
Equilibrium Polycondensations with Several Monomers Instead of the elegant but often error-prone Gordon’s notation, we will introduce a more straightforward description, hopefully easier to translate into computer pro-grams. The goal is to obtain a set of formulae for predicting average molecular weights, molecular weight distributions, and other polymer properties, valid for ge-neric chemical systems.These computations assume there is some way of predicting how molecular fragments (usually monads, unless higher-order substitution effects have to be tackled) are mutually connected. More specifically, it is necessary to know the dis-tributions of the numbers of bonds connecting the fragments, for each kind of frag-ment. In fact, one may need only some of the moments of the aforementioned dis-tributions for making a few calculations.A kinetic method may be used for this prediction, but mainly as a means to avoid solving equations derived from mass action laws for concentrations of frag-ments. This has been one of our motivations for presenting the formalism in Sec-tion 3.4.2.Each directed bond Z i is supposed to start a pendent chain Vi ðxZ ; xA Þ with counts of end groups and directed bonds xA and xZ , respectively. Notice that the molecular graphs have to be considered as digraphs, otherwise xZ would be mean-ingless: it would be impossible to know the counts of the monomer units accord-ing to their chemical nature.All isomeric trees with the same counts of groups are lumped into the same chemical species leading to vector count distributions with NZA ¼ NZ þ NA inde-pendent variables.Vectors of dummy Laplace variables sA and sZ will be associated with the counts of unreacted groups and directed bonds. Variables sA and sZ will be often ranged together as subvectors of a vector s, of size NZA .

Vector xX containing the counts of the monomer units can be obtained from xZ through Eq. (72).xX ¼ ðZ ZX Þ t xZ ð72ÞZZX is a matrix containing the incidence vectors z defined by Eq. (73).

1 if j ¼ zi ZijZX ¼ ð73Þ0 if j 0 zi The chemical system is further described through knowledge of the molar frac-tions of monomers or monomer units, yX i , (summing to 1) and of the molecu-lar weights of monomer units, unreacted groups, and half-bonds, respectively MX i ; MA i and MZ i . Hence, the molecular weight M½Vi ðxZ ; xA Þ of a tree Vi ðxZ ; xA Þcan be computed through Eq. (74).
X
NZ X
NA
M½Vi ðxZ ; xA Þ ¼ ðMX½ jZ þ MZ j ÞxZ j þ MA j xA j ð74Þ
j¼1 j¼1
X i ðxZ ; xA Þ is, according to the concept introduced in Section 3.1.5, a monad with vectors of group counts xZ and xA . Its concentration, normalized by the concentra-tion of repeating units X i , can be thought of as a probability: the probability that a certain repeating unit is attached to those counts of groups, as stated in Eq. (75).PfX i is connected to xZ ; xA groupsg ¼ ½X i ðxZ ; xA Þ/½X i  ð75ÞLet F X i ðsZ ; sA Þ; F A i ðsZ ; sA Þ and F Z i ðsZ ; sA Þ be the probability generating functions(pgf ) of the counts of the different kinds of connecting and unreacted functional groups directly linked to a unit X i , an unreacted group A i or a directed bond Z i[Eq. (76)].
X
y X
y X
y X
F J ðsZ ; sA Þ ¼  xZ1 ¼0 xZN ¼0 xA1 ¼0 xAN ¼0
Z A
X
xZ xAN
PfJ is connected to xZ ; xA groupsgsZ1 1    sAN A
xAN ¼0
J ¼ Xi; Ai; Zi ð76ÞThese pgf values are mutually related through Eqs. (77) and (78), in which 1N means a vector with N components equal to 1.

qF X½ jA qF X½ jA X X
F Aj
ðsÞ ¼ s
1
Aj ¼ s
1 ½ jA ½ jAA j LA j ðsÞ/lA j ð77Þq log sA j q log sA j js¼1
NZA

3.4 Modeling of Complex Polycondensation Reactions 119

F Z j ðsÞ ¼ s
1 Zj ¼ s
1 ½ jZ ½ jZZ j LZ j ðsÞ/lZ j ð78Þq log sZ j q log szj js¼1
NZA
Notice the use of L with lower indexes for the derivatives of the pgf values with respect to the logarithms of dummy Laplace parameters, as well as of l for their moments, a useful convention which will be encountered often in the rest of this chapter.If a pgf relative to the count of monomer units is desired, for a vector of dummy Laplace variables sX , it can be found by obtaining the pgf with respect to the counts of directed bonds with Eq. (79).sZ ¼ ZZX sX ð79ÞThe average numbers of bonds Z j ; lZX ij , and of unreacted functional groups A j ; lAX ij ,attached to a monomer unit X i , will be often needed, and can be obtained using Eqs. (80) and (81).
qF X i
lZX ij ¼ ð80Þq log sZ j js¼1
NZA
qF X i
lAX ij ¼ ð81Þq log sA j js¼1
NZA
The mass of polymer per mole of monomer units, MP , can therefore be computed using Eq. (82).
NX X
NZ X
NA
MP ¼ y X i MX i þ lZX ij MZ j þ lAX ij MA j ð82Þi¼1 j¼1 j¼1 For the classic self-polycondensation of XA f , with equal and independent groups A; p, the conversion of A groups, is introduced in Eqs. (83).F X ¼ ½ð1
pÞsA þ psZ  f F Z ¼ F A ¼ ½ð1
pÞsA þ psZ  f
1
ð83Þ
lZX ¼ f p lAX ¼ f ð1
pÞMP ¼ MX þ f pMZ þ f ð1
pÞMA Expressions for the less trivial case XA f þ YBg C illustrated diagrammatically in Figure 3.12, A reacting with B or C, B not reacting with C (such as adipic acid þ glycerol, f ¼ g ¼ 2), with constant reactivity of end groups A; B, or C, are presented in Table 3.3.

Fig. 3.12. Example of proposed notation: polycondensation XA2 þ YB2 C.Tab. 3.3. Probability generating functions F describing polycondensation XA f þ YBg C and its gelation condition.X F XX ½ð1
pA ÞsA þ pAB sZAB þ pAC sZAC  f
F ¼
F XY ½ð1
pB ÞsB þ pB sZBA  g ½ð1
pC ÞsC þ pC sZCA F ZAB ½ð1
pB ÞsB þ pB sZBA  g
1 ½ð1
pC ÞsC þ pC sZCA 6 ZAC 7 6 ½ð1
p Þs þ p s  g 76F 7 6 B B B ZBA 7 FZ ¼6 7 6 74 F ZBA 5 4 ½ð1
pA ÞsA þ pAB sZAB þ pAC sZAC  f
1 5
f
1
F ZCA ½ð1
pA ÞsA þ pAB sZAB þ pAC sZAC 
2 AA 3 2 3
F ½ð1
pA ÞsA þ pAB sZAB þ pAC sZAC  f
1
6 A 7 6 7
F ¼4 F B 7
5 4 ½ð1
pB ÞsB þ pB sZBA  g
1 ½ð1
pC ÞsC þ pC sZCA  5
g
F AC ½ð1
pB ÞsB þ pB sZBA Gelation condition gpBg pACg þ ðg
1Þ pBg pABg þ pCg pABg ¼
f
1
Group counts have multinomial distributions for these simple systems and can be easily related to conversions of functional groups (which are the probabilities of reaction).The theory of branching processes leads to a system of NZ algebraic equations[Eqs. (84)] for the pgf of pendent trees of the different kinds:Vi ðsZ ; sA Þ ¼ sZ i F Z i ½VðsZ ; sA Þ; sA  i ¼ 1; . . . ; NZ ð84ÞThe vector v of the NZ probabilities of extinction (the fractions of finite pendent chains) v ¼ Vð1NZA Þ contains the solutions of system (84) for s ¼ 1NZA .Gelation occurs when system (84) has a double root v ¼ 1NZA for s ¼ 1NZA, imply-ing that its Jacobian becomes nil [Eq. (85)].
qF Z i

I ¼ j½LzZj i ð1NZA Þ
Ij ¼ 0 ð85Þ
qnj
For the polycondensation of a single monomer, this leads to the well-known result of Eq. (86).

3.4 Modeling of Complex Polycondensation Reactions 121
f
1
Prediction of the gel point for XA f þ YBg C is also presented in Table 3.3.The pgf values of trees starting with a prescribed monomer unit X i , an unreacted group A i , or a directed bond Z i , are obtained from the theory of branching pro-cesses through Eqs. (87) and (88).GYi ðsZ ; sA Þ ¼ sYi F Yi ½VðsZ ; sA Þ; sA  Y ¼ Z; A ð87ÞG X i ðsx ; sA Þ ¼ sX i F X i ½VðZZX sX ; sA Þ; sA  ð88ÞThe chain length or molecular weight distribution of polymer is usually described primarily with the help of the molar concentration of species with x monomer units, ½Px , in which x is the degree of polymerization. Since we are dealing with several kinds of units, an overall degree of polymerization can be defined, which is the sum of the degrees of polymerization corresponding to the various monomer units (the components of vector xX ). Notice that vector xZ with the counts of di-rected bonds has more information about the molecular composition.There is a basic difficulty when trying to use Eqs. (87) and (88): they provide pgf values of monomer units or groups, not directly molar concentrations of polymer molecules, even as generating functions. These latter will have to be computed rel-ative to the molar concentrations of the various groups or monomer units, and thus will come multiplied also by the number of those groups in the distributions.For instance, if there is only one kind of monomer unit and so just one kind of directed bond, G X as computed by Eq. (88) will provide the generating function of x½Px /½X  with respect to x; for an alternating polycondensation of two monomers,G X 1 ðsZ1 ; sZ2 Þ as computed by Eq. (88) will provide the generating function of x1 ½Pðx1 ; x2 Þ/½X1  with respect to x1 and x2 .Fractions of units and groups of the various kinds in finite molecules (in sol),after gelation, ySX i ; ySZ i and ySA i , can be computed by replacing s ¼ 1NZA in the above equations to give Eq. (89).ySYi ¼ F Yi ðv; 1NA Þ Y ¼ X; Z; A ð89ÞPrediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions(EANJs) and chains (EANCs), respectively me and ne [260, 261]. EANJs are the inter-section of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13).These concentrations can be easily predicted, given the probabilities of extinction and the moments with respect to the numbers of pendent chains as previously de-fined. Defining xZy as the count of infinite pendent chains and the correspondent dummy Laplace variable as sZy , its pgf for the chains stemming out of a monomer unit X i is F X i ½v þ sZy ð1NZ
vÞ; 1NA , and so me and ne can be computed using Eqs.
(90) and (91).

Finite pendent chain Infinite pendent
chain
EANJ
Junction
EANC
EANJ
Chain connecting to gel Fig. 3.13. Elastically active and inactive junctions and chains in a polymer network.
X
NX X
NZ
me ¼ ½X i  1
F X i ðv; 1NA Þ
nj ð1
nj ÞLZXji ðv; 1NA Þ
i¼1 j¼1
1X NZ XNZ
Xi

nj nk ð1
nj Þð1
nk ÞLZ j Zk ðv; 1NA Þ ð90Þ
2 j¼1 k¼1
1X NX XNZ
ne ¼ ½X i  ½lZX ij
nj ð1
nj ÞLZXji ðv; 1NA Þ
2 i¼1 j¼1
X
NZ X
NZ

nj nk ð1
nj Þð1
nk ÞLZXjiZk ðv; 1NA Þ ð91Þ
j¼1 k¼1
For instance, assuming the so-called ‘‘phantom network’’ model, shear modulus Ge would be predicted for gaussian chains to be given by Eq. (92).Ge ¼ RTðne
me Þ ð92ÞIn the presence of gel, it is convenient to introduce the pgf of finite pendent chains,named V^i ðsZ ; sA Þ, and the pgf of counts of finite pendent chains connected to units or groups, which can be found using Eqs. (93) and (94).

3.4 Modeling of Complex Polycondensation Reactions 123
F^Yi ðsZ ; sA Þ ¼ F Yi ðn1 s Z1 ; . . . ; nNZ sZNZ ; sA Þ/ySYi ¼ F Yi ðv5sZ ; sA Þ/ySYi V^i ðsZ ; sA Þ ¼ sZ i F^Z i ½V^ ðsZ ; sA Þ; sA  i ¼ 1; NZ ð94ÞSo, the various pgf values with respect to the different kinds of groups in the^ X i ðsZ ; sA Þ; G molecules of the sol, G ^Z i ðsZ ; sA Þ; G^A i ðsZ ; sA Þ, can be computed using Eqs. (95).^Yi ðsZ ; sA Þ ¼ F^Yi bV G ^ ðsZ ; sA Þ; sA c Y ¼ X; Z; A ð95ÞAfter computing the probabilities of extinction, the moments in Eq. (96) can be evaluated.l^YZij ...Zk ¼ LYZi j ...Zk ðv; 1NA Þ/ySYi Y ¼ Z; A ð96ÞPgf values of finite pendent chains with respect to molecular weight, for which the dummy Laplace variable associated with molecular weight is sM , can be found from Eq. (97) (notice the conventional use of a power of a scalar to a vector):MZ þM ZN M þMX ½NZ Z MA MAN M þMX z MA V^Mi ðsM Þ ¼ V^i ðsM 1 X½1Z ; . . . ; sM Z ; sM 1 ; . . . ; sM A Þ ¼ V^i ðsM z ; sM Þ
ð97Þ
Prediction of average molecular weights is now possible by introducing GM ðsM Þ,the pgf values of the mass fractions of the polymer molecules in the sol, wi , with respect to their molecular weight Mi , defined below; the index i in the infinite sum sweeps all finite polymer molecules.
Mi
X X
NY
MY MA
GM ðsM Þ ¼ sM wi ¼ wYi sM i F^Yi ½V^ M ðsM Þ; sM  ð98Þi¼1 Y ¼X; Z; A i¼1 The mass fractions of the units and groups in Eq. (98) above are relative to the mass of the sol.The weight fraction of the sol wS , relative to the overall mass of the polymer computed by Eq. (82), is therefore given by Eqs. (99)–(102).
NX X
NZ X
NA
wS ¼ y X i ySX i MX i þ l^ZX ij MZ j þ l^AX ij MA j MP ð99Þi¼1 j¼1 j¼1 y X i ySX i MX i wX i ¼ ð100Þ
MP wS
NX
MZ i y X j ySX j l^Z ji
j¼1
wZ i ¼ ð101Þ
MP wS

X
NX
X
MA i y X j ySX i l^A ji
j¼1
wA i ¼ ð102Þ
MP wS
Before carrying out the evaluation of distributions and average molecular weights,a special reasoning must be carried out in order to compute number-average mo-lecular weight and degrees of polymerization. When it is taken into account that,with the reaction of every pair of end groups in finite molecules, one polymer mol-ecule is consumed, the number of moles of polymer molecules per mole of mono-mer units before gelation is given by Eq. (103).
1X NX XNZ
yP ¼ 1
yXi lZX ij ð103Þ
2 i¼1 j¼1
After gelation, the more general expression [Eq. (104)] is needed.
X
NX
1X NZ
yP ¼ y X i ySX i 1
l^X i ð104Þ
i¼1
2 j¼1 Z j
The number-average molecular weight of the sol can thereafter be computed through Eq. (105).
X
NX X
NZ X
NA
y X i ySX i MX i þ MZ i l^ZX ij þ MA i l^AX ij i¼1 j¼1 j¼1 Mn ¼ ! ð105Þ
X
NX
1X NZ
y X i ySX i 1
l^X i
i¼1
2 j¼1 Z j
An expression for the number-average degree of polymerization of the sol follows from Eq. (105) by setting the molecular weights of monomer units equal to one and the molecular weights of bonds and unreacted groups equal to zero [Eq. (106)].
X
NX
y X i ySX i
i¼1
xn ¼ N ! ð106Þ
X X
1X NZ
y X i ySX i 1
l^X i
i¼1
2 j¼1 Z j
The moments with respect to molecular weight lM ; lMM , and so on, can be now obtained through differentiation of Eq. (98) with respect to log sM and setting sM ¼ 1. First of all, the systems of linear algebraic equations (107)–(113) must be solved.

3.4 Modeling of Complex Polycondensation Reactions 125
½miZ  ¼ ½di
l^ZZ ij 
1 ½MZ i þ MX½iZ  ð107Þ½miA  ¼ ½di
l^ZZ ij 
1 ½l^ZAji ½MA i  ð108Þ
X
NZ NZ X
X NZ
½miZZ  ¼ ½di
l^ZZ ij 
1 2ðMZ i þ MX½iZ Þ mZl l^ZZli þ Z ^Z i mZl mm lZl Zm ð109Þl ¼1 l ¼1 m¼1
X
NA NZ X
X NA
½miZA  ¼ ½di
l^ZZ ij 
1 ðMZ i þ MXX½iZ Þ MAm l^AZki þ mZl MAm l^ZZliAm ð110Þm¼1 l¼1 m ¼1
X
NZ X
NA NA X
X NA
½miAA  ¼ ½di
l^ZZ ij 
1 mlA MAm l^ZZliAm þ MAl MAm l^AZliAm ð111Þl¼1 m¼1 l¼1 m¼1½miMZX  ¼ ½di
l^ZZ ij 
1 ½ðMX½iZ þ MZ i Þ 2  ð112Þ½miMA  ¼ ½di
l^ZZ ij 
1 ½MA2 i  ð113ÞWeight-average and z-average molecular weights are now obtained explicitly through Eqs. (114) and (115).
qGM
lM ¼ ¼ Mw
q log sM jsM ¼1
X X NY X
NZ X
NA
A ^Yi Y
¼ w Yi M Yi þ Zðmj þ mj ÞlZ j þ MA j l^A ji ð114ÞY ¼X; Z; A i¼1 j¼1 j¼1 lMM ¼ Mw MZ
X X
NY X
NZ X
NA
¼ w Yi MY2i þ 2MYi ðmjZ þ mjA Þl^YZij þ MA j l^YZij Y¼X; Z; A i¼1 j¼1 j¼1
X
NZ X
NZ
þ ðmjZ þ mjA ÞðmkZ þ mkA Þl^YZij Zk
j¼1 k¼1
X
NZ
þ ðmjZZ þ 2mjZA þ mjAA þ mjMXZ þ mjMA Þl^YZij
j¼1
NZ X
X NA
þ2 ðmjZ þ mjA ÞMAk l^YZij Ak
j¼1 k¼1
NA X
X NA X
NA
þ MA j MAk l^AYjiAk þ MA2 j l^AYji ð115Þj¼1 k¼1 j¼1 Except for extremely simple polycondensations, formulae for predicting average molecular weights are very cumbersome and numerical evaluation is a must.

The weight-average and z-average degrees of polymerization, x w and x z , are obtained in the same way as for number-average degrees of polymerization: the molecular weights of the monomer units are set equal to one and the molecular weights of bonds and unreacted groups are set equal to zero. For the polyconden-sation of a single monomer XA f , this leads to Eqs. (116) and (117).1 þ pð2v
1Þ
xw ¼ ð116Þ
1
p½1 þ vð f
2Þ2f pv þ f p 2 vð f
1Þð1
p þ pvÞ f
2
xz ¼ 1 þ
ð1
p þ 2pvÞ½1
pð f
1Þð1
p þ pvÞ f
2 ð f
1Þð f
2Þ p 2 v 2
þ ð117Þ
ð1
p þ 2pvÞ½1
pð f
1Þð1
p þ pvÞ f
2  2 The number-average degree of polymerization x n is obtained through a stoichio-metric reasoning as previously discussed [Eq. (118)].
1
p þ pv
xn ¼ ð118Þ
1
p þ pvð1
f /2ÞValues of average degrees of polymerization versus conversion of end groups p shown for f ¼ 2 and f ¼ 3 in Figure 3.1 have been computed using the above expressions.Pgf values of the various distributions with respect to the counts of monomer units, bonds, or unreacted functional groups can be obtained from Eq. (98) by set-ting equal to one the molecular weight of the species in question and to zero the molecular weights of all the other species. Analytical inversion by computing deriv-atives with respect to dummy Laplace variables on s ¼ 0 is feasible with the sim-plest chemical systems, for which the resulting recurrence formulae are not too complex, as happens with the Stockmayer distribution.Numerical inversion of generating functions of the concentration distributions is usually a better way to predict them. Unless the cost of evaluating GðsÞ ¼ s x ½Px 
x ¼0
is too high and more sophisticated methods based upon Laplace transform inver-sion are needed, an accurate evaluation can be obtained using the method inde-pendently developed by Mills [262] and ourselves [263] (see also Ref. 264 for a thor-ough analysis of round-off and truncation errors of similar approaches). It consists in computing the inversion contour integral on a circle C in a complex plane cen-tered on the origin with radius jsj slightly below 1, using the trapezium rule and Fast Fourier Transform for evaluating the sums [Eq. (119)].
 
jsj
x X
N
1
2pimx Xy
½Px  ¼ exp
Gðsm Þ
½PxþNn jsj Nn ð119ÞN m¼0 N n¼1

3.4 Modeling of Complex Polycondensation Reactions 127
The second term in Eq. (119) can be neglected for large enough N or small enough jsj. A recent surge on this approach has led to exploitation of other more complex but hopefully more efficient methods [265], mainly developed for Laplace trans-form inversion, and so more adapted to high average molecular weights. The error of those formulae is more complex to control, unlike Eq. (119).Besides molecular weight distribution, it is also possible to access readily some information about the distribution of molecular sizes and other polymer proper-ties, such as the angular dependence of light scattering intensity. The evaluation of averages involving the distances of every pair of monomer units is required,and a starting point for that purpose is the evaluation of the trail generating func-tions [31–33], allowing the counting of path lengths.Equilibrium polycondensation of a single monomer XA f taking FSSEs into account has been analyzed by Gordon and Scantlebury [268] using TBP, and exper-imental results concerning the POCl3 /P2 O5 system have been successfully de-scribed. More general calculations are better carried out using the method de-scribed by Kuchanov et al. [267], summarized below for the polycondensation of XA f (but only in the absence of gel).Chemical equilibrium will be attained in two hypothetical stages:
1. All the rings are formed, but no fused rings, such as naphthalene, are allowed
and molecules look like ‘‘cactus’’ [266]. The fraction of repeating units X in rings of size n ðn ¼ 1; yÞ at the end of this stage is assumed to be y Xc ðnÞ, sum-ming 1
yc , which will be found afterwards from mass action laws. No other reactions of functional groups A occur.
2. The unreacted monomer and the rings start a polycondensation with an infinity
of monomers with a single group A and different functionalities, which are f for the unreacted monomer coming from stage 1 and nð f
2Þ
1 for the rings.Defining sCn as the dummy Laplace variables associated to the counts of rings(including sC0 for the count of units X in the chains connecting rings) and sZl as the variable counting the bonds not belonging to rings, the generating func-tion of the trees with either a ring or a unit not belonging to any ring can be found from TBP as shown in Eqs. (120).
X
V ¼ F Zl ðVÞ ¼ b 0 sC0 ½sA ð1
ar Þ þ ar V f
1 þ b n sCn ½sA ð1
ar Þ þ ar V nð f
2Þ
1
n¼1
X
G ¼ ð1
yc ÞsC0 ½sA ð1
ar Þ þ ar V f þ y Xc ðnÞsCn ½sA ð1
ar Þ þ ar V nð f
2Þ
n¼1
f ð1
yc Þ ð f
2Þ y Xc ðnÞb0 ¼ ; bn ¼ ð120Þf
2yc f
2yc A pgf for the counts of repeating units X will result from substituting sC0 in Eq.
(120) by sX and sCn by sXn. Variable ar in the expressions (120) is the probability of

reaction of the unreacted functional groups after ring formation in stage 1. It is necessary to eliminate it, and to introduce the mass action laws for the rings. This last step is not straightforward, as it requires the application of graph theory in or-der to compute the concentrations of linear polymer molecules ½L n  in Eq. (23).The final result, which reduces to the distribution found by Jacobson and Stock-mayer in their classic paper [21] for f ¼ 2, is Eq. (121).
 
½xð f
1Þ!arx
1 ð1
ar Þ xð f
2Þþ2 f ð1
yc Þ x½Px  ¼ ½X ð f
2yc Þx!½xð f
2Þ þ 2! f
2yc nK c ðnÞ n f ð f
1Þar ð1
yc Þy Xc ðnÞ ¼ q ; q¼½X  f
2yc
X
f ð p
ar ÞnK c ðnÞq n ¼ ½X  yc ; yc ¼ ð121Þ
n¼1
2ð1
ar Þ
A solution is also known for the analogous system of the alternating polycondensa-tion XA f þ YBg [267].Dilution with an inert solvent will make [X] decrease without affecting the cycli-zation constants much, and the fraction of rings will increase, until it becomes practically unity, a phenomenon which has experimental support obtained using polysiloxanes as model polymers. In bulk systems, the fraction of rings is usually only a few per cent.With nonlinear polycondensations of aliphatic monomers, gel points in bulk are affected by a few per cent due to cyclizations, and elastic properties are also af-fected (that effect using TBP has been modeled by Dušek et al. [269]). Application of the approach described above could lead to improved modeling where small numbers of rings are present.Taking cyclizations into account raises the question of introducing information about the spatial location of atoms in models of network formation. The classic ge-lation theory described here considers a uniform distribution in space of all the chemical properties. Stauffer, one of the main contributors to the progress of per-colation theory, has strongly criticized this view, claiming that Gordon’s theory is inapplicable in the vicinity of gel point [270]. Gordon has not accepted that argu-ment [271], claiming his theory to be universal and capable of describing gel for-mation in any homogeneous system.A new theory, starting with classic gelation theory and using TBP as a particular case, has been developed by Kuchanov [272, 273], and considers the molecular graphs to be embedded in ordinary three-dimensional space – not in a lattice, as is usually done in percolation theory. Generating functions are replaced by generat-ing functionals of the ensemble of positions of the functional groups and repeating units. Dependence of space coordinates is eventually eliminated by averaging, so the algebraic equations become integral equations. In spite of its power, it has not become a widely used instrument for dealing with ‘‘real’’ complex chemical sys-

3.4 Modeling of Complex Polycondensation Reactions 129
tems, and we leave it here only for reference, as even a basic but comprehensible description would be too extensive.Kinetic Modeling of Irreversible Polycondensations In his pioneering study of nonlinear polycondensation [7], Stockmayer has already checked his statistical solution [Eq. (5)] by solving the mass balance equations in a batch reactor for the concentrations of functional groups A and the set of isomeric polymer molecules Px with x repeating units X [Eqs. (122)].d½Px  1Xx
1¼k ½ yð f
2Þ þ 2½ðx
yÞð f
2Þ
dt 2 y¼1
þ 2½Py ½Px
y 
½A½xð f
2Þ þ 2½Px  ð122Þ
d½A
¼
k½A 2 ; ½Px jt¼0 ¼ ½X ; ½Ajt¼0 ¼ f ½X 
dt
The two solutions are identical. Hence, for a long time no importance was attrib-uted to the use of a kinetic approach for describing batch polycondensations start-ing from monomers, and the statistical approach was preferred. Of course, chemi-cal engineers had to deal with semi-batch and continuous stirred tank reactors where the statistical approach, although possible, is cumbersome and error-prone,so a few papers appeared in the 1960s dealing with kinetically controlled linear polycondensations [274–276].In reality, Kuchanov [277, 278] has shown that,with polycondensations present-ing FSSEs, kinetic and statistical approaches give distinct results for average mo-lecular weights and gel points. Dušek [279] has pointed out that this behavior is even more visible when dealing with polyadditions (linear polyaddition leads to a Poisson CLD, whereas a simplistic use of a statistical approach would lead to a geometrical/Schulz–Flory CLD) and has confirmed this result using Monte Carlo simulation of XA f polycondensation with FSSEs [280]. Sarmoria and Miller [35]have tried to extend the ‘‘recursive approach’’ to systems presenting FSSE by con-sidering network building starting from dyads and larger fragments. But chemical systems can be found in which this latter idea does not provide useful results[281], so that use of statistical approaches outside the description of chemical equi-librium now seems more like a waste of time.Kuchanov’s kinetic approach divides polymer molecules into classes PðxÞ having a vectorial count of groups x. In this approach, ‘‘groups’’ An should include not only the unreacted functional groups, but also the bonds and repeating units, and even larger molecular fragments when needed. We will use NXZA as the number of kinds of groups in that generalized sense.

It is possible to obtain a rate equation for the members of each class by adding the contributions of the various condensation reactions, leading to a version of Smoluchowski’s coagulation equation.Ring-forming reactions involving functional groups in the same monomer can be described by a simple extension of the preceding FSSE model, just by consider-ing unimolecular reactions and new fake functional groups, which are pairs of groups [Eq. (123)].
ki X
NXZA
A½i   ! nin An i ¼ 1; NR ð123Þ
n¼1
The rate of formation of groups requires a modification of Eq. (68), since it is no longer possible to include breakage or exchange reactions. Adding the unimolecu-lar reactions defined above, the new general rate equation becomes Eq. (124).

X
NR NR
X
R An ¼ ðn

in þ n þ
in Þki ½A½i
 ½A ½iþ  þ nin ki ½A½i    ð124Þ
i¼1 i¼1
The multiple sums in the rate of formation of PðxÞ, which will not be presented,are simplified through consideration of its generating function [Eqs. (125), (126)]
X
NR  
qG qG qG qG GRP ðsÞ ¼ ki G
þni Gni
½A½iþ 
½A½i
 
i¼1
q log s½i
 q log s½iþ q log s½i
 q log s½iþ

NR
X qG
þ ki ðG 
1Þ ð125Þ
i¼1
q log s½i   ni
where
NXZA
Gni ¼ snin J ¼ þ;
;  ð126Þ
n¼1
Equation (125) replaces a similar expression in Ref. 282 with the advantage of considering only the reactions actually taking place, and not every combination of pairs of unreacted groups, which does not make sense now, as repeating units are also An moieties.Insertion of Eq. (125) into mass balance equations, such as a continuous stirred tank reactor (CSTR) with constant volume, leads to a nonlinear first-order PDE[Eq. (127)].

3.4 Modeling of Complex Polycondensation Reactions 131
qG X NR
qG qG qG qG¼ ki G
G þni ni
½A½iþ 
½A½i
 
qt i¼1
q log s½i
 q log s½iþ q log s½i
 q log s½iþ

NR
X qG GF ðsÞ
GðsÞþ ki ðGni
1Þ
RV G þ
i¼1
q log s½i 
 t
GðsÞjt¼0 ¼ G0 ðsÞ ð127ÞSolution of the above equation by the method of characteristics [283] is described in Ref. 282, earlier examples being found in Refs. 284–286. They will not be repro-duced here, for the sake of brevity. If GðsÞ is to be evaluated, in order to take advan-tage of the numerical inversion formula Eq. (119), or if average degrees of polymer-ization in the presence of gel have to be predicted, numerical solution of Eq. (127)leads to a two-point boundary solution problem with twice as many unknowns as the number of derivative terms log sn (the number of active groups in the polymer).A much simpler problem, as Galina was apparently the first to remark [287], is the prediction of the moments of ½Px  with respect to the counts of groups, lmn... , in the absence of gel. Differentiation of Eq. (126) with respect to log sn and setting s ¼ 1NZXA leads to an ODE system with known initial conditions, which has a straightforward numerical solution [Eq. (128)].
dljkP X
NR
¼ k m ½ðn
þ
þmk þ nmk Þðnmj þ nmj Þ½A½m
 ½A½mþ 
dt m¼1
þ ðn
þ P P mj þ nmj Þðl½m
k ½A½mþ  þ l½mþk ½A½m
 Þþ ðn
þ P P P P P P mk þ nmk Þðl½m
 j ½A½mþ  þ L½mþ j ½A½m
 Þ þ l½m
 j l½mþk þ l½mþ j l½m
k 

NR
X ljkPF
ljkP
þ km ðl½m
P  P     j nmk þ l½m  k nmj þ ½A½m   nmj nmk Þ þ
RV ljkP ð128Þ
m¼1
As there is no gel, a rate equation for the overall concentration of polymer ½P(zeroth-order moment) is obtained from Eq. (126), setting s ¼ 1NZXA [Eqs. (129)].q½P XNR ½Pf
½P¼
ki ½A½i
 ½A½iþ 
RV ½P þ
qt i¼1
½Pjt¼0 ¼ ½P0 ð129ÞNumber-average and weight-average molecular weights are found by the well-known expressions (130).

132 3 Polycondensation
X
NZXA X
N X
ZXA NZXA
MAn ½An  MAm MAn lmn
n¼1
Mn ¼ Mw ¼ m¼1 Nn¼1 ð130Þ
½P X ZXA
MAn ½An 
n¼1
This approach can only deal with ring-forming reactions either for a limited num-ber of the smallest rings, or alternatively, for linear polycondensations. The impor-tant practical case of the irreversible polycondensation of AXA þ BYB þ BYC (C being an inert group) leads to the rate laws in Eqs. (131) for the molecules with the six possible combinations of end groups PnAA ; PnAB ; . . . PnCC and rings Cn , where index n counts the most frequent kind of repeating units in the molecule:
X
n
1 X
n
1
RPnAA ¼ k 4 ½PmAA ½Pn
m
AA
þ2 ½PmAA ½Pn
m
AB

2½PnAA ½B
m¼1 m¼1
X
n
1 X
n
1
RPnBB ¼ k 4 ½PmBB ½Pn
m
BB
þ2 ½PmBB ½Pn
m
AB

2½PnBB ½A
m¼1 m¼1
X
n X
n
1
RPnAB ¼ k 4 ½PmAA ½Pn
mþ1
BB
þ ½PmAB ½Pn
m
AB

½PnAB ð½A þ ½BÞ
kc ðnÞ½PnAB 
m¼1 m¼1
RCn ¼ kc ðnÞ½PnAB 
X
n X
n
1
RPnAC ¼ k 2 ½PmAA ½Pn
mþ1
BC
þ ½PmAB ½Pn
m
½PnAC ½B
m¼1 m¼1
X
n
1 X
n
1
RPnBC ¼ k 2 ½PmBB ½Pn
mþ ½PmAB ½Pn
m
BC

½PnBC ½A
m¼1 m¼1
X
n
1
RPnCC ¼ k ½PmAC ½Pn
m
BC
m¼1
X
y
RA ¼ RB ¼
k½A½B
kc ðnÞ½PnAB  ð131Þ
n¼1
Modern computers will not have much difficulty with the ‘‘brute force’’ approach of solving the mass balances after inserting the above relationships for n ¼ 1 up to an upper value N of a few hundreds or even thousands (notice that this implies 11N 2 þ OðNÞ multiplications and sums every time this set of rates of reaction is evaluated). A relatively large value of N is needed if the mole ratio is close to one,in order that extrapolation of CLD and evaluation of the infinite sum using the last

3.4 Modeling of Complex Polycondensation Reactions 133
equation of Eqs. (131) may be done accurately. A more elegant method uses gener-ating functions [288] and avoids the problem of the lack of closure of the above equations for any finite N.When the kinetic approach was at an early stage, it was thought that it could pro-vide no information about polymer or network properties. More recently, a descrip-tion of batch polycondensation using TBP which is rigorously equivalent to the one obtained by the kinetic approach was found [289], taking into account the times of birth of molecules, so that the fundamental restriction does not hold. Through more elementary reasonings, it is nevertheless possible to estimate probabilities of extinction and thus network elastic properties [282] or average radius of gyration[290].
3.4.5
Kinetic Modeling of Linear Reversible Polycondensations The reversible alternating polycondensation with FSSEs in both monomers (see Section 3.1.5), disregarding exchange reactions, is a convenient case study for dis-cussing problems of modeling this kind of systems. It can be described by the rate laws of Eqs. (132) and (133).RP1AA ¼
4k1 ½P1AA ½P1BB 
2k2 ½P1AA ð½P1AB  þ ½ZB Þ þ k1Z ½P1AB  þ k2Z ½ZA RP1BB ¼
4k1 ½P1AA ½P1BB 
2k3 ½P1BB ð½P1AB  þ ½ZA Þ þ k1Z ½P1AB  þ k3Z ½ZB RP1AB ¼ 4k1 ½P1AA ½P1BB 
½P1AB ½2k2 ½P2AA  þ 2k3 ½P1BB  ð132Þþ k4 ð2½P1AB  þ ½ZA  þ ½ZB Þ
k1Z ½P1AB  þ 2k2Z ½P2AA  þ 2k3Z ½P2BB þ 2k4Z ð½ZA  þ ½ZB 
2½P2AA 
2½P2BB Þ
n b 2:
X
n
2
RPnAA ¼ 2k2 ½P1AA ½Pn
1
AB
 þ 2k4 ½PmAB ½Pn
m
AA

2½PnAA ½2k3 ½P1BB  þ k4 ð½P1AB  þ ½ZB Þ
m¼1
X

½PnAA ½2k2Z þ 2k4Z ðn
2Þ þ k3Z ½PnAB  þ k4Z ½ZA 
ð½PmAB  þ 2½PmAA Þ
m¼2
X
n
2
RPnBB ¼ 2k3 ½P1BB ½Pn
1
AB
 þ 2k4 ½PmAB ½Pn
m
BB

2½PnBB ½2k2 ½P1AA  þ k4 ð½P1AB  þ ½ZA Þ
m ¼1
X

½PnBB ½2k3Z þ 2k4Z ðn
2Þ þ k2Z ½PnAB  þ k4Z ½ZB 
ð½PmAB  þ 2½PmBB Þ
m¼2

RPnAB ¼ 4k2 ½P1AA ½PnBB  þ 4k3 ½P1BB ½PnAA 
X
n
1 X
n
1
þ k4 ½PmAB ½Pn
m
AB
þ4 ½PmAA ½Pn
mþ1
BB

m¼1 m¼2

½PnAB ½2k2 ½P1AA  þ 2k3 ½P1BB  þ k4 ð2½P1AB þ ½ZA  þ ½ZB Þ þ k2Z þ k3Z þ k4Z ð2n
3Þ þ 2k2Z ½Pnþ1
AA
 þ 2k3Z ½Pnþ1
BB

X
n X
nþ1
þ k4Z ½ZA  þ ½ZB 
2 ½PnAB 
2 ð½PmAA  þ ½PmBB Þ ð133Þ
m¼2 m¼2
It may be observed from these expressions that, in general, it is not possible to ob-tain a closed finite set of rate equations for the first oligomers – rate equations for oligomers with chain length n always depend on concentrations of oligomers with chain length n þ 1. Only when the rate constants of reverse equations are equal do the above expressions simplify (FSSE for reverse reaction), and the above-mentioned difficulty disappears.On introduction of generating functions of the CLD, defined in Eq. (134), the concentrations of end groups, trimers, and tetramers conform with Eqs. (134) and
(135).
X
y X
y X
GAA ¼ s n
2 ½PnAA  GBB ¼ s n
2 ½PnBB  GAB ¼ s n
2 ½PnAB  ð134Þn¼2 n¼2 n¼2½ZA  ¼ 2GAA ð1Þ þ GAB ð1Þ ½ZB  ¼ 2GBB ð1Þ þ GAB ð1Þ
ð135Þ
½P2AA  ¼ GAA ð0Þ ½P2BB  ¼ GBB ð0Þ ½P2AB  ¼ GAB ð0ÞThe rate equations in terms of those generating functions become Eqs. (136).RGAA ¼ 2k2 ½P1AA ðsGAB þ ½P1AB Þ þ k4 s 2 GAA GAB
2GAA ½2k3 ½P1BB 
qGAA
þ k4 ð½P1AB  þ ½ZB Þ
2k2Z GAA
2k4Z s þ k3Z GAB
qs
½ZA 
2GAA
GAB
þ k4Z
1
s
RGBB ¼ 2k3 ½P1BB ðsGAB þ ½P1AB Þ þ k4 s 2 GBB GAB
2GBB ½2k2 ½P1AA 
qGBB
þ k4 ð½P1AB  þ ½ZA Þ
2k3Z GBB
2k4Z s þ k2Z GAB
qs
½ZB 
2GBB
GAB
þ k4Z
1
s

3.4 Modeling of Complex Polycondensation Reactions 135
RGAB ¼ 4k2 ½P1AA GBB þ 4k3 ½P1BB GAA þ k4 ½2½P1AB ðsGAB þ ½P1AB Þþ 4sGAA GBB 
½2k2 ½P1AA  þ 2k3 ½P1BB  þ k4 ð2½P1AB  þ ½ZA  þ ½ZB ÞqGAB ½ZA  þ ½ZB 
2ðGAB þ GAA þ GAB Þþ k2Z þ k3Z þ k4Z GAB
2k4Z s þ k4Z
qs 1
s
GAA
½P2AA  GBB
½P2BB þ 2ðk2Z
k4Z Þ þ 2ðk3Z
k4Z Þ ð136Þ
s s
Rate laws for ½ZA  and ½ZB  become Eq. (137):RZA ¼ 2k2 ½P1AA ð½ZB  þ 2½P1AB Þ
2k3 ½P1BB ½ZA  þ k4 ð2½P1AB 2
½ZA ½ZB Þ
k2Z ð½ZA  þ 2½P2AA Þ
k3Z ð½ZB 
2½P2BB Þ
 
þ k4Z ½X þ ½Y þ ð½ZA  þ ½ZB Þ
½P1AA 
½P1BB 
2½P1AB RZB ¼
2k2 ½P1AA ½ZB  þ 2k3 ½P1BB ð½ZA  þ 2½P1AB Þ þ k4 ð2½P1AB 2
½ZA ½ZB Þ
k2Z ð½ZA 
2½P2AA Þ
k3Z ð½ZB  þ 2½P2BB Þ
 
þ k4Z ½X þ ½Y þ ð½ZA  þ ½ZB Þ
½P1AA 
½P1BB 
2½P1AB  ð137ÞA rate law for by-product [Eq. (138)] is usually required.RW ¼ 4k1 ½P1AA ½P1BB  þ 2k2 ½P1AA ½ZA  þ 2k3 ½P1BB ½ZA  þ k4 ½ZA ½ZB 
k1Z ½P1AB 
X
y

k2Z ½ZA 
k3Z ½ZB 
k4Z ½ð2n
4Þð½PnAA  þ ½PnBB Þ þ ð2n
3Þ½PnAB 
n¼2
¼ 4k1 ½P1AA ½P1BB  þ 2k2 ½P1AA ½ZA  þ 2k3 ½P1BB ½ZA  þ k4 ½ZA ½ZB 
k1Z ½P1AB 
k2Z ½ZA 
k3Z ½ZB 
 

k4Z ½X  þ ½Y 
ð½ZA  þ ½ZB Þ
½P1AA 
½P1BB 
2½P1AB  ð138ÞInsertion of these rate laws in mass balances of ideal reactors (batch/plug flow or transient CSTR) leads to systems of semi-linear, first-order, partial differential equations, with a single family of characteristics [Eq. (139)].¼ 2sk4Z ð139Þ
dt
The relationship of Eq. (138) allows integration along characteristics, relating ini-tial values GAA ð0; s 0 Þ; GAB ð0; s 0 Þ, and GBB ð0; s 0 Þ to GAA ðt; sÞ; GAB ðt; sÞ, and GBB ðt; sÞ,

but a serious problem remains: integration along characteristics and prediction of concentration of first oligomers have to be done at the same time.Only the solution with equal reactivity has been published [291]. A sketch of an iterative method to solve this more complex problem is described below (no actual calculations have been published at the time of writing).
1. Two auxiliary characteristics sþ ðtÞ ¼
s
ðtÞ, starting at sþ ¼ es js0 j and s
¼

es js0 j, respectively, for t ¼ 0, are introduced in order to provide an accurate nu-merical estimation of oligomer concentrations through Eq. (140).GI ðsþ Þ þ GI ð
s
ÞGI ð0Þ ¼ þ Oðes2 Þ I ¼ AA; BB; AB ð140ÞParameter es should be a small value (say 10
6 to 10
8 ). Owing to Eq. (139), it is easy to relate the auxiliary characteristics to the ‘‘main’’ characteristic sðtÞ start-ing with s ¼ s0 through Eq. (141).sþ ðtÞ ¼
s
ðtÞ ¼ es sðtÞ ð141Þ
2. The resolution of a system of ODE comprising Eq. (139), the balances of end
groups ZA , ZB , by-product W and differential equations for GAA ðt; sÞ, GAB ðt; sÞ,GBB ðt; sÞ, GAA ðt; sþ Þ, GAB ðt; sþ Þ, GBB ðt; sþ Þ, GAA ðt; s
Þ, GAB ðt; s
Þ and GBB ðt; s
Þ,will eventually allow the evaluation of all unknown variables of this problem.As s0 is to be found iteratively in order that the characteristic starting with s ¼ s0 will attain the prescribed value of s at the prescribed time t, this is a boundary value problem, but very straightforward to solve.If there are more than two functional groups per repeating unit, the approach de-scribed above is no longer useful, as there is no simple way of describing forma-tion of chemical species through breaking of larger ones, owing to their branched structure.Nevertheless, in most situations resembling industrial practice, such as CSTRs in series, the CLD of these chemical systems will stay close to equilibrium (see Ref. 291). The distributions of monads depend only on the equilibrium ratios and it is not necessary to use models with SSSEs for the reverse reactions (the source of the most challenging mathematical problems), as the same distributions will be found using equal values of the rate constants of the reverse reaction.
Notation
1N vector with N components equal to one av film area per unit volume [m
1 ]b length of a Kuhn segment connecting two half-bonds in the same repeating unit [m]

Notation 137 bA empirical parameter in Eq. (56) (dimensionless)Cd drag coefficient (dimensionless)Cg parameter in Eq. (12) [s
1 ]D binary diffusivity [m 2 s
1 ]db bubble diameter [m]dR diameter of ring (in a RDC devolatilizer) [m]dT inner diameter of tube, or of the barrel of an extruder [m]Ec activation energy of amidation reaction [J mol
1 ]ei index of by-product formed by reaction leading to bond Z iR ; no by-product if it is nil
þ
e

i ; ei indices of ‘‘half-bonds’’ of bond Z iR F X i ðsZ ; sA Þ; F A i ðsZ ; sA Þ; F Z i ðsZ ; sA Þ probability generating functions (pgf ) of the counts of the different kinds of connecting and unreacted functional groups directly linked to, respectively, a monomer unit X i , an un-reacted group A i , or a half-bond Z i f monomer functionality (number of functional groups it contains)Ge equilibrium zero strain shear modulus of polymer network [Pa]GðsÞ moment generating function of molar concentrations of polymer mol-ecules with respect to their counts of monomer units and unreacted functional groups Gni generating functions of stoichiometric coefficients of reaction i ð J ¼ þ;
;  Þ, as defined by Eq. (126)G ðsX ; sA Þ; G A i ðsX ; sA Þ; G Z i ðsX ; sA Þ pgf of trees starting, respectively, with a pre-
Xi
scribed monomer unit X i , an unreacted group A i , or a half-bond Z i GM ðsM Þ pgf of mass fractions of polymer molecules in a sol g acceleration due to gravity [m s
2 ]gA ratio mA /bA [m 3 mol
1 ]

þ
gi ; gi indices of functional groups A gi
; A giþ reacting to create bond Z i H depth of liquid in the channel of a vented extruder [m]He ; Hi partition coefficients of hydrophilic monomer in interfacial polymer-ization relating, respectively, the bulk concentration in the water phase and the concentration in the outer interface of the polymerfilm, and the concentrations in organic phase in the two interfaces of polymer film (dimensionless)HW Henry constant of component W in terms of pressure versus weight fraction [Pa]h clearance between the screw and barrel in a vented extruder [m]hi index of a repeating unit carrying functional group A i hR height of the liquid above the gas injection point in an RDC [m]J number of compartments in a staged model of an RDC devolatilizer Jj mass transfer rate per unit volume of component j [mol m
3 s
1 ]K apparent equilibrium ratio of bond formation from end groups(dimensionless)K0 thermodynamic equilibrium constant of bond formation from end groups (dimensionless)

Ka apparent equilibrium ratio of amidation (dimensionless)K a0 reference value of the apparent equilibrium ratio of amidation(dimensionless)K c ðnÞ apparent equilibrium ratio of cyclization [mol m
3 ]k apparent second-order rate constant [m 3 mol
1 s
1 ]k0 parameter in Eq. (12) [m 3 mol
1 s
1 ]kA empirical parameter in Chen–Wu relation, Eq. (53) [mol m
3 s
1 ]k aE rate constant of breakage of a bond in a ring molecule through reac-tion with a functional group [mol m
3 s
1 ]k c ðnÞ first-order rate constant of L n cyclization [s
1 ]k cE ðnÞ rate constant of formation of Cn by the back-biting cyclization reaction
[s
1 ]
kcr rate coefficient of crystallization in Avrami equation [s
ncr ] (see below)k cZ apparent first-order rate constant of breakage of a bond in a ring molecule [s
1 ]kc0 third-order pre-exponential factors of amidation rate constant[kg 2 mol
2 s
1 ]kfYi mass transfer coefficient of species Yi in terms of molar concentra-tions [m s
1 ]
kfYi
mass transfer coefficient of species Yi in terms of activities[mol m
2 s
1 ]ki0 noncatalytic contribution to the rate constant in Eq. (61) [mol m
3 s
1 ]kijE rate constant of exchange reaction forming bond Z iR and destroying
bond Z jR
kijE
; kijEþ rate constants of exchange reaction forming bond Z iR and replacing bond Z jR , when the attacking group is, respectively, A½i
 or A½iþkiH acid-catalyzed reaction constant in Eq. (61) [m 6 mol
2 s
1 ]kiOH catalytic term of hydroxyls in Eq. (61) [m 6 mol
2 s
1 ]km water mass transfer rate constant [s
1 ]k nuc; n rate coefficients of nucleation of polymer species with chain length n
[s
1 ]
kscat ; kfcat rate constants of self-catalyzed and foreign acid-catalyzed esterification[m 6 mol
2 s
1 ]ku0 second-order pre-exponential factor of amidation rate constant[m 3 mol
1 s
1 ]kZ apparent first-order rate constant of bond group breakage [s
1 ]kiZ apparent first-order rate constant of breakage of bond group ZiR [s
1 ]ki first-order rate constant of intramolecular reaction i destroying group A½i   [s
1 ]L film thickness [m]Lp thickness of polymer film in interfacial polymerization [m]LR thickness of reaction zone [m]LW width of the channel in an extruder [m]Lx film perimeter [m]

Notation 139 M relative molecular weight Mi relative molecular weight of generic polymer species i, where i is an
arbitrary
 y index
y P
Mn ¼ Mi ½Pi  ½Pi  number-average molecular weight of polymer
i¼1 i¼1
MP mass of polymer per mole of monomer units [kg mol
1 ]MW relative ymolecular weight of by-product W
Py P
Mw ¼ Mi2 ½Pi  Mi ½Pi  weight-average molecular weight of polymer
i¼1 i¼1

y P
Mz ¼ Mi3 ½Pi  Mi2 ½Pi  z-average molecular weight of polymer
i¼1 i¼1
mA empirical parameter in Eq. (56) [mol
1 m 3 ]NA number of functional (or end) groups NAL Avogadro–Löschmidt number [mol
1 ]Nb number of bubbles N_ j flux of component j [mol m
2 s
1 ]NR number of distinctive chemical bonds NRs number of symmetrical chemical bonds NX number of kinds of monomer units NW number of by-products NY number of volatile (or precipitating) components NZ number of half-bonds NZA ¼ NZ þ NA overall number of kinds of functional groups and half-bonds NZXA ¼ NZ þ NX þ NA overall number of kinds of functional groups, monomer units, and half-bonds n_ screw rotational speed (as rotations per unit time) [s
1 ]ncr Avrami exponent Pb pressure inside a bubble [Pa]Pme local pressure over a bubble [Pa]PW vapor pressure of W [Pa]PW vapor pressure of pure W [Pa]p conversion of reference functional groups A pc conversion of functional groups A having given rise to rings Q volumetric flow rate of polymer [m 3 s
1 ]Qg volumetric flow rate of gas [m 3 s
1 ]R ideal gas constant [J mol
1 K
1 ]Re Reynolds number for the rising movement of a bubble Rj rate of formation by chemical reaction of species j [mol m
3 s
1 ]RV combined relative rate of change of reaction volume by chemical reac-tion and transfer of by-product [s
1 ]r stoichiometric ratio in alternating polycondensations rb radius of a bubble [m]s vector of dummy Laplace variables associating sA and sZ (defined
below)

sA vector of dummy Laplace variables associated with the counts of un-reacted functional groups sCn dummy Laplace variables associated with the counts of rings (includ-ing sC0 for the count of units X in the chains connecting rings) for the equilibrium polycondensation of a single monomer XA f sM dummy Laplace variable associated with molecular weight sZ vector of dummy Laplace variables associated with the counts of half-
bonds
s Zy dummy Laplace variable associated with xZy sZl dummy Laplace variable associated with the counts of bonds not be-longing to rings for the equilibrium polycondensation of a single monomer XA f T absolute temperature [K]T0 reference temperature [K]
t time [s]
tf exposure time [s]t fP exposure time of liquid in the pool of a vented extruder [s]ub bubble rising speed [m s
1 ]ux ; uz average velocity in directions x; z [m s
1 ]V volume of reacting mixture [m 3 ]Vb volume of a bubble [m 3 ]Vi ðsZ ; sA Þ pgf of pendent chains Vi V^i ðsZ ; sA Þ pgf of finite pendent chains Vi Vj molar volume of species j [m 3 mol
1 ]VMi ðsM Þ pgf of finite pendent chains Vi with respect to molecular weight Vm overall melt volume [m 3 ]Vmj volume of liquid in compartment j in a staged model of an RDC devo-latilizer [m 3 ]v vector of the NZ probabilities of extinction (the fractions of finite pen-dent chains)vT average transverse velocity of the screw dragging the film [m s
1 ]wcr weight fraction of crystalline material wj weight fraction of j wS weight fraction of the sol, relative to the overall mass of the polymer x degree of polymerization, the number of repeating units in a polymer
molecule
x coordinate along the perimeter, when discussing mass transfer from a liquid flowing over a cylindrical wall [m]xA vector storing the counts of functional groups xi degree of polymerization of generic polymer species i, where i is an
arbitrary
 y index
y P
xn ¼ x i ½Pi  ½Pi  number-average degree of polymerization of polymer
i¼1 i¼1

Py Py
xw ¼ xi2 ½Pi  x i ½Pi  weight-average degree of polymerization of polymer
i¼1 i¼1

Notation 141 xX vector containing the counts of monomer units xZ vector storing the counts of half-bonds xZ y number of infinite pendent chains connected to a junction y Cartesian coordinate normal to the interface [m]yc overall molar fraction of repeating units X in rings for a single mono-mer polycondensation yP number of moles of polymer molecules per mole of monomer units before gelation ySX i ; ySZ i and ySA i fractions of units and groups of the various kinds in finite molecules (in a sol), after gelation y Xc ðnÞ molar fraction of repeating units X in rings of size n for a single monomer y X i molar fractions of monomers or monomer units ZZX matrix defined in Eq. (73) from incidence vectors z defined below z Cartesian coordinate parallel to the interface [m]zi index of the monomer unit to which half-bond Z i points
Greek
a empirical parameter of Chen–Wu relation [Eq. (53)] (dimensionless)ar probability of reaction of unreacted functional groups after ring for-mation in the first stage (absence of open-chain molecules) of a hypo-thetical two-stage equilibration of XA f b g ; b g0 parameters in Eq. (11) describing the glass effect [K]bj fraction of volume flow rate in compartment j going to the preceding compartment j
1 gj activity coefficient of species j wij Flory–Huggins parameter (multicomponent mixture)DH enthalpy of reaction [J mol
1 ]DS entropy of reaction [J mol
1 K
1 ]d upstream distance of re-entry of film wiped by the screw in a vented extruder [m]es small starting value of s in characteristic line y screw angle k c ðnÞ ratio of rate constants of similar intramolecular and intermolecular reactions [mol m
3 ]q . . . qF I LIJ...K ðsÞ ¼ derivatives of pgf F I ðsÞ (or other moment gener-q log sJ . . . q log sK ating functions) with respect to logarithms of dummy Laplace
parameters
I q . . . qF I P
y Py
lJ...K ¼ ð1N Þ ¼ ... xJ . . . xJ IðxÞ moments of distribu-q log sJ . . . q log sK x1 ¼0 xN ¼0 tions, where F ðsÞ is the moment generating function of vector distri-bution IðxÞ

lM ; lMM first and second moments of molecular weight of distribution of
polymer
m dynamic viscosity [N m
1 s
1 ]me concentrations of elastically active network junctions (EANJ)
[mol m
3 ]
n stoichiometric coefficient (the change in number of moles caused by chemical reaction)ninA
; ninZ
stoichiometric coefficients of functional groups An and oriented bonds Zn coming out of the monomer unit in which A½i
 was situated for the reaction forming Z iR
Aþ Zþ
nin ; nin stoichiometric coefficients of functional groups An and oriented bonds Zn coming out of the monomer unit in which A½iþ was situated for the reaction forming Z iR
AE

ZE


nijn ; nijn changes in numbers of functional groups An and bonds Zn , respec-tively, connected to the root unit X
where either A½i
 or A½iþ were attached, for the exchange reaction forming Z iR at the expense of Z jR AE
þþ ZE
þþ AEþþþ ZEþþþnijn ; nijn ; nijn ; nijn changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X þþ where the living group A½ j
 or A½ jþ was situated, for the exchange reaction forming Z iR at the expense of Z jR AE
þ
ZE
þ
AEþþ
ZEþþ
nijn ; nijn ; nijn ; nijn changes in numbers of groups attached to the root unit X þ
which gets connected to the unit where the attacking group was situated, for the exchange reaction forming Z iR at the ex-pense of Z jR
W
nin stoichiometric coefficients of by-products for the reaction forming Z iR

nin stoichiometric coefficient of group An for the ith intramolecular
reaction
ne concentration of elastically active network chains [mol m
3 ]rj density of j [kg m
3 ]s surface tension [N m
1 ]t space time based upon feed flow rate, Vm /Q F for a CSTR [s]tr structural relaxation time [s]t0 parameter in Eq. (11) [s]tD tortuosity for by-product diffusion in a solid polymer particle(dimensionless)fcr volume fraction of crystalline material fj volume fraction of j fL fraction of channel extruder filled with liquid w Flory–Huggins parameter (binary mixture) (dimensionless)WW weight fraction activity coefficient of W Chemical symbols A; B functional groups A gi
1 A½i
 ; A giþ 1 A½iþ functional groups reacting in bond Z iR formation

Notation 143 Aj active component (molecular species or fragment) j; nonvolatile,nonprecipitating Cn ring molecule with n bonds Ln linear polymer molecule with n
1 bonds and a pair of active end
groups
LT length of tube Pi generic polymer species, where i is an arbitrary index Pn linear polymer molecule with end groups I and J and n repeating units of the most frequent kind½U  molar concentration of species U Vi ðxZ ; xA Þ pendent chain starting with a half-bond Z i and containing vector counts of bonds and functional groups, respectively, xZ and xA W by-product of condensation reaction We i 1 W½i by-product of condensation reaction leading to bond Z iR X; Y monomer units Xh i 1 X½iA monomer unit carrying functional group A i Xz i 1 X½iZ monomer unit to which each half-bond Z i points Yj volatile (or precipitating) component j Z bond group; Z iR is the ith distinctive bond in the chemical system ZA bond group in fragment AXZA YZ (polycondensation AXA þ BYB)ZB bond group in fragment BYZB XZ (polycondensation AXA þ BYB)ZD bond group in dimer AXZD YB (polycondensation AXA þ BYB)Ze
i 1 Z½i
 ; Zeþi 1 Z½iþ ‘‘half-bonds’’ of bond Z iR ZP bond group in fragment ZXZP Z (polycondensation AXA þ BYB)
Subscripts
aq aqueous
b of a bubble
blk bulk
c critical
cr crystalline
F in feed
g at gel point org organic sat at saturation Superscripts* used for describing values of concentration at the interface x average value of x
Acronyms
BHET bis(2-hydroxyethyl) terephthalate BPA bisphenol A

CLD chain length distribution CSTR continuous stirred tank reactor DEG diethylene glycol DMT dimethyl terephthalate DPC diphenyl carbonate EANC elastically active network chains, the chains connecting EANJs (see
below)
EANJ elastically active network junctions, the intersection of at least three chains leading to a gel ECH epichlorohydrin EG ethylene glycol FSSE first-shell substitution effect ODE ordinary differential equation PBT poly(butylene terephthalate)PDE partial differential equation PET poly(ethylene terephthalate)pgf probability generating function RDC rotating disk contactor RIM reaction injection molding SSP solid-state polycondensation SSSE second-shell substitution effect TBP theory of branching processes TPA terephthalic acid UF urea–formaldehyde polymer WFR wiped-film reactor
References
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Free-radical Polymerization: Homogeneous1 Robin A. Hutchinson Free-radical polymerization (FRP) is one of the most important commercial pro-cesses for preparing high molecular weight polymers. It can be applied to almost all vinyl monomers under mild reaction conditions over a wide temperature range and, although requiring the absence of oxygen, is tolerant of water. Multiple mono-mers can be easily copolymerized via FRP, leading to the preparation of an endless range of copolymers with properties dependent on the proportion of the incorpo-rated comonomers. This chapter will provide an overview of the kinetics and mech-anisms, and the techniques used to construct mathematical representations of bulk and solution FRP. The description will also serve as a good base for the sus-pension and emulsion FRP chapters that follow.
4.1
FRP Properties and Applications Polymers produced via free-radical chemistry include the following major families:
Low-density polyethylene (LDPE) and copolymers, used primarily in films and packaging applications. LDPE has density of <0.94 g cm3 , and is produced via high-pressure free-radical polymerization; polyethylenes of higher density (and polypropylene) are produced via transition metal catalysis, as described else-where (see Chapter 8).
Poly(vinyl chloride) and copolymers, used primarily to produce pipe and fittings,flooring material, and films and sheet.
Polystyrene and its co- and terpolymers with acrylonitrile and butadiene. Homo-polymer is used for packaging and containers, while the acrylonitrile-containing polymers are used for various molded products in the appliance, electronics, and
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

154 4 Free-radical Polymerization: Homogeneous automotive industries. Styrene–butadiene is the most widely used synthetic
Acrylic- and methacrylic-based polymers. Poly(methyl methacrylate) (PMMA),due to its transparency and weatherability, is used extensively in signs, lightingfixtures, and windows. Polyacrylates and copolymers are used extensively in the adhesives and coatings markets, and are also combined with acrylonitrile to make acrylic fibers.
Poly(vinyl acetate) and copolymers, used extensively in adhesives, coatings, and paper and textile treatment.
Fluoropolymers including polytetrafluoroethylene and copolymers, used widely in the wire and cable industry. They also have many specialty applications as coatings due to their inertness and low-surface tension.In addition to these major product families, there exist many other smaller-volume(but often high-value) polymeric products synthesized via free-radical chemistry.These products are manufactured via heterogeneous (emulsion, suspension) or homogeneous (bulk, solution) polymerization in a wide range of reactor configura-tions ranging from tubular to well-mixed tanks (and everything in between) in pro-cesses that may be continuous, batch, or semi-batch. FRP kinetics together with re-actor design and operating conditions controls the composition and architecture of the polymer produced. As with other chemistries, the final product consists of a mixture of polymer chains with varying length, composition, and structural proper-ties such as branch points and branch lengths. The processing and end-use proper-ties of the polymer depend upon the distributions of these characteristics, not just the average values. Thus it is necessary to develop methods to relate the fundamen-tal kinetic mechanisms of free-radical polymerization to these distributed charac-teristics.
4.2
Chain Initiation Free-radical polymerization, like other chain growth mechanisms, involves the se-quential addition of vinyl monomers to an active center. The defining characteristic of FRP is that the active centers are free radicals. Each growing polymeric radical increases in size rapidly; a typical chain is initiated, grows to high molecular weight, and is terminated in the time scale of, at most, a few seconds. When the macromolecule stops growing it cannot generally react further (barring side reac-tions), and is considered ‘‘dead’’. These dead chains have a residence time of mi-nutes or hours in the polymerization reactor, and the final polymer product is an intimate mixture of chains formed under time and/or spatially varying conditions.The free radicals that initiate polymerization are in most cases generated by ther-mal or photochemical homolytic cleavage of covalent bonds. Commercial initiators include azo and peroxy compounds. The driving force for the dissociation of azo

156 4 Free-radical Polymerization: Homogeneous polymer) in the system, or recombine to form inactive compounds. The relative rates of these processes are dependent on both the nature of the primary radicals and the monomer system. The mechanistic pathway followed will influence end-group structures of the polymeric products, affecting final properties such as ther-mal stability. Further details on initiator decomposition kinetics and mechanistic pathways can be found in the excellent monograph by Moad and Solomon [3].The above discussion highlights the fact that the simple initiation process of most polymer texts is the exception rather than the rule, with the fraction of pri-mary radicals that initiate a new polymer chain a complex function of the reaction system. Nonetheless, the kinetic treatment is usually simplified by the introduction of a fractional initiator efficiency ( f ), formally defined by Eq. (1), where n is the number of moles of primary radicals generated per mole of initiator; n ¼ 2 for most common initiators.Initiation Rate of Propagating Chains
f ¼ ð1Þ
nðRate of Initiator DisappearanceÞThe initiator efficiency is normally in the range 0.4–0.9, with a low value indicating inefficient usage of the initiator and potentially a high formation rate of undesired by-products.The kinetic descriptions in this chapter are developed for thermal unimolecular scission of a compound to yield two radicals, as this is the most common means of generating radicals in industrial systems. Thermal initiation of monomers is an ad-ditional mechanism capable of forming primary radicals at higher temperatures, as discussed for styrene (see Section 4.3.1.3). Other initiation systems are also avail-able, and bear a brief mention at this point. Initiators with multiple peroxide link-ages ðn > 2Þ have been the subject of recent academic study [4]. Photoinitiators that produce radicals by ultraviolet irradiation are commonly used to initiate cross-linking and curing reactions in polymeric systems, as the rate of initiation can be controlled through the intensity and location of the light source. And finally, a re-dox (reduction–oxidation) process is often used to initiate chains in emulsion poly-merization (see Chapter 6).
4.3
Polymerization Mechanisms and Kinetics It is important to achieve an understanding of how the basic FRP mechanisms control polymerization rate and average polymer chain length. This section starts with the derivation of appropriate kinetic expressions for a single monomer system. Complicating (but industrially important) secondary reactions are then introduced, followed by the extension to multi-monomer systems. Dispersed throughout are up-to-date estimates for important free-radical polymerization rate coefficients, and descriptions of how they are obtained experimentally.

4.3 Polymerization Mechanisms and Kinetics 157
Homopolymerization Many commercial polymers, including polystyrene, PMMA, and LDPE, are synthe-sized via homogeneous free-radical polymerization of a single monomer. Homo-polymer properties are controlled by average chain length and chain-length distri-bution as well as, in some cases, structural characteristics such as branching level.
4.3.1.1 Basic Mechanisms
The basic set of FRP mechanisms includes initiation, propagation, termination,and transfer to monomer and solvent or transfer agent, as shown in Scheme 4.3.
k
Initiator Decomposition I 
→2 f I ∗
k
Chain Initiation I ∗ + M 
→ P1
kp
Chain Propagation Pn + M → Pn +1 Chain Termination
k
By Combination Pn + Pm 
tc
→ Dn+m
k
By Disproportionation Pn + Pm 
td
→ Dn + Dm
Chain Transfer
k mon
Pn + M 
tr
→ Dn + M *
To Monomer k mon M * + M 
→ P1
k sol
Pn + S →
tr
Dn + S *
To Solvent or Agent k sol S * + M →
P1
Scheme 4.3. Basic free-radical homopolymerization mechanism.The subscript n denotes the number of monomeric units in growing polymer radicals (Pn ) and dead polymer chains (Dn ). Each reaction has an associated kinetic rate law expression and a specific rate coefficient. The free-radical initiator (I) un-imolecularly decomposes (with rate coefficient kd ) to form two primary radicals (I  )with efficiency f. Chain initiation occurs when the primary radical adds to mono-mer M, and chain growth continues via successive addition of monomer units to the radical center (chain propagation, with rate coefficient k p ). Bimolecular cou-pling of two growing chains results in the loss of two radicals from the system and the formation of either one (termination by combination, k tc ) or two (termina-tion by disproportionation, k td ) dead polymer chains. Chain stoppage may also oc-cur via a transfer mechanism, where the growing radical abstracts a weakly bonded

158 4 Free-radical Polymerization: Homogeneous atom (usually hydrogen) from monomer or other molecules (solvent or chain-transfer agent, denoted by S) in the system to generate a dead polymer chain as well as a new radical that initiates another polymer chain.A key assumption implicit in the formulation of Scheme 4.3 is that the rates of propagation, transfer, and termination reactions are independent of n, the length(s) of the radical(s) involved. It is known that propagation and likely transfer reactions involving very short chains (n ¼ 1; 2; 3) are faster by a factor of 10 than addition to long-chain radicals [5], but this effect diminishes rapidly with chain length and has a negligible effect on the overall kinetics of the system. Chain ter-mination, the coupling of two polymeric radicals, is a very fast chemical process that is controlled by the rate at which the two radicals find each other in the reac-tion system. The nature of this diffusion control makes termination the most com-plex reaction in the polymerization process since the apparent rate coefficient can vary greatly with system conditions such as monomer conversion and solution vis-cosity (see Section 4.3.3). Although termination may also exhibit some chain-length dependence, most engineering treatments of FRP neglect this complex de-pendence. For further discussion of the individual mechanisms of Scheme 4.3 and their rate coefficients, see Section 4.3.1.2.The set of rate laws that can be written from Scheme 4.3 is given by Eqs. (2)–(6).Initiator Decomposition R d ¼ kd ½I ð2ÞChain Initiation R init ¼ 2f kd ½I ð3ÞChain Propagation R p ¼ k p ½M½Ptot  ð4ÞChain Termination R term ¼ ðk tc þ k td Þ½Ptot  ¼ k t ½Ptot  ð5ÞChain Transfer R tr ¼ ðktrmon ½M þ ktrsol ½SÞ½Ptot  ð6ÞHere Ptot represents the concentration of all polymer radicals in the system [Eq.
(7)].
X
y
½Ptot  ¼ ½Pn  ð7Þ
n¼1
In some literature the right-hand side of the termination rate expression [Eq. (5)] is written as 2k t ½Ptot  2 . The mode of termination – combination or disproportionation– has no effect on the overall termination rate, and thus the two events can also be expressed by the nomenclature of Eq. (8), where d is the fraction of the termination events that occur by disproportionation.
k td
k t ¼ k tc þ k td ; d¼ ð8Þk tc þ k td The following assumptions are widely accepted and usually valid in FRP systems:

4.3 Polymerization Mechanisms and Kinetics 159

The small radical species I  ; M  , and S  are not consumed by side reactions and do not accumulate in the system, but are converted to polymeric radicals with 100% efficiency. Thus, the total rate of polymer radical formation is given by(R init þ R tr Þ. The net formation of polymeric radicals is R init , since transfer events both consume and create a polymeric radical species.
With a continuous source of new radicals in the system, an equilibrium is achieved instantaneously between radical generation and consumption, such that R init ¼ R term . This characteristic, proven to be true for almost all FRP condi-tions [6], is a result of the fast dynamics of radical reactions compared to that of the overall polymerization system. Often referred to as radical stationarity or the quasi-steady-state assumption (QSSA), it leads to the well-known analytical expres-sion for total radical concentration [Eq. (9)].



R init 1=2 2f kd ½I 1=2½Ptot  ¼ ¼ ð9Þ
kt kt

The consumption of monomer by chain-initiation or transfer events is negligible compared to that by propagation. This result, called the long-chain hypothesis(LCH), must be true if high molecular weight polymer is being produced. Thus the rate of polymerization (disappearance of monomer) can be taken as equal to the rate of propagation (R pol ¼ R p ) with the rate of heat generation proportional to the rate of this exothermic reaction.Under these (generally valid) assumptions, the classic expressions for rate of poly-merization (R pol ), kinetic chain length (u, the average number of monomer units on a living chain), and instantaneous degree of polymerization (DPninst , the average number of monomer units on a dead polymer chain formed at any instant) are given in Eqs. (10)–(12), respectively.


2f kd ½I 1=2 R pol ¼ k p ½M½Ptot  ¼ k p ½M ð10Þ
kt
Rp k p ½M
u¼ ¼ ð11Þ
R term þ R tr k t ½Ptot  þ ktrmon ½M þ ktrsol ½S
k p ½M
DPninst ¼ ð12Þðk td þ 0:5k tc Þ½Ptot  þ ktrmon ½M þ ktrsol ½SThe difference between Eqs. (11) and (12) arises because termination by combina-tion yields a single polymer chain such that the chain length of dead polymer formed (DPninst ) is greater than the chain length of polymer radicals (u) in the sys-tem at the same instant.Table 4.1 lists the range of concentration and rate coefficient values typically en-countered in homogeneous free-radical polymerization systems at low conversion.These can be combined with Eqs. (9)–(12) to illustrate the tradeoffs involved be-

160 4 Free-radical Polymerization: Homogeneous Tab. 4.1. Typical values of coefficients and concentrations in low-conversion homogeneous FRP systems.Coefficient/Concentration Typical range kd [s1 ] 106 –104
f 0.4–0.9
k p [L mol1 s1 ] 10 2 –10 4 k t [L mol1 s1 ][a] 10 6 –10 8 ktrmon =k p 106 –104 ktrsol =k p 106 –103[I] [mol L1 ] 104 –102[M] [mol L1 ] 1–10[S] [mol L1 ] 1–10[a] At low conversion.tween the desire for high throughput (R pol ) and the need to produce high MW(DPn ) polymer.
The denominator of Eq. (12), the rate of dead chain formation, must be of the order 105 –108 chains L1 s1 in order to produce polymer with a DPn of 10 2 –10 4 . Individual polymer radicals exist on average only for a fraction of a sec-ond, as calculated by the expression u=ðk p ½M). Thus after the first few seconds of polymerization, the concentration of dead polymer chains is higher than that of polymeric radicals, and by the end of a typical polymerization the concentration of dead chains is orders of magnitude higher than [Ptot ]. Final polymer MW and MWD (molecular weight distribution) are controlled by how the concentrations and kinetic coefficients in Eq. (12) vary with polymer conversion.
The theoretical MW limit for a system is controlled by transfer events, and is cal-culated by setting [Ptot ] to zero in Eq. (12). For bulk FRP with no solvent, limiting values of DPn are 10 4 –10 6 . However, this theoretical limit can only be ap-proached in homogeneous FRP by reducing rates of polymerization to extremely low levels. For most systems both termination and transfer events play an impor-tant role in controlling polymer MW.
To produce high MW polymer (DPn of 10 2 –10 4 ) it is necessary to keep total rad-ical concentration low, so that [Ptot ] is of the order 108 –106 mol L1 . This dic-tates the choice of initiator such that R init (the product of kd and [I]) is also of order 108 –106 mol L1 s1 .
Transfer can occur to monomer, solvent or any other species in the system. In some cases, chain-transfer agents are added deliberately to limit and control poly-mer DPn . These agents are generally chosen such that the rate of abstraction is much higher than that which occurs with monomer or solvent (k tr =k p ¼ 103 –10 0 ) and thus can be added in trace quantities (f 1 mol L1 ). The use of transfer agents allows for independent manipulation and control of R pol and DPn , but is only possible if the desired MW is less than that achieved for the corresponding transfer-free reaction.

4.3 Polymerization Mechanisms and Kinetics 161

Initial rates of polymerization (monomer consumption rates) at low conversion are of order 104 –102 mol L1 s1 . Approximately 10 3 –10 5 s are required to take a batch system to complete conversion. Faster rates can be achieved by in-creasing R init , but at the expense of decreased polymer MW. Achievable values of R pol are also often limited by the heat removal capabilities of the reactor sys-tem, as the heat released by monomer addition is of the order 50–100 kJ mol1 .It should be cautioned that these statements are generalities for a typical FRP sys-tem. Rate coefficients vary with monomer, initiator, and solvent choice (see Section
4.3.1.2) as well as polymerization conditions, and the kinetic treatment is compli-
cated by the occurrence of side reactions (Section 4.3.1.3) and the variation of k t with conversion and other system conditions (Section 4.3.3). These factors necessi-tate the use of more-powerful modeling techniques to quantitatively describe FRP systems (Section 4.4). Nonetheless, Eqs. (9)–(12) provide an idea of the factors con-trolling rate and MW, and are very useful for a qualitative examination of FRP systems.
4.3.1.2Kinetic Coefficients
The coupling of polymer MW and polymerization rate is further illustrated via re-arrangement of Eq. (12) to give Eq. (13).1 0:5k tc ½Ptot  k td ½Ptot  ktrmon ktrsol ½S
inst
¼ þ þ þ
DPn k p ½M k p ½M kp k p ½M0:5k tc R pol k td R pol k mon k sol ½S¼ þ þ tr þ tr ð13Þkp2 ½M 2 kp2 ½M 2 kp k p ½MR pol [Eq. (10)] and DPn are both dependent on k 2p =k t , with DPn also a function of mode of termination (disproportionation versus combination) and chain transfer.Although R pol and DPn are easily measured experimentally, it is not possible to re-solve the quantities into estimates for individual rate coefficients. Even the estima-tion of the ratio k 2p =k t from R pol requires independent knowledge of initiator char-acteristics f and kd , and the assumption that radicals are not being consumed or retarded by adventitious impurities in the system. These factors have led to consid-erable scatter in rate coefficients reported in the literature (for example, Ref. 7), es-pecially for individual rate coefficients [8, 9]. Yet individual values, and knowledge of how they vary with temperature, are required for model development and an ac-curate representation of multi-monomer systems. The emergence of specialized experimental techniques since 1988 has greatly improved this situation and led to an improved understanding of free-radical polymerization kinetics. The following discussion highlights some of these advances, as well as summarizing key FRP rate coefficients as expressed by the Arrhenius law [Eq. (14)].k i ¼ A i expððEi þ 0:1DVi PÞ=RTÞ ð14Þ

162 4 Free-radical Polymerization: Homogeneous Activation energies (Ei ) and volumes (DVi ) are reported with units of kJ mol1 and cm 3 mol1 respectively, with T in K and P in bar. All second-order rate coefficients are reported with units of L mol1 s1 .Initiation Thermal scission of an initiator is the most common means of generat-ing radicals in FRP (see Section 4.2). This unimolecular reaction is characterized by a first-order rate coefficient (kd , s1 ) so that, for a constant-volume batch system,Eq. (2) may be integrated to yield Eq. (15), with ½I0 the initial concentration at½I ¼ ½I0 expðkd tÞ ð15ÞThe decomposition kinetics is often expressed by the half-life of the initiator, the time needed for the concentration to decrease to half of its initial value [Eq. (16)].t 1=2 ¼ ðln 2Þ=kd ð16ÞThe requirements for an initiator can vary widely: an initiator with a half-life of 10 h might be used in an academic study so that [I] does not change significantly dur-ing the course of an experiment, while an initiator used for high-pressure ethylene polymerization typically has a half-life on the order of a few seconds. Activation en-ergies for thermal homolysis of peroxide and azo compounds are in the range of 100–150 kJ mol1 , and thus decomposition rates are very temperature-sensitive:the t 1=2 of benzoyl peroxide drops from 13 h at 70 C to 0.4 h at 100 C. The depen-dence on pressure is much less, in the range of 0 to 15 cm 3 mol1 [1, 2], but still important for high-pressure LDPE systems. Special care must be taken in handling and transport of thermal initiators, especially those with faster decomposition rates.Values of kd at a particular temperature and pressure can be determined by mea-suring initiator concentration as a function of reaction time using a technique such as infrared spectroscopy. The experimental difficulty increases as half-life shortens,where special care must be taken to eliminate transient nonisothermal effects. De-composition kinetics are summarized for a wide range of initiators in the Polymer Handbook [7] and in trade literature available from commercial suppliers. Of spe-cial note is the recent work of Buback and co-workers that systematically examines not only how Ed and DVd vary with alkyl substituent for the peroxyester family(Scheme 4.2), but also how the substituent choice affects the decomposition path-way and initiator efficiency [1, 2].Propagation In general, chain growth or propagation proceeds in a highly selec-tive manner to yield a polymer chain consisting of head-to-tail linkages (Scheme
4.4). In order for high polymer to be formed the propagating radical must not be
too stable: addition must occur at a sufficiently high rate in comparison with com-peting transfer and termination events.

4.3 Polymerization Mechanisms and Kinetics 163
CH2 C + H2C CH2 C CH2 C R2 R2 R2 R2 Scheme 4.4. Free-radical chain propagation.A number of nonsteady polymerization rate techniques have been introduced to measure k p , many prone to significant error [8, 9]. The introduction of a new method, that couples pulsed-laser-induced polymerization (PLP) with size exclu-sion chromatographic (SEC) analysis of the resulting polymer [10], has greatly im-proved the reliability of k p data. In this technique, a mixture of monomer and pho-toinitiator is illuminated by short laser pulses separated by a time of t 0 , typically
0.01–0.2 s. Propagation and termination (but no initiation) occur between pulses,
with the radical concentration and the rate of termination both decreasing with time according to Eq. (5). Those growing macroradicals formed by a laser pulse which escape termination will all have the same chain length (within a narrow Poisson distribution) that increases with time. There is a high probability that these surviving radicals will be terminated at t 0 , when a new population of short radicals are generated by the next laser pulse. Thus, a significant fraction of dead chains formed have a chain length DP0 corresponding to a lifetime of t 0 seconds[Eq. (17)].DP0 ¼ k p ½Mt 0 ð17ÞSince radicals have a certain probability of surviving the laser flash and of terminat-ing at a later pulse, the relative concentration of polymer with chain lengths 2DP0 ; 3DP0 ; . . . is also increased. As a result, PLP produces a well-structured MWD with peaks at chain lengths of DP0 and its multiples, as shown in Figure
4.1. With known values for t 0 and [M] (kept constant, as samples are pulsed only
long enough to allow a conversion of 1–3%), Eq. (17) yields a direct estimate of k p from the experimentally-determined value of DP0 .PLP-SEC has proven to be a very simple and robust experimental technique for determining k p and its temperature dependence, provided adequate care is taken with SEC analysis. Data collected from various research laboratories around the world have been compiled in a series of papers for styrene [11] and methacrylates[12–14]. These papers provide benchmark k p data for these monomer systems, il-lustrate the good agreement between facilities (typically 10–20%), and make rec-ommendations for best experimental practices.Table 4.2 summarizes the Arrhenius parameters for free-radical propagation of a wide range of monomers, as determined by PLP-SEC. An extensive review by Beuermann and Buback [15] contains further discussion and data for additional monomers. The data for acrylates are limited, due to measurement difficulties aris-ing from additional secondary mechanisms that occur (see Section 4.3.1.3). What

164 4 Free-radical Polymerization: Homogeneous
1.2
2DP0
0.8
wt log(MW) 0.6 DP0
0.4
0.2
3.5 4 4.5 5 5.5 6 6.5
Log MW
Fig. 4.1. A typical PLP-generated MWD, as measured by SEC for poly(butyl methacrylate) produced by PLP of bulk butyl methacrylate at 25 C and a laser repetition rate of 10 Hz. DP0 is determined from the inflection point of the peak, obtained by differentiating the distribution.is striking is not only the large variation in k p and activation energies between monomer families, but also the similar behavior within a monomer family. All methacrylates exhibit a similar temperature (Ep of 21 to 23 kJ mol1 ) and pressure(DVp of 15 to 17 cm 3 mol1 ) dependence, and the k p values for alkyl methacry-lates at 50 C increase with increasing size of the alkyl ester group (methyl to do-decyl) by a factor of less than 2. The alkyl acrylate family exhibits exactly the same trends, although the activation energies (17 to 18 kJ mol1 ) and volumes (10 to12 cm 3 mol1 ) are lower than for methacrylates, and the values of k p at 50 C Tab. 4.2. Arrhenius k p parameters for various monomers determined by PLP-SEC.[a]Monomer Ep DVp Ap ˚k p at 50 C/1 atm[kJ molC1 ] [cm 3 molC1 ] [L molC1 sC1 ] [L molC1 sC1 ]Ethylene 34.3 27.0 1:88  10 7 54 Styrene 32.5 12.1 4:27  10 7 238 Methyl methacrylate 22.4 16.7 2:67  10 6 648 Butyl methacrylate 22.9 16.5 3:78  10 6 757 Dodecyl methacrylate 21.0 16.0 2:50  10 6 995 Glycidyl methacrylate 22.9 15.0 6:19  10 6 1230 Cyclohexyl methacrylate 23.0 16.2 6:29  10 6 12042-Hydroxypropyl 20.8 n.d. 3:51  10 6 1504 methacrylate Vinyl acetate 20.7 10.7 1:47  10 7 6625 Methyl acrylate 17.7 11.7 1:66  10 7 22 900 Butyl acrylate 17.4 n.d. 1:81  10 7 27 900 Dodecyl acrylate 17.0 11.7 1:79  10 7 32 000[a] All values taken from Ref. 15; n.d., not determined.

4.3 Polymerization Mechanisms and Kinetics 165
are 30 to 40 times greater. The propagation behavior is affected more significantly by electron-donating or -withdrawing substituents present in functional monomers such as glycidyl and 2-hydroxypropyl methacrylates. The PLP-SEC method has also been utilized to show that there is little to no solvent influence on propagation ki-netics of most monomers [15]. However, the propagation kinetics of acrylic acid in water is a strong function of concentration, a result that may be explained by a change in local monomer concentration around the propagating radical center[16].With accurate estimates of k p now available, the possibility of correlating the co-efficient with the structural characteristics of the propagating radicals and mono-mers is receiving attention. While some progress has been made in relating activa-tion energy to reaction enthalpy corrected for polar and resonance factors [17],more work is required for quantitative predictions.Termination With independent measures of k p available, k t can now be estimated from the lumped ratio of k 2p =k t . Specialized techniques involving pulsed-laser-induced polymerization have also been developed to yield accurate estimates of the ratio k p =k t [15]. Termination rates in FRP are always diffusion-controlled so that the apparent value of k t depends on the conditions under which it has been measured, including the lengths of the radicals involved in the reaction. Nonethe-less, the assumption of chain-length independence is usually made for modeling of commercial FRP systems, since the errors introduced are not large. For a more detailed discussion of the diffusion-controlled nature of the termination reaction,including its strong conversion dependence, see Section 4.3.3.Most measurements of k t have been carried out at low monomer conversion,and thus it is useful to tabulate available data in this regime. Significant scatter,as much as an order of magnitude, is found in the data contained in the Polymer Handbook [7], as reflected in the vinyl acetate k t data in Table 4.3. The uncertainty Tab. 4.3. Arrhenius kt parameters for various monomers.[a]Monomer Et DVt At ˚k t at 50 C/1 atm[kJ molC1 ] [cm 3 molC1 ] [L molC1 sC1 ] [L molC1 sC1 ]Ethylene 4.6 15.6 1:6  10 9 2:9  10 8 Styrene 6.5 14 2:2  10 9 2:0  10 8 Methyl methacrylate 4.1 15 4:3  10 8 9:4  10 7 Butyl methacrylate 4.1[b] 15[b] 8:5  10 7 1:9  10 7 Dodecyl methacrylate 4.1[b] 15[b] 2:8  10 7 6:2  10 6 Vinyl acetate – – – (5–50)  10 7 [c]Methyl acrylate 6.7 20 6:0  10 9 5:1  10 8 Butyl acrylate 4.0 16 5:1  10 8 1:2  10 8 Dodecyl acrylate 1.7 21 2:1  10 7 1:1  10 7[a] All values taken from Ref. 15, unless otherwise indicated.[b] Assumed equal to MMA value.[c] From Ref. 7.

166 4 Free-radical Polymerization: Homogeneous arises from a number of measurement and interpretation factors [18]. The rest of the data in Table 4.3 are based on PLP studies, and have a higher level of accuracy(within a factor of 2); these are taken from the recent review by Beuermann and Buback [15], adjusted to conform to the convention used in Eq. (5) for the ter-mination rate. Note the large difference in magnitude between values for sty-rene, methyl methacrylate, and methyl acrylate (k t of 1  10 8 –5  10 8 L mol1 s1 )and values for dodecyl acrylate and dodecyl methacrylate (@0:5  10 7 –1  10 7 L mol1 s1 ). This has been attributed to differences in segmental diffusion rates and/or steric hindrance of the large ester side groups. The reported activation energies and volumes are consistent with the diffusion-controlled nature of the reaction.The mode of termination is also of importance, as it affects the molecular archi-tecture of the polymer formed, and thus some of its properties. Polymer polydis-persity (Mw/Mn ) is 1.5 if all chains are terminated by combination of two radicals,and 2 if chains are terminated by disproportionation or chain transfer. As shown in Scheme 4.5, termination by combination results in head-to-head linkages in the polymer chain, whereas disproportionation results in the formation of an unsatu-rated end group that may undergo further reaction.CH2 C C CH2
R2 R2
CH2 C + C CH2
R2 R1 R1
R2
CH2 CH + C CH
R2 R2
Scheme 4.5. Free-radical chain termination by combination and disproportionation.The relative importance of termination by disproportionation versus combina-tion expressed by d [Eq. (8)] is difficult to measure. The value depends largely on the structure of the monomer: d is in the range of 0.5–0.8 for a-methylvinyl mono-mers such as the methacrylates and 0.05–0.2 for styrene and acrylates [3]. A weak temperature dependency has been proposed, but is difficult to verify within the scatter of the data [19].Chain transfer Chain transfer in radical polymerization involves the transfer of the radical center from a polymeric radical to another molecule. It can occur to all of the substances present in the polymerization system, and always causes a reduc-tion in u and DPninst . Scheme 4.3 includes only transfer to monomer and solvent (or added transfer agent), but transfer to initiator and dead polymer can also occur.The former is often neglected since [I] is small relative to the concentration of other species, while the latter can be an important reaction impacting MWD and

4.3 Polymerization Mechanisms and Kinetics 167
final polymer properties (see Section 4.3.1.3). As well as reducing chain length, the fragments from the transfer reactions (I  and M  in Scheme 4.3) are incorporated as end groups in the final polymer product.The rate and MW equations in Section 4.3.1.1 were derived assuming that the radicals formed by transfer reinitiate new polymeric radicals quickly, within about the same time period as that required for a propagation step (k imon A k isol A k p ).This assumption is valid for transfer to most species, and can be verified by exam-ining whether addition of the transfer agent has an effect on polymerization rate.If the low-conversion polymerization rate is significantly decreased, the species is classified as a retarding agent or inhibitor, and Eqs. (9)–(13) no longer describe the kinetics of the system; see Section 4.3.1.3.The chain-transfer ability of a compound can be studied by varying its concentra-tion while holding all other variables fixed. By stopping the reaction at very low conversion such that concentrations and diffusion-controlled k t values (and thus R pol and DPninst ) are kept constant, Eq. (13) can be rearranged as Eq. (18).1 1 k sol ½S¼ þ tr ð18ÞDPn ðDPn Þ0 k p ½MA plot of 1=DPn against ½S=½M should yield a straight line with slope k trsol =k p and intercept 1=ðDPn Þ0 , where ðDPn Þ0 is the average chain length measured in the ab-sence of transfer agent. The form of this equation leads to the definition of a chain-transfer constant C for each species, including monomer, in Eq. (19).ktrsol ktrmon Ctrsol ¼ ; Ctrmon ¼ ð19Þ
kp kp
It is these ratios and their temperature dependence that are tabulated in standard references (for example, Ref. 7).Solvent/transfer agent For some homogeneous FRP systems, solvent is added not for its chain-transfer ability, but as a diluent to control viscosity and/or heat trans-fer. In other systems, a small amount of chain-transfer agent (CTA) is added spe-cifically to control and reduce the MW of the polymer. Thus values for Ctrsol may vary widely from a low of 106 –104 to a high of 101 –10 1 , depending on the number of weakly bonded atoms (generally hydrogen or halogen) on the transfer agent and their ease of abstraction. For very active compounds (for example, thiols and halogenated compounds) with Ctrsol > 101 it is necessary to account for the consumption of CTA during the course of the polymerization, since a changing ra-tio of ½S=½M will cause a corresponding drift in polymer MW. In such cases care-ful control of CTA addition to the system is required. If it is added at higher con-centrations, telomerization (formation of low-MW species) will occur rather than polymerization.Table 4.4 summarizes values for a few typical solvents and CTA compounds at 50 C. Most organic compounds have low transfer rates (Ctrsol of 106 –104 ) so that

168 4 Free-radical Polymerization: Homogeneous Tab. 4.4. Transfer constants (k trsol =k p ) to solvents and transfer agents at 50 C.[a]Styrene Methyl Methyl Vinyl Ethylene[b]methacrylate acrylate acetate Benzene 2  106 4  106 2  105 1  104 9  104 [b]Toluene 1  105 2  105 1  104 2  103 1:3  102 [b]Ethyl acetate 5  104 1  105 6  105 2  104 5  103 [b]Triethylamine 5  104 8  104 4  102 4  102 1:8  102 [b]CCl 4 1  102 2  104 2  104 0.8 1.0[b]1-Butanethiol 20 0.6 1.5 50 6.0[b][a] Representative values at 50 C [7]; there can be an order of magnitude range in literature data.[b] Ethylene values at 130 C and 1360 atm [20].transfer is important only when they are present in high concentration (that is, as solvent). They tend to be more reactive toward a radical such as ethylene or vinyl acetate than a resonance-stabilized radical such as that of styrene. For a given radical type, aliphatic compounds that yield tertiary radicals will be more effec-tive transfer agents than those that produce secondary radicals. Halomethanes and thiols, used as MW-controlling agents due to their high transfer rates, re-act faster with nucleophilic radicals such as styrene or vinyl acetate than with(meth)acrylates. Abstraction reactions from organic solvents generally have higher activation energies than propagation, with (Etr  Ep ) in the range of 20–50 kJ mol1[21, 22]. The Polymer Handbook [7] and the monograph by Moad and Solomon [3]provide a summary of available data for a wider range of monomer–CTA (solvent)pairings.Monomer Transfer to monomer cannot be avoided, and the maximum upper limit of chain length that can be achieved in a polymerization is Ctrmon , assuming the ab-sence of all other transfer and termination events. The details of the mechanism depend upon the specific monomer involved. For monomers that contain aliphatic hydrogens such as vinyl acetate and (meth)acrylates, the transfer process involves H-atom abstraction to form an unsaturated new radical, as shown for vinyl acetate in Scheme 4.6. The polymer chain that grows from this radical thus contains an unsaturated end group that may undergo further reaction. Transfer to monomer rates are generally very low, and are difficult to measure experimentally since the ratio with propagation (R trmon =R p ) is independent of [M]. Table 4.5 summarizes
O O O O
O C CH3 O C CH3 O C CH3 O C CH2 CH2 C + H2C C CH2 CH2 + H2C C
H H H
Scheme 4.6. Free-radical chain transfer to monomer (vinyl acetate).

4.3 Polymerization Mechanisms and Kinetics 169
Tab. 4.5. Transfer constants (k trmon =k p ) to monomer.[a]C trmon E tr C E p [kJ molC1 ] Reference Methyl methacrylate 2  105 23.7 23 Butyl acrylate 6  105 15.2 24 Vinyl acetate 2  104 – 7 Ethylene[b] 4  104 [b] 43.9 25[a] Representative values at 50 C, unless otherwise noted; values reported in Ref. 7 can show an order of magnitude range.[b] At 250 C and 2000 bar.the range of values found in the literature. Ctrmon increases with temperature, with(Etrmon  Ep ) in the range of 10–40 kJ mol1 .
4.3.1.3 Additional Mechanisms
The basic mechanisms of Scheme 4.3 are common to and occur in every FRP system. Other mechanisms are more system specific, dependent not only on the choice of monomer but also the process operating conditions. These additional mechanisms complicate the kinetic analysis and often play an important role in controlling polymerization rate and polymer structure under typical industrial op-erating conditions.Thermal initiation Primary radicals in most FRP systems are generated by scis-sion of an added initiator. Free-radical polymerizations can also be initiated by the monomer itself, or by reactions involving trace impurities in the system. Generally the rate of radical generation by these processes is very low – negligible compared with R init from the added initiator. Styrene, however, exhibits significant thermal polymerization at temperatures of 100 C or more. Indeed, it is not necessary to add initiator to styrenic systems at high temperatures, as the rate of thermal initia-tion is sufficient to make an industrially viable process [26]. Acrylates and metha-crylates have also been reported to undergo thermal polymerization, but at a signif-icantly slower rate than styrene.The mechanism behind the thermal polymerization of styrene is still under de-bate, but is believed to involve the reversible formation of a dimeric species by a Diels–Alder reaction, followed by the subsequent hydrogen transfer to a third sty-rene molecule to form two radicals that can initiate polymerization (Scheme 4.7).This complex mechanism can be approximated by a third-order dependency on sty-rene concentration [Eq. (20)] [27].
k therm
3M ! 2I 


L2
R therm ¼ k therm ½M 3 ; k therm ¼ 2:2  10 5 expð114:8=RTÞ ð20Þ
mol 2  s

170 4 Free-radical Polymerization: Homogeneous Scheme 4.7. Thermal initiation of styrene.This expression from the mid-1970s has stood the test of time, although the pre-exponential factor has been increased by a factor of 2–3 to fit 260–340 C data ob-tained in a more recent study [26].Retardation and inhibition Some substances retard or suppress free-radical poly-merization by reacting with primary or polymer radicals to yield nonradical prod-ucts or radicals that are too stable to add further monomer. By decreasing the concentration of reactive radicals in the system, polymerization rate is slowed (re-tardation) or stopped completely (inhibition).Phenolic inhibitors (for example, hydroquinone, monomethylhydroquinone) are commonly added to monomers at parts-per-million levels in order to prevent poly-merization during transport and storage by rapidly and effectively scavenging any radicals (Eq. (21), in which R  represents I  or Pn ) that may form.
kinhib
R  þ Z ! dead products; R inhib ¼ kinhib ð½Ptot  þ ½I  Þ½Z ð21ÞIdeally an inhibitor (Z) stops all polymerization until completely consumed(R inhib g R p ), at which time polymerization proceeds at a normal rate, as shown schematically in Figure 4.2. In many academic studies, monomer is distilled or passed over a column to remove inhibitor before polymerization. This purification is not done in industry if the concentration of inhibitor in the monomer is much lower than the concentration of initiator added to the system, since the excess of initiator-generated radicals quickly consumes the inhibitor.Retardation, the slowing of polymerization rate by consumption of radicals, can take different forms and thus must be considered on a case-by-case basis. If R inhib is of similar magnitude to R p , polymerization will proceed at a lower rate until all of the retarding species is consumed (curve a in Figure 4.2). The kinetic ex-pressions (Section 4.3.1.1) cannot be applied since termination is no longer the

4.3 Polymerization Mechanisms and Kinetics 171
Polymerization
Inhibition
Conversion
Retardation
Time
Fig. 4.2. Conversion–time plots for normal, retarded, and inhibited free-radical polymerizations. Retardation cases a and b are described in the text.sole mechanism of radical consumption; application of radical stationarity yields R init ¼ R term þ R inhib .A different form of retardation occurs when a radical species formed from trans-fer (S  in Scheme 4.3) reinitiates at a slow rate. In addition to the slower reaction rate with monomer to form a polymer radical, the termination of S  with other radicals in the system may also need to be considered (Scheme 4.8). Explicit bal-ances must be written for S , and the extra mechanisms must be included when deriving expressions for [Ptot ], R pol , and DPn . As solvent/transfer agent is generally not completely consumed, the retardation effect will last the duration of the poly-merization (curve b in Figure 4.2). The degree of retardation depends on the value of k isol , which can vary with monomer type; many carbon-centered radicals show much lower reactivity toward vinyl esters (for example, vinyl acetate) than(meth)acrylates [3].
k sol
Pn + S →
tr
Dn + S *
k sol
S *+ M →
P1
k sol
S *+ Pn →
Dn
Scheme 4.8. Retardation of polymerization by solvent or transfer agent.Oxygen inhibits or retards vinyl polymerizations by the formation of peroxy rad-icals that are generally unreactive. However, subsequent monomer addition can oc-cur in some systems; this copolymerization forms polymeric peroxides that affect the thermal stability of the final polymer product. Good practice requires the re-moval of air by sparging or freeze–thaw cycles before a reaction is started, and ex-clusion of air during polymerization by operating under an inert atmosphere (for example, nitrogen) or refluxing solvent.

172 4 Free-radical Polymerization: Homogeneous Depropagation The addition of a radical to a double bond – the propagation reaction – is potentially a reversible process [Eq. (22)].Pn þ M Ð Pnþ1 ; R dep ¼ kdep ½Ptot  ð22Þ
kdep
The relative importance of the reverse reaction is governed by the free energy change, DGp ¼ DHp  TDSp , where DHp represents the enthalpy and DSp repre-sents the entropy change upon propagation; polymerization can only occur sponta-neously when DGp is negative. With most common monomers, depropagation is negligible at typical reaction temperatures. For some systems, however, the en-thalpy and entropy terms for propagation are delicately balanced so that depropa-gation has a significant effect on polymerization rate (R pol ) at higher temperatures[Eqs. (23)].R pol ¼ R p  R dep ¼ ðk p ½M  kdep Þ½Ptot  ¼ kpeff ½M½Ptot 
kdep ð23Þ
½M
Examining the reaction from a thermodynamic viewpoint leads to the definition of½Meq , the monomer concentration for a particular temperature at which the effec-
eff
tive propagation rate coefficient (k p ) and polymerization rate become zero [Eq.
(24)].



K eq ¼ ¼ exp ¼ exp  ¼ ð24Þkdep RT RT R ½Meq This equation can also be rearranged to Eq. (25), to define the ceiling temperature Tc at which, for a given monomer concentration, the polymerization rate becomes
zero:
DHp
Tc ¼ ð25Þ
DSp þ R ln½MIt also provides a link between the thermodynamic and kinetic coefficients accord-ing to Eqs. (26).


Ap
DHp ¼ Ep  Edep ; DSp ¼ R ln þ R lnð½MÞ ð26Þ
A dep
Values of DHp for common monomers are tabulated in Table 4.6; an expanded list may be found in Ref. 7. DSp is more difficult to measure experimentally, but is typ-ically in the range of 100 to 140 J mol1 K1 . Table 4.6 also contains estimates of Tc calculated at ½M ¼ 1 mol L1 . Ethylene, vinyl acetate, and acrylates have very high values, and depropagation does not occur under viable polymerization condi-

4.3 Polymerization Mechanisms and Kinetics 173
Tab. 4.6. Depropagation behavior of common monomers.CDH p [kJ molC1 ] [a] ˚
Tc [ C][b]
a-Methylstyrene 35 19 Styrene 73 335 Methyl methacrylate 56 194 Methyl acrylate 80 394 Vinyl acetate 88 460 Ethylene 102 577[a] From Ref. 7.[b] Calculated for ½M ¼ 1 mol L1 , assuming DS 1 1 p ¼ 120 J mol K .tions. Styrene has a slightly lower ceiling temperature and depropagation must be considered above 250 C. This temperature range, while higher than that generally employed for styrene polymerization, is used for some commercial processes [26].The addition of a methyl group to a monomer greatly reduces Tc , as seen by comparing values for a-methylstyrene to styrene and methacrylates to acrylates.
eff
Figure 4.3 is a plot of k p measured for dodecyl methacrylate (bulk monomer) by
eff
the PLP-SEC technique [28]. At 180 C, the highest temperature examined, k p is1 1 1 14000 L mol s , less than half of the k p value of 9400 L mol s extrapolated from the lower-temperature data. With known Arrhenius coefficients for propaga-tion (Table 4.2), the temperature dependence of kdep may be estimated directly from
3.8
3.6
log[kpeff (L/mol-s)]
3.4
3.2
2.8
2.6
2.4
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6
1000/T (K -1)Fig. 4.3. Effective propagation rate coefficient for bulk dodecyl methacrylate measured by PLP. The line indicates the forward rate coefficient (k p ) calculated according to the Arrhenius parameters of Table 4.2. Experimental data are taken from Ref. 28.

174 4 Free-radical Polymerization: Homogeneous these data. The corresponding values for DHp (54 kJ mol1 ) and DSp (123 J mol1 K1 ) agree well with thermodynamic estimates. The depropagation behav-ior of many methacrylate monomers is similar [28], and depropagation must be considered at temperatures above 120 C, especially for systems with low monomer Long-chain branching All of the mechanisms presented to this point do not change the basic linear architecture of the polymer chains, with each repeat unit linked to two others. Branched polymers, those in which the repeat units are not linked solely in a linear array, can have significantly different physical properties than their linear counterpoints, depending upon the number and distribution of the branches along the polymer backbone as well as their length. Understanding the mechanisms by which these branches are formed, therefore, is a key compo-nent to manipulation and control of polymer structure. Branches can be formed either by intramolecular reactions, discussed in the next subsection, or intermolec-ular reactions. The reaction of a polymeric radical with another polymer chain, or long-chain branching, can occur by three distinct mechanisms. The common feature is the re-activation of a dead polymer chain. As well as creating branched struc-tures, these reactions significantly broaden the polymer chain-length distribution.Intermolecular transfer to polymer The transfer of a radical center from a polymeric radical to another polymer chain is shown in Scheme 4.9. Addition of monomer to the mid-chain radical produces a polymer with a branch point, with the length controlled by the number of propagation events before chain transfer or radical–radical termination occurs. An additional subscript can be added to track the num-ber of long-chain branches formed [Eq. (27)].
pol
mk tr pol pol Pn; b þ Dm; c ! Dn; b þ Pm; cþ1 ; R tr ¼ k tr ½Ptot m1 ð27ÞThis reaction does not change the number of monomer units that have been poly-merized or the number of polymer chains in the system, and thus has no effect on DPn . The rate expression in Eq. (27) is written assuming that all repeat units on all R1 R1 R1 R1 CH2 C + CH2 C CH2 CH2 CH2 + CH2 C CH2
H H
R1
R1
CH2 C CH2
+ H2C CH CH2 C CH2
R1 H2C
R1 CH
Scheme 4.9. Long-chain branch formation by chain transfer to polymer.

4.3 Polymerization Mechanisms and Kinetics 175
dead polymer chains have an equal probability of reaction, as represented by m1, the total concentration of polymerized monomer units in the system [Eq. (28)].
X y
m1 ¼ n½Dn; b  ð28Þ
n¼1 b¼0
Note that the reaction is not proportional to the number of chains in the system,but to the number of repeat units contained in the chains. This has two important consequences. First, the importance of transfer to polymer increases with polymer conversion (x p ) in the system, as seen in Eq. (29) by looking at the ratio of branch-ing to monomer consumption.pol pol pol R tr k ½Ptot m1 k tr x p¼ tr ¼ ð29ÞR pol k p ½Ptot ½M k p ð1  x p ÞThus polymerizations operating at high monomer conversion will have signifi-cantly higher branching than low-conversion systems. Second, longer polymer chains – those with more repeat units – are more likely to participate in a branch-ing reaction than short chains. Since the re-activated chains increase in length through subsequent propagation, this leads to a broadening of the molecular weight distribution reflected by an increase in the weight-average MW (Mw ); even low levels of branching can increase polydispersity values (Mw/Mn ) to 5–15 com-pared to 2–3 for linear polymers.As well as conversion, the importance of transfer to polymer depends upon the monomer system. The reaction can be important in systems with very reactive rad-icals such as ethylene [30–32], vinyl acetate [33–35], and acrylate [36, 37] polymer-izations, but seldom occurs in styrene and methacrylate systems. Transfer to poly-mer usually occurs via abstraction of a methine hydrogen as shown in Scheme 4.9,but may also involve other easily abstracted H-atoms, such as the acetate methyl
pol pol
hydrogens on poly(vinyl acetate). Transfer constants to polymer (Ctr ¼ k tr =k p )are not as readily determined as other transfer constants because the process does not decrease DPn . Long-chain branching (LCB) levels are usually quite low, less than 2 per 1000 repeat units, making it difficult to employ NMR. Indirect methods such as multi-detector SEC [32, 38] are often used, leading to a significant scatter
pol
in reported Ctr values [7]. Like other transfer events, the relative importance in-creases with temperature.Reaction with unsaturated polymer chains Termination by disproportionation, trans-fer to monomer, and chain scission (discussed later in this section) create polymer chains with terminal unsaturation (denoted in Eq. (30) by D ¼ ). These reactive chains, sometimes called macromonomers, can add to a growing radical to form a long-chain branch.
pol
kp
¼ ¼
Pn; b þ Dm; c ! Pnþm; bþcþ1 ; R pol pol p ¼ k p ½Ptot ½Dtot  ð30Þ

176 4 Free-radical Polymerization: Homogeneous CH2 C + H2C C CH2 C CH2 C
R2 R2
Scheme 4.10. Long-chain branch formation by addition to a terminally unsaturated polymer chain.This reaction is fundamentally different than transfer to polymer in several re-spects. It is an addition rather than a transfer (H-abstraction) reaction (Scheme
4.10). The reaction rate is dependent on the number of unsaturated chain ends
rather than the number of repeat units in the chains so that its importance relative
¼
to propagation is controlled by the ratio ½Dtot =½M according to Eq. (31).
pol pol ¼
Rp k p ½Ptot ½Dtot 
¼ ð31Þ
R pol k p ½Ptot ½MUnlike transfer to polymer, the mechanism combines two polymer chains (and all of their repeat units) into one, affecting both DPn and DPw . Reaction with terminally unsaturated chains can be important in vinyl acetate [33] and higher-temperature methacrylate [3] polymerizations.Reaction with multifunctional monomers Another way to introduce branching is through addition of a multifunctional monomer to the polymer system. An exam-ple is the addition of small levels of ethylene glycol dimethacrylate (EGDMA) to methyl methacrylate (MMA), as shown in Scheme 4.11. Reaction of the first dou-
CH3
O O CH2CH2 O O C CH2
O OCH3
C C C C
O O CH2CH2 O O CH2 C + H2C C C CH2 C H3C H3C CH3
CH2 C
H3C
CH3
CH3 C CH2
C CH2 C
C O O
O O H2C
CH2 CH2
O OCH3 CH2 O O C O O O CH3 C
O C
CH2 C + H2C C CH2 C CH2 C
H3C
CH3 H3C CH3 Scheme 4.11. Addition of ethylene glycol dimethacrylate to a methyl methacrylate radical. The pendent double bond is attacked by another polymer radical to form a crosslink.

4.3 Polymerization Mechanisms and Kinetics 177
ble bond incorporates EGDMA in the PMMA chain, with the second reactive site remaining as a pendent double bond. Reaction of this pendent bond with another polymer radical creates a branched structure termed a crosslink, since the branch is tetrafunctional rather than trifunctional in nature.The addition of a small amount of difunctional monomer such as EGDMA to MMA [39, 40] or divinylbenzene to styrene is a means to increase polymer MW without decreasing radical concentration. Addition of higher levels leads to an in-terconnected branched, or network, polymer.Short-chain branching (intramolecular transfer to polymer) Intramolecular H-atom abstraction, often called back-biting, occurs via the formation of a six-membered ring, as shown in Scheme 4.12 for butyl acrylate (nBA). Monomer addition to the resulting interior radical leaves a short-chain branch (SCB) consisting of two repeat units. This mechanism has long been known important for high-pressure LDPE production at 150–300 C [30], where the number of short-chain branches is in the range of 20–50 per 1000 ethylene repeat units. This level of SCB significantly decreases the polyethylene crystallinity and gives LDPE some of its unique proper-ties; LDPE density is 0.92 g cm3 compared to 0.98 g cm3 for linear polyethylene.Multiple back-biting events lead to other SCB structures; in addition to the com-mon butyl branch in LDPE, ethyl and 2-ethylhexyl branches have been identified using 13 C NMR [30, 41, 42].COOBu COOBu COOBu COOBu COOBu COOBu COOBu
COOBu
R CH
CH C R CH
CH CH C
CH
H
HC H
CH
COOBu
COOBu
Scheme 4.12. Formation of a mid-chain radical by intramolecular chain transfer to polymer. Monomer addition to the new radical structure creates a short-chain branch in the polymer.In ethylene/nBA copolymerization it is observed that the methine hydrogens on nBA units in the polymer chain are much more susceptible to abstraction than hy-drogen abstraction from a CH2 unit in the backbone [43]. This result, along with the observed high concentration of branch points formed at conditions of very low polymer concentration during nBA homopolymerization [36], indicates that the in-tramolecular (back-biting) mechanism is also important in acrylate polymeriza-tions. There is strong evidence to suggest that monomer addition to the resulting mid-chain radical, due to its higher stability, is much lower than addition to a chain-end radical [44]. Thus the mechanism affects nBA polymerization rate as well as polymer structure [37]. The same type of back-biting reaction has also

178 4 Free-radical Polymerization: Homogeneous been shown to occur during styrene polymerization at high temperatures (260–Chain scission The radical structure formed by intra- or intermolecular transfer to polymer is less reactive than a chain-end radical. Under higher-temperature con-ditions, the radical can undergo b-fragmentation (chain scission) as shown in Scheme 4.13 for butyl acrylate [45]. The scission can occur in either direction,yielding a short-chain radical or a short-chain unsaturated trimer species. As well as lowering polymer MW, the scission event produces an unsaturated chain end that can react further, as discussed previously (see Scheme 4.10).COOBu COOBu COOBu COOBu COOBu
CH2
CH CH C + HC CH2 COOBu COOBu COOBu
COOBu
CH C CH
CH
H COOBu COOBu CH COOBu COOBu COOBu
COOBu CH
+ H2C
C HC CH2
CH
Scheme 4.13. b-Scission of butyl acrylate mid-chain radical.Scission events can occur in any system where mid-chain radicals are formed.However, scission is more temperature-activated than H-abstraction and thus be-comes important only at elevated temperatures. The reaction is not believed to oc-cur during butyl acrylate polymerization at 75 C [37], but is shown to be important at 140 C [29, 45]. Scission is a dominant mechanism in styrene polymerizations at 260–340 C [26], and also occurs during LDPE production [30]. Scission of mid-chain radicals formed via intermolecular transfer to polymer can have a significant effect on the breadth and the shape of polymer MWD [46].Kinetic treatment of these more complex mechanisms is often difficult. Equa-tions (32)–(34) are a network of reactions developed to treat intramolecular trans-fer, short-chain branch formation, and b-scission for butyl acrylate polymerization[47].
kbb
Pn ! Q n ; R bb ¼ kbb ½Ptot  ð32Þ
kptert
Q n þ M ! Pnþ1 þ SCB; RSCB ¼ kptert ½M½Q tot  ð33Þ
0:5kb¼
Q n ! Dn2 þ P2
0:5kb ð34Þ
Q n ! Pn3 þ D3¼ ; R b ¼ kb ½Q tot The kinetic coefficients are estimated by measuring the level of quaternary carbons(equated to short-chain branch level) and terminal unsaturations (D ¼ ) by NMR.

4.3 Polymerization Mechanisms and Kinetics 179
Termination of the mid-chain radical (Q n ) is also considered in the scheme. While the mechanism provides improved understanding of this complex system, many questions still remain: does the mid-chain radical terminate at the same rate as chain-end radicals; what is the reactivity of the unsaturated chain end; and does the mid-chain radical scission with equal probability in each direction? The reac-tion engineering challenge is to consider the set of mechanisms required to repre-sent the basic rate behavior and polymer architecture of the system without intro-ducing unneeded complexity.
4.3.2
Copolymerization Reaction of two or more monomers by free radical polymerization is an effective way of altering the balance of properties of commercial products. Addition of the polar monomer acrylonitrile to styrene (or methyl acrylate to ethylene) produces a polymer that combines the strength of the base homopolymer with much im-proved oil and grease resistance. The adhesive and cohesive properties of coatings resins are balanced by controlling the mix and relative proportions of monomers in the recipe.FRP leads to the formation of statistical copolymers, where the arrangement of monomers within the chains is dictated purely by kinetic factors. However, reactiv-ity of a monomer in copolymerization cannot be predicted from its behavior in ho-mopolymerization. Vinyl acetate polymerizes about 30 times more quickly than styrene (see Table 4.2), yet the product is almost pure polystyrene if the two mono-mers are copolymerized together in a 50:50 mixture. a-Methylstyrene cannot be ho-mopolymerized to form high-MW polymer due to its low ceiling temperature (see Table 4.6), yet is readily incorporated into copolymer at elevated temperatures.These and other similar observations can be understood by considering copolymer-ization mechanisms and kinetics.
4.3.2.1 Basic Mechanisms
In copolymerization, the presence of more than one type of monomer adds an ex-tra degree of complexity to the kinetics. The different monomers form different radical structures, and the relative rates of chain growth depend on the structure of both monomer and radical. It is these propagation mechanisms that control polymer composition (the relative amounts of each monomer unit incorporated into the copolymer) and sequence distribution (the way in which these monomer units are arranged within the chain). Developing a set of mechanisms to describe how radical structure affects termination and transfer rates is required to represent copolymer chain length and molecular weight distributions.The most common treatment of free radical copolymerization kinetics assumes that the reactivity of the polymer radical depends solely on the nature of its termi-nal monomer unit; that is, that the identity of the penultimate unit on the radical does not affect its reactivity. This assumption provides a good representation of polymer composition and sequence distribution, but not necessarily polymeriza-

180 4 Free-radical Polymerization: Homogeneous tion rate (see Section 4.3.2.2). This so-called terminal model is widely used to model free-radical copolymerization according to the set of mechanisms in Scheme 4.14.
k
→2 f I ∗
ki
Chain Initiation I ∗ + M j 
→ P1 j
kp
Chain Propagation Pni + M j →
Pn +j 1
Chain Termination
ktc
By Combination Pni + Pmj →
Dn + m
ktd
By Disproportionation Pni + Pmj →
Dn + Dm
Chain Transfer
ktrmon
To Monomer Pni + M j → Dn + P1 j
ktrsol
Pni + S →
Dn + S *
To Solvent or Agent k isol S * + M j →
P1 j
Scheme 4.14. Basic free-radical copolymerization mechanism,assuming terminal radical kinetics.In this scheme monomer-j (denoted by Mj ) adds to the initiator primary radical to form a polymer radical of type-j and unit length. The dead polymer and radical-i chains of length n (Dn and Pni ) are made up of a mixture of the monomer types in the system, with their relative amounts governed by the copolymer composition equation developed below. Chain growth occurs by addition of Mj to radical-i (Pni )with the propagation rate coefficient k pij dependent on both radical and monomer type. The rate coefficients for transfer and termination reactions can also be depen-dent on the nature of the radical center, as indicated by subscripts. However, since radical–radical termination is a diffusion-controlled reaction, the rate coefficient
cop
can usually be assumed to be independent of radical type, such that kt ¼ k tij for all i and j.Most of the kinetic coefficients in Scheme 4.14 are binary parameters, dependent on the radical and monomer type. Thus the polymerization behavior of three or more monomers can be estimated reliably from knowledge of the corresponding binary copolymerizations. For a two-monomer system assuming the long-chain hy-pothesis, the consumption rates of the two monomers are written as in Eqs. (35).R pol1 ¼ k p11 ½Ptot ½M1  þ k p21 ½Ptot ½M1 
ð35Þ
R pol2 ¼ k p12 ½Ptot ½M2  þ k p22 ½Ptot ½M2 

4.3 Polymerization Mechanisms and Kinetics 181
where Ptot represents the concentration of all polymer radicals of type-i in the sys-
X
y
½Ptot ¼ ½Pni  ð36Þ
n¼1
The ratio of the two consumption rates dictates the instantaneous composition
(Fpinst
) of the polymer being formed [Eq. (37)].
Fpinst
R pol1 k p11 ½Ptot 2½M1  þ k p21 ½Ptot ½M1 ¼ ¼ 1 2 ½M 
ð37Þ
Fpinst
R pol2 k p12 ½Ptot ½M2  þ k p22 ½Ptot 2 Application of the quasi-steady-state assumption yields the ratio of the radical types as in Eq. (38), and substitution and rearrangement leads to the well-known poly-mer composition equation [Eq. (39)], where f1 and f2 are the mole fractions of M1 and M2 in the monomer mixture, and monomer reactivity ratios r1 and r2 are de-fined as k p11 =k p12 and k p22 =k p21 .½Ptot  k p21 ½M1 2  ¼ k ½M 
½Ptot
ð38Þ
p12 2
r1 f12 þ f1 f2 Fpinst ¼ ð39Þr1 f1 þ 2 f1 f2 þ r2 f22 Equation (39) defines the composition of the copolymer formed at any instant dur-ing polymerization, and is dependent only on the ratios of the propagation rate co-efficients and not on their absolute values.Copolymer properties depend on the distribution of the monomer units along the chain as well as the average composition. Reactivity ratios also control copoly-mer sequence distribution. The probability P11 that an M1 unit follows an M1 unit in the copolymer is equal to the rate of M1 M1 formation divided by the sum of the rates of all additions to radical-1 [Eq. (40)].k p11 ½Ptot ½M1  r1 f1 P11 ¼ 1 1 ½M 
¼ ð40Þ
k p11 ½Ptot ½M1  þ k p12 ½Ptot 2 r1 f1 þ f2 The probability P12 that an M2 unit follows an M1 is given by Eq. (41).
f2
P12 ¼ 1  P11 ¼ ð41Þ
r1 f1 þ f2
Similar expressions, Eqs. (42) and (43), can be derived for addition to radical-2:

182 4 Free-radical Polymerization: Homogeneous k p22 ½Ptot ½M2  r2 f2 P22 ¼ 2 2 ½M  ¼ r f þ f ð42Þk p22 ½Ptot ½M2  þ k p21 ½Ptot 1 2 2 1
f1
P21 ¼ 1  P22 ¼ ð43Þ
r2 f2 þ f1
These probabilities can be used to calculate NðM1 ; n i Þ, the fraction of all M1 se-quences that are exactly n i units long. This is simply the probability of havingðn i  1ÞM1 M1 linkages followed by an M1 M2 linkage, according to Eq. (44).n i 1 n 1 NðM1 ; n i Þ ¼ P11 P12 ; NðM2 ; nj Þ ¼ P22j P21 ð44ÞThus the fraction of M1 sequences that consists of an isolated M1 unit is P12 , the fraction that consists of isolated M1 M1 diads is P11 P12 , the fraction of triads is P11 P12 , and so forth. The number-average length of M1 sequences (N 1 ) is given by Eq. (45) and the fraction of all M1 units contained in a sequence of length n i is
n i 1 2 2
n i P11 P12 ; that is, the fraction of M1 contained in isolated diads is 2P11 P12 .N1 ¼ ¼ ; N2 ¼ ¼ ð45Þ1  P11 P12 1  P22 P21 Implicit in these expressions is the approximation, valid for long-chain polymer,that the number of M1 M2 linkages in a chain is equal to the number of M2 M1 link-ages. Thus the ratio of M1 to M2 units in the chain must equal the ratio of the re-spective average sequence lengths [Eq. (46)].
Fpinst
1 N1
¼ ð46Þ
Fpinst
N2
Substitution and rearrangement of this equation yields the polymer composition equation, Eq. (39). Thus it is possible to estimate reactivity ratios for binary copoly-mers from triad distributions measured by NMR analysis.While copolymer composition and sequence distribution are only functions of the reactivity ratios, the same is not true for polymerization rate. The overall rate of monomer consumption is given by Eq. (47), where [Mtot ] indicates the total monomer concentration (½M1  þ ½M2 ).R pol ¼ k p11 ½Ptot ½M1  þ k p21 ½Ptot ½M1  þ k p12 ½Ptot ½M2  þ k p22 ½Ptot ½M2 
XX2 2
¼ k pij fri fj ½Ptot ½Mtot  ð47Þ
i¼1 j¼1
The fraction of radical-i in the system, fri [Eq. (48)] can be calculated from Eq. (38).

4.3 Polymerization Mechanisms and Kinetics 183
fri ¼ 1 2
¼ tot ð48Þ
[Ptot ], the total radical concentration, is calculated from an overall radical balance similar to Eq. (9) and given by Eq. (49).



R init 1=2 2f kd ½I 1=2½Ptot  ¼ cop ¼ cop ð49Þ
kt kt
The form of Eq. (47) is analogous to the homopolymerization rate expression [Eq.
(10)], with a copolymer-averaged rate coefficient for propagation (generalized for a
system with Nmon different monomers) defined in Eq. (50).
Nmon X
X Nmon
k pcop ¼ k pij fri fj ð50Þ
i¼1 j¼1
For a two-monomer system, application of the QSSA [Eq. (38)] and simplification lead to Eq. (51).r1 f12 þ 2f1 f2 þ r2 f22 k pcop ¼ ð51Þðr1 f1 =k p11 Þ þ ðr2 f2 =k p22 ÞUsing Eqs. (47)–(51), the copolymerization reaction rate can be analyzed as for
cop
homopolymerization (see Section 4.3.1.4), with k p now a function of monomer composition.
4.3.2.2 Kinetic Coefficients
The traditional method for determining reactivity ratios involves determination of copolymer composition for a range of monomer feeds at very low conversion; that is, measuring Fp1 as a function of f1 . NMR measurement of sequence distributions provides additional information about chain microstructure, but suffers from greater experimental noise and signal assignment uncertainty from tacticity ef-fects. There is a large body of published r1 –r2 data for monomer pairings, summar-ized in Ref. 7. The scatter for these ratios is much less than found in k p and k t data, but care still must be exercised when extracting values from this and similar compilations. Some error can arise from the methodology used to estimate r1 and r2 from Fp1 versus f1 data. The estimation is best accomplished by nonlinear least squares techniques [48, 49], and a statistical analysis also provides a guide to the optimal monomer compositions at which experimentation should be performed to improve the estimates [48]. Error can also occur if polymer conversion is suffi-ciently high for f1 to deviate significantly from the zero-conversion value (in which case an integrated form of Eq. (39) must be used [50]), or if the experimental sys-tem does not remain homogeneous (often observed with acid monomers).

184 4 Free-radical Polymerization: Homogeneous Tab. 4.7. Monomer reactivity ratios at 50 C[a], r1 is tabulated horizontally and r2 vertically.Monomer-1 Monomer-2 Styrene Methacrylate Acrylate Vinyl acetate Styrene – 0.6 0.8 40 Alkyl methacrylate 0.4 – 2.2 20 Alkyl acrylate 0.2 0.4 – 6 Vinyl acetate 0.02 0.03 0.03 –[a] Representative values Ref. 7.There are only a few systems for which the terminal model upon which Eq. (39)is based does not provide a good description of copolymer composition. The result-ing polymers exhibit an alternating structure (for example, styrene–maleic anhy-dride), or the observed reactivity ratios in the system vary with monomer concen-tration or solvent choice. Since they often include a polar monomer with strong electron-withdrawing or electron-donating properties, alternative kinetic models that include the formation of monomer complexes have been developed to repre-sent these systems [3, 51]. The majority of systems, however, are well behaved and well represented by the terminal model. Table 4.7 summarizes values of reac-tivity ratios for styrene, alkyl methacrylate, alkyl acrylate, and vinyl acetate systems.Reactivity ratios for alkyl methacrylates and acrylates (for example, methyl, butyl,dodecyl) exhibit a family type behavior, with the composition data of various sys-tems fitted well by a single curve [52]. Values for ethylene copolymer systems at typical production conditions are summarized in Table 4.8: While r1 (1 ¼ ethylene)values for alkyl methacrylates, alkyl acrylates, acrylic acid, and methacrylic acid could not be distinguished within experimental error [53], there were significant differences in the r2 values [54].It is informative to consider some of the implications of these values. In Figure
4.4 the relationship between polymer and monomer composition is plotted for var-
ious copolymer systems:Tab. 4.8. Monomer reactivity ratios for ethylene (monomer-1)systems at 240 C and 2000 bar [53, 54].Monomer-2 r1 r2 Alkyl methacrylates 0.058 7 Methacrylic acid 0.058 11 Alkyl acrylates 0.058 4 Acrylic acid 0.058 8 Vinyl acetate [a] 1.0 1.0[a] From Ref. 7.

4.3 Polymerization Mechanisms and Kinetics 185
0.9
0.8
0.7
0.6
Fp 1
0.5
0.4
0.3
MMA-Sty
0.2 MMA-MA
MMA-VAc
0.1
Eth-VAc
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
f1
Fig. 4.4. Relationship between monomer composition f1 and corresponding instantaneous polymer composition Fpinst
for
various monomer pairings (MMA ¼ methyl methacrylate,Sty ¼ styrene, MA ¼ methyl acrylate, VAc ¼ vinyl acetate,Eth ¼ ethylene).
For the case where r1 ¼ r2 ¼ 1:0 (ethylene–vinyl acetate; methacrylate–methacrylate), the monomers have equal reactivity to propagation and will be incorporated into polymer at the same ratio as they are in the monomer phase(Fp1 ¼ f1 ).
Where r1 > 1 and r2 < 1, the copolymer will always be richer in monomer-1 than in the monomer phase, so that monomer-1 will become depleted in a batch poly-merization. The further the reactivity ratios deviate from unity, the greater the deviation between polymer and monomer composition. Systems that exhibit this behavior include methacrylate–acrylate polymerizations, and styrene, meth-acrylates, or acrylates polymerized with vinyl acetate or ethylene.
With both r1 and r2 less than unity (styrene–acrylate, styrene–methacrylate),cross-propagation is favored over homopropagation and the copolymer tends toward an alternating structure. The system has an azeotropic composition at which the copolymer composition is exactly equal to the monomer composition.Reactivity ratios exhibit a weak temperature dependence that is often difficult to measure. With increasing temperature, the ratios tend to approach unity as dem-onstrated for styrene–butyl acrylate [55], butyl acrylate–methyl methacrylate [56],and ethylene copolymer systems [53, 54]. The temperature dependence of the latter agrees well with activation energies reported for addition of monomers to small radicals [57].

186 4 Free-radical Polymerization: Homogeneous(L/mol-s) 1000.0
cop
kp
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
f 1 (MMA)
Fig. 4.5. Relationship between monomer composition f1 and corresponding copolymer-averaged propagation rate coefficient,
cop
k p . Lines are calculated assuming terminal-model kinetics and points are experimental data for: MMA–nBA (—, e) [62] and MMA–styrene (---, C) [59] at 20 C.While the terminal model effectively represents copolymer composition for most systems, it does not provide as good of a description of polymerization propaga-tion behavior. Rate abnormalities have previously been attributed to termination reactions and represented by introducing physically unrealistic cross-termination mechanisms [58]. Careful experimental work by Fukuda and co-workers [51], how-ever, demonstrated that the rate deviations are due to the inadequacy of the ter-minal model to describe propagation, with further evidence obtained by applica-
cop
tion of the PLP-SEC technique to measure k p . Many common systems show deviation from terminal propagation kinetics, including styrene–methacrylate [51,59], styrene–acrylate [60, 61], and acrylate–methacrylate [52, 62] systems. As
cop
shown in Figure 4.5, the measured k p values can be higher or lower than the ter-minal model predictions, with the deviation substantial in some cases.The ‘‘implicit penultimate unit effect’’ model, which accounts for the influence of the penultimate monomer unit of the growing polymer radical on the propaga-tion kinetics [51], provides a good representation of this behavior [Eqs. (52)].r1 f12 þ 2f1 f2 þ r2 f22
k pcop ¼
ðr1 f1 =k p11 Þ þ ðr2 f2 =k p22 Þ
ð52Þ
k p ½r1 f1 þ f2  k p ½r2 f2 þ f1 k p11 ¼ 111 k p22 ¼ 222 r1 f1 þ ½ f2 =s1  r2 f2 þ ½ f1 =s2 

4.3 Polymerization Mechanisms and Kinetics 187
The extra parameters s1 and s2 , called radical reactivity ratios, capture the effect of the penultimate unit on the addition rate of monomer [Eqs. (53)].s1 ¼ s2 ¼ ð53Þk p111 k p222 A value of greater than unity for si indicates that a comonomer unit-j in the penul-timate position increases the addition rate of monomer-i to radical-i compared to the homopolymerization case.The kinetics of diffusion-controlled termination in copolymerization is also diffi-cult to study. The original interpretation of low-conversion rate data was based on a chemically controlled model utilizing a cross-termination factor [Eq. (54)].
cop
Nmon X
X Nmon
k t 12
kt ¼ k tij fri frj ; F¼ ð54Þ
i¼1 j¼1
k t 11 k t22 Assuming terminal propagation kinetics, the best fit for F was found to be much greater than unity for styrene–methacrylate, styrene–acrylate, and other systems
cop
such that kt was greater than either homotermination value. When the deviation of propagation kinetics from the terminal model is taken into account, however,
cop
the estimates for kt become well behaved and bounded by the homotermina-tion values [51]. A penultimate model [51, 63] that accounts for the influence of polymer composition on segmental diffusion is required to fit recent acrylate–
cop
methacrylate low-conversion kt data [63, 64]. In order to use this representation(Eq. (55), with the cross-termination coefficients k t 12; 12 and k t21; 21 fitted to experi-mental data), it is necessary to track the four possible penultimate radical types in the system, with frij ði; j ¼ 1; 2Þ the penultimate radical fraction.
cop
ðkt Þ 0:5 ¼ kt0:5 f þ kt0:5
11; 11 r11
f þ kt0:5
21; 21 r21
f þ kt0:5
22; 22 r22
f
12; 12 r12
ð55Þ
While the terminal model and reactivity ratios provide a good description of copoly-mer composition, the kinetic studies summarized in this section indicate that ad-
cop
ditional parameters and penultimate radical fractions are required to represent k p
cop
and kt . (For a discussion of transfer to monomer in copolymer systems, see Ref.
65.) These mechanistic complexities are often not considered when developing
FRP models for polymer reaction engineering applications. It is expected that this situation will change as more data become available.
4.3.2.3 Additional Mechanisms
The secondary mechanisms presented for homopolymerization in Section 4.3.1.3 also occur in multi-monomer systems. The kinetics of depropagation, chain trans-fer to polymer, and chain scission are strongly influenced by not only the nature of the monomer and terminal radical, but also the penultimate unit on the polymer radical.

188 4 Free-radical Polymerization: Homogeneous
kp11
→→ ~~~ M 1M 1 .~~~ M 1 ⋅ + M 1 

kdep 11
~~~ M 1 ⋅ + M 2 → ~~~ M 1M 2 .~~~ M 2 ⋅ + M 2 → ~~~ M 2 M 2 .~~~ M 2 ⋅ + M 1 → ~~~ M 2 M 1 .Scheme 4.15. Depropagation in copolymerization for the case where M2 does not depropagate and M1 depropagates only when an M1 -unit is in the penultimate position.Depropagation It is not possible to produce high MW homopolymer close to the ceiling temperature of the monomer (see Eq. (25) and Table 4.6). Addition of a non-depropagating monomer (M2 ) to the system disrupts this behavior, as illus-trated by Scheme 4.15. Depropagation of radical-1 becomes a competitive process with addition of monomer-2, with monomer-1 units at the radical end becoming irreversibly trapped in the growing chain as soon as monomer-2 adds. Further-more, depropagation of monomer-1 will occur only if an M1 unit is also in the pen-ultimate position; depropagation to the less-stable M2 radical can be assumed to be unlikely (kdep21 A 0). Thus MMA can be successfully copolymerized with ethylene at 290 C [57], and a-methylstyrene with various comonomers at temperatures well above its ceiling temperature [66, 67].Depropagation in copolymerization affects polymer composition, sequence prob-abilities, and polymerization rate. With Scheme 4.15, the relative rates of monomer consumption are given by Eq. (56) (compare with Eq. (37) in the absence of depro-pagation).
fr11
Fpinst
k p11 ½Ptot 2½M1  þ k p21 ½Ptot ½M1   kdep11 ½P 1 R pol1 fr11 þ fr21 tot¼ ¼ 1 ½M  þ k ½P 2 ½M 
ð56Þ
Fpinst
R pol2 k p12 ½Ptot 2 p22 tot 2 The ratio fr11 =ð fr11 þ fr21 Þ accounts for the fraction of radical-1 that ends in a 11 diad. The effect of depropagation becomes larger as total monomer concentration decreases and as the fraction of the depropagating monomer in the system in-creases. Lowry [68] first derived the composition expressions for the situation where only one monomer depropagates, and general expressions have been devel-oped for the situation where all four of the propagation reactions are reversible[66, 69]. Depropagation must be considered when examining the kinetics of starved-feed semi-batch copolymerization involving methacrylates at the higher-temperature conditions typically used to produce acrylic coatings [29, 70].Chain transfer to polymer The rate of chain transfer to polymer is dependent on both the reactivity of the radical and the abstractability of the hydrogen atom on the monomer unit in the polymer chain. For the case of intermolecular chain transfer (long-chain branching), this is represented by Eq. (57), where m1 repre-sents the total concentration of polymerized monomer-j units in the system and

4.3 Polymerization Mechanisms and Kinetics 189
nj represents the number of monomer-j units on a particular chain of length n
pol
mj k tr
ij j pol
Nmon X
X Nmon
pol j
Pn;i b þ Dm; c ! Dn; b þ Pm; cþ1 ; R tr ¼ i k trij ½Ptot m1 ð57Þ
i¼1 j¼1
X
y X
y X
Nmon
m1 ¼ nj ½Dn; b ; n¼ nj ð58Þn¼1 b¼0 j¼1 Active radicals (ethylene, acrylate, vinyl acetate) are more likely to abstract from a polymer chain than styrenic or methacrylate radicals, and acrylate and vinyl acetate monomer units on a chain are more likely to have an H-atom abstracted. Thus it is not uncommon for the intermolecular transfer to polymer rate of one pairing (for example, acrylate radical to acrylate monomer unit) to dominate the system, with
pol
the overall transfer rate R tr decreasing rapidly with increasing content of the less-reactive monomer.The situation is more complicated in the case of intramolecular transfer, which occurs through the formation of a six-membered ring. In the case of acrylate (1)/methacrylate (2), it can be assumed that the methacrylate radical is not reactive enough to back-bite and that the acrylate radical can only abstract hydrogen if the antepenultimate unit on the chain is also an acrylate unit. Thus back-biting can oc-cur only for two monomer sequences (M1 M1 M1 and M1 M2 M1 ) at the radical end,as shown in Scheme 4.16 [70]. The overall back-biting rate must be corrected for the sequence probabilities [Eqs. (40)–(43)] at the chain end, according to Eq. (59).COOBu COOBu COOBu COOBu COOBu
COOBu
C C CH
C C CH
R H
R H
HC
CH
COOBu
COOBu
COOBu COOBu COOBu COOBu COOBu COOBu
C C C C
H3C C C H3C
R H R
HC H
CH
COOBu COOBu Scheme 4.16. Back-biting reaction in high temperature copolymerization of butyl acrylate and butyl methacrylate(R ¼ H or CH3 ).

190 4 Free-radical Polymerization: Homogeneous R bb ¼ kbb ½Ptot ðP11 P11 þ P12 P21 Þ ð59ÞBack-biting mechanisms have also been examined for styrene/acrylate [71] and ethylene copolymer systems [43].Chain scission There is evidence that the rate of b-fragmentation of a mid-chain radical is affected by the nature of the units adjacent to the mid-chain radical. Har-ada et al. [72] studied the copolymerization of cyclohexyl acrylate and methyl meth-acrylate at high temperatures. As in Scheme 4.16, H-abstraction only happens to the acrylate unit. If the adjacent units are acrylate and methacrylate, two types of fragmentation with different reaction rates are possible, as illustrated in Scheme
4.17. It was observed that the number of the unsaturated end groups per chain is
increased by increasing the methacrylate content in the polymerization. Thus it was deduced that fragmentation greatly favors the generation of tertiary (methacry-late chain end) radical species, a result in agreement with the high rate of fragmen-tation observed in methacrylate macromonomer systems [73].
CH3
CH3
C C HC fast H2C CH2 CH2 CH2 CH2
CH2
+ C HC
CO2CH3 CO2CH3 CO2CH3 CO2CH3 CO2CH3 CO2CH3
CH3
slow
C C
CH2 CH2
CH2
+ HC
CO2CH3 CO2CH3
CO2CH3
Scheme 4.17. Fragmentation of a mid-chain radical with adjacent MA and MMA units (after Ref. 72).Fragmentation after intramolecular transfer results in the formation of a long-chain radical and trimer species or a dimer radical and an unsaturated dead chain(Scheme 4.13). Consideration of all possible pathways and structures becomes complex, but the resulting model requires no additional parameters from the ho-mopolymerization back-biting/scission case and provides a good representation of high temperature acrylate–methacrylate copolymerization [70].
4.3.3
Diffusion-controlled Reactions The kinetic schemes in this chapter have been written assuming that k t is indepen-dent of the sizes of the radicals involved in the termination reaction. This is not

4.3 Polymerization Mechanisms and Kinetics 191
1.0E+09
1.0E+08
1.0E+07
kt (L/mol-s)
1.0E+06
1.0E+05
1.0E+04 MMA
nBA
1.0E+03
DA
1.0E+02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Conversion
Fig. 4.6. Typical variation of k t with conversion for methyl methacrylate (MMA), butyl acrylate (nBA), and dodecyl acrylate(DA). Based upon data from [15].strictly true, since the termination rate is limited by the rates at which the radical ends can encounter each other. For a diffusion-controlled reaction, the apparent rate coefficient is affected not only by pressure and temperature, but also by system viscosity (a function of solvent, polymer concentration, and MW) and the lengths of the two terminating radicals. This complex behavior, as well as experimental dif-ficulties in measuring k t , has led to a large scatter in reported values, even at low conversion [7]. Through the application of pulsed-laser experimental techniques[15] and a critical examination of available data [18], the situation is starting to improve.For most commercial free radical polymerization, the errors involved by neglect-ing the dependence of k t on radical chain length are not large. The change in k t with conversion (increasing viscosity), however, cannot be neglected. Figure 4.6 shows the three to four orders of magnitude decrease in k t typically observed. The shape of the curve reflects the changing nature of the rate-controlling diffusion mechanism. The usual division is as follows:
Low conversion: the system viscosity is still low, and the two chains diffuse to-gether quickly. The rate of reaction is controlled by segmental diffusion, the in-ternal reorganization of the chain that is required to bring the reactive ends to-gether. In this region k t is of the order of 10 8 L mol1 s1 for many common monomers (Table 4.3), with the value remaining relatively constant up to 10–20% conversion. Solvent choice can have a significant effect on the value [18].

192 4 Free-radical Polymerization: Homogeneous Lower values of 10 6 L mol1 s1 for dodecyl (meth)acrylate is attributed to shield-ing of the radicals by the long-chain dodecyl ester groups; for these monomers k t remains relatively constant up to 60% conversion [64]. Even lower k t values are measured for termination during polymerization of sterically hindered mono-mers such as the itaconates [74] and acrylate trimer species [44]. The variation of low-conversion k t with polymer composition in copolymer systems can be rep-resented by Eq. (55).
Medium to high conversion: the large increase in system viscosity with polymer formation leads to a change in the controlling mechanism. The rate of reaction is controlled by how quickly the two chains find each other among the tangle of dead polymer chains in the system. This so-called center-of-mass or translational diffusion mechanism is complex, affected by the lengths of the reacting chains as well as the system viscosity. The value of k t can drop by several orders of mag-nitude in this regime.
Very high conversion: at high conversion, the system may become so viscous that the polymer radicals move more quickly through propagation (addition of new monomer units) than by translation. This phenomenon, called reaction diffu-sion, leads to a second plateau region in the k t versus conversion plot, with k t proportional to k p ð1  x p Þ. If the glass transition temperature of the reaction mixture exceeds the reaction temperature, the propagation reaction and apparent initiator efficiency [75] may also become diffusion-controlled.The overall behavior of k t with conversion is often modeled as a composite of the various diffusional processes [Eq. (60)].
¼ þ ð60Þ
k t k t; SD k t; TD þ k t; RD The subscripts SD; TD, and RD refer to segmental diffusion, translational diffu-sion, and reaction diffusion. k t; SD can be set to the low conversion values summar-ized in Table 4.3, and k t; RD is set proportional to propagation, with proportionality coefficient CRD fitted to experimental data [Eq. (61)].k t; RD ¼ CRD k p ð1  x p Þ ð61ÞMany semi-empirical approaches have been used to model k t; TD as a function of system viscosity, conversion, or free volume. The latter treatment relates the chain diffusivity to the system free volume vf by Eq. (62), with parameters Ci fitted to ex-perimental data.k t; TD z C1 expðC2 vf Þ ð62ÞThis effect of polymer MW on system viscosity may be captured by expressing C1 as a function of Mw . It is observed experimentally that addition of a powerful chain-transfer agent to MMA lowers polymer MW and system viscosity, thereby in-

4.4 Polymer Reaction Engineering 193
0.9
0.8
0.7
Conversion
0.6
0.5
0.4
0.3
0.2
MMA
0.1
nBA
0 500 1000 1500 2000
time (s)
Fig. 4.7. Typical time–conversion plots for methyl methacrylate and butyl acrylate batch polymerizations. The sharp increase in rate seen for MMA, known as the gel effect, is due to the large decrease in k t with conversion.creasing k t , while addition of a small amount of EGDMA crosslinking agent has the opposite effect [39]. Details and variations of this modeling approach can be found in the literature [40, 76–79].A good model for k t is necessary to capture the time–conversion behavior in ho-mogeneous batch FRP systems. The large decrease in k t at intermediate conver-sion results in an increase in radical concentration [Eq. (9)] and a corresponding increase in R pol [Eq. (10)] that causes an upward curvature in the time–conversion plot (Figure 4.7). This accelerated rate is accompanied by a large heat release that can be difficult to remove from the viscous reaction system. The severity of the gel effect is directly related to the magnitude of the decrease in k t , as seen by compar-ing the nBA and MMA rate profiles in the figure. The decrease also leads to an in-crease in DPninst [Eq. (13)] for systems where MW is controlled by termination.
4.4
Polymer Reaction Engineering The design of an industrial polymerization process begins with a clear under-standing of objectives and an appreciation of constraints. Design and operation requirements are very different for a process manufacturing several grades of a high-volume commodity homopolymer, and one that produces dozens of differ-

194 4 Free-radical Polymerization: Homogeneous ent (and often new) low-volume higher-value products of varying composition and structure. Typical product specifications for a homogeneous FRP process may include average molecular weight/molecular weight distribution, copolymer composition/copolymer composition distribution, degree of branching/branching distribution, and sequence length distribution. Depending on the nature of the product, any of these properties can simultaneously be product specifications.However, the polymer is ultimately not sold on basic structural characteristics but rather on end-use properties. This poses the challenge of relating structural fea-tures to properties. Invariably the end-use properties are a product of not one but several structural features; therefore establishing relationships is a complex task,and the ensuing relationships are usually restricted to a narrow range of materials.Establishing structure–property relationships remains an active area of research.Together with an understanding of the key properties and their desired values is a need to understand quantitatively how much variation is acceptable for each property. In an industrial polymerization environment, there will naturally be some degree of process variability that will translate into product variability. Know-ing the extent to which deviations from the target value of a property affect the manufacturer’s ability to sell that product is a critical piece of design information.Unlike many chemical systems, off-spec polymeric material cannot be easily re-cycled or altered by downstream unit-operations. The difficulty of characterizing polymer structure on-line makes design of a robust easy-to-operate process espe-cially important.Design of a process involves several decisions such as the type of reactor used,the flow and contacting patterns for the reagents, the choice of homogeneous ver-sus heterogeneous process types, and so on. Heat removal and mixing issues are two key factors that strongly influence design and operation of homogeneous FRP processes.Heat removal Free-radical polymerizations are highly exothermic, with adiabatic temperature rises for bulk monomers typically @200–500 C. (Values of DHp for common monomers are tabulated in Table 4.6.) In addition, the overall activation energy for a radical polymerization, calculated from the activation energies of initiation, propagation, and termination [see Eq. (10)] is on the order of 80 G 15 kJ mol1 . The sudden and dramatic increase in the heat produced in the gel-effect region can result in loss of effective temperature control. If there is a process dis-turbance leading to a thermal runaway condition, the heat generation rate can ex-ceed the heat removal rate to such an extent that the reaction behaves close to adia-batically. The resulting temperature increase can pose serious safety concerns such as reactor overpressurization and possible explosion, requiring processes to be designed to safely release the pressure prior to failure (rupture) of the process equipment.Heat removal is also an important issue to consider during process scale-up. As reactor size increases, the system dynamics become increasingly slow, and there-fore it takes longer for desired changes (for example decreasing the reactor temper-ature) to occur. This may in itself not be a serious problem, provided the reactor

4.4 Polymer Reaction Engineering 195
cooling system is able to maintain control, albeit at a higher than desired tempera-ture. However, large variations in the temperature profile can translate into differ-ences in the final product properties, particularly molecular weight distribution.The monograph by Biesenberger and Sebastian [80] provides an excellent discus-sion of thermal effects, including thermal runaway.Mixing Mixing can directly affect the kinetics, molecular weight, and composition in radical polymerizations by homogenizing local concentration gradients, but it can also indirectly play an important role through its role in reducing thermal gra-dients in a reactor. In small-scale experiments, most transport phenomena (heat transfer, mixing) occur sufficiently fast for overall behavior to be dictated primarily by reaction kinetics. However, as scale increases, kinetic and transport effects be-come increasingly coupled. Homogeneous FRP reactions offer a particularly chal-lenging problem because of the large increase in viscosity accompanying the conversion from bulk monomer (@1 cp for liquids) to polymer (> 10 5 cp). The in-crease in viscosity can greatly affect the reaction kinetics (see Section 4.3.3) as well as the heat removal and quality of mixing in the system. Some processes are de-signed to not require mixing. For example, PMMA can be polymerized in large sheets. By having large surface areas available for heat transfer, adequate tempera-ture control is achieved without the need to provide mixing during polymerization.Other bulk polymerizations (for example, styrene) employ more than one reactor in series, since different reactor configurations and agitators will be required as the viscosity increases. Solution polymerizations offer low viscosity but the trend in industry is to eliminate (or greatly reduce) the use of organic solvents. Mixing can also be an issue at lower viscosities such as those found in LDPE systems,where the high-temperature conditions make for very fast reactions (for example,initiator half-life of <1 s) on the order of characteristic mixing times.
4.4.1
Types of Industrial Reactors There are three major classifications of chemical processes, categorized by the method by which the reactants are added to the reaction vessel. Varying the contacting pattern can dramatically alter the local reaction conditions (for exam-ple, concentrations of individual species, including monomers, initiators, chain-transfer agents, and so on), and is therefore a potentially powerful design tool for controlling properties such as molecular weight distribution, copolymer composi-tion distribution, and degree of branching. Because of the ability to manipulate lo-cal monomer concentrations, the rate of polymerization can also be controlled,thereby providing safer and more robust operation.
4.4.1.1 Batch Processes
All of the reactants are added to the reactor prior to starting the polymerization. No material is added to or removed from the reactor during operation. When the poly-merization is complete, the contents are discharged and the reactor prepared for

196 4 Free-radical Polymerization: Homogeneous the next batch. Batch polymerizations are the simplest to run, but offer the least control over the polymerization. For polymerizations with more than one mono-mer, the relative consumption rates of the different monomers will be governed by their respective reactivities, possibly resulting in broad copolymer composition distributions and inhomogeneous product. Another feature of batch polymeriza-tions is that reactant concentrations change throughout the polymerization. Molec-ular weight distribution drift is therefore a common phenomenon and can lead to very broad distributions in the final product. From an economic perspective, batch polymerizations suffer from downtime between batches, although much progress has been made in automating many of the reactant weighing, charging, and dis-charging steps to minimize this interval. Automation has also improved reproduci-bility of batch reactions. For operations where changes to the formulation or the polymerization conditions are common, batch processes have the advantage of be-ing flexible and readily adaptable to new products.
4.4.1.2 Semi-batch Processes
These processes (also called semi-continuous) are similar to batch processes, ex-cept that reactants can be added and/or products removed during the polymeriza-tion. Usually only a portion of the total reactant charge is initially fed into the reac-tor. The polymerization is then started, and reactants are added during reaction in order to control a desired property (for example molecular weight distribution, co-polymer composition distribution) or the reaction rate. Any reactant can be fed,and it is common practice to add monomer(s), initiators, and/or chain-transfer agents. Two of the most common applications for semi-batch operation in homo-geneous FRP are control of copolymer composition distribution and control of re-action rate. In a batch reaction, copolymer composition drifts according to the in-herent reactivities of the monomers. However in semi-batch operation, drift can be substantially reduced by maintaining a (near) constant concentration ratio of the respective monomers in the reactor. Production of low molecular weight co- or ter-polymers (for example coatings) is also readily done using this type of approach.Initiator and monomers are continuously fed in the desired ratio to provide com-position and MW control while maintaining starved conditions (low monomer con-centration) in the reactor. This mode of operation also ensures that at any time the monomer concentration in the reactor is low, and therefore the maximum poten-tial hazard in the event of a thermal runaway reaction is minimized. A potential concern with operating in a starved mode is that polymer concentrations are high,resulting in higher rates of transfer to polymer and branching reactions.
4.4.1.3 Continuous Processes
Reactants are fed, and products and unconsumed reactants are removed, continu-ously. The process may take place in a single reactor or in a train (series) of reac-tors in which the monomer conversion is gradually increased. Most continuous processes are operated at ‘‘steady-state’’ conditions, meaning all reactant concentra-tions and process conditions (temperature, pressure, and so on) are time-invariant.This can be an enormous advantage for certain types of properties if the reactor is

4.4 Polymer Reaction Engineering 197
also well mixed (no spatial variations); because concentrations are constant, there is no molecular weight distribution drift, and no composition distribution drift.Thus, narrower molecular weight distributions can be produced in a well-mixed continuous reactor for linear polymer systems than in batch systems. Branching reactions, however, broaden the MWD in a well-mixed continuous reactor to a greater extent than in a batch system due to the residence time distribution [80,81]. For large-volume polymers with a limited number of variations to the polymer-ization conditions (for example, formulation changes), continuous processes are favored because of their low operating cost, high throughput rates, more uniform product quality, and simplicity of operation.
4.4.2
Mathematical Modeling of FRP Kinetics Mathematical modeling is a powerful methodology to improve the understanding and operation of polymer processes. A good process model can be used to predict the influence of operating conditions on reaction rate and polymer properties, to guide (along with appropriate experimentation) the selection and optimization of standard operating conditions for existing and new polymer grades, to guide pro-cess development from lab to pilot-plant to full-scale production, to help discrimi-nate between kinetic and physical (for example, heat and mass transfer) effects, to perform design and safety studies, to train plant personnel, and to understand and optimize transitions and other dynamic behavior (that is, process control).The modeling approach and level of detail should be dictated by the application.Whereas an empirical model linking measured inputs and outputs may be the best solution for control of an existing industrial reactor, it would be totally inappropri-ate for design of a new process or to choose operating conditions for a new poly-mer grade. On the other hand, it makes little sense to develop a model that can predict detailed polymer architecture for control purposes when the only measure of polymer structure is a melt index value obtained from the lab two hours after the sample was produced. While empirical modeling has its uses, the focus of this section is the development of fundamental models based upon first-principles descriptions of chemical and physical phenomena. Although a perfect description of an actual process is, in the end, an unattainable goal, the attempt often leads to valuable insights that can aid process and product development, scale-up, and opti-mization. Techniques to model the FRP mechanisms and kinetics of Section 4.3 are presented first, followed by a discussion of issues related to reactor modeling.Wherever possible, examples have been selected to emphasize industrial applica-tion. Although a few historical references are included, the main focus is on advan-ces made since the mid-1990s.The objective of kinetic modeling is to build a description of how polymer archi-tecture and polymerization rate depend on reaction conditions (temperature, pres-sure) and species concentrations from a defined set of kinetic mechanisms; a dy-namic model is required to examine how properties change as a function of time.The mechanisms to be included in a model depend upon its end-use. For simple

198 4 Free-radical Polymerization: Homogeneous mass and energy balances, it is only necessary to consider those that consume monomer, initiator, and radicals – initiation, propagation, and radical–radical ter-mination. To track polymer molecular weight, all mechanisms that include radical transfer must also be included. Additional balances are needed to follow other mo-lecular properties, such as the density of short- or long-chain branches, end-group functionality, and the creation and reaction of terminal double bonds.
4.4.2.1 Method of Moments
The equations in Section 4.3, while useful for examining rate of polymerization and instantaneous chain length, are not written in an appropriate form for substi-tuting into a generalized reactor model. In addition, they do not provide a means of tracking higher MW averages that are strongly affected by long-chain branching reactions. One of the challenges in modeling polymerization systems is how to re-duce a very large number of individual species (living and dead chains with lengths from 1 to >10 5 , often with other distributed attributes such as the number of branch points) to a tractable solution. The classical approach to this problem is to reduce the system of equations through definition of the principal moments of the various distributions [82]. Construction of moment balances allows the tracking of average polymer properties: for molecular weight this would be Mn(number-average), Mw (weight-average), and possibly Mz , and for branched sys-tems it is possible to track the number-average (B n ) and weight-average (B w ) num-ber of branches per chain. Only the basics of the mathematical treatment for a ho-mopolymerization system will be given here: more details can be found in recent comprehensive reviews [83, 84].Consider a free radical system that includes the basic set of mechanisms shown in Scheme 4.3 as well as long-chain branching [Eq. (27)]. The moments for the rad-ical (l j ) and dead (zj ) polymer distributions are defined in Eqs. (63) and (64).
X
y
lj ¼ n j ½Pn  ð63Þ
n¼1
X
y
vj ¼ n j ½Dn  ð64Þ
n¼1
It is also helpful to define moments for the bulk polymer (mj ), the total polymer in the system including live radicals [Eq. (65)].
X
y
mj ¼ n j ð½Dn  þ ½Pn Þ ð65Þ
n¼1
With ½Dn  g ½Pn  there is little difference in magnitude between mj and zj . Its intro-duction, however, eliminates the moment closure problem created by the LCB mechanism [40, 81, 85]. Many of the moments have precise physical meanings.

4.4 Polymer Reaction Engineering 199
The zeroth live moment, l 0 , is the concentration of polymer radicals in the system(denoted by [Ptot ] in Section 4.3), and the first live moment, l1 , is the concentration of monomer units contained in all growing radicals. Similarly, m 0 is the concentra-tion of all polymer chains in the system, and m1 is the concentration of monomer units bound in all polymer chains. These moment definitions collapse the infinite set of equations for polymeric species into a manageable subset used to calculate MW averages, where m w is the molecular weight of the monomeric repeat unit
m1 m2 m3
Mn ¼ m w ; Mw ¼ m w ; Mz ¼ m w ð66Þ
m0 m1 m2
For this example, equations for the kinetic expressions for m 0 ; m1, and m 2 will be developed for the calculation of Mn and Mw .The first step is to formulate balances for live radicals, dead chains, and total chains of length n, accounting for all of the consumption and generation terms from the kinetic mechanisms [Eqs. (67)–(69)].
X
y X
y
RPn ¼ 2f kd ½I þ ktrmon ½M sol½Pj  þ ktr ½S ½Pj  dðn  1Þ
j¼1 j¼1
þ k p ½Mð½Pn1   ½Pn Þ
X
y
 ktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þ ½Pj  ½Pn 
j¼1
pol
X
y
pol
X
y
þ k tr n½Dn  ½Pj   k tr ½Pn  j½Dj  ð67Þ
j¼1 j¼1
Xy
1 X n1
R Dn ¼ ktrmon ½M þ ktrsol ½S þ k td ½Pj  ½Pn  þ k tc ½Pj ½Pnj 
j¼1
2 j¼1
pol
X
y
pol
X
y
þ k tr ½Pn  j½Dj   k tr n½Dn  ½Pj  ð68Þ
j¼1 j¼1
X
y X
y
RPn þDn ¼ 2f kd ½I þ ktrmon ½M sol½Pj  þ ktr ½S ½Pj  dðn  1Þ
j¼1 j¼1
þ k p ½Mð½Pn1   ½Pn Þ
Xy
1 X n1
 k tc ½Pj  ½Pn  þ k tc ½Pj ½Pnj  ð69Þ
j¼1
2 j¼1
The origin of the various terms in these balances should be evident by looking at the mechanisms of Scheme 4.3 and Eq. (27). The Kronecker delta function[dðxÞ ¼ 1 if x ¼ 0 and sðxÞ ¼ 0 if x 0 0 ] accounts for the generation of new poly-meric radicals (P1 ), and the terms for transfer to polymer account for the probabil-

200 4 Free-radical Polymerization: Homogeneous ity that transfer to a certain chain Dn is proportional to chain length n. The expres-sion for termination by combination accounts for the possibility of creating Dn from any combination of two smaller radical fragments whose lengths sum to n.The next step in the procedure is to substitute these species balances into the moment definitions in Eqs. (63)–(65). The use of generating functions [82, 86, 87]eliminates the tedium (and possible errors) of performing the required series sum-mations, and leads to Eqs. (70)–(78) for the moments.Rl 0 ¼ 2 f kd ½I  ðk td þ k tc Þl02 ð70ÞRl1 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Ml 0
pol
 fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl1 þ k tr ðl 0 v2  l1 v1 Þ ð71ÞRl2 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Mðl 0 þ 2l1 Þ
pol
 fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl2 þ k tr ðl 0 v3  l2 v1 Þ ð72ÞDead moments:R v0 ¼ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k td l02 þ k tc l02 ð73Þ
pol
R v1 ¼ fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl1  k tr ðl 0 v2  l1 v1 Þ ð74Þ
pol
R v2 ¼ fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl2 þ k tc l12  k tr ðl 0 v3  l2 v1 Þ ð75ÞBulk moments:Rm 0 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0  k tc l02 ð76ÞRm1 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Ml 0 ð77ÞRm 2 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Mðl 0 þ 2l1 Þ þ k tc l12 ð78ÞThe set of moment expressions to be considered consists of either the live and dead moments [Eqs. (70)–(75)] or the live and bulk moments [Eqs. (70), (71) and (76)–
(78)], substituting m1 and m 2 for z1 and z2 in Eq. (71). Choice of the former, while it
is common practice in the literature, suffers from the problem that the equations for l2 and z2 depend on z3 . The Hulburt and Kutz [88] method is often used to ap-proximate z3 , assuming that the molecular weight distribution can be represented by a truncated series of Laguerre polynomials by using a gamma distribution weighting function [Eq. (79)].
v2
v3 ¼ ð2v0 v2  v12 Þ ð79Þ
v0 v 1

4.4 Polymer Reaction Engineering 201
Using the bulk moments not only eliminates the need for this approximation, but also reduces the set of equations by one, since Eq. (72) is not required to solve for Mw . An additional balance [Eq. (80)] can be added to either set of equations to track the concentration of LCB formed by the transfer to polymer mechanism,RLCB ¼ k tr l 0 m1 ð80ÞThus, a set of six equations [Eqs. (70), (71), (76)–(78), and (80)] can be used to col-lapse the molecular weight distribution into its principle averages to calculate Mn ,Mw , and LCB density.The set of moment equations developed here can be extended to include addi-tional complex mechanisms, such as reactivity of terminal double bonds [Eq. (30),Scheme 4.10] [81], crosslinking (Scheme 4.11) [40], and chain b-scission following intermolecular H-abstraction [89]. The methodology can also be extended to co-polymerization systems, either by defining copolymer-averaged rate coefficients as in Eqs. (50) and (54) [6, 85–87], or by defining additional moment quantities (for examples, see Refs. 40, 81, 85). Furthermore, since it is easy to implement as part of larger-scale reactor modeling, it is the standard methodology used in process simulation packages (see, for example, Refs. 90, 91). For discussion regarding thefinal step of model development, substitution of the kinetic expressions for the mo-ments into reactor balances, see Section 4.4.3.
4.4.2.2 Modeling of Distributions
The major limitation of models based on the method of moments is that they only track average quantities. While adequate for most situations, more detail may be needed if the objective of the study is to improve our knowledge of the kinetics– for example, to examine the combined effect of chain-scission and long-chain branching on polymer architecture, or to incorporate chain-length dependent ter-mination kinetics into the mechanistic scheme. In such cases, the kinetic and modeling assumptions can be tested more rigorously through a detailed compari-son with full molecular weight distributions (MWDs) measured experimentally.Recent advances in modeling tools now make it possible to simulate the complete MWD, as well as how a second distributed quantity (for example, LCB) varies with chain length.The modeling of complete MWDs has long been possible for linear polymer sys-tems, that is, those without any branching. In the absence of long-chain branching and employing the QSSA, a recursive relationship can be derived for Pn and Dn from Eqs. (67) and (68), leading to Eqs. (81) and (82) for the weight-fraction distri-bution of polymer formed at any instant in time [92].

n
wninst ¼ ðt þ 0:5ðn  1Þbðt þ bÞÞðt þ bÞn ð81Þ
1þbþt
k td ½l 0  þ ktrmon ½M þ ktrsol ½S k tc ½l 0 t¼ ; b¼ ð82Þk p ½M k p ½M

202 4 Free-radical Polymerization: Homogeneous Assuming the formation of long-chain polymer (t þ b f 1), the values for instanta-neous DPn and DPw are given by Eqs. (83).DPninst ¼ ; DPwinst ¼ ð83Þt þ 0:5b ðt þ bÞ 2 The instantaneous PDI is 2 for a system where there is no termination by combi-nation (b ¼ 0), and 1.5 if termination by combination is the only chain-ending mechanism (t ¼ 0). These expressions can be integrated to follow the evolution of the MWD with time or conversion and compare against experimental data, as was done by Balke and Hamielec [92] for batch isothermal FRP of MMA.Unfortunately, the methodology cannot be easily extended to branched systems,due to the interaction of the polymer radical and dead polymer chain distributions through reaction (for example, H-abstraction, terminal double bond polymeriza-tion, crosslinking). The methods for modeling MWDs with branching can be di-vided into three main groups. The first, utilizing Monte Carlo techniques, has been greatly advanced through the efforts of Tobita. Assuming a given set of mechanisms, the probabilities of connections between primary polymer molecules(the linear chain that would exist if all of its branch points were severed) is calcu-lated, and the resulting MWD solved using Monte Carlo techniques [93].A second group of models is based upon the ‘‘numerical fractionation’’ concept developed by Teymour and Campbell [94]. This seminal work identifies a succes-sion of branched polymer generations based on the degree of complexity of their architecture, tracking the population of chains in each generation using the method of moments. The complete MWD is approximated by combining the MWDs for individual generations which themselves are reconstructed from the leading moments assuming a distributional form. Numerical fractionation was specifically developed to examine the problem of gel formation in polymer sys-tems. Thus, the generations were defined to follow the geometric progression in chain length caused by connection of two molecules in the same generation: while chains from the zeroth generation progress to the first generation by participating in a branching event, a chain from the first generation can only progress to the sec-ond by joining (through crosslinking, terminal double bond polymerization, or ter-mination by combination) with another molecule from the same generation [94]. It has been shown that this classification scheme leads, in certain cases, to errors in the shape of the overall MWD: through comparison with distributions calculated by rigorous numerical solution, Butté et al. [95] have shown that the definition of generations proposed by Teymour and Campbell can create an artificial high MW shoulder due to the accumulation of chains with a wide distribution of the number of branches (and thus high polydispersity) in the first branched generation. The authors conclude that a more accurate approximation is obtained by classifying the chains according to their number of branches. Both Monte Carlo [96] and a modified numerical fractionation technique [95] can also be used to calculate the LCB number as a function of chain length, an important quantity often presented experimentally.

4.4 Polymer Reaction Engineering 203
The commercial software Predici2 package uses yet another numerical tech-nique, calculating MWDs using a discrete Galerkin technique with variable grid and variable order [97]. In 2000, the package was extended to follow branch point concentrations as a function of chain length through the introduction of balance equations [98]. The possibility of performing these tasks – calculation of complete MWD as well as LCB distribution – in a commercial software package is especially noteworthy because it makes it possible for a wider range of practitioners to per-form detailed kinetic modeling. These new modeling capabilities, in combination with improved characterization techniques, will hasten progress to a better under-standing and representation of complex polymer architecture [46, 99].
4.4.3
Reactor Modeling The kinetic expressions of Section 4.4.2 are substituted into overall material and energy balances to construct a model to represent an FRP process. For a well-mixed reactor system, Eqs. (84)–(89) comprise the general system of equations for homopolymerization.
dðV½MÞ
¼ ðqin ½Min Þ  ðqout ½MÞ  VR pol ð84Þ
dt
dðV½IÞ
¼ ðqin ½Iin Þ  ðqout ½IÞ  VR d ð85Þ
dt
dðV½SÞ
¼ ðqin ½Sin Þ  ðqout ½SÞ  VRtrsol ð86Þ
dt
dðV½l i Þ
¼ ðqin ½li; in Þ  ðqout ½l i Þ þ VRl i ð87Þ
dt
dðV½m i Þ
¼ ðqin ½m i; in Þ  ðqout ½m i Þ þ VRmi ð88Þ
dt
dðVrc p ðT  Tref ÞÞ¼ ðqin ðrc p Þin ðTin  Tref ÞÞ  ðqout ðrc p Þout ðT  Tref ÞÞ
dt
þ ðDHp ÞVR pol  Q ð89ÞIn these balances, which must be accompanied by appropriate initial conditions and a volume balance, V is the reactor volume, qin and qout are the inlet and outlet volumetric flow rates, r and c p are the density and heat capacity of the mixture, and Q is the rate of heat removal from the reactor system. The various source terms have been presented earlier: Eq. (10) for the monomer and energy balance, Eq. (2)for the initiator balance, Rtrsol ¼ ktrsol ½Sl 0 in the solvent balance, Eqs. (70) and (71)for the live moment balances, and Eqs. (76)–(78) for the bulk moment balances.

204 4 Free-radical Polymerization: Homogeneous
4.4.3.1Batch Polymerization
Inflow and outflow terms are set to zero, but the change in volume with conver-sion must be considered for the constant-mass system [Eq. (90)].
dV V dr
¼ ð90Þ
dt r dt
Volume contraction due to the difference in polymer and monomer densities can be as high as 20% for bulk polymerization, and should not be neglected when solv-ing the material balances. Solution of the initiator balance and substitution into the monomer balance lead to a differential equation for conversion, Eq. (91), where the volume contraction factor is defined as e ¼ ðrfinal  r0 Þ=rfinal where rfinal is the system density at 100% conversion of monomer to polymer and r0 is the initial density of the system.
! !1=2

dx p kp 2f kd ½I0 kd t
¼ 1=2
ð1  x p Þ exp  ð91Þdt kt 1  ex p 2
4.4.3.2 Continuous Polymerization
Equations (84)–(89) can be solved to provide a description of the dynamic behavior of a continuous well-stirred system. Analytical solutions may be derived for the steady-state case by setting all derivative terms to zero. Assuming no inflow of rad-icals, Eq. (92) results for l 0.ðqout ½l 0 Þ ¼ Vð2f kd ½I  k t ½l 0  2 Þ ð92ÞFor most cases, the radical lifetime is much shorter than the average residence time. Thus outflow of radicals can be neglected, resulting in the familiar expres-sion, Eq. (93), for l 0.


2f kd ½I 1=2½l 0  ¼ ð93Þ
kt
[I] can be calculated from the steady-state solution of Eq. (83), namely Eq. (94),where y ¼ V=qout is the reactor residence time and e ¼ ðrout  rin Þ=rout is the frac-tional density change between inlet and outlet streams.qin ½Iin  ½Iin ½I ¼ ¼ ð94Þqout þ Vkd ð1 þ ykd Þð1  eÞSubstitution of these expressions into the monomer balance leads to a nonlinear relationship between conversion and y [Eq. (95)].

4.4 Polymer Reaction Engineering 205
kp 2f kd ½Iin x p ¼ yk p l 0 ¼ y 1=2
ð95Þ
kt ð1  eÞð1 þ ykd ÞAssuming no inflow of polymer into the reactor and the long-chain hypothesis, the
pol
steady-state values for DPn and DPw in the absence of LCB (k tr ¼ 0) are given by Eq. (83). LCB and reaction with terminal double bonds broaden the MWD signifi-cantly [81].
4.4.3.3 Complex Flowsheets
These are often constructed to represent systems with nonideal mixing and fast re-action. A classic example is the high-pressure high-temperature free radical pro-duction of ethylene copolymers, generally conducted in a homogeneous phase con-sisting largely of supercritical ethylene monomer. These conditions make for very fast reactions (for example, initiator half-life of <1 s), promote numerous side re-actions (long-chain branching, short-chain branching, and chain scission), and in-troduce the potential of thermal runaway. Recent models of these systems combine a detailed description of polymerization kinetics with a complex flowsheet of con-tinuous well-mixed tanks in series with recycle to represent mixing in a multizone,multifeed autoclave reactor [100–102]. Models for multifeed tubular systems also include heat transfer and pressure drop along the length of the system [103]. The general strategy is to ‘‘tune’’ the model (based upon a set of proposed kinetic mechanisms captured by the method of moments) by fitting kinetic coefficients and mixing and heat transfer parameters to a set of industrial data, and then to use the model to interpret and optimize industrial operating conditions. Advances in computing power have allowed the complexity of these models to increase.
4.4.3.4 Computational Fluid Dynamics (CFD)
CFD simulation is emerging as an alternative and more fundamental approach to examining polymerization systems with complex mixing and reaction. Once again,much of the work is focused on high-pressure ethylene polymerization systems.A major challenge is incorporating both macromixing (turbulent diffusion and convection) as well as micromixing (molecular diffusion) into the representation[104]. The first efforts in this area [104, 105] concern themselves with the predic-tion of temperature and conversion profiles in the reactors; to simplify the calcula-tional load they consider only initiation, propagation, and termination reactions.More recently, Kolhapure and Fox [106] incorporated a more complete kinetic scheme to allow the prediction of polymer MW, polydispersity, and average branch-ing number. These CFD studies can point the way to improved reactor design and operation; for example, by examining the importance of initiator distribution at the injection point, and defining conditions for stable reactor operation [106]. An arti-cle in 2000 discussed the implementation of CFD calculations within a process simulation package [107]. Although not yet applied to polymerization systems,this advancement shows enormous promise.

206 4 Free-radical Polymerization: Homogeneous Model-based control of polymerization systems has also garnered its share of atten-tion. The goal of these efforts is the development of robust strategies to guide and control the manufacture of polymer safely and reproducibly in the face of unmea-sured disturbances and frequent product grade transitions. The main challenge in controlling polymerization systems is the lack of on-line measurements of polymer structure. A review by Congalidis and Richards [108] provides a good summary of literature focusing on this difficult issue; for further discussion, see Chapter 12 of this book. In most cases, the implementation of detailed fundamental models is not warranted for control application. However, simple models can often be com-bined with limited on-line measurements (for example, in Refs. 109–112) to im-prove control performance. Fundamental models can also be used to test empirical models developed for control purposes [113].
4.5
Summary
Polymer reaction engineering issues can pose major challenges for industrial-scale polymer synthesis. As reactor size increases, transport phenomena such as heat and mass transfer become more difficult, and large-scale homogeneous FRP pro-cesses are often limited by transport effects. In many cases, the final reaction rate,molecular weight, and copolymer composition are determined by the coupled ef-fects of reaction kinetics and transport phenomena. Mathematical modeling, in conjunction with a strong experimental program, is a powerful means to improve our mechanistic and process understanding. The use of empirical models has util-ity in the control of polymerization reactors, and can be an important contributor to product quality and process robustness. Continuing development of new mea-surement techniques, together with an ability to relate measurements to functional properties, will be a critical area of future research.While many current free-radical polymerization processes have been in indus-trial use for years, the next several years may see the emergence of new industrial technologies. Promising technologies for homogeneous FRP include living radical polymerization chemistries and polymerization in supercritical carbon dioxide.The adoption of new technology requires identifying an application or product for which the new technology is clearly advantaged, and successfully overcoming the numerous scaleup challenges in converting the process to industrial scale. These processes will require a detailed understanding of polymerization kinetics com-bined with fundamental reaction engineering principles.
Notation
Ai pre-exponential factor for mechanism-i CtrXXX ratio of chain-transfer rate coefficient (k trXXX , XXX ¼ mon, pol, sol) to propagation [Eq. (19)]

Notation 207 cp heat capacity [kJ kg1 K1 ]Dn¼ dead polymer chain of length n with terminal unsaturation Dn; b dead polymer chain of length n with b branch points DPn number-average degree of polymerization DPw weight-average degree of polymerization Ei activation energy for mechanism-i [kJ mol1 ]
Fpinst
mole fraction of monomer-i contained in polymer chains being formed f initiator efficiency [Eq. (1)]fi mole fraction of monomer-i in the monomer mixture fri mole fraction of polymer radicals ending in monomer unit-i frij mole fraction of polymer radicals ending in monomer unit-j with pen-ultimate unit-i[I] initiator concentration [mol L1 ]K eq equilibrium propagation–depropagation constant kbb rate coefficient for intramolecular H-abstraction [s1 ]kd rate coefficient for initiation decomposition [s1 ]kdep rate coefficient for depropagation [s1 ]ki rate coefficient for primary-radical initiation [L mol1 s1 ]k inhib rate coefficient for inhibition [L mol1 s1 ]kp rate coefficient for propagation [L mol1 s1 ]k pij rate coefficient for addition of monomer-j to radical-i [L mol1 s1 ]k pijk rate coefficient for addition of monomer-k to penultimate radical-ij[L mol1 s1 ]
cop
kp copolymer-averaged propagation rate coefficient [L mol1 s1 ]
eff
kp effective propagation rate coefficient, corrected for depropagation[L mol1 s1 ]
pol
kp rate coefficient for radical addition to a terminally-unsaturated chain[L mol1 s1 ]
kptert
addition rate of monomer to mid-chain radical [L mol1 s1 ]kt rate coefficient for termination [combination þ disproportionation)[L mol1 s1 ]
cop
kt copolymer-averaged termination rate coefficient [L mol1 s1 ]k tc rate coefficient for termination by combination [L mol1 s1 ]k td rate coefficient for termination by disproportionation [L mol1 s1 ]k therm rate coefficient for thermal initiation of styrene [L 2 mol2 s1 ]k trXXX rate coefficient for transfer to species XXX (mon, pol, sol) [L mol1 s1 ]kb rate coefficient for radical b-scission [s1 ][M] monomer concentration [mol L1 ][M]eq equilibrium monomer concentration [mol L1 ]Mn number-average molecular weight [g mol1 ]Mw weight-average molecular weight [g mol1 ]mw molecular weight of the reaction mixture [g mol1 ]Ni number-average length of monomer-i sequences NðMi ; nj Þ fraction of monomer-i sequences that are exactly nj units in length Pij probability of unit-j following unit-i in a polymer chain

208 4 Free-radical Polymerization: Homogeneous Pn polymer radical of length n Pn; b polymer radical of length n with b branch points[Ptot ] total concentration of radicals [mol L1 ][Ptot ] total concentration of polymer radicals of type-i [mol L1 ]Qn mid-chain polymer radical of length-n (formed by intramolecular or in-qin reactor inlet volumetric flow [L s1 ]qout reactor outlet volumetric flow [L s1 ]Rd rate of initiator disappearance [mol L1 s1 ]R dep rate of depropagation [mol L1 s1 ]R inhib rate of inhibition [mol L1 s1 ]R init rate of radical generation from initiator [mol L1 s1 ]RLCB rate of long-chain branch formation [mol L1 s1 ]Rp rate of propagation [mol L1 s1 ]
pol
Rp addition rate of terminally unsaturated polymer chains mol L1 s1 ]R pol total rate of monomer consumption [mol L1 s1 ]R term rate of radical–radical termination [mol L1 s1 ]R therm rate of thermal initiation of styrene [mol L1 s1 ]R trXXX rate of transfer to species XXX (mon, pol, sol) [mol L1 s1 ]Rl i rate of change of moment l i [mol L1 s1 ]ri monomer reactivity ratio for radical-i (for example, r1 ¼ k p11 =k p12 )[S] solvent (transfer agent) concentration [mol L1 ]si radical reactivity ratio for monomer-i in the penultimate copolymeriza-tion model (for example, s1 ¼ k p211 =k p111 )T temperature [K]Tc ceiling temperature of monomer [ C]t 1=2 initiator half-life [s1 ]V reactor volume [L]wn weight fraction of chains of length-n xp fractional monomer conversion to polymer[Z] inhibitor concentration [mol L1 ]
Greek
b ratio of chain termination (combination) to propagation [Eq. (82)]DGp free energy change of polymerization [kJ mol1 ]DHp heat of polymerization [kJ mol1 ]DSp entropy of polymerization [J mol1 K1 ]DVi activation volume for mechanism-i [cm 3 mol1 ]d fraction of termination by disproportionation e volume contraction factor zk kth moment of the dead polymer chains [mol L1 ]y average residence time [s]lk kth moment of the polymer radicals [mol L1 ]mk kth moment of the total (dead þ radical) polymer chains [mol L1 ]

References 209 n kinetic chain length of polymer radicals r density [kg L1 ]t ratio of chain termination (disproportionation) and transfer to propaga-tion [Eq. (82)]u average number of monomer units on a living chain F copolymerization cross-termination factor
Acronyms
AIBN 2,2 0 -azobisisobutyronitrile BMA butyl methacrylate CFD computational fluid dynamics CTA chain-transfer agent DMA dodecyl methacrylate EGDMA ethylene glycol dimethacrylate FRP free radical polymerization LCB long-chain branching LCH long-chain hypothesis LDPE low-density polyethylene MA methyl acrylate MMA methyl methacrylate MW molecular weight MWD molecular weight distribution nBA Butyl acrylate PDI polydispersity index (Mw/Mn )PLP pulsed-laser polymerization PMMA poly(methyl methacrylate)QSSA quasi-steady-state assumption SCB short-chain branching SEC size exclusion chromatography Sty styrene
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212 4 Free-radical Polymerization: Homogeneous Macromol. Sci. – Rev. Macromol. Chem. 99 M. Busch, Macromol. Theory Simul.Phys. 1992, C32, 183–234. 2001, 10, 408–429.84 M. A. Dubé, J. B. P. Soares, A. 100 W.-M. Chan, P. E. Gloor, A. E.Penlidis, A. E. Hamielec, Ind. Eng. Hamielec, AIChE J. 1993, 39, 111–Chem. Res. 1997, 36, 966–1015. 126.85 D. J. Arriola, PhD Thesis, University 101 H. Nordhus, O. Moen, P. Singstad,of Wisconsin-Madison, 1989. J. Macromol. Sci. – Pure Appl. Chem.86 T. Xie, A. E. Hamielec, Macromol. 1997, A34, 1017–1043.Chem. Theory Simul. 1993, 2, 421–454. 102 P. Pladis, C. Kiparissides, J. Appl.87 T. Xie, A. E. Hamielec, Macromol. Polym. Sci. 1999, 73, 2327–2348.Chem. 1993, 2, 455–483. 103 A. Brandolin, M. H. Lacunz, P. E.88 H. M. Hulburt, S. Katz, Chem. Eng. Ugrin, N. J. Capiati, Polym. React.Sci. 1964, 19, 555–574. Eng. 1996, 4, 193–241.89 R. C. Zabisky, W. M. Chan, P. E. 104 K. Tsai, R. O. Fox, AIChE J. 1996, 42,Gloor, A. E. Hamielec, Polymer 1992, 2926–2940.33, 2243–2262. 105 N. K. Read, S. X. Zhang, W. H. Ray,90 S. Ochs, P. Rosendorf, I. Hyanek, AIChE J. 1997, 43, 104–117.S. X. Zhang, W. H. Ray, Comp. Chem. 106 N. H. Kolhapure, R. O. Fox, Chem.Eng. 1996, 20, 657–663. Eng. Sci. 1999, 54, 3233–3242.91 A. Pertsinidis, E. Papadopoulos, C. 107 F. Bezzo, S. Macchietto, C. C.Kiparissides, Comp. Chem. Eng. 1996, Pantelides, Comp. Chem. Eng. 2000,20, S449–S454. 24, 653–658.92 S. T. Balke, A. E. Hamielec, J. Appl. 108 J. P. Congalidis, J. R. Richards,Polym. Sci. 1973, 17, 905–949. Polym. React. Eng. 1998, 6, 71–111.93 H. Tobita, J. Polym. Sci.: Part B: 109 R. K. Mutha, W. R. Cluett, A.Polym. Phys. 1993, 31, 1363–1371. Penlidis, Ind. Eng. Chem. Res. 1997,94 F. Teymour, J. D. Campbell, Macro- 36, 1036–1047.molecules 1994, 27, 2460–2469. 110 S.-M. Ahn, M.-J. Park, H.-K. Rhee,95 A. Butté, A. Ghielmi, G. Storti, M. Ind. Eng. Chem. Res. 1999, 38, 3942–Morbidelli, Macromol. Theory Simul. 3949.1999, 8, 498–512. 111 Y. Yabuki, T. Nagasawa, J. F.96 H. Tobita, Macromol. Theory Simul. MacGregor, Comp. Chem. Eng. 2000,1996, 5, 129–144. 24, 585–590.97 M. Wulkow, Macromol. Theory Simul. 112 M.-J. Park, H.-K. Rhee, Chem. Eng.1996, 5, 393–416. Sci. 2004, 58, 603–611.98 P. D. Iedema, M. Wulkow, H. C. J. 113 S.-S. Na, H.-K. Rhee, J. Appl. Polym.Hoefsloot, Macromolecules 2000, 33, Sci. 2000, 76, 1889–1902.498–512.

Free-radical Polymerization: Suspension1
5.1
Key Features of Suspension Polymerization Many important polymers are made commercially via suspension polymerization of vinyl monomers. These include poly(vinyl chloride), poly(methyl methacrylate),expandable polystyrene, styrene–acrylonitrile copolymers and a variety of ion-exchange resins and specialist materials. The annual polymer production from sus-pension processes is very high.
5.1.1
Basic Ideas There are two key requirements for any commercial free radical polymerization process. The polymerization rate must be reasonably high and the polymer prod-uct must have the correct molecular weight distribution. The conditions that are necessary to achieve those goals can be predicted from the kinetics of homoge-neous free radical polymerization (see Chapter 4). But those conditions cannot be obtained easily in large-scale production when bulk processes are used. Uniform mixing and temperature control are difficult to achieve because the high heat of polymerization is combined with a large viscosity of the reaction medium. As most polymers are poor thermal conductors, heat transfer from large reactors is usually poor because the ratio of surface area to volume decreases as the reaction mass increases.In suspension polymerization, drops of a monomer-containing phase are dis-persed in a continuous liquid phase. Monomer solubility in the continuous phase is often low and polymer is produced inside the drops. Although the drop viscosity increases with monomer conversion, the effective viscosity of the suspension re-mains low and efficient agitation is possible. The ratio of surface area to volume for small drops is relatively high and local heat transfer is good. If the continuous
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

214 5 Free-radical Polymerization: Suspension phase is aqueous and well mixed, heat transfer from the reactor is also good. That permits effective control of temperature and of those variables which depend on temperature, which include reaction rates, polymer molecular weight, and copoly-Suspension stability is maintained by good agitation and by the use of drop stabilizers. Removal of the stabilizing agents, after polymerization, may not be complete and some contamination of the final product is sometimes inevitable.Suspension polymerization usually requires larger reactor volumes than bulk pro-cesses because the vessels often contain about 50% of the continuous phase. Sus-pension polymerization has been reviewed previously by Munzer and Trommsdorff[1], Bieringer et al. [2], Warson [3], Brooks [4], Yuan et al. [5], Vivaldo-Lima et al.[6], and Arshady [7].
5.1.2
Essential Process Components The chemical events that occur inside drops of the dispersed phase are similar to those found in bulk polymerization. The drops contain monomer (or monomers),radical generators (often called initiators), and polymer. Sometimes chain-transfer agents are added also. The continuous phase is often regarded as chemically inert,but drop stabilizers are usually present in it and, in some cases, those stabilizers participate in the polymerization process. For a discussion of stabilizer types, see Section 5.2.1.
5.1.3
Polymerization Kinetics Higher conversions of monomers can be accommodated more readily in suspen-sion processes than in bulk processes because suspensions are more mobile than molten polymers. Therefore, simple rate expressions may not be applicable be-cause the values of some rate coefficients diminish at high polymer concentrations(see Section 5.3.2).It is often assumed that the polymerization chemistry which occurs in the dis-persed phase is identical to that which occurs in the equivalent bulk process. That assumption may be valid if the monomers and initiator are virtually insoluble in the continuous phase. Then, polymerization rates, molecular weight distributions,and copolymer compositions can be predicted from conventional kinetic schemes.But drop stabilizers may react with species inside the drops (for example, to form graft copolymers).When all the monomers in a suspension polymerization are virtually immiscible with the continuous phase, then the instantaneous copolymer composition can be predicted from idealized relationships which apply to homogeneous systems. How-ever, the use of those relationships is not straightforward if one, or more, of the monomers is partially soluble in the continuous phase, because the actual compo-sition of the drops may then be unknown. The effective monomer concentrations,

5.1 Key Features of Suspension Polymerization 215
Tab. 5.1. Apparent reactivity ratios in solution and suspension copolymerization (reproduced with permission from Ref. 4).Monomer 1 AS AN VC MMA Monomer 2 S MA VA MAA r1 (soln) 1.18 1.02 1.68 0.35 r2 (soln) 0.85 0.7 0.23 1.63 r1 (susp) 1.0 0.75 2.47 0.63 r2 (susp) 1.0 1.54 1.99 1.07 Temp [
C] 60 50 70 69 Abbreviations: AS, p-acetoxystyrene; S, styrene; AN, acrylonitrile; MA,methyl acrylate; VC, vinyl chloride; VA, vinyl acetate, MMA, methyl methacrylate; MAA, methacrylic acid.which should be used in these relationships, might be predicted if the appropriate partition coefficients for the two phases can be obtained. That is not often possible but models that allow for water solubility of monomers have been developed for the copolymerization of vinyl chloride and vinyl acetate [8] and for the copolymer-ization of styrene and acrylonitrile [9].Apparent reactivity ratios obtained directly from suspension polymerization ex-periments may not be identical to those expected from the equivalent bulk pro-cesses if some monomer migrates to the continuous phase. Ashady et al. [10]found values for reactivity ratios that were not expected from results observed in bulk or solution copolymerization. Izumi and Kitagawa [11] showed that reactivity ratios for suspension copolymerization, of acrylonitrile and methyl acrylate, were different from those obtained from either solution or emulsion polymerization.Table 5.1 compares reactivity ratios obtained from solution copolymerization with those observed in suspension copolymerization.If the initiator in suspension polymerization is slightly soluble in water, then simultaneous emulsion polymerization may occur when free stabilizer remains in the continuous phase [12, 13].
5.1.4
Fluid–Fluid Dispersions and Reactor Type Batch, or semi-batch, reactors are often used for suspension polymerization on an industrial scale. Dispersions in tubular flow reactors are difficult to maintain and a continuous stirred tank would produce drops containing partially polymerized ma-terial that would coalesce in the receiving equipment. However, new types of flow reactors are being developed for suspension polymerization (see Section 5.4.1).Many of the published studies on drop behavior in agitated liquid–liquid disper-sions are applicable to suspension polymerization. But sometimes their use is lim-ited because they do not account for changing physical properties of drops or for the presence of drop stabilizers.

216 5 Free-radical Polymerization: Suspension Uses of Products from Suspension Polymerization Advantages of suspension polymerization are not restricted to temperature control.Some polymers, such as poly(vinyl chloride) (PVC), are immiscible with their monomers. Subsequent polymer coagulation makes bulk processes difficult to con-trol. In suspension polymerization, however, that problem is avoided. There, coag-ulation is largely confined to the drop interiors and aggregation of polymerizing drops is restricted. That is why suspension polymerization is used for the large-scale production of PVC. In that case, the initial drop diameters, and the final par-ticle sizes, range between 10 and 100 mm.If a polymer product is required in particulate form, then suspension polymer-ization is especially suitable. The energy required to disperse monomer drops is lower than that required to break up the finished polymer. Also, drop size control before polymerization is easier to achieve than particle size manipulation of granu-lated polymer. Suspension polymerization provides a good route to functionalized particles such as those used in ion-exchange resins. Expandable polystyrene beads are also made by suspension polymerization. When a product is to be used in‘‘bead form’’, initial drop diameters can be as large as 1–2 mm [1].
5.2
Stability and Size Control of Drops For many applications, the size range of the final product particles is very impor-tant. For example, bead diameters affect flow rates through ion-exchange columns.But particle size can also be important when the polymer is to be converted to a macroscopic object. Heat transfer rates to polymer particles during extrusion and mass transfer rates of plasticizers in particulate polymers both depend on particle size.During suspension polymerization, drop size depends on the physical properties of the two phases, the phase ratio, the nature of the suspension flow, and the con-dition of the phase interface. Interfacial tension and drop stability depend largely on the nature of the drop stabilizer. If no stabilizer were used to protect the drops,the suspension would be unstable and the final polymer particles would reach an undesirable size.The adsorption of stabilizer molecules at the interface between monomer and the continuous phase reduces the interfacial tension and hence reduces the energy required to form drops. Drop stability against coalescence depends largely on the ability of the stabilizer to form a thin protective film at the interface. That gives the drops better elastic properties [14]. The effect of the elastic properties is en-hanced by increasing the concentration of the suspending agent, until a certain surface coverage of the drops is reached. At that point, a ‘‘critical surface coverage’’is established and a further increase in the suspending agent concentration will have a very little effect on the drop stability [15].

5.2 Stability and Size Control of Drops 217
Stabilizer Types Many drop stabilizers in suspension polymerization are water-miscible polymers.These materials are sometimes called protective colloids. They include naturally oc-curring substances, such as gelatin and pectin, and a wide range of synthetic poly-mers such as partially hydrolyzed poly(vinyl acetate)s. Modified natural products such as cellulose ether derivatives are also widely used. Munzer and Trommsdorffprovide a detailed list of patented stabilizers [1]. Polymeric stabilizers do not all act in the same way but steric effects are often important [16]. Although transfer of stabilizer molecules from the continuous phase to the drop surfaces can some-times be fast, the development of drop stability may be slow [17]. That may arise because rearrangement of stabilizer molecules on the drop surface is necessary.Water-miscible polymers are not expected to be good drop stabilizers when the con-tinuous phase is nonaqueous (see Section 5.5).Partially hydrolyzed poly(vinyl acetate) (PVA), a common stabilizer, is often called poly(vinyl alcohol), but that is a misnomer because not all the acetate groups are hydrolyzed. The extent of hydrolysis (DH) has significant effects on the behav-ior of the PVA. This is especially important in the suspension polymerization of vinyl chloride. PVA with a DH between 70% and 80% is a good stabilizer for drops in aqueous media. Drops retain their integrity even when agitation levels are re-duced [18]. But when the DH is less than 60%, drop sizes become sensitive to re-duction in agitation intensity [18]. Although PVAs with a low DH are poor drop stabilizers in aqueous media, they are still useful because they affect product mor-phology by influencing events inside the vinyl chloride drops (see Section 5.3.3).Attempts have been made to measure the distribution of the stabilizer between the two phases and the interface [19]. PVA often becomes grafted onto polymer that is formed inside the drops, leading to the formation of a ‘‘skin’’ on the final particle surface. This is important because subsequent removal of the skin is not easy.Although organic substances are commonly used as drop stabilizers, it is well known that some particulate inorganic solids can also stabilize drops in suspen-sion [20]. There are many reports of inorganic solids (such as calcium phosphate,aluminium hydroxide, and zinc phosphate) being used in the suspension polymer-ization of styrene [1]. In some cases, small amounts of surfactants (such as sodium alkylsulfonates) are added to assist the dispersion of those solids. The effect of elec-trolytes can also be important [21]. Inorganic stabilizers are advantageous when only low levels of contamination are required because they can sometimes be re-moved effectively from the final polymer particles. Also, inorganic particles are able to stabilize relatively large drops, such as those formed in the manufacture of expandable polystyrene beads.Although some inorganic solids are effective stabilizers for drops in suspen-sion polymerization, other solids are not stabilizers and may even be destabilizers.O’Shima and Tanaka [22] suggested that the contact angle between dispersed liq-uid and inorganic solid is a crucial factor in determining whether an inorganic

218 5 Free-radical Polymerization: Suspension solid is a stabilizer or a destabilizer in suspension polymerization. Solids that pro-vide a relatively large contact angle (such as aluminium hydroxide) would be stabil-izers in aqueous media. In contrast, those which have a relatively small contact angle (such as carbon black) would tend to be destabilizers. In practical operation,the contact angle will probably vary with any given inorganic solid if different monomers are used. In many cases it is not easy to measure contact angles.Two theories, screen theory and coverage theory, have been suggested to explain the mechanism of drop stabilization by inorganic solids [21]. According to the screen theory, finely divided inorganic solids which are dispersed in water form a screen.Dispersed monomer drops smaller than the mesh of the screen are free to move through the meshes, while those bigger than the mesh cannot pass through the meshes and are stopped from coalescing further. In the coverage theory, it is sug-gested that dispersed inorganic solids cover the surfaces of monomer drops and form a layer which prevents drop coalescence. Both theories are plausible but many workers believe that the stabilization is obtained mainly by a coverage effect.However, not all inorganic solids are adsorbed by the monomer drops. When Wol-ters et al. [23] used hydroxyapatite and calcium carbonate in the suspension poly-merization of styrene, the adsorption equilibrium was found to be far on the side of desorption. Also, Wang and Brooks [24] noticed that many of their stabilizing particles settled on the bottom of the vessel when stirring of a liquid–liquid disper-sion ceased.If some particles remain in the continuous phase, it is not feasible for the cover-age theory to provide a description for every case of suspension polymerization in which an inorganic solid is used as a stabilizer. Wang and Brooks suggested a crowding theory which takes account of the dynamic effects of stabilizer particles[24]. In that model, particles are not required to be permanently adsorbed on drop surfaces but they become effective when two drops come close together. Although crowding theory does not explain all the observations, it is compatible with a wide range of experimental results. The addition of surfactants often improves the dis-persion action of the fine particles, but it is possible to stabilize dispersions solely with mono-sized spherical colloidal particles [25].
5.2.2
Drop Breakage Mechanisms In an agitated suspension, the dispersed phase can be broken into small drops when its surface is disrupted. That disruption can be caused either by frictional forces (via viscous shear) or by inertial forces (via turbulence) [26]. The ratio of the external disrupting force to the interfacial tension force is often expressed as the Weber number, We. Drop deformation increases as We increases. When We ex-ceeds a critical value, a drop will break into smaller drops.The fluids in agitated vessels are often turbulent. If the turbulence in local re-gions can be regarded as isotropic, a criterion for the drop breakage mechanism can be developed [26, 27]. In turbulent flow, random eddies are superimposed on the main flow. Eddy sizes are influenced by the location of the vessel walls and are

5.2 Stability and Size Control of Drops 219
restricted by the impeller diameter [28]. Kinetic energy is transferred to smaller eddies in a sequential fashion, until it reaches the smallest ones. This transfer is assumed to occur without energy dissipation. However, when the kinetic energy reaches the small eddies, it is dissipated as heat to overcome the viscous forces.In theories of local isotropy, it is assumed that the small eddies are statistically independent of each other. Velocity fluctuations are determined by the local rate of energy dissipation per unit mass of fluid (e) and by the kinematic viscosity (n).Kolmogorov [29], by dimensional reasoning, derived an expression [Eq. (1)] for the length of the smallest eddy (h).h ¼ n 3=4 e1=4 ð1ÞRushton et al. [30] suggested that local isotropy could exist when the Reynolds number is >10 4 . The macroscale of turbulence, L (approximately equal to the im-peller diameter), is then much larger than the microscale of turbulence. Here, the Reynolds number (Re) is defined by Eq. (2), where N is the impeller speed, D is the impeller diameter, and n is the kinematic viscosity of the dispersion.Re ¼ ND 2 =n ¼ ND 2 r=m ð2ÞPressure fluctuations can deform the drops and they may break if the inertial forces exceed the interfacial tension forces. Kolmogoroff [27] and Hinze [31] de-rived an expression for the maximum drop diameter, d max , that should be observed when turbulence is isotropic. Here, d g h and the viscous forces may be neglected in comparison with the inertial forces; d max , can then be related to a critical Weber number, We crit , by Eq. (3).We crit ¼ rc u 2 ðdÞd=s ð3ÞThe inertial forces are related to e. Thus, We crit is given by Eq. (4).We crit ¼ ðC1 rc e 2=3 d max
5=3
Þ=s ð4Þ
Rushton et al. [30] showed that Eq. (5) holds, so that Eqs. (6) [32] and (7) follow.e ¼ C3 N 3 D 2 ð5Þ3=5 6=5 4=5 d max ¼ C4 ðs=rc Þ N D ð6Þd max =D ¼ C4 ðWeÞ0:6 ð7ÞIn order to use this expression, it is often necessary to relate d max (which is difficult to measure) to the Sauter mean diameter, d32 (which is more readily determined).A linear relationship between d32 and d max has been demonstrated by Sprow [33],Coulaloglou and Tavlarides [28], Kuriyama et al. [34], and Zerfa and Brooks [35].

220 5 Free-radical Polymerization: Suspension Volume fraction of dispersed phase d 32 /D ( x 10 )
0.05
0.2
0.3
0.4
0 10 20 30
- 0.6 4
0.027(1 + 3.06 )We ( x 10 )
Fig. 5.1. Correlation of vinyl chloride drop size with volume fraction j and Weber number We. Reproduced by permission of Elsevier Science Ltd. From Ref. 35.In the suspension polymerization of vinyl chloride, Zerfa and Brooks found the relationship in Eq. (8).
0:58
d32 ¼ d max ð8ÞEquation (6) has been verified with a large number of liquid–liquid dispersions where the volume fraction of the dispersed phase is not high, that is, in non-coalescing systems [28]. But, unlike many laboratory studies, real suspension poly-merizations are operated with a high volume fraction of dispersed phase, j. Then,drop coalescence and breakage happen simultaneously and damping of the turbu-lent fluctuations may occur. To allow for this, a factor f ðjÞ is often introduced into Eq. (7). Experimental expressions for f ðjÞ are reviewed by Yuan et al. [5], and by Zerfa and Brooks [35]. The latter workers showed that, in the suspension polymer-ization of vinyl chloride, d32 was given by Eq. (9).d32 =D ¼ 0:027ðþ3:06jÞWe0:6 ð9ÞEquation (9) applied when values of j ranged between 0.01 and 0.4, as shown in Figure 5.1. An expression with similar form is given by Borwankar et al. [15].When drop coalescence is significant, Shinnar [32] showed that there will be a minimum drop size, d min , that is proportional to N 0:75 . The extent of drop coales-cence is reduced by addition of an appropriate drop stabilizer. If the surface cover-age of the drops by the stabilizer exceeds a critical value then the dispersion can remain stable after agitation ceases [15].It must be remembered that correlations between average drop sizes and We only apply at steady state. The steady-state drop size distribution (DSD) can take some time to become established [18, 36–38]. That may not be a serious problem with nonreacting dispersions but, in suspension polymerization, the physical prop-

5.2 Stability and Size Control of Drops 221
erties change with time so that the DSD can continue to change during the process[37].When the diameter of the drops is less than the Kolmogorov length h, stresses from viscous shear will be much larger than those from inertial effects. Drop breakage is then the result of viscous shear [26]. Here, again, a drop will break if the deformation variable, sometimes described as a generalized Weber group [31],exceeds a critical value [39]. Taylor showed [40] that the extent of drop distortion,before breakup, depended on the ratio of phase viscosities (Eq. (10), where md and mc are the viscosities of the dispersed and the continuous phase, respectively).We crit ¼ mc ðqu=qrÞðd=sÞ ¼ f1 ðmd =mc Þ ð10ÞShinnar and Church [26, 32] demonstrated that for locally isotropic flow whereðqu=qrÞ 2 ¼ e=n, d max is given by Eqs. (11) and (12).d max ¼ ðsnc1=2 =mc e 1=2 Þ f ðmd =mc Þ ð11Þd max z ðsnc1=2 =mc ÞN 3=2 D1 f ðmd =mc Þ ð12ÞWhen viscosity-increasing agents were present in the continuous phase for the sus-pension copolymerization of styrene and divinylbenzene, Jegat et al. [41] found that drop breakage could occur via viscous shear when flow in the suspension was turbulent. In that case, the Kolmogorov length would have been larger that that found when low-viscosity aqueous solutions are used.The form taken by f ðmd =mc Þ depends on the nature of the flow. In laminar flow,the contributions of rotational and elongational components can be expressed via the value of a parameter a, which ranges between 0 and 1. For simple shear a ¼ 0,and for pure elongational (hyperbolic) flow a ¼ 1 [43]. With low a values, We crit(now equivalent to the critical capillary number Cacrit ) passes through a minimum as md =mc increases. After the minimum is reached, relatively small increases in md =mc are accompanied by large increases in We crit . When a ¼ 0, We crit tends to an infinitely high value if md =mc > 4 so that drop breakage no longer occurs [42,43]; see Figure 5.2.Correlations that are found in idealized laboratory studies only provide rough guides to events that occur in commercial reactors. Local flow in large stirred ves-sels is often ill defined. Also, drop breakage can produce more than two new drops.Chatzi and Kiparissides [44] suggested that the formation of a daughter drop is ac-companied by the formation of a number of smaller satellite drops. Drop stabi-lizers influence turbulence near drop surfaces and non-Newtonian behavior leads to further complications. Therefore, it is not surprising that the many published studies on drop behavior in liquid–liquid suspensions lead to a variety of different results. Leng and Quarderer [39] found that d max depended on N 0:8 . By using data from the literature, for the suspension polymerization of methyl methacrylate, they also showed that d max depended on mc0:5 when drop breakage occurred by viscous shear. Presumably, the drop size was not able to respond to changes in md =mc that

222 5 Free-radical Polymerization: Suspension
1.2 A
Ca crit
0.8
0.4
0 1 2 3 4
log10 (viscosity ratio) + 3 Fig. 5.2. Critical capillary number against log10 of viscosity ratio (md =mc ) for laminar flow. (A) a ¼ 0 (simple shear flow);(B) a ¼ 1 (hyperbolic flow). Data from Refs. 42 and 43.would have occurred during polymerization. However, average drop size was found to be a linear function of interfacial tension, as expected, when viscous shear was important. Borwankar et al. [15] suggested that the relationship between d max and agitator speed may depend on agitator type.When polymer-containing drops are broken, their elastic properties must be taken into account. Arai et al. [45] derived a correlation for drop breakup from a Voigt model to represent the elastic properties. The validity of the correlation was confirmed experimentally using a dispersed phase with a wide range of viscosity.Wang and Calabrese [46] carried out experiments with drops made from well-characterized model fluids. They showed that the influence of interfacial tension on drop breakage decreased as the dispersed-phase viscosity increased.
5.2.3
Drop Coalescence The volume fraction of drops in commercial suspension polymerization reactors is usually high and drop coalescence cannot be ignored. In liquid–liquid dispersions the drop size distributions (DSDs) depend on the breakage and on coalescence processes.From experiments in which both drop breakage and coalescence occurred, Kuri-yama et al. [34] found that drop sizes reached a steady value within an hour, when the initial drop viscosity was low. But with a high drop viscosity, the drop size re-duction continued for longer periods of time and the final drop size was higher.Although model, nonreacting, fluids were used for those experiments, the results are relevant to suspension polymerization. In their study of drop coalescence in the suspension polymerization of styrene, Konno et al. [47] found that the Sauter mean diameter increased as the polymer viscosity increased. They also concluded that the stabilizer does not effectively prevent the coalescence of drops with diame-ters larger than d max .The overall rate of drop coalescence is related to the collision frequency of the drops and to the coalescence efficiency. By comparing drop collisions in agitated

5.2 Stability and Size Control of Drops 223
dispersions with molecular collisions in homogeneous fluids, previous workers have developed expressions for collision rates between drops of different sizes [48,49]. If drops adhere for sufficient time to allow them to deform, and to permit drainage of the continuous phase that is trapped between them, then coalescence may occur [26]. By taking account of these events, expressions can be obtained for the coalescence efficiency [50]. Expressions, for coalescence and for collision rates are not easy to use because they often contain parameters that are difficult to quan-tify. Alvarez et al. [50] constructed a model for drop breakage and coalescence, in the suspension polymerization of styrene, which takes account of viscosity effects.That model assumes that breakage of a drop, exposed to a turbulent flow field, is a result of fluctuations with a wavelength equal to the drop diameter. Fluctuations with wavelengths that are smaller or larger than the drop diameter do not lead to drop breakage.
5.2.4
Drop Size Distributions Even when expressions for drop breakage and coalescence rates are available, their successful use must allow for variations of turbulence intensity within the reactor.Maggioris et al. [51] describe a two-compartment population balance model for an agitated suspension polymerization reactor. That model distinguishes between the impeller region and the remainder of the reactor. Both regions are assumed to be well mixed and CFD simulations predicted drop size distributions that were com-patible with experimental results from nonpolymerizing model liquid–liquid dis-persions. For some combinations of stabilizer type and stirrer speed, bimodal drop size distributions were predicted by the model and found in the experiments.Yang et al. [52] showed how the size distribution of nonreacting styrene drops changed with agitation time. Bimodality developed and the relative size of the two peaks in the distribution depended on the amount of drop stabilizer that was used.Calabrese et al. [46, 53] showed that, in dilute agitated suspensions for which co-alescence is negligible, the equilibrium drop size distribution broadened consider-ably as dispersed-phase viscosity increased.If no chemical reaction occurs, then the rates of drop breakage and drop coales-cence eventually become equal and a stable DSD is obtained [54]. Considerable periods of time might be required for that to occur [18], especially when the drops have a high viscosity [55]. But, in the case of suspension polymerization, the phys-ical properties of the drops change with monomer conversion. In a batch process,early increases in drop viscosity reduce the rates of breakage and coalescence be-fore a steady-state DSD can be established. Then, the DSD continues to change,and the average drop size increases, as the polymer content of the drops increases.Consequently, the final particle size distribution (PSD) is quite different from the steady-state DSD that would be expected from nonpolymerizing drops. That was shown to be the case by Jahanzad et al. [37] in the suspension polymerization of methyl methacrylate. There, the DSDs of polymerizing drops were broader than that of the DSDs in the corresponding nonpolymerizing monomer drops. But the

224 5 Free-radical Polymerization: Suspension average drop size in the nonpolymerizing drops was only slightly smaller than that in the polymerizing suspension when a large amount of stabilizer was used, be-cause excess stabilizer reduced drop coalescence in both systems. Similar results were obtained by Konno et al. [47]. With smaller amounts of stabilizer, a growth stage exists in which drop coalescence continues when drop breakage rates are low [50]. The increase in average drop size can be substantial and continues until the identification point is reached when drop viscosity is too high to permit further drop coalescence. Jahanzad et al. [37] also showed that, although the diameters of most drops and particles ranged between 10 and 300 mm, a second peak appeared in the particle size distribution. The average diameter of particles that accounted for that second peak was about 1 mm. Most of those smaller particles probably de-veloped from satellite drops which are formed during the breakage process. The existence of small satellite drops is compatible with the ideas of Chatzi and Kiparis-sides [44] and with the work of Sathyagal et al. [56]. But some of the very small particles could have been formed directly in the aqueous phase because the water solubility of the free radical initiator (lauroyl peroxide) may be high enough to in-duce some emulsion polymerization. That can by important with monomers that are more water-soluble, such as methyl methacrylate, vinyl acetate, and vinyl chlo-ride. If emulsion polymerization becomes prevalent, then a significant amount of the drop stabilizer will be adsorbed on the surfaces of the small particles. The amount of stabilizer that is available to stabilize the drops is then reduced [57].In batch reactors, the rate of drop coalescence is affected by changes in drop viscosity. Therefore, any induced kinetic effects that alter the polymer molecular weight (and, thereby, change the viscosity) could influence the coalescence rate and the DSD. Promoting chain transfer is one way for that to happen.When drop viscosity remains low, coalescence events can depend on the nature of the drop stabilizer. Low viscosities may be encountered when the polymers, or copolymers, are immiscible with their monomers. The polymers then precipitate inside the drops and their apparent viscosity does not increase substantially until appreciable conversions have been attained. That is the case with the suspension polymerization of vinyl chloride. When hydrolyzed poly(vinyl acetate) (PVA) is used as the drop stabilizer, Zerfa and Brooks [18] showed that DSDs depended on stirrer speed. With ‘‘good’’ PVA stabilizers, that had a high degree of hydrolysis, a reduction in stirrer speed had little effect on the DSD, as shown in Figure 5.3. But when a ‘‘poor’’ PVA stabilizer, with a low degree of hydrolysis, was used, a reduc-tion in stirrer speed led to a broadening of the DSD. There, the DSD became sim-ilar to that expected when the stirrer speed was maintained at its lowest value,as shown in Figure 5.4. Clearly, the ‘‘good’’ stabilizer protected the drops from coalescence.
5.2.5
Drop Mixing Most of the coalescence events described in Section 5.2.3 occur when drops have a uniform chemical composition, but sometimes it is necessary to add material to a

5.2 Stability and Size Control of Drops 225
Drop number 200
150 A
15 35 55 75 95 Drop diameter (microns)Drop number 15 35 55 75 95 Drop diameter (microns)Drop number
150 C
15 35 55 75 95 Drop diameter (microns)Fig. 5.3. Change in drop size with stirrer speed with PVA
72.5% hydrolyzed. (A) N ¼ 350 rpm (5.8 s1 ); (B) N ¼ 650
rpm (10.8 s1 ); (C) N reduced from 650 to 350 rpm. Data from Ref. 18.reactor in which a suspension already exists. That can happen if control of copoly-mer composition is important. In batch operation, copolymer composition usually changes with overall conversion because monomers react at different rates [58].This drift in copolymer composition may be limited by adding one of the mono-

226 5 Free-radical Polymerization: Suspension
150 A
Drop number 10015 35 55 75 95 Drop diameter (microns)Drop number 15 35 55 75 95 Drop diameter (microns)Drop number 15 35 55 75 95 115 Drop diameter (microns)Fig. 5.4. Change in drop size with stirrer speed with PVA 55%hydrolyzed. (A) N ¼ 350 rpm (5.8 s1 ); (B) N ¼ 650 rpm
(10.8 s1 ); (C) N reduced from 650 to 350 rpm. Data from
Ref. [18].mers to the reactor incrementally (that is, by using a semi-batch procedure). But,when new dispersible material is added to an existing suspension, the new mate-rial and existing drops can remain segregated for a significant period of time.Then, new drops may form with a monomer composition that differs from that of

5.2 Stability and Size Control of Drops 227
the original drops. The new monomer cannot mix with existing drops (which con-tain the initiator and have adsorbed most of the drop stabilizer). Adding extra drop stabilizer, with the new monomer, might not be helpful because there is the dan-ger of stabilizing new drops with the ‘‘wrong’’ monomer composition that contain little, or no, radical generator.Hashim and Brooks [59] studied the addition of styrene to a suspension of stabi-lized drops. The drops were composed of polystyrene solutions in styrene. The ini-tial drop viscosity affected the drop size and the rate of coalescence between drops.As the dispersed-phase viscosity increased, the drop size distribution broadened; at some stirrer speeds, the mixing rate increased. It appears that there is a critical drop size which determines the coalescence efficiency. Above that size, the drop mixing rate increases as the drop viscosity decreases. Below the critical drop size,the mixing rate is influenced noticeably by the drop size; as the drop size increases,the coalescence rate also increases.Drop mixing may become an issue when volatile monomers are used. The en-thalpy of polymerization for most of the vinyl monomers that are used in suspen-sion polymerization ranges between 30 and 90 kJ mol1 . Therefore, a high heat re-moval rate is usually necessary to maintain a constant reactor temperature. This is difficult to achieve by heat transfer through the reactor walls in commercial opera-tions because large reactors have a relatively small surface area to volume ratio.Heat removal can be improved by allowing the monomer to vaporize. The vapor is then condensed, cooled, and returned to the reactor as a liquid. If the polymer-ization process is to be maintained, drops of returning monomer must gain access to initiator and drop stabilizer. With the suspension polymerization of vinyl chlo-ride, Zerfa and Brooks [60] found that monomer returning from a reflux con-denser formed drops that acquired initiator without the need for coalescence with existing stabilized drops. The presence of small particles, formed by simultaneous emulsion polymerization, appeared to provide a mechanism for transfer of initiator(bis(4-t-butylcyclohexyl) peroxy dicarbonate) through the continuous phase. With high reflux rates, the drop size distribution became bimodal whereas, in the ab-sence of reflux, a monomodal DSD is expected [60]. New drops, from refluxed monomer, had limited access to the drop stabilizer (PVA) and were larger than the ‘‘old’’ drops.A special drop mixing problem arises with the suspension polymerization of vinyl chloride. Because the monomer is very reactive and has a high enthalpy of polymerization, operators are reluctant to mix initiator in the monomer before a suspension is formed. Therefore, as a safety precaution, the initiator is often dis-persed in the aqueous phase of a stabilized suspension. Then the subsequent mix-ing of monomer and initiator can be quite slow. Zerfa and Brooks [61, 62] showed that many monomer drops remained ‘‘uninitiated’’ when monomer in other drops had polymerized to a considerable extent. That did not happen when initiator was dissolved in the monomer before it was dispersed: see Figure 5.5. This nonuni-formity in drop behavior affected the final polymer properties. Also, addition of ini-tiator via the aqueous phase promoted simultaneous emulsion polymerization and modified the PSD. Drop mixing rates were quantified by using dyed monomer

228 5 Free-radical Polymerization: Suspension Fig. 5.5. Effect of the method of initiator 1, 5 min; 2, 20 min; 3, 60 min. N ¼ 350 rpm addition to vinyl chloride: (A) initiator (5.8 s1 ); j ¼ 0:1; PVA concentration ¼ 0.06%.predissolved in drop phase; (B) initiator Reproduced by permission of John Wiley &predispersed in continuous phase. Panels: Sons, Inc. From Ref. 62.drops in the experiments. The mixing rate was found to depend on both the con-centration of the PVA and the grade that was used [61].Events at High Monomer Conversion Suspension polymerization provides a practical method of achieving high mono-mer conversions. Therefore, studies of polymerization and drop behavior that deal with events at low monomer conversion may not be applicable to suspension poly-merization.

5.3 Events at High Monomer Conversion 229
Breakage of Highly Viscous Drops In batch reactors drop breakage and coalescence are affected by the polymerization process because the viscosity of the polymerizing fluid often increases. Many as-pects of the interaction of drop behavior with the polymerization process have been discussed already (see Section 5.2).
5.3.2
Polymerization Kinetics in Viscous Drops Free radical polymerization kinetics has received much attention and many aspects of the process are well understood (see Chapter 4). Most academic investigations have been carried out in ‘‘idealized’’ conditions where the extent of monomer con-version is low. The classical expression for the rate of polymerization (R p ), in a single-phase reaction, is Eq. (13), where k p is the propagation rate coefficient, CM is monomer concentration, R i is the initiation rate and k t is the termination rate coefficient [63].R p ¼ k p CM ðR i =k t Þ 1=2 ð13ÞIf radicals are generated by the thermal decomposition of an added initiator then,at steady-state conditions, Eq. (14) applies, where f is an efficiency factor, kd is the initiator decomposition rate coefficient and CI is the initiator concentration.R i ¼ 2f kd CI ð14ÞChain termination is often diffusion-controlled and the value of k t diminishes sub-stantially as the polymer concentration increases. At high polymer concentrations,k p decreases also [64]. The reduction in k t leads to an increase in the polymeriza-tion rate, a phenomenon often described as a ‘‘gel effect’’.Although most workers agree that the chain termination reaction is diffusion-controlled, reliable quantitative relationships between the rate coefficients and measurable properties of the reaction medium are not generally available. Radical diffusion can depend on solution viscosity, polymer volume fraction and polymer molecular weight. The latter three entities are interrelated in complicated ways;meaningful experiments, which may relate them to the rate coefficients, are not easy to devise [65]. However, Brooks et al. showed that effects of viscosity changes on polymerization rate could be distinguished from the effects of polymer volume fraction [66]. In some cases, the value of f may also depend on polymer content[67].The value of kd can often be determined independently but traditional steady-state experiments give values for k p =k t0:5 and do not provide separate values for k p and k t . This restriction becomes a problem for reactor modeling because, in the presence of polymer, both k p and k t diminish (even in isothermal conditions).

230 5 Free-radical Polymerization: Suspension Pulse laser techniques, combined with accurate molecular weight measurement,have been used to determine independent values for k p but those techniques are only feasible in the absence of preformed polymer [68, 69].When a gel effect is observed, kinetic analysis can be difficult because k t be-comes dependent on chain length [70] and the pseudo-steady state assumption may not be valid [71]. Tefera et al. [72] showed that, when a high monomer conver-sion is attained, a number of different models for the gel and glass effects gave an adequate description of isothermal time–conversion data over the entire conver-sion range for a single type and loading of initiator. But models that ignored the effect of polymer molecular weight on the diffusion of macro radicals failed to de-scribe the time–conversion data if the concentration of the initiator varied. It has been shown that autoacceleration in free radical polymerization can be observed even when low molecular weight polymer is being formed, indicating that chain entanglements are not necessary for a gel effect to occur [73].In practice, isothermal conditions are not always maintained in suspension poly-merization and temperature increases can lead to a marked reduction in initiator concentration. As the monomer conversion increases, the glass transition temper-ature of the polymer solution also increases and the drops can become glassy. The chain propagation reaction then becomes diffusion-controlled and the value of k p decreases significantly. An increase in reaction temperature may become necessary in order to achieve complete monomer conversion. Although most workers agree that k p is reduced at high monomer conversions, Faldi et al. [74] have suggested that diffusion control may not be the only reason for that to occur. Qin et al. [75]developed a three-stage model to account for the gel and glass effects. The match with experimental data, for methyl methacrylate and styrene polymerizations, up to high monomer conversion was satisfactory for isothermal conditions. Achilias and Kiparissides [76] showed, however, that data for polymerization of those mono-mers were compatible with a model that did not require ‘‘break points’’ for the gel and glass effects. The initiator efficiency, f , has been shown to depend on the size of the initiator radicals at high monomer conversion [76].If crosslinking or copolymer precipitation occurs, bulk polymerization may be difficult to handle. A suspension process may then be the only feasible way in which the copolymerization can be carried out [4]. Suspension processes also pro-vide a means of investigating copolymerization kinetics at high conversion. The monomer sequence in styrene–methyl methacrylate copolymers at high conver-sion have been found to differ from those observed at low conversion [77].
5.3.3
Consequences of Polymer Precipitation Inside Drops Polymers that are insoluble in their monomers will precipitate during polymeriza-tion. The resulting fouling problems that occur in bulk polymerization are greatly reduced by using suspension polymerization. This is one of the reasons for choosing a suspension process for PVC manufacture, as discussed previously (see Section 5.1.5). Manipulation of polymer precipitation, inside the drops, during

5.3 Events at High Monomer Conversion 231
Fig. 5.6. Schematic representation of polymer formation inside vinyl chloride drops.polymerization can influence the properties of the final product, especially the po-rosity, which in the case of PVC affects the ability of the polymer to take up plasti-cizers. Structural changes that occur inside the drops of polymerizing vinyl chlo-ride monomer (VCM) have been discussed by a number of authors [78, 79].PVC starts to precipitate from the monomer phase when the conversion exceeds
0.1% conversion, forming a separate polymer-rich phase inside the drops. The pre-
cipitating polymer aggregates to form unstable microdomains, which aggregate further to give domains. Subsequent aggregation of domains results in the for-mation of primary particles. The primary particles grow by polymerization within them and by buildup of microdomains or domains on their surfaces. Eventually multiple contacts lead to the formation of a continuous network of primary par-ticles throughout the polymer particle/monomer droplet; see Figure 5.6. At about 70% conversion the monomer-rich phase disappears and further polymerization occurs in the polymer-rich phase [80, 81]. The final polymer grains have irregular shapes, unlike particles that form from polymers that are completely miscible with their monomers. Some components of the drop stabilizers are chosen because they are miscible with the monomer and they can influence the agglomeration of pri-mary particles [82]. Those components are sometimes called secondary suspend-ing agents (SSAs).As a direct consequence of the process described above, and of the density differ-ence between PVC and VCM, PVC grains have a complicated morphology and they can be highly porous. The particle size, the PSD, the morphological characteristics,and the degree of porosity of PVC grains depend on polymerization conditions.These include the agitation in the reactor, the type and concentration of suspend-

232 5 Free-radical Polymerization: Suspension ing agent(s), the secondary suspending agents, the polymerization temperature,the monomer conversion, and the ratio of monomer to water.Effects of agitation on the mean particle size of PVC resins have been studied by many researchers, using different reactor capacities, and many correlations have been developed [83, 84]. The effects of agitation on PVC porosity have also received much attention [80]. As with the suspension polymerization of other monomers,the choice of suspending agent(s) affects the particle size distribution of the final polymer. In the case of PVC, however, the suspending agents also affect the sub-structure and porosity of the particles. The primary suspending agent system is often based on PVA, substituted cellulose, or a mixture of the two. Wolf and Schuessler [85] concluded that the plasticizer absorption (which depends on po-rosity) of the resulting PVC was related to the surface activity of the suspending agent, regardless of type. Ormondroyd [86] demonstrated the effect of PVA struc-ture on the particle size, cold plasticizer absorption, and bulk density of the PVC produced. Cheng [87] and Cheng and Langsam [88] used hydroxypropyl methylcel-lulose (HPMC) as a suspending agent and analyzed the influence of molecular weight and chemical structure of the cellulose on the particle morphology of the resulting PVC. Cebollada et al. [81] showed how HPMC structure and concentra-tion influenced the particle properties of PVC. SSAs are regularly used in the pro-duction of suspension PVC to impart higher porosity to the PVC grains. That improves the subsequent uptake of plasticizer and also promotes the removal of unreacted VCM at the end of polymerization. PVA with a low degree of hydrolysis and nonionic surfactants, such as sorbitan monolaurate, have been used commer-cially as SSAs [89]. The mechanism by which such SSAs function is not entirely clear [80, 90–92].When nonionic surfactants, such as Span85, Span60 and Span20, are used as SSAs, with PVA as the main stabilizer, the mean particle size of PVC resins in-creases quickly as the concentration of nonionic surfactant increases [82]. The degree of agglomeration of primary particles increases with polymerization tem-perature and with conversion. A nonionic surfactant with a greater hydrophile-lyophile balance (HLB) value is more effective in raising the particle size [82].The increase in the mean particle size of the final PVC is probably caused by in-creased coalescence of VCM/PVC droplets during the polymerization process, as a result of a lowering of water-drop interfacial tension by the SSA. Nonionic surfac-tants such as Span20, Span60, and Span85 have more affinity with VCM than the PVA has, and they will be absorbed faster than PVA on the VCM/water interface.Part of the interface, which would otherwise be occupied by PVA molecules, will become occupied by nonionic surfactant molecules. Thus, the colloid protective ability of the composite suspending agent would decrease because the nonionic surfactants with a low molecular weight have a lower colloid protective ability than that of PVA. In some cases, graft copolymers of PVA and PVC form a skin(or membrane) around the polymer particles [93]. Zerfa and Brooks [60] showed that, at low monomer conversion, PVC porosity increases if a high monomer reflux rate is used. Porosity, surface roughness, and particle shape were found to depend on the origin of the PVA [62]. Particle shape can also be influenced by the method

5.3 Events at High Monomer Conversion 233
0 20 40 60 80 100 Polymerisation conversion (%)Fig. 5.7. Effect of polymerization conversion on the cold plasticizer absorption (CPA) of PVC. Reproduced by permission of John Wiley & Sons, Inc. From Ref. 82.of initiator addition [62]. High conversions of VCM are not always desirable be-cause the porosity of PVC particles usually decreases linearly with monomer con-version, as shown in Figure 5.7 [60, 82].When a nonionic surfactant is used as an SSA, in conjunction with PVA, the porosity of PVC increases as the concentration of nonionic surfactant increases.Here, increased porosity may be the result of incomplete drop coalescence creating voids at the sites where droplets are not well contacted [82]. Nonionic surfactant with a lower HLB value (hence with a higher affinity for VCM and a higher solubil-ity in VCM) is more effective in raising the product porosity. The surface of pri-mary particle aggregates becomes coarser as surfactant is added [82]. This could be the consequence of an altering interfacial tension between the PVC-rich phase and the monomer. That might be expected to decrease the contact deformation of the primary particles and increase the pore space. When nonionic SSAs are used,PVC porosity decreases linearly with an increase of polymerization temperature[82]. The porosity increases as the concentration of nonionic surfactant increases,and a surfactant with a lower HLB value is more effective in raising the porosity.The increase in porosity may be caused by a combination of increased coalescence of VCM/PVC droplets and the nonionic surfactant’s steric effect inside the droplets[82]. Bao et al. showed that particle morphology and PVC properties can be con-trolled by using blends of PVA suspending agents with differing degrees of hydro-lysis [94]. The use of PVA, with a low degree of hydrolysis, as an SSA increased particle porosity in the suspension copolymerization of vinyl chloride and propy-lene [95]. In that case, the SSA was more soluble in the organic phase than the pri-mary stabilizer (a cellulose ether).In vinyl chloride polymerization, particle porosity facilitates the subsequent up-take of plasticizers, but in other cases particle porosity is induced to enhance access to functional groups within the polymer. That is useful when polymer beads are required for use in ion-exchange columns, or in analytical instruments. A diene comonomer can be added to a monomer, to promote crosslinking and phase sepa-

234 5 Free-radical Polymerization: Suspension ration inside the drops [96–99]. Appropriate functional groups can be incorporated
5.4
Reaction Engineering for Suspension Polymerization The discussion above (see Sections 5.2 and 5.3) provides guidelines for the design of suspension polymerization processes on an industrial scale. However, extrapo-lating from idealized small-scale studies to full-scale commercial operation is not straightforward. Maintenance of uniform reactor conditions cannot be guaranteed inside large reactors and controlled heat transfer from reactors can be difficult.
5.4.1
Dispersion Maintenance and Reactor Choice Many suspension polymerization processes employ stirred vessels. Oldshue et al.[100] showed that agitator type and vessel geometry have an important influence on internal liquid flow. This must be taken into account in the design of large ves-sels for suspension polymerization. Maggioris et al. [51] and Vivaldo-Lima et al.[101] devised two-compartment population balance models to account for the large spatial variations of the local turbulent kinetic energy. These permit the prediction of drop size evolution in a suspension polymerization reactor with a high volume fraction of dispersed phase. Basic expressions were modified to allow for evolving physical properties of the suspension. Maggioris et al. [51] showed that it was fea-sible to estimate the volume ratio of the impeller and circulation regions, the ratio of turbulent dissipation rates, and the exchange flow rate of the two compartments at different agitation rates and continuous-phase viscosities. Although many indus-trial suspension polymerizations are described as ‘‘batch’’ processes, they are not genuine batch operations because some material enters the reactor after polymer-ization has started. In some cases, substantial reflux of monomers occurs and con-densed monomer returns to the reactor continuously. In others, a semi-batch pro-cess is used to control product composition. In both these situations, unconverted monomer with low viscosity is being mixed with drops of higher viscosity. The com-plexities of these mixing processes have been discussed above; see Section 5.2.5.Reactor configurations other than conventional stirred tanks have been proposed for suspension polymerization. Draft tubes, or ‘‘internal loops’’, can be used for suspension polymerization [102], but drop size changes can occur [103] and flow patterns may be complicated [104]. Tanaka et al. [105] used a loop reactor for the suspension polymerization of styrene. They employed a double agitation method to control the transient droplet diameter distribution and the final particle size dis-tribution. Ni et al. [106] developed an oscillatory baffled reactor for batch suspen-sion polymerization of methyl methacrylate. Fluid mixing was achieved by eddies that are generated when a fluid passes through a set of equally spaced, stationary,orifice baffles that are located inside a tube. Periodically formed vortices were con-

5.4 Reaction Engineering for Suspension Polymerization 235
trolled by choosing appropriate values for oscillation frequency, oscillation ampli-tude, baffle diameter, and baffle spacing. The final particle size distribution and polymer molecular weight were comparable with those of a product obtained from a conventional stirred tank reactor.Although most industrial polymerization processes are batch or semi-batch oper-ations, some continuous-flow processes have been proposed [107]. Use of flow re-actors may affect the nature of the drop size distribution and the molecular weight distribution. If flow through a back-mixed region occurs, the drops will be subject to a distribution of residence times and the nature of the mixing for the dispersed phase is important. Baade et al. [108] and Taylor and Reichert [109] showed that complete segregation of drops, in the suspension polymerization of vinyl acetate,produced an increase in polymer molecular weight. Also, the distribution of mo-lecular weights became broader than that expected in a batch reactor. Complete mixing of the drops gave a molecular weight distribution which was narrower than that found in batch reactors. An oscillatory baffled reactor can also be used for continuous-flow suspension polymerization. Ni et al. [110] describe the effect of superimposed oscillations on the main flow through a tube. Radial mixing of thefluid occurs but near plug-flow is obtained. A constant level of turbulence intensity in the reactor can lead to particle size distributions similar to those expected from a batch reactor.
5.4.2
Agitation and Heat Transfer in Suspensions In single-phase processes, reactor agitation influences the rate of heat transfer to,and from, the surroundings and determines the quality of mixing. With suspen-sion polymerization reactors, the agitation must also be good enough to generate,and to maintain, a two-phase dispersion. When conventional stirred tanks are used, a ‘‘standard’’ reactor geometry may be adequate to promote convective heat transfer. Here, the height of the total suspension (H), the impeller diameter (D)and the clearance underneath the impeller (G) are related to the tank diameter(T) as follows:
 HAT
 D A T=3
 G A T=3
The use of baffles limits nonuniformity in the turbulence and restricts vortex for-mation. Vortices are undesirable because the centrifugal effect favors drop congre-gation and may promote unwanted drop coalescence. That can lead to polymer deposition either on the agitator or the reactor walls (depending on the relative density of the aqueous and nonaqueous phases). When vertical baffles are close to the walls their width (B) is often given by:
 B A T=10

236 5 Free-radical Polymerization: Suspension With suspension polymerization, reactor fouling is sometimes a problem. In such cases, it can be helpful to have a small gap between the outer baffle edge and the reactor wall. That may reduce accumulation of coagulated solids at the wall–baffle Some form of turbine impeller is often suitable for suspension polymerization when the apparent viscosity of the suspension is not very high. But the power in-put to the suspension, via the impeller, must be sufficient to create and to maintain the required drop size distribution. The required stirrer speed for the appropriate turbulence level can be estimated from Eqs. (6)–(7) (see Section 5.2.2). But, in real-ity, the turbulence will not be uniform. Drop breakage and drop coalescence may occur in different regions of the reactor [5]. The necessary power input can be pre-dicted from correlations between the power number and the Reynolds number[30]. However, those correlations are usually developed for single-phase fluids and care must be taken when using them for suspensions, where the effective viscosity of the suspension may be unknown. The Reynolds number in most large-scale sus-pension polymerization reactors is usually high enough to generate turbulent flow,and in that region the use of baffles increases the required power input.Heat removal from suspension polymerization reactors can be a problem, espe-cially after the startup period in batch operation when a gel effect causes auto-acceleration in the polymerization. If the monomers are volatile at the working pressure of the reactor, then some heat can be removed relatively quickly by mono-mer vaporization (as discussed above for vinyl chloride polymerization; see Section
5.2.5). When most of the heat transfer occurs via a cooling jacket, special strategies
may be necessary to manage the heat transfer. Pinto and Giudici [111] suggested that a mixture of initiators, with different half-lives and activation energies, could be used. Then it may become possible to match the heat generation rate with the cooling capacity of the reactor jacket. The feasibility of such an approach depends on the accuracy of the reactor model that is used. Mejdell et al. [112] showed that the reaction rates predicted from models for the industrial suspension polymeriza-tion of vinyl chloride were subject to significant noise because the heat capacity of the suspension was large and the resolution of temperature measurements was limited. If the reactor cooling capacity is chosen to cope with heat generation rates in the later stages of batch operation, when autoacceleration occurs, then the heat removal capacity of the reactor will not be fully utilized in the early stages of the polymerization. Longeway and Witenhafer [113] suggested that the reactor could be operated at a higher temperature in the early stages, to make full use of the cooling capacity. They constructed a model that predicted a reduction in total reac-tion time for a given monomer conversion. It should be remembered that the over-all heat transfer coefficient for a jacketed reactor depends on the nature of the wall material. Transfer coefficients for stainless steel reactors are approximately double the coefficients found for glass-lined carbon steel reactors [114].It is commonly assumed that, in suspension polymerization, heat transfer be-tween polymerizing drops and the continuous phase is rapid and both phases have the same temperature. However, Lazrak and Ricard [115] showed that, with methyl methacrylate polymerization, the internal temperature of drops could be

5.4 Reaction Engineering for Suspension Polymerization 237
higher than that of their surroundings when drop diameters exceeded 1 mm. Tem-perature differences were large enough to cause changes to polymer quality.Scaleup Limitations with Suspension Polymerization Many aspects of scaleup (or scaledown) of suspension polymerization reactors are similar to those encountered with other polymerization processes. These concern parameters that control polymerization rate, product composition, and polymer molecular weight. In the case of suspension polymerization, however, special con-sideration must be given to maintaining the required drop/particle size distribu-tion. In the case of VCM polymerization, the maintenance of particle porosity is also an issue. In principle, correlations between dimensionless entities, such as the Weber number correlations (see Section 5.2.2), should help reactor designers to relate observations in laboratory experiments to events that occur on a large scale. But, even if fundamental requirements for drop breakage and drop coales-cence are known (which is not always the case), it is difficult to ensure similarity of conditions inside reactors with greatly differing sizes. Some key relationships are discussed by Yuan et al. [5] and by Leng and Quarderer [39]. Scaleup criteria should allow for any variations of volumetric phase ratio that may occur. Some-times those can be included in modifications to the Weber number correlations(see Section 5.2).In addition to the problem of matching turbulence homogeneity in reactors of different sizes, there can be a serious difficulty in maintaining constant average tur-bulence intensities. This arises because a high Reynolds number (Re ¼ ND 2 r=m) is easier to achieve in a large reactor than in a small reactor. Often, it not practical to use a stirrer speed, in a small reactor, which is high enough to compensate for the relatively small value of D 2 . A good estimate for the physical properties of the whole suspension is necessary in order to predict values for the Reynolds number.When the suspension viscosity is high, it is possible for the fluid in a large reactor to be fully turbulent but for the fluid in a small reactor to be in the transition re-gion (between laminar and turbulent flow). In such cases, the drop breakage mech-anism can change with reactor size. Scaleup is most likely to be successful when the reactors have a similar basic shape. An unbaffled spherical laboratory reactor will not behave in the same way as a large baffled cylindrical reactor. When drop formation depends on reactors with a special geometry, such as that required for a crossflow membrane [16] or a pulsed column [106], then new scaleup criteria may become necessary.Heat transfer rates per unit mass of suspension increase as reactor size decreases. Consequently, it may not be possible to achieve the same time–temperature profile in a large reactor as is obtained with a small reactor, especially when autoacceleration occurs. If some heat transfer occurs via monomer evapora-tion, then the rate of monomer reflux can depend on reactor size. In the case of VCM polymerization, this can result in particle porosity changing with reactor size (see Section 5.3.3).

238 5 Free-radical Polymerization: Suspension Reactor Safety with Suspension Processes The potential for thermal runaway exists with most commercial free-radical poly-merization processes that employ batch reactors. This arises because the heat of reaction is high and there is no effective steady state. Although temperature con-trol in suspension polymerization reactors is often better that that found in bulk polymerization, the possibility of thermal runaway cannot be ignored. This was demonstrated by Nemeth and Thyrion [116] who showed that very sharp tempera-ture rises could occur in the suspension polymerization of styrene, largely because spontaneous free-radical generation occurs with that particular monomer as the temperature increases. When models are used to predict temperature rises in poly-merization reactors, it is important to allow for small changes that occur in the physical properties of the polymerizing fluid. Otherwise, large overestimates of temperature rises are obtained [117].Axial mixing throughout the reactor is particularly important when the density of the continuous phase lies between the density of the monomer and the density of the polymer. In those cases the polymerizing drops may gradually sink during the process. If good mixing is not maintained, highly viscous drops can accumu-late and coalesce at the bottom of the reactor. The process then reverts to a bulk polymerization, with all its inherent disadvantages. Such a potential risk arises with suspension polymerization of styrene in the production of pre-expanded beads. Even though styrene has a relatively high boiling point, the reactor is main-tained at a positive pressure because a volatile blowing agent (for example, pen-tane) must remain in the drop phase. If the suspension fails, heat transfer from the coalesced drops is quickly reduced and the temperature of the polymerizing mass begins to rise. The subsequent increase in vapor pressure of the water can cause the total pressure to reach unsafe values. Disaster may be avoided if a pres-sure release device (for example, a bursting disk) functions well, but it is essential that the design, and location, of such a device do not permit excessive fouling by polymer deposits. Regular reactor cleaning is important.In small-scale suspension polymerization, inhibitors are often removed from the monomer before it is dispersed in the continuous phase. In large-scale operation,however, the inhibitor may not be removed. Its presence lowers the risk of prema-ture polymerization, which can be dangerous when large amounts of monomer are being handled. Consequently, the particle size distribution obtained from the large reactor may differ from that in the small reactor because drop breakage can occur before significant amounts of polymer are formed (that is, when the drop viscosity is still relatively low).
5.4.5
Component Addition during Polymerization Scaleup often involves more than an increase in reactor size. Even when geometric similarity is maintained, it may be difficult to reproduce exact laboratory proce-

5.5 ‘‘Inverse’’ Suspension Polymerization 239
dures when a large reactor is used. Many scaleup criteria do not apply to tran-sient events because they employ relationships that assume steady-state conditions.Thus, the time required to disperse added material may depend on reactor size.That can be important in suspension polymerization where new material is added to the continuous phase but it is required to reach the dispersed phase or the phase interface. The new material may include fresh monomer(s), blowing agents, or modifiers for polymer properties. At startup conditions, the initial dispersion of drop stabilizers and initiator may require different amounts of time with reactors of different sizes. Consequently, the DSD and final PSD may depend on the scale of operation, because polymerization (and a change in drop properties) occurs dur-ing the drop creation process.
5.5
‘‘Inverse’’ Suspension Polymerization When a water-miscible polymer is to be made via a suspension process, the con-tinuous phase is a water-immiscible fluid, often a hydrocarbon. In such circum-stances the adjective ‘‘inverse’’ is often used to identify the process [118]. The drop phase is often an aqueous monomer solution which contains a water-soluble initi-ator. Inverse processes that produce very small polymer particles are sometimes referred to as ‘‘inverse emulsion polymerization’’ but that is often a misnomer because the polymerization mechanism is not always analogous to conventional emulsion polymerization. A more accurate expression is either ‘‘inverse microsus-pension’’ or ‘‘inverse dispersion’’ polymerization. Here, as with conventional sus-pension polymerization, the polymerization reaction occurs inside the monomer-containing drops. The drop stabilizers are initially dispersed in the continuous(nonaqueous phase). If particulate solids are used for drop stabilization, the sur-faces of the small particles must be rendered hydrophobic. Inverse dispersion poly-merization is used to make water-soluble polymers and copolymers from mono-mers such as acrylic acid, acylamide, and methacrylic acid. These polymers are used in water treatment and as thickening agents for textile applications. Beads of polysaccharides can also be made in inverse suspensions but, in those cases,the polymers are usually preformed before the suspension is created. Physical changes, rather than polymerization reactions, occur in the drops. Conventional stirred reactors are usually used for inverse suspension polymerization and the drop size distribution can be fairly wide. However, Ni et al. [119] found that good control of DSD and PSD could be achieved in the inverse-phase suspension poly-merization of acrylamide by using an oscillatory baffled reactor.
5.5.1
Initiator Types The thermal decomposition of single-component initiators (such as inorganic per-sulfates) can be used to generate the free radicals in the aqueous drops, but in

240 5 Free-radical Polymerization: Suspension some cases a redox initiator system is used. Redox reactions produce free radicals at relatively low temperatures, which is advantageous when the polymerization is very fast at higher temperatures. This type of polymerization is not always strictly analogous to ‘‘simple’’ conventional suspension polymerization. In some circum-stances, there is evidence to suggest that the polymerization rate is diminished when large amounts of an oil-soluble surfactant are used to stabilize the monomer drops, and that a small amount of polymerization occurs in the oil phase [120].The polymerization kinetics can be complicated when redox systems are used and the mechanism of radical generation may depend on the specific reductant–oxidant pair that is chosen [121].
5.5.2
Drop Mixing with Redox Initiators At least one of the two major components of a redox initiator (reductant or oxidant)must be segregated from the monomer while the suspension is being formed.Otherwise, polymerization would begin before the desired DSD was attained.Often, drops of an aqueous solution of monomer and oxidant are initially dis-persed in an oil and stabilized with an appropriate agent. Then aqueous reductant is added to start the reaction [122, 121]. Therefore, the initial suspension has two types of aqueous drops which must become mixed before polymerization can be-gin. Liu and Brooks [123] used a freeze–fracture technique with electron micros-copy to show how ‘‘large’’ reductant drops became mixed with ‘‘small’’ monomer-containing oxidant drops in the early stages of acrylic acid polymerization (Figure
5.8). The actual volumes and concentrations used for the two aqueous solutions af-
fect the rate of viscosity increase in the drops and the polymerization rate [124].
5.6
Future Developments Future work on suspension polymerization will be concerned with new products and with new processes. Much of the new work on recipes and product properties is material-specific and will continue to appear in the patent literature. The devel-opment of new processes requires further fundamental investigation. Improve-ments to existing processes need closer attention to ways in which reactors are operated. These should include the following aspects.
5.6.1
Developing Startup Procedures for Batch and Semi-batch Reactors Suspension polymerization can rarely be described as a conventional batch process because uniform physical conditions cannot be created instantly. The way in which the ingredients are brought together can affect the polymer properties. Therefore,improvements in the control of product quality may depend on developing better

242 5 Free-radical Polymerization: Suspension a large surface area and they can distort the distribution of drop stabilizer during the initial drop development stage.Maintaining Turbulence Uniformity in Batch Reactors Turbulence in conventional stirred vessels is far from uniform and prediction(or control) of DSD is difficult. Thus, the final PSD is often very wide. That is un-desirable when the final particles are used directly (for example, in ion-exchange resins). In those cases, a narrow PSD is often required. New reactor geometries and agitation methods could ensure the correct values for the overall Reynolds and Weber numbers and also give more uniform distribution of turbulence condi-tions. That will lead to a lower proportion of ‘‘off-specification’’ material. Here, the use of new reactors, such as the oscillatory baffled reactor, could be helpful.
5.6.3
Developing Viable Continuous-flow Processes Continuous-flow reactors are often developed to produce high tonnages with low reactor maintenance but, with suspension polymerization, continuous-flow reac-tors could offer other advantages. If ‘‘near plug-flow’’ conditions could be attained,the drop formation and polymerization stages could be separated to give better control of the DSD. Also, temperature programming would become feasible so that variation of drop viscosity with monomer conversion could be manipulated to achieve better control of drop coalescence and of the final PSD. For a continuous-flow reactor to operate successfully the following conditions must be attained simultaneously: a suitable residence time (often a few hours); small re-gions of localized turbulence; restricted overall back-mixing; and good heat trans-fer. New designs of oscillatory baffled reactors [106] may make such processes fea-sible on an industrial scale.
5.6.4
Quantitative Allowance for the Effects of Changes in the Properties of the Continuous
Phase
In suspension polymerization, the continuous phase is often regarded as ‘‘inert’’.Consequently, events that occur in that phase are sometimes neglected, even when they are potentially important. The overall Reynolds number is often high enough to ensure that drop breakage is via turbulence and that ‘‘conventional’’ Weber num-ber correlations apply. But when the volume fraction of drops is high, or when a significant fraction of the drop stabilizer remains in the continuous phase, the effective viscosity of the suspension becomes high. Then, the Kolmorgorov length may exceed the drop diameter, and drop breakage via viscous shear becomes im-portant. More work is required to determine the relationships between drop sizes and agitation conditions when the fluid flow is not fully turbulent. Maintenance

Notation 243 of uniform viscous shear (as opposed to maintenance of uniform turbulence)throughout the reactor will bring a new set of challenges.As more complex copolymers are developed, the range of monomer properties in any one recipe will widen. The chances of all monomers being ‘‘insoluble’’ in the continuous phase are small. Therefore the effects of monomer distribution be-tween the phases, especially on the copolymer composition, will be important. Pre-diction of phase distribution is not easy in systems that are thermodynamically nonideal and it may become necessary to develop new sensors that monitor the composition of the continuous phase.
5.6.5
Further Study of the Role of Secondary Suspending Agents The porosity and structure of polymer particles are important for many product applications. Then, the use of SSAs is often important. However, to gain a better understanding of their role, it is necessary to determine what fraction of those stabilizers is inside drops of the dispersed phase and how they affect structure in-side the particles. The importance of grafting of SSAs to the polymer should be clarified.
5.6.6
Further Characterization of Stabilizers from Inorganic Powders Particulate stabilizers will continue to have a place in suspension polymerization because they can often be removed after polymerization, but our understanding of their action is far from complete. Some of the theories do not always agree with experimental results. Further systematic studies are required to determine the ef-fect of the size and surface condition of the stabilizer particles on their ability to stabilize drops. Also, the location of those particles at the end of polymerization is important. In some cases, it may be advantageous to treat stabilizer particles so that they remain with the polymer product in order to confer some desirable property.
Notation
B baffle width [m]C1 ; C3 ; C4 constants (dimensionless)CI initiator concentration [mol m3 ]CM monomer concentration [mol m3 ]d drop diameter [m]d max maximum drop diameter [m]d min minimum drop diameter [m]d32 Sauter mean diameter [m]D impeller diameter [m]

244 5 Free-radical Polymerization: Suspension f efficiency factor (dimensionless)G clearance underneath impeller [m]H height of the total suspension [m]kd initiator decomposition rate coefficient [s1 ]kp propagation rate coefficient [m 3 mol1 s1 ]kt termination rate coefficient [m 3 mol1 s1 ]N impeller speed [s1 ]Re Reynolds number (dimensionless)Ri initiation rate [mol m3 s1 ]Rp polymerization rate [mol m3 s1 ]T tank diameter [m]We Weber number (dimensionless)We crit critical Weber number (dimensionless)
Greek
a parameter to define flow (dimensionless)e local energy dissipation rate per unit mass of fluid [m 2 s3 ]h Kolmogorov length [m]m viscosity [kg m1 s1 ]n kinematic viscosity [m 2 s1 ]r density [kg m3 ]s interfacial tension [kg m2 ]j volume fraction of dispersed phase (dimensionless)
Subscripts
c continuous phase d dispersed phase
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José C. de la Cal, José R. Leiza, José M. Asua, Alessandro Buttè,Giuseppe Storti, and Massimo Morbidelli
6.1
Introduction Emulsion polymerization leads to the production of a fine dispersion of a polymer in a continuous medium, which most often is water; the dispersion is called latex.With a worldwide annual production in excess of 20 million tonnes, emulsion polymerization is used in the production of a wide range of specialty polymers,including adhesives, paints, binders for nonwoven fabrics, additives for paper, tex-tiles, and construction materials, impact modifiers for plastic matrices, and diag-nostic tests and drug delivery systems [1–4]. The development of this industry has been due to both the possibility of producing polymers with unique properties and the environmental concerns and governmental regulations relating to substitu-tion of solvent-based system by waterborne products.In a scenario of reduction of margins, increasing competition, and public sensi-tivity to environmental issues, emulsion polymer producers are forced to achieve efficient production of high-quality materials in a consistent, safe, and environ-mentally friendly way. Emulsion polymers are ‘‘products-by-process’’ whose main properties are determined during polymerization. A critical point is to know how these process variables affect the final properties of the product. One possibility is to consider the reactor as a ‘‘black box’’ and to develop empirical relationships between process variables and product properties. Long-term success can only be guaranteed by applying knowledge-based strategies that use the polymer micro-structure (here, the term microstructure is used in a broad sense, including aspects such as copolymer composition, molecular weight distribution (MWD), branching,crosslinking, gel fraction, particle morphology, and particle size distribution (PSD)of the dispersion) as a link between the reactor variables and the final properties(Figure 6.1).The implementation of the approach illustrated in Figure 6.1 has important eco-
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

250 6 Emulsion Polymerization Fig. 6.1. Polymer properties as a function of formulation and process type.nomic implications and is scientifically challenging. On the one hand, the needs/opportunities of the market, expressed as desired properties of the final product,should be translated in terms of the desired polymer microstructure. This requires quantitative microstructure/properties relationships. On the other hand, this poly-mer microstructure should be achieved in the reactor. This involves a deep under-standing of the emulsion polymerization process, the highest level of under-standing being the development of predictive mathematical models. In addition,efficient, safe, and consistent production requires accurate on-line monitoring, op-timization, and control. Last but not least, efficient methods for removal of residual monomer and volatile organic compounds (VOCs) should be developed to produce environmentally friendly products.This chapter is focused on the challenge of achieving the efficient, safe, and consistent production of emulsion polymers of the desired microstructure. First,the main features of emulsion polymerization are discussed in Sections 6.2 and
6.3. Sections 6.4–6.7 are devoted to the understanding of the kinetics of the pro-
cess. Finally, the emulsion polymer reaction engineering is addressed in Sections
6.8–6.11.
6.2
Features of Emulsion Polymerization
6.2.1
Description of the Process Table 6.1 presents a typical formulation for emulsion polymerization. The polymer is mainly made out of a mixture of ‘‘hard’’ monomers (leading to polymers with a high glass transition temperature, Tg ; for example, styrene) and ‘‘soft’’ monomers(low Tg; for example, butyl acrylate) of low solubility in water. In addition, small amounts of functional monomers such as acrylic and methacrylic acids are in-cluded in the formulation as they provide some special characteristics, such as im-proved adhesion. Crosslinking agents and chain-transfer agents (CTAs) are used to control the molecular weight distribution of the polymer.

6.2 Features of Emulsion Polymerization 251
Tab. 6.1. Typical formulation for emulsion polymerization.Ingredient Content [wt.%]Monomer(s) 50–55
styrene
methyl methacrylate vinyl chloride vinyl acetate
butadiene
butyl acrylate 2-ethylhexyl acrylate
Veova 10
ethylene
(meth)acrylic acid crosslinking monomers Deionized water 45 Initiators 0.5 Emulsifiers 0.5–3 Chain-transfer agents Typically, emulsion polymerization is carried out in stirred-tank reactors, which commonly operate in a semicontinuous mode, although both batch and continu-ous operations are also used.In a batch emulsion polymerization, the mixture of monomers is dispersed in water using emulsifiers. The monomer droplets are stabilized by the surfactant adsorbed on their surface. In principle, any type of surfactant may be used, but in practice anionic surfactants, nonionic surfactants, and mixtures thereof account for the great majority of the systems used. The available surfactant partitions between the surface of the monomer droplets and the aqueous phase; in most formula-tions, the amount of surfactant exceeds that needed to completely cover the mono-mer droplets and saturate the aqueous phase. The excess of surfactant forms micelles that are swollen with monomer (Figure 6.2(a)).Polymerization is commonly initiated by water-soluble initiators (both ther-mal, for example, potassium persulfate, and redox, for example, tert-butyl hydroperoxide/ascorbic acid) although oil-soluble initiators (for example, AIBN)may also be used. When a water-soluble initiator such as potassium persulfate is added to the monomer dispersion, radicals are formed and as these radicals are too hydrophilic to enter the organic phases of the systems, they react with the monomer dissolved in the aqueous phase, forming oligoradicals. The growth rate of the oligoradicals is generally modest because of the low concentration of mono-mer in the aqueous phase. After adding some monomer units, the oligoradicals be-come hydrophobic enough to be able to enter the micelles (entry into the mono-mer droplets is not likely because their total surface area is about three orders of magnitude smaller than that of the micelles). Because of the high concentration of monomer in the micelle, the oligoradical that has entered the micelle grows

252 6 Emulsion Polymerization Fig. 6.2. Intervals in the batch emulsion polymerization.fast, forming a polymer chain. The new species formed upon entry of a radical into a micelle is considered to be a polymer particle. The process of formation of poly-mer particles by entry of radicals into micelles is called heterogeneous nucleation[5]. The oligoradicals that do not enter into micelles will continue growing in the aqueous phase, and upon reaching some critical length they become too hydropho-bic and precipitate. The emulsifier present in the system will adsorb onto the newly formed interface, thus stabilizing the polymer. Then, monomer will diffuse into the new polymer particle. The process of formation of polymer particles by precipitation of oligoradicals is called homogeneous nucleation [6]. Both homoge-

6.2 Features of Emulsion Polymerization 253
neous and heterogeneous nucleation may be operative in a given system. In gen-eral, homogeneous nucleation is predominant for monomers of relatively high water solubility (for example, methyl methacrylate – 1.5 g/100 g of water – and vinyl acetate – 2.5 g/100 g of water) and heterogeneous nucleation is predominant for water-insoluble monomers (for example, styrene – 0.045 g/100 g of water).Irrespective of the mechanism of particle nucleation (heterogeneous or homoge-neous) the newly formed particles are very small and suffer a tremendous increase in surface area upon particle growth. It is arguable that the emulsifier molecules may diffuse fast enough to adsorb on the surface of these fast-growing particles,stabilizing them [7–9]. Therefore, in the so-called mechanism of coagulative nucle-ation, the species formed by entry of radicals into micelles and by precipitation of growing radicals in the aqueous phase are considered to be precursor particles that only become stable upon growth by coagulation and polymerization.During nucleation, monomer droplets, monomer swollen micelles, and mono-mer swollen polymer particles coexist in the reactor (Figure 6.2(b)). Polymer particles compete efficiently for radicals, and hence monomer is consumed by po-lymerization inside the polymer particles. It is worth mentioning that polymeriza-tion proceeds according to the same mechanisms as are discussed in Chapter 4 for free-radical polymerization in homogeneous systems. The monomer that is con-sumed by polymerization in the polymer particles is replaced by monomer that dif-fuses from the monomer droplets through the aqueous phase. Therefore, the size of the particles increases and that of the monomer droplets decreases. The number of micelles decreases because they become polymer particles upon entry of a radi-cal and also because they are destroyed to provide surfactant to stabilize the in-creasing surface area of the growing polymer particles. After some time, all the micelles disappear. This is considered to be the end of the nucleation; only limited formation of new particles may occur after this point because heterogeneous nucle-ation is not possible and there is no free surfactant available in the system to stabi-lize the particles formed by homogeneous nucleation. The stage of the batch emul-sion polymerization in which particle nucleation occurs is called Interval I. At the end of Interval I, which typically occurs at a monomer conversion of about 5–10%(depending on the surfactant/monomer ratio), 10 17 –10 18 particles per liter are formed. Unless coagulation occurs, the number of particles remains constant dur-ing the rest of the process.In Interval II, the system is composed of monomer droplets and polymer par-ticles (Figure 6.2(c)). The monomer consumed by polymerization in the polymer particles is replaced by monomer that diffuses from the monomer droplets through the aqueous phase. The mass-transfer rate of monomers with water solu-bility equal or greater than that of styrene (0.045 g/100 g of water) is substantially higher than the polymerization rate, and hence monomer partitions between the different phases of the system according to the thermodynamic equilibrium.Therefore, in the presence of monomer droplets, the concentration of the mono-mer in the polymer particles reaches a maximum value. As discussed below (see Section 6.4.1), this saturation value arises from the energy (interfacial tension)needed to increase the surface area of the polymer particles upon swelling. Conse-

254 6 Emulsion Polymerization quently, the monomer concentration in the polymer particles is roughly constant during Interval II. The transport of reactants (monomers, chain-transfer agents)with water solubility lower than that of the styrene from monomer droplets to poly-mer particles may be diffusionally limited.Because of the polymerization and monomer transport, the polymer particles grow in size, and after some time the monomer droplets disappear. This marks the end of Interval II. The monomer conversion at which Interval II ends depends on the capability of the polymer particle to be swollen by monomer. The higher the maximum swelling, the earlier the monomer droplets disappear. In general, the more water-soluble the monomer, the higher the maximum swelling, and hence the lower the monomer conversion at the end of Interval II. Thus, the transition from Interval II to Interval III occurs at about 40% conversion for styrene and at about 15% conversion for vinyl acetate. This means that most monomer polymer-izes in Interval III (Figure 6.2(d)). In this interval, monomer concentration in the polymer particles decreases continuously. The final product is a waterborne, con-centrated (50–60 wt.% solids) dispersion of tiny (80–500 nm in diameter) polymer particles called latex.In a semicontinuous reactor in which monomers, surfactant, initiator, and water may be continuously fed into the reactor, emulsion polymerization does not follow the sequence of events described above. Thus, slow monomer feed and fast surfac-tant feed may lead to a system composed of polymer particles and micelles (Figure
6.3(a)). The system will contain only monomer-swollen polymer particles if both
monomer and surfactant are fed slowly (Figure 6.3(b)). On the other hand, a fast monomer feed and a low surfactant feed will lead to a system containing monomer droplets and polymer particles (Figure 6.3(c)).
6.2.2
Radical Compartmentalization From a mechanistic point of view, the compartmentalization of radicals among a huge number of polymer particles is likely to be the most distinctive feature of emulsion polymerization. This refers to the fact that the radicals are distributed among the different particles, and hence radicals in different particles cannot ter-minate between them. Consequently, the overall concentration of radicals is higher than in homogeneous (bulk and solution) free-radical polymerization, leading to a higher polymerization rate. In addition, the compartmentalization of radicals in a large number of particles results in a longer lifetime of the radicals, which leads to higher molecular weights.
6.2.3
Advantages of Emulsion Polymerization Free-radical polymerization is a very exothermic process, liable to suffer thermal runaways. Compared with bulk and solution polymerization, emulsion polymeriza-

6.2 Features of Emulsion Polymerization 255
Fig. 6.3. Species present in semicontinuous emulsion polymerization.tion is advantageous because the low viscosity of the latex allows high heat re-moval rates. In addition, because of its high heat capacity, the water absorbs large amounts of heat with only a moderate increase in temperature. This allows combi-nation of high polymerization rates and good temperature control. Compared with solution polymerization, emulsion polymerization is environmentally friendly, as water is used as the reaction medium. In addition, the low viscosity of the system allows easy removal of both unreacted monomer and VOCs. Because of the com-partmentalization of the radicals, emulsion polymerization overcomes the limita-

256 6 Emulsion Polymerization tions imposed by the kinetics of bulk and solution polymerization, and high poly-merization rates and high molecular weights can be achieved simultaneously.Some valuable products with applications in paper coating, leather treatment,binders for nonwoven fabrics, additives for paper, textiles and construction ma-terials, impact modifiers for plastic matrices, and diagnostic tests and drug delivery systems, can only be produced by emulsion polymerization. In addition, when needed (for example, for rubber for tires) latexes are easy to process into dry poly-mer. The main disadvantage of emulsion polymerization is that the product con-tains emulsifier and residues of initiator, which give it water sensitivity.Alternative Polymerization Techniques Dispersed polymers are also achieved by inverse emulsion polymerization, minie-mulsion polymerization, dispersion polymerization, and microemulsion polymer-ization. Inverse emulsion polymerization is used in the preparation of solvent-based dispersions of polymers produced from water-soluble monomers. The process is similar to conventional emulsion polymerization but the dispersed phase is an aqueous solution of the monomer and the continuous phase is an or-ganic solvent [10]. In miniemulsion polymerization [11], the size of the monomer droplets is substantially reduced (dd ¼ 50–1000 nm) by combining a suitable emul-sifier and an efficient emulsification apparatus, and stabilizing the resulting mono-mer miniemulsion against diffusional degradation by using a costabilizer (hydro-phobic, low molecular weight compound). The available surfactant adsorbs on the large surface area of the droplets, and hence no micelles are formed. When the initiator is added to the system, the radicals enter the monomer droplets,which become polymer particles. Droplet nucleation minimizes the diffusional limitations encountered in conventional emulsion polymerization and allows the incorporation of water-insoluble compounds (monomers, polymers, catalysts, cata-lytic chain-transfer agents, inorganic materials, agents for controlled radical poly-merization) in the reaction loci. In the last few years, plenty of new applications have been discovered. These applications include:
production of high-solids, low-viscosity latexes;
stable operation in continuous polymerization reactors;
controlled radical polymerization in dispersed media;
catalytic polymerization;
encapsulation of inorganic solids;
incorporation of hydrophobic monomers;
production of hybrid polymer particles;
miniemulsion polymerization in nonaqueous media;
anionic polymerization;
step polymerization in aqueous dispersed media;

6.3 Alternative Polymerization Techniques 257
Fig. 6.4. Dispersion polymerization.
production of low molecular weight polymers in dispersed media;
preparation of latexes with special particle morphology.In dispersion polymerization [12], the monomers, the initiator and the stabilizer(or stabilizer precursor) are dissolved in a solvent which is not a solvent for the polymer (Figure 6.4(a)). Polymerization starts in a homogeneous phase and the polymer is precipitated, forming unstable nuclei (Figure 6.4(b)). The nuclei are sta-bilized by the stabilizer present in the system (Figure 6.4(c)). This stabilizer may be included in the formulation or formed in the reactor by grafting onto the stabilizer precursor (the case shown in Figure 6.4). Nucleation ends when the number of sta-ble polymer particles increases to a point at which all new nuclei are captured by the existing stable particles. Because of the compartmentalization of the radicals among the polymer particles, the polymerization locus changes from the continu-ous to the dispersed phase. Dispersion polymerization allows the production of monodispersed micron-size particles, which are too large for emulsion polymeriza-tion and too small for suspension polymerization.Microemulsion polymerization [13] involves the polymerization of oil-in-water and water-in-oil monomer microemulsions. Microemulsions are thermodynami-cally stable and isotropic dispersions, whose stability is due to the very low in-terfacial tension achieved using appropriate emulsifiers. Particle nucleation occurs upon entry of a radical into a microemulsion droplet. Microemulsion polymeriza-tion allows the production of particles smaller than those obtained by emulsion po-lymerization. This leads to a higher number of polymer particles, which results in a more compartmentalized system. Under these conditions the lifetime of the poly-mer chains increases, leading to ultra-high molecular weights.

258 6 Emulsion Polymerization Kinetics of Emulsion Polymerization Conventional emulsion polymerization occurs in a three-phase system (polymer particles, aqueous phase, and monomer droplets) and polymerization may, in prin-ciple, occur in any phase. However, the concentrations of both monomer and radi-cals in the polymer particles are much higher than those in the aqueous phase (see below), and hence the extent of the polymerization in the aqueous phase is in most cases negligible. On the other hand, the concentration of the radicals in the mono-mer droplets is very low because the monomer droplets are not efficient at captur-ing the radicals formed in the aqueous phase. The polymerization in the polymer particles follows the same mechanisms as in bulk polymerization and the rate of polymerization per polymer particle Rpp is given by Eq. (1), where k p is the propa-gation rate constant [m 3 mol1 s1 ], ½Mp the concentration of monomer in the polymer particles [mol m3 ], n the average number of radicals per particle and NA Avogadro’s number.Rpp ¼ k p ½Mp ðmol/particle sÞ ð1Þ
NA
The overall polymerization rate Rp takes into account the existence of many poly-mer particles in the system according to Eq. (2), where Np is the number of poly-mer particles in the reactor, and V the volume of the reactor.
n Np
Rp ¼ k p ½Mp ðmol/m 3 sÞ ð2Þ
NA V
Most emulsion polymerization processes involve several monomers. However,there is considerable evidence that for many polymerization systems, propagation depends on the nature of the monomer and on the last two units of the growing chain [14]. The lack of available values for the large number of parameters involved in this model makes it advisable to use simpler models, such as the terminal model [15], in which polymerization is governed by the nature of the monomer and the terminal unit of the growing chain. In this case, the polymerization rate Rpi of a given monomer is expressed as Eq. (3), where kpji is the propagation rate constant of radicals with terminal unit j with monomer i [m 3 mol1 s1 ], and Pj the time-averaged probability of finding an active chain with ultimate unit of type j that for a two-monomer system is given by Eq. (4) [16].
X n Np
Rp i ¼ kpji Pj ½Mi p ðmol/m 3 sÞ ð3Þ
NA V
kp21 ½M1 p P1 ¼ ; P2 ¼ 1  P1 ð4Þkp21 ½M1 p þ kp12 ½M2 p In order to calculate the polymerization rate, ½Mi p ; n and Np should be available.

6.4 Kinetics of Emulsion Polymerization 259
Monomer Partitioning During Intervals I and II of a batch emulsion polymerization, monomers partition among monomer droplets, aqueous phase, and polymer particles. The monomer that is consumed by polymerization in the polymer particles is replaced by mono-mer that diffuses from the monomer droplets through the aqueous phase. In Inter-val III, there are no droplets and the monomer is mostly located in the polymer particles. In semibatch processes, monomers are continuously fed into the reac-tor, usually under starved conditions, that is to say at high fractional conversions(polymer/monomer ratios close to 90:10 (by weight). Under these circumstances,only the newly fed monomer droplets are present in the reactor and the lifetime of these droplets is short because the monomers diffuse through the aqueous phase to the polymer particles, where they are consumed by polymerization. The concentration of monomer in the polymer particles depends on the relative values of mass-transfer and polymerization rates. Except for poorly emulsified, highly water-insoluble monomers, the rate of mass-transfer is faster than the polymeriza-tion rate, and hence the concentrations of the monomers in the different phases are given by the thermodynamic equilibrium.The amount of monomer that can be absorbed in the polymer particles is limited and the excess of monomer forms droplets. The limiting swelling of the polymer particles is due to the contribution of the surface energy to the total free energy of the system. Because of the interfacial tension, the surface energy increases as par-ticles swell and, at some point, this compensates for the decrease in the free energy due to the monomer–polymer mixing. From values of the maximum swelling for homopolymerization [17] (Table 6.2) it can be seen that the higher the water solu-bility of the monomer, the higher the equilibrium swelling. The swelling equilib-rium depends on the particle size, but for particle sizes common in emulsion poly-merization this effect is negligible [18, 19].For a multimonomer system, the calculation of the concentrations of the monomers in the different phases involves the simultaneous solution of the ther-modynamic equilibrium equations and the material balances. Several equilibrium equations may be used, those based on partition coefficients [20] and on the Tab. 6.2. Swelling equilibrium data for homopolymerization[17].Monomer fM [a]Styrene 0.6 n-Butyl methacrylate 0.6 n-Butyl acrylate 0.65 Methyl methacrylate 0.73 Vinyl acetate 0.85 Methyl acrylate 0.85
[a] fp
M ¼ volume fraction of monomer in the polymer particles.

260 6 Emulsion Polymerization Morton–Flory–Huggins equation [21] being the most commonly applied. For a multimonomer system, the parameters of the Morton–Flory–Huggins are not usu-ally available, and it has been shown [22] that for solids contents typical of com-mercial latexes (> 50 wt.%) the use of the Morton–Flory–Huggins equation does not provide significant advantages over the use of partition coefficients. In the lat-ter case, the system of algebraic equations (5) and (6) needs to be solved, where Ki is the partition coefficient of monomer i between phase j and the aqueous phase,fi the volume fraction of monomer i in phase j, fww the volume fraction of water in the aqueous phase, fpp the volume fraction of polymer in the polymer particles,Vp the volume of polymer particles, Vd the volume of monomer droplets, Vw the volume of the aqueous phase, and Vi ; Vpol , and W are the volumes of monomer i,polymer, and water, respectively.Equilibrium equations:
j fi
Ki ¼ j ¼ polymer particles; droplets ð5Þ
fiw
Material balances:
X p
fpp þ fi ¼ 1
X
fww þ fiw ¼ 1
X
fid ¼ 1 ð6ÞVp fi þ Vd fid þ Vw fiw ¼ Vi
Vw fww ¼ W
Vp fpp ¼ Vpol Efficient methods for solving Eqs (5) and (6) are given elsewhere [19, 20].
6.4.2
Average Number of Radicals per Particle The average number of radicals per particle n is given by Eq. (7).
X
nNn
n¼ X ð7Þ
Nn
Figure 6.5 illustrates the mechanisms controlling n. In most emulsion polymeriza-tion systems, radicals are produced in the aqueous phase from thermal or redox

6.4 Kinetics of Emulsion Polymerization 261
Fig. 6.5. Processes controlling the average number of radicals per particle.initiators. Often, these radicals are too hydrophilic and cannot directly enter the hy-drophobic phases (polymer particles, monomer droplets, micelles). Therefore, they must stay in the aqueous phase until they polymerize, thus adding a number of monomer units and becoming hydrophobic enough to enter the organic phases.As the concentration of monomer in the aqueous phase is low, the period needed to reach the critical length for entry may be long and a substantial fraction of the radicals may terminate in the aqueous phase, resulting in a low initiation effi-ciency. In conventional emulsion polymerization, the total area of the polymer par-ticles is much larger than that of the monomer droplets; therefore most of the radicals are captured by the polymer particles. There is some debate about the mechanism for radical entry; diffusional [23] and propagational [24]. The diffu-sional model is more general, whereas the propagational model is a simpler approach applicable to many homopolymerizations. The rate of radical entry can also be expressed as Eq. (8), where ka is the entry rate coefficient [maqueous phase mol1 s1 ] and ½Rw the concentration of radicals in the aqueous phase[mol m3 aqueous phase ].Rate of entry ¼ ka ½Rw ðradicals/particle sÞ ð8ÞEquation (8) is a simplification because it assumes that all radicals are able to enter the polymer particles, but the radicals directly produced from inorganic initiators are too hydrophilic to be able to enter a hydrophobic phase. On the other hand,the radicals generated from radical desorption (see below) are hydrophobic and able to enter the polymer particles regardless of their length.Once inside the polymer particles, the radicals undergo the classical mecha-

262 6 Emulsion Polymerization nisms of free-radical polymerization: propagation, termination, chain transfer, and so on. Obviously, in order to suffer a bimolecular termination reaction, the particle should contain two or more radicals, as radicals in different particles cannot termi-nate between them. The rate of termination in a particle with n radicals is given by


2kt radicals Rate of Termination ¼ nðn  1Þ ¼ 2cnðn  1Þ ð9Þvp NA particle s One consequence of the compartmentalization of radicals in the particles is that the overall concentration of radicals in the system is much greater than in solution and bulk polymerization, and hence the polymerization rate is higher.Chain-transfer reactions to monomers and chain-transfer agents lead to the for-mation of small and mobile radicals that can exit the polymer particle. Radical de-sorption leads to a decrease in the average number of radicals per particle. Equa-tion (10), where kd is the rate coefficient for radical exit [Eq. (11)] [25], gives the rate of radical exit from a population of particles with an average number of radi-cals per particle n. In Eq. (11), l is an overall mass-transfer rate coefficient, g=h the ratio between the rate of generation of small radicals by chain transfer and the rate of consumption of these radicals (mostly by propagation), m the partition coeffi-cient of the small radicals between the polymer particles and the aqueous phase,½Mw the concentration of monomer in the aqueous phase, k tw the termination rate constant in the aqueous phase, and ½Rw the concentration of radicals in the aqueous phase.Rate of Exit ¼ kd n ðradicals/particle sÞ ð10Þ


gNA lNp
kd ¼ l 1 ðs1 Þ ð11Þhm lNp þ k p ½Mw þ 2k tw ½Rw The rate of radical exit from particles with n radicals is the product of kdðnÞ and n[Eq. (12)], and kdðnÞ is given by Eq. (13).Rate of Exit ¼ kdðnÞ n ðradicals/particle sÞ ð12ÞgNA @ lNp A kdðnÞ ¼ l 1 n ðs1 Þ ð13Þhm lNp þ k p ½Mw þ 2k tw ½Rw In Eq. (13), the term n=n takes account of the fact that in particles with n radicals,the desorption rate is proportional to n but the re-entry of newly desorbed small radicals is proportional to n.Because radical entry, exit, and termination are stochastic events, particles have a different number of radicals and the number of radicals in a given particle varies stochastically with time. Equation (14) gives the population balance of particles

6.4 Kinetics of Emulsion Polymerization 263
with n radicals; it includes the concentration of radicals in the aqueous phase, and hence the material balance for these radicals is needed from Eq. (15), where f is the efficiency factor of the initiator radicals, kI the rate coefficient for initiator de-composition [s1 ], [I] the concentration of the thermal initiator in the aqueous phase [mol m3 aqueous phase ], and k tw the termination rate in the aqueous phase[maqueous phase mol s ].
dNn
¼ ka ½Rw Nn1 þ kdðnþ1Þ ðn þ 1ÞNnþ1 þ cðn þ 2Þðn þ 1ÞNnþ2
dt
 ðka ½Rw Nn þ kdðnÞ nNn þ cnðn  1ÞNn Þn ¼ 0; 1; 2; 3 . . . ð particles/sÞ ð14Þd½Rw Np Np¼ 2f kI ½I þ kd n  2k tw ½Rw2  ka ½Rw 3ðmol/maqueous phase sÞ ð15Þdt NA Vw NA Vw For most practical cases, the pseudo steady-state assumption can be applied to the radicals in the polymer particles and in the aqueous phase. Therefore, Eqs. (14)and (15) are converted into algebraic equations by making the left-hand sides equal to zero. The exact solution for n, under pseudo steady-state conditions, has been given in terms of Bessel functions [26], but their use is not friendly. A simpler and still accurate equation for n is Eq. (16), with C defined in Eq. (17). [27].
2ka ½Rw
n¼ ð16Þ
kd þ ðkd2 þ 4ka ½Rw CcÞ 0:52ð2ka ½Rw þ kd Þ
C¼ ð17Þ
2ka ½Rw þ kd þ c Equations (16) and (17) should be solved together with Eq. (15). The solution of this system of algebraic equations includes the three limiting cases of the pioneer-ing work of Smith and Ewart [28]. When the exit rate of radicals is zero (kd ¼ 0)and the termination of the entering radical and that existing in the polymer parti-cle is instantaneous, the average number of radicals per particle is n ¼ 0:5. This is known as Smith–Ewart Case 2 and corresponds to systems in which (1) there is no chain transfer to small molecules (that is, monomers and CTAs) or these small molecules are highly water insoluble, and (2) the polymer particles are relatively small (typically dp < 200 nm). Because for Smith–Ewart Case 2 n is constant, the polymerization rate is directly proportional to the number of polymer particles.In systems in which the entry rate per particle is limited and the rate of radical desorption is relatively high (large kd ), n f 0:5. This is known as Smith–Ewart Case 1 and corresponds to systems with (1) relatively water-soluble monomers or relatively water-soluble CTAs, (2) small particles (dp < 100 nm), (3) a low rate of generation of radicals from the initiator, and (4) a large number of particles. Under these circumstances, n can be easily calculated by means of Eq. (18).

264 6 Emulsion Polymerization
n¼ ð18Þ
Under Smith–Ewart Case 1 conditions, for a given solids content and if termina-tion in the aqueous phase is negligible, n o 1=Np . Therefore, the polymerization rate is independent of the number of polymer particles. If termination in the aque-ous phase is significant, the polymerization rate increases with the number of poly-mer particles.For large particles (dp > 200 nm), high initiator concentrations or redox initia-tors, and slow termination rates (gel effect), n g 0:5 (Smith–Ewart Case 3) and n can be calculated as follows:


n¼ ð19Þ
2c
For a given solids content under Smith–Ewart Case 3 kinetics, n is inversely pro-portional to the number of polymer particles, and hence the polymerization rate is independent of the number of polymer particles if aqueous-phase termination is negligible. Otherwise, the polymerization rate increases with Np .
6.4.3
Number of Polymer Particles Particle nucleation may occur through heterogeneous nucleation, homogeneous nucleation, and coagulative nucleation (Figure 6.6). The rate of particle generation depends on the mechanism considered.
6.4.3.1 Heterogeneous Nucleation
The rate of formation of polymer particles by heterogeneous nucleation is ex-pressed by Eq. (20), where R nuc is the nucleation rate, kam the rate coefficient for radical entry into the micelles, and Nm the number of micelles in the reactor given by Eq. (21).
1 dNp Nm
¼ R nuc ¼ kam ½Rw ð particles/m 3 sÞ ð20Þ
V dt V
ðSw  CMCVw ÞNA Nm ¼ ðmicellesÞ ð21Þ
nm
In Eq. (21), CMC is the critical micellar concentration [mol m3 aqueous phase ] and hence CMC  Vw is the amount of surfactant dissolved in the aqueous phase; nm is the aggregation number (average number of molecules of surfactant per mi-celle); and Sw [mol] is the amount of surfactant that is in the aqueous phase form-ing micelles and dissolved in such a phase. Sw can be calculated by means of the overall material balance for the surfactant, Eq. (22), where ST [mol] is the total

6.4 Kinetics of Emulsion Polymerization 265
Fig. 6.6. Particle nucleation mechanisms in emulsion polymerization.amount of surfactant in the reactor, as [m 2 mol1 ] the parking area – that is, the area of the saturated surface of the polymer particles covered by 1 mol of surfactant– and A p [m 2 ] the total surface area of the polymer particles given by Eq. (23).
Ap
Sw ¼ ST  ð22Þ
as
!0:66
Vpol
A p ¼ 4:83Np0:33 ð23Þ
fpp

266 6 Emulsion Polymerization Assuming no termination of radicals in the aqueous phase and that during nucle-ation n ¼ 0:5, Smith and Ewart [28] solved Eqs. (20)–(23), obtaining Eq. (24) for the dependence of the number of particles on surfactant and initiator concentra-tions, where rv [m 3 s1 ] is the volumetric growth rate of one polymer particle.Np 2f kI ½INA fpp 0:4 as ST 0:6
o ð24Þ
Vw rv Vw
The fulfillment of this equation, in particular the 0.6th power dependence of Np with respect ST , is often considered as a proof of the occurrence of micellar nucle-ation. However, one should be aware of the assumptions used in the derivation of Eq. (24). Actually, when radical desorption is taken into account, the solution of Eqs (20)–(23) leads to Eq. (25) [29, 30], where z approaches unity as the water solubility of the monomer increases.Np 2f kI ½INA fpp 1z as ST z o 0:6 a z a 1 ð25Þ
Vw rv Vw
6.4.3.2 Homogeneous Nucleation
In homogeneous nucleation, particles are formed by precipitation of the growing oligoradicals once they exceed the critical length that makes them insoluble in water. The water solubility of the oligoradicals depends on their composition. In this context, the critical length of the oligoradicals formed from an inorganic water-soluble initiator such as potassium persulfate (that is, one containing an inorganic fragment) will be longer than that of the oligoradicals formed from desorbed radicals that are much more hydrophobic. Therefore, the rate of forma-tion of particles by homogeneous nucleation, R nuc , is the rate of polymerization of oligoradicals of critical length as expressed by Eq. (26), where ½Mw is the concen-tration of monomer in the aqueous phase, and R jcrit and Micrit are the number of oligoradicals of critical length (with jcrit > icrit ) formed from the initiator and from desorbed radicals, respectively.ðR jcrit þ Micrit ÞR nuc ¼ k p ½Mw ð particles/m 3 sÞ ð26Þ
V
R jcrit and Micrit are calculated from the balances of radicals of type R and M in the aqueous phase assuming that pseudo steady-state conditions apply [Eqs. (27) and
(28)].
jcritdþ1 d1 2f kI INA R jcrit ¼ a1 a2 ðradicalsÞ ð27Þ
k p ½Mw
kd nNp icrit Micrit ¼ a ðradicalsÞ ð28Þ
k p ½Mw 1

268 6 Emulsion Polymerization which different particles may have a different number of radicals. In addition, the number of radicals in a given particle varies with time. Therefore, the length of the macromolecules formed in a given particle depends on the number of radicals in the particle at the moment in which the macromolecule was formed. Furthermore,the architecture of the polymer formed depends on both the formulation and the kinetics of the process. Thus, for monofunctional monomers with a polymeriza-tion scheme that does not include chain transfer to polymer, linear polymers are obtained. On the other hand, branched and crosslinked polymers are obtained when multifunctional monomers are included in the formulation and when chain transfer to polymer is operative.Mathematical models for the calculation of the molecular weight distribution of linear [36] and nonlinear [37–40] polymers are available, but a detailed discussion of this issue is out of the scope of the present chapter. Instead, some simplified equations for the calculation of the molecular weights of linear polymers in the limiting cases of Smith and Ewart [28] will be presented.
6.5.1
Linear Polymers As discussed above, the MWD depends on the number of radicals per particle.Smith and Ewart [28] distinguished three limiting cases: In Case 1 n f 0:5; in Case 2 n ¼ 0:5; and in Case 3 n g 0:5. In Cases 1 and 2, the probability of having particles with more than one radical is almost negligible, and hence the system may be considered to be formed by particles with no radicals and particles with one radical (zero–one system). In Case 3, the average number of radicals is large and the kinetics is close to bulk polymerization.
6.5.1.1 Zero–One System (Smith–Ewart Cases 1 and 2)
In a zero–one system, the inactive polymer chains are formed in particles contain-ing one radical by chain transfer to monomer and by instantaneous termination upon entry of one radical.The balance for inactive chains of length m in particles with one radical (N1 ) is given by Eq. (34), where R m is the total number of radicals of length m in particles N1 .
dMm
¼ k tr; M ½Mp R m þ ka ½Rw R m m ¼ 1; 2; 3; . . . ðmacromolecules/sÞ ð34Þ
dt
Integration of Eq. (34) allows calculation of the whole molecular weight distribu-tion. However, this is computationally demanding and often the MWD is repre-sented in terms of the moments of the distribution, the kth-order moment of the distribution being defined by Eq. (35).
X
y
nk ¼ m k Mm ð35Þ
m¼1

6.5 Molecular Weights 269
Combination of Eqs. (34) and (35) yields Eq. (36), where Ruk is the generation rate of the kth-order moment of the distribution of inactive chains and mk is the kth-order moment of the distribution of active chains.Ruk ¼ ðk tr; M ½Mp þ ka ½Rw Þ k ð36Þ
V
The balance for active chains of length m in particles with one radical (N1 ) is ob-tained from Eq. (37).
dR m  X 
¼ ka ½Rw N0 þ k tr; M ½Mp R m  kd N1 dm¼1 þ k p ½Mp ðRm1  R m Þ
dt
 k tr; M ½Mp R m  ka ½Rw R m ¼ 0 ð37ÞThe first moments of the distribution of active polymer chains can be calculated from Eq. (37) by applying the pseudo steady-state assumption to the active chains[Eqs. (38)].
X
m0 ¼ R m ¼ N1 ¼ nNp ka ½Rw N0 þ ðk tr; M ½Mp þ k p ½Mp  kd ÞN1
m1 ¼ ð38Þ
ka ½Rw þ k tr; M ½Mp
2k p ½Mp
m 2 ¼ m1 1 þka ½Rw þ k tr; M ½Mp The cumulative number-(Mn ) and weight-(M w ) average molecular weights can be calculated from the moments of the MWD by means of Eqs. (39), where PM is the average molecular weight of the repeating unit in the polymer chain.
n1 n2
Mn ¼ PM ; Mw ¼ PM ð39Þ
n0 n1
On some occasions, it is interesting to know the average molecular weights pro-duced at a given moment in the process. These instantaneous average molecular weights can be calculated by means of Eqs. (40).
Rv1 Rv2
Mni ¼ PM ; Mwi ¼ PM ð40Þ
Rv0 Rv1
Combination of Eqs. (36), (38), and (40) yields Eqs. (41).

270 6 Emulsion Polymerization m1 ka ½Rw N0 þ ðk tr; M ½Mp þ k p ½Mp  kd Þm0 ka ½Rw þ k tr; M ½Mp
k p ½Mp
A PM ð41Þ
ka ½Rw þ k tr; M ½Mp m2 2k p ½Mp M wi ¼ PM ¼ 1 þ PM A 2M ni m1 ka ½Rw þ k tr; M ½Mp For the Smith–Ewart Case 2, ka ½Rw g k tr; M ½Mp and then Eq. (42) holds.
k p ½Mp
M ni A PM ð42Þ
ka ½Rw
If radical termination in the aqueous phase is negligible, from Eq. (15) we obtain Eq. (43) and hence Eq. (44).
NA Vw
ka ½Rw ¼ 2f kI ½I ð43Þ
Np
k p ½Mp Np PM M ni A ð44Þ2f kI ½INA Vw Consequently the molecular weight, as well as the polymerization rate, increases with the number of polymer particles, and hence it is possible to increase both Rp and the molecular weights at the same time by conveniently adjusting the formu-lation to increase the number of polymer particles. This is a specific feature of emulsion polymerization that cannot be achieved in any homogeneous (bulk or so-lution) free-radical polymerization.For the Smith–Ewart Case 1, k tr; M ½Mp g ka ½Rw and then Eq. (45) applies.
kp
M ni A PM ð45Þ
k tr; M
Therefore the molecular weights are controlled by chain-transfer reactions and are independent of Np .
6.5.1.2 Pseudo Bulk System (Smith–Ewart Case 3)
For Smith–Ewart Case 3, the number of radicals per particle is large and the kinet-ics approaches bulk polymerization. In this case, the concentration of radicals in the polymer particles is given by Eq. (19) and the molecular weight is mainly con-trolled by chain transfer and bimolecular termination. The rate of generation of polymer of length m in particles with i radicals is given by Eq. (46).

6.5 Molecular Weights 271
dMmi cc X
n1
¼ k tr; M ½Mp Rmi þ 2cd ði  1ÞRmi þ ði  1Þ Rki Rmkðmacromolecules/sÞ
dt iNi k¼1
ð46Þ
From Eq. (46) the moments of the inactive chains can be calculated according to Eqs. (47), and the moments of the active chains can be calculated by applying the pseudo steady state in the material balance of the active radicals [Eqs. (48) and
(49)].
m02 m
Rn0 ¼ ð2cd þ cc Þ þ k tr; M ½Mp 0
Np V V
m 0 m1 m
Rn1 ¼ ð2cd þ 2cc Þ þ k tr; M ½Mp 1 ð47Þ
Np V V
m0m2 m2 m
Rn2 ¼ ð2cd þ 2cc Þ þ 2cc 1 þ k tr; M ½Mp 2 Np V Np V V
dR m  X 
¼ ka ½Rw Np þ k tr; M ½Mp R m dm¼1 þ kp ½Mp ðRm1  R m Þ
dt
ðcc þ cd Þ X k tr; M ½Mp R m  2 Rm Rm ¼ 0 ð48Þ
Np
!1=2
X ka ½Rw Np2 m0 ¼ R m ¼ nNp ¼
2cd þ 2cc
ka ½Rw Np þ k p ½Mp m 0 þ k tr; M ½Mp m 0 k p ½Mp m 0 m1 ¼ m0 A m ð49Þð2cd þ 2cc Þ þ k tr; M ½Mp ð2cd þ 2cc Þ 0 þ k tr; M ½Mp
Np Np
ka ½Rw Np þ k p ½Mp ð2m1 þ m 0 Þ þ k tr; M ½Mp m 0 2k p ½Mp m1 m2 ¼ m0 A mð2cd þ 2cc Þ þ k tr; M ½Mp ð2cd þ 2cc Þ 0 þ k tr; M ½Mp
Np Np
The instantaneous number and weight-average molecular weights are then given by Eqs. (50).
k p ½Mp
Mni ¼ m PM
ð2cd þ cc Þ 0 þ k tr; M ½Mp
Np
ð50Þ
m m m2
ð2cd þ 2cc Þ 0 2 þ 2cc 1 þ k tr; M ½Mp m 2
Np Np
Mwi ¼ m 0 m1 PMð2cd þ 2cc Þ þ k tr; M ½Mp m1
Np

272 6 Emulsion Polymerization If bimolecular termination is predominant, then the instantaneous average molec-ular weights reduce to Eqs. (51).
M ni ¼ PM
ð2cd þ cc Þm 0
ð51Þ
 


m cc m1 2cd þ cc cc M wi ¼ 2 þ PM A Mni 1þ PM m1 ðcd þ cc Þm 0 cd þ cc 2ðcd þ cc Þ
Therefore:
M wi ¼ 2Mni if bimolecular termination is only by disproportionation; and M wi ¼ M ni if bimolecular termination is only by combination:Because for a given polymer content cc and cd are proportional to Np , the molecular weights are independent of the number of particles.On the other hand, if chain-transfer reactions are predominant over bimolecular termination, the instantaneous molecular weights are given by Eqs. (52).
M ni ¼ PM
k tr; M
ð52Þ
M wi ¼ 2Mni For this case, the molecular weights are also independent of Np and depend only on the ratio k p =k tr; M , and polydispersity is equal to 2.
6.5.2
Nonlinear Polymers: Branching, Crosslinking, and Gel Formation In contrast to linear polymers, which, once they are formed, preserve their struc-ture (molecular size) during the rest of the process, the molecular weight of the nonlinear polymers evolves throughout the whole process because they may be-come active and inactive several times during that process. Thus, inactive non-linear chains formed earlier in the process may become active through either a chain-transfer reaction to polymer or a propagation to pendent double bonds, fur-ther polymerizing and modifying their structure. The microstructure of these non-linear polymers is more complex than that of linear polymers, and besides the MWD of the polymer, other properties such as the level of branching and cross-linking density are required to fully characterize the polymer. In addition, large polymer networks (gel) are formed in some cases.The accurate calculation of the structure of these nonlinear polymers is not an easy task in homogeneous systems [41, 42], and it is even more complex in emul-

274 6 Emulsion Polymerization Fig. 6.7. Development of particle morphology.of the constituent polymers in a synergetic way. The properties of these materials largely depend on the particle morphology.Figure 6.7 illustrates the processes occurring during formation of the particle morphology during seeded semicontinuous emulsion polymerization. The reactor is initially charged with previously formed latex (seed). Then, the new monomer is fed into the reactor and the conditions are adjusted so that polymerization occurs inside the existing polymer particles. The position at which each polymer chain is formed depends on the radical concentration profile inside the polymer particles. If the entering radicals are anchored to the surface of the polymer particle, the new polymer chains will be mainly located in the outer layer of the particle. As the con-centration of the newly formed polymer chains increases, phase separation occurs,leading to the formation of clusters. Polymerization occurs in the clusters as well as in the polymer matrix, and therefore both the size and the number of clusters increase. The resulting system is not thermodynamically stable because of the high surface energy associated with the large polymer–polymer interfacial area. In order to minimize the free energy, the clusters migrate toward the equilibrium morphol-ogy. During this migration, the size of the clusters may vary because of (1) poly-merization in the cluster, (2) diffusion of polymer into or from the cluster, and (3)coagulation with other clusters. The motion of the clusters is ruled by the balance between the van der Waals attraction/repulsion forces and the resistance to flow that arises from the viscous drag. The van der Waals forces between clusters are always attractive. On the other hand, the van der Waals forces between clusters and the aqueous phase may be either attractive, bringing the clusters toward the surface of the particle, and repulsive, bringing the clusters toward the center of the polymer particle. It is worth mentioning that the van der Waals forces are pro-

6.7 Living Polymerization in Emulsion 275
portional to the interfacial tensions. The final morphology heavily depends on the kinetics of cluster migration [47–49]. Metastable morphologies can be achieved by working under starved conditions (high internal viscosity of the particles) and pro-moting grafting reactions (low interfacial tensions). Equilibrium morphologies may be attained if the internal viscosity of the particle is low, and if the polymers are very incompatible (high interfacial tensions resulting in high van der Waals forces). The equilibrium morphology is the one that minimizes the interfacial en-ergy of the system; it depends on the polymer–polymer and polymer–water inter-facial tensions [47].
6.7
Living Polymerization in Emulsion Free-radical polymerization (FRP) is a widespread technique for preparing long-chain polymers (see Chapter 4). Its success is mainly due to the broad range of monomers that can be readily polymerized by this technique, to its tolerance to-ward functional monomers, and to the mild reaction conditions required to run the polymerization. Moreover, as discussed in the preceding sections in this chap-ter, emulsion polymerization (and in general every polymerization reaction carried out using water as the continuous medium) has alleviated all the residual incon-veniences of FRP. In fact, the corresponding processes do not involve a large de-mand for organic solvents, they easily achieve large monomer conversions, and they exhibit reduced problems of heat removal, thus making the process intrinsi-cally safer.On the other hand, careful control of the polymer microstructure (in the broad sense defined in Section 6.1) is still an issue. With reference to the macromolecule structure (that is, average molecular weight, polydispersity, composition, monomer sequences, chain architecture, degree of crosslinking, and chain-end functionaliza-tion), a novel process has recently been proposed with much greater potential in terms of control capacity: living radical polymerization (LRP). By this specific process it becomes possible to obtain block copolymers by FRP, filling the gap be-tween the free-radical polymerization technique and other polymerization techni-ques such as living anionic or cationic polymerization.
6.7.1
Chemistry of LRP In conventional FRP, termination mainly by bimolecular combination limits the chain lifetime to a small fraction of the entire process time. Therefore, changes in the operating conditions (such as monomer concentration and composition, viscos-ity, temperature, and so on) affect the structure of the polymer chains produced at different stages of the process, thus increasing the product heterogeneity. In a liv-ing polymerization, instead, these changes are equally distributed among all the polymer chains, which grow uniformly during the whole reaction. Moreover, as

276 6 Emulsion Polymerization the living chains are still able to restart propagating when the monomer is com-pletely depleted, living polymerization represents a route to the production of block copolymers by successive additions of monomers, an approach clearly not possible So far, the only living processes industrially available are anionic and cationic polymerization [50, 51], which generally suffer little or no termination. In these processes, the initiation step is very fast compared to the process time and, hence,all the chains start growing almost simultaneously. The degree of polymerization,DP, increases linearly with monomer conversion and is inversely proportional to the initiator concentration. At the same time, Poisson-like distributions of the poly-mer chain length are obtained with final polydispersity values close to the ideal value of (1 þ 1/DP). Finally, the polymer retains the ionic end groups till the end of the polymerization and the reaction is simply restarted by further addition of monomer. However, this kind of polymerization is often impractical from the in-dustrial viewpoint, since the main requirements are high purity of all the reactants,very low temperatures, and the use of solvents. Moreover, it does not work with several widely used monomers, such as styrene.On the other hand, as previously pointed out, FRP does not suffer the same lim-itations. It can be applied to a broad range of monomers, nearly all vinyl and vinyl-idene monomers, it is easily operated in the presence of impurities, such as resid-ual inhibitor residues and traces of oxygen, and over a wide temperature range [52,53]. Therefore, it is quite natural to attempt to establish living conditions in such a process. However, it is not practical to approach such conditions by simply reduc-ing the radical concentration so as to minimize the rate of bimolecular termination(a second-order reaction with respect to radical concentration, the propagation be-ing a first order reaction). Besides the drawback of the corresponding reduction of the polymerization rate, this would lead to the production of polymer chains with extremely high molecular weight, since the instantaneous DP is given by the ratio between the frequencies of propagation and termination (compare Chapter 4). Us-ing LRP, this control is instead restored by introducing an additional but reversible termination reaction with a ‘‘capping’’ species, generically indicated as X. In this case, each chain experiences a sequence of activation–deactivation steps and the in-stantaneous DP grown during the generic active step is given by the ratio between the rates of propagation and reversible termination (k p ½M=kt ½X, where kt indi-cates the rate constant of the reversible termination). If the rate of this new termi-nation is so large as to be dominant with respect to that of the irreversible termina-tion reactions and comparable with that of propagation, polymer growth will be distributed all over the process.Living polymerization is typically started by introducing into the system an ‘‘ini-tiator’’ providing the capping species, indicated as RX. This initiator reactivates it-self many times and adds some monomer units before going back to the so-called dormant state in the form R–ðMÞn –X, where ðMÞn indicates a polymer chain made of n monomer units. In other words, the living process can be regarded as the in-sertion of a well-defined number of monomer units between the groups R and X,which are always acting as polymer chain ends and, thus, defined a priori. The av-

6.7 Living Polymerization in Emulsion 277
erage DP of the polymer is given by the ratio between the converted monomer,M0 XT (where XT is the monomer conversion and M0 the initial amount of mono-mer), and the total number of polymer chains. Note that some irreversibly termi-nated chains are produced anyhow, since these reactions are always taking place in competition with the activation/deactivation reactions. If the fraction of terminated chains remains negligible compared to the initial amount of initiator, RX, this last value corresponds also to the concentration of the dormant chains in the system and the DP grows linearly with conversion (DP ¼ M0 XT =RX). Moreover, if the number of active periods each chain experiences is large enough (that is, if the number of monomer units added per active period is small enough), the polymer growth is distributed throughout the duration of the process and all the chains grow uniformly, leading to low polydispersity of the chain length distributions.Different living mechanisms are available and the main ones are briefly enumer-ated in the following sections.
6.7.1.1 Nitroxide-mediated Polymerization (NMP)
The ability of stable nitroxide radicals to react with carbon-centered radicals and to act as radical inhibitors was already known at the beginning of the 1980s [54],when Solomon and co-workers showed that the reversible reaction of nitroxides with growing polymer chains can be used to produce low-DP polymers [55]. But it is only in the 1990s with the work of Georges and co-workers [56, 57] that this novel polymerization technique, and in general LRP, received the attention they deserved.This living mechanism [Eq. (a)] consists of the reversible combination of the growing radical chains, R n , and the so-called ‘‘persistent radical species’’, X (the ni-troxide radical group), to form dormant polymer chains, R n X:
kde
Rn þ X S RnX ðaÞ
kac
Today, many different routes are known which use different persistent radicals[54–58]. Among these, TEMPO is by far the most widely used, even though it suf-fers very limited applicability to monomers unlike styrene, and requires high oper-ating temperatures (about 120–140 C). More recent studies were aimed to reduce the operating temperature and to broaden the monomer applicability so as to en-large the spectrum of block copolymers accessible by this technique [58]. Despite these efforts, the range of application remains quite limited.
6.7.1.2 Atom-transfer Radical Polymerization (ATRP)
Atom-transfer radical polymerization (ATRP) was first reported by Kato et al. [59]and by Wang et al. in 1995 [60, 61]. This mechanism is based on the so-called atom-transfer radical addition reaction, and it is catalyzed by a metal: the homolytic cleavage of the bond in an organic halide occurs through transfer of the halogen to the metal complex, accompanied by oxidation of the metal atom. The catalytic cycle is closed by back-transfer from the transition metal to the final adduct of the halo-gen. It is clear that, if the radical produced can undergo a few propagation steps

278 6 Emulsion Polymerization before participating in the back-transfer, and if this product is still able to undergo another transfer cycle, this reaction can be used to produce the same exchange be-tween active and dormant states as is found in NMP. The resulting reversible reac-tion is represented by Eq. (b), where X indicates the halogen atom, MeðnÞ the metal with the corresponding oxidation state and Li the ligand.R n þ X –Meðnþ1Þ =Li S R n X þ MeðnÞ =Li ðbÞ
kac
ATRP owes most of its success to its high compatibility with many different mono-mers, such as styrene, acrylates, methacrylates, (meth)acrylamides, and acryloni-trile, which made this technique readily available for the production of several new block copolymers [62]. Even though the majority of the work has been done with copper as the transition metal, styrene ATRP has been carried out using Fe-,Ru-, Ni-, Pd-, and Co-based systems [62]. Note that ATRP does not require the high reaction temperature typical of NMP and this is also part of the success of this polymerization technique. Different ligands have been used to solubilize the cop-per atom and it has been noticed that they not only prepare the copper for the re-action, but can also modify the reactivity of the metal toward both the activation and deactivation reactions. Actually, the presence in the system of a metal, the need for complex ligands to solubilize it, and the deep color typically imparted by this complex to the final polymer (if not removed) represent the major drawbacks of this process. Thus far, chlorine and bromine have been used successfully as the halogen atoms, whereas iodine gives rise to side reactions [63].
6.7.1.3 Degenerative Transfer (DT)
As mentioned above, in both NMP and ATRP the exchange between the active and the dormant state is based on a reversible (although different) termination mecha-nism. Therefore, the exchange directly affects the radical concentration. In LRP by DT, instead, this exchange is carried out by direct transfer of the o-end group be-tween an active and a dormant chain. When an iodine atom is used as end group,the reaction can be summarized by Eq. (c), where R n I indicates the generic dor-mant species with iodine.
kex
Rn þ Rm I ! Rm þ RnI ðcÞTherefore, the main difference from the previous two systems is that this living mechanism does not form new radicals and a conventional initiator is needed to start and ‘‘sustain’’ the reaction. The initial amount of this species has to be prop-erly selected. As a matter of fact, since the living reaction of Eq. (c) is not affecting the radical concentration, the final concentration of the chains terminated by bimo-lecular combination will be half of the initial concentration of initiator. Therefore,the initial concentration of the species carrying the iodine group (in the following simply called the ‘‘transfer agent’’) determines the final DP of the polymer, pro-vided that the initiator concentration is small compared to that of the transfer agent.

6.7 Living Polymerization in Emulsion 279
Only a few papers have appeared in the literature dealing with LRP by DT [64–66], and the applications are almost completely limited to the homopolymerization of styrene. In this case, it was possible to obtain good control of the final CLD, with polydispersity values as low as 1.3–1.4. Better performances are difficult to obtain with styrene, mainly because of the limited transfer activity of the iodine atoms.This is the main reason for the very poor results obtained when applying this pro-cess to the polymerization of acrylates (for example, n-butyl acrylate), and for the complete lack of control reported for other monomers [64–66].
6.7.1.4 Reversible Addition–Fragmentation Transfer (RAFT) Polymerization
The RAFT process can be regarded as a special case of degenerative transfer. As shown in Eq. (d), the reaction proceeds through the direct interaction of an active and a dormant chain with the formation of a reaction intermediate involving both chains [67, 68]. At this stage, the reaction can either go back, forming the initial radical again, or proceed forward with the transfer of the Y ¼ CðZÞ–Y moiety from the dormant to the active chain, which is now identified as the transfer species.
ðþÞ
kadd kfrag
R n þ R m YCðZÞY ! R m YC . ðZÞYR n ! R m þ R n YCðZÞY ðdÞ
ðÞ
kfrag
Note that the best results have been reported when using a sulfur atom as the Y group [67–70]. The process is started by introducing into the system a so-called RAFT agent, the structure of which can be described as R–Y –CðZÞ ¼ Y. The cor-rect choice of the R group, or leaving group, is of paramount importance, not only because this is going to be one polymer chain end (the other end being occupied by the RAFT group), but mainly because it influences the initial reactivity of the RAFT agent. From Eq. (d), it is in fact clear that in the first addition reaction an intermediate species is formed, having the R group on one side and the polymer radical on the other. The fate of such a species – that is, the probability that the following fragmentation reaction proceeds backward or forward – depends mainly upon thermodynamic considerations, namely upon which, the polymer radical or the leaving-group radical, is the more stable. In other words, if the leaving group generates a radical that is too unstable compared to the polymer radical, the RAFT is always going to proceed backward, and thus no control is actually achieved. Similar considerations apply in block polymerization of two monomers,where the first block can be regarded as a special case of the R group. Accordingly,care must be given to the correct choice of which monomer has to be polymerizedfirst.Even though the most satisfactory results have been obtained by RAFT polymer-ization of styrene (Figure 6.8 shows an example of the application of RAFT to bulk styrene polymerization, indicating the good control of polymer growth), the pro-cess is also very effective for many other monomers, such as acrylates and metha-crylates [67–70]. Moreover, the operating temperatures used to carry out this poly-merization are usually close to those typical of conventional radical polymerization

280 6 Emulsion Polymerization Fig. 6.8.Example of application of RAFT to bulk polymerization of styrene. Left: degree of polymerization (DP)and polydispersity versus conversion; right: evolution of molecular weight distribution as measured by GPC [71].[67–70]. The low temperature, along with the wide range of compatible mono-mers, makes this mechanism one of the most promising techniques to be applied on an industrial scale for the production of new materials, in competition with ATRP. Once more, a significant drawback is the need to remove the sulfur atoms,which confer to the final polymer a deep color ranging from yellow to red, from the product.
6.7.2
Polymerization of LRP in Homogeneous Systems As already pointed out, the final aim of LRP is to strictly control the architecture of the polymer chain by minimizing the fraction of dead chains in the system while maintaining uniform growth of the whole population of chains. The homogeneity of chain lengths is the key factor in homopolymerization and it is usually ex-pressed in terms of polydispersity ratio, Pd (defined as the ratio between weight and number-average molecular weight; Pd ¼ 1 when all chains are of the same length). The fraction of dead chains becomes even more important when the pro-cess is intended to produce block copolymers, in which case terminated homo-polymer chains represent a significant drawback with respect to the product qual-ity. To minimize terminations, different strategies are effective for the different living mechanisms, as briefly reviewed below.For clarity, let us start by discussing the application of NMP to a bulk system. As previously pointed out, a successful LRP requires the termination reaction between the growing radical chain and the nitroxide (referred to simply as the deactivation reaction hereafter) to be dominant with respect to bimolecular irreversible termina-tion. It has been exhaustively demonstrated that, in spite of the fact that these bi-molecular reactions, deactivation and termination, are both very fast and controlled

6.7 Living Polymerization in Emulsion 281
by diffusion, deactivation soon becomes the favored reaction path anyway. This is often referred to as the ‘‘persistent radical effect’’ [72, 73]. To explain this behavior,let us first point out that, because of the nature of NMP, where new radical chains can be created from dormant ones by activation, this mechanism does not require a radical initiator. Accordingly, when the reaction starts, dormant chains start to activate, building up a radical concentration. At the same time, the two bimolecular termination reactions, termination and deactivation, start to operate. But, while de-activation simply produces a dormant chain again, termination subtracts polymer chains from the activation/deactivation equilibrium and, according to simple stoi-chiometric arguments, trapping radicals (nitroxides) are accumulated. The ulti-mate effect of this accumulation of trapping radicals is that deactivation becomes faster and faster, thus shortening the active lifetime of the active radicals and their concentration, but also decreasing the final amount of irreversibly terminated chains. Fischer et al. derived Eq. (53) to evaluate the concentration of radicals for an NMP [72], where ðRXÞ0 is the initial concentration of dormant chains.


kac ðRXÞ0 1=3
R¼ ð53Þ
This equation confirms that the radical concentration steadily decreases in time.Albeit the living action in ATRP takes place by a different reaction mechanism,the same arguments presented above for NMP can be used. It can be easily verified that the concentration of the metal in the reduced form, MeðnÞ [compare the corre-sponding reaction scheme, Eq. (b)], remains roughly constant during the whole process, and therefore the activation process can be approximated as a monomolec-ular process as in NMP.On the other hand, degenerative transfer and RAFT are characterized by com-pletely different kinetics when the polymerization is performed in a homogeneous system. Given the nature of the living mechanism of these two systems, based on a transfer reaction, radical concentration is not affected by the living system. Thus,the persistent radical effect is totally absent and the kinetics of DT and RAFT is identical to that of a conventional nonliving system. Accordingly, an initiator is necessary to sustain the reaction and the concentration of radical chains is set by the equilibrium between initiation and bimolecular termination; that is, the well known formula in Eq. (54), where Ri represents the rate of radical generation,holds in this case also.R ¼ ðRi =kt Þ 1=2 ð54ÞNo matter which is the living mechanism under consideration, it is fundamental to keep the radical concentration as low as possible to ensure a final fraction of dead chains which is negligible with respect to the dormant polymer. In a bulk or solution polymerization, the final concentration of dead chains is a function of the radical concentration only: high polymerization rates correspond to high dead

282 6 Emulsion Polymerization chain concentrations. It can be shown [72] that the time needed to obtain 90% con-version after proper tuning of the process parameters so as to obtain a defined frac-tion of dead chains, f, is given by Eq. (55), where the factor C is different for the different living mechanisms (NMP/ATRP: C ¼ 4=3; RAFT/DT: C ¼ 1).t 90; f ¼ C ð55Þ
fkp2 ðRXÞ0
Using typical parameter values for styrene homopolymerization at 80 C, reaction times of the order of magnitude of 100 h are needed to have f ¼ 0:05. Even if it can be shown that low polydispersity values can be achieved also at higher f values(> 0.2–0.3) [72, 73], these values cannot be accepted in copolymerization. While increased reaction temperatures and thus propagation rates generally should pro-mote smaller dead chain contents [compare Eq (55)], in the case of styrenic copoly-mers this leads to limited improvements only, since undesired side reactions nega-tively affecting the polymer quality (such as chain transfer and thermal initiation)become more and more important.
6.7.3
Kinetics of LRP in Heterogeneous Systems A possible way out of this problem comes from the application of LRP to segre-gated systems, namely emulsion polymerization. Let us focus on a system charac-terized by the presence of very small polymer particles and a water-soluble initiator.When a radical is formed and, after a few propagation steps, absorbs or enters a particle without radicals, this same radical goes on propagating until the particle experiences a second entry. At this point, given the very small size of the particle and the diffusion-limited nature of the termination reaction, the two radicals react with each other almost instantaneously to produce a dead chain. Thus, the particle goes back to a state without radicals until another entry takes place. This particular kinetic condition is often referred to as a zero–one system (compare Section 6.5)and, under the assumption of negligible radical desorption, the corresponding av-erage number of active chains per particle is equal to 0.5 (Smith–Ewart Case 2).Kinetic behavior similar to that of a zero–one system is readily established with DT and RAFT as living mechanisms [74]. Since a transfer reaction is taking place in both cases, the same kinetics as in a nonliving process is again operative. There-fore, the fraction of dead chains in the system can be adjusted by tuning the fre-quency of entry properly, while the transfer reaction rate independently controls the homogeneity of polymer growth. It is again important to notice that in emul-sion polymerization the rate of formation of dead chains by irreversible termina-tion is regulated by the frequency of entry only, while the polymerization rate is not affected. Accordingly, in principle it is possible to maintain the same high polymerization rates typical of emulsion systems, while keeping under control the ratio between dormant and dead chains, and thus the final quality of the polymer.

6.7 Living Polymerization in Emulsion 283
Fig. 6.9. Expected kinetics for LRP in emulsion: (a) RAFT/DT;(b) NMP/ATRP. i ¼ number of radicals per particle;r; fc ; fac ; fde ¼ frequencies of radical entry, bimolecular termination, activation, and deactivation, respectively.The corresponding time evolution of the number of active chains per particle, i, is therefore as sketched in Figure 6.9(a). Note that the living reactions are not in-volved at all. Accordingly, the average number of active chains per particle remains Comparable propagation rates are not found when using NMP or ATRP [74, 75].Referring once more to NMP for simplicity, it is not possible to have one radical per particle for a significant time period, since each activation event will produce a transient and a persistent radical: keeping in mind the extremely small size of a typical polymer particle produced in emulsion, the radicals recombine almost im-mediately. In other words, the principle behind radical segregation cannot hold when NMP is active, since two radicals are generated each time an activation reac-tion occurs in the particle. Note that the average time particles spend with zero and one radical is 1=fa and 1=fd , respectively, the frequencies of activation and deactiva-tion being fa ¼ kac ½RXp and fd ¼ kde ½Xp . This reduction of the polymerization rate cannot be counteracted by increasing the rate of the activation reaction: when fa approaches fd , a second activation is more likely to take place instead of a deactiva-tion, the probability of this event being equal to fa =ð fa þ fd Þ. Accordingly, the two transient radicals may terminate with each other and accumulate two persistent radicals in the particle, thus increasing the frequency of deactivation. In other words, each particle accumulates persistent radicals till the rate of the second acti-vation becomes small enough (that is, when fa f fd ), and the average number of radicals per particle has a value much smaller than 0.5, that typical of DT and RAFT in emulsion. Note that, when fa f fd , this average number of radicals can be readily estimated as fa =fd . Actually, it has been shown that the process kinetics approaches that of the corresponding bulk process, thus canceling the advantages of operating in emulsion from a kinetic point of view [75]. The corresponding sketch of the time evolution of the number of active chains per particle is shown in Figure 6.9(b). An average value of active chains per particle, much smaller than
0.5, is readily verified.

284 6 Emulsion Polymerization Application of LRP in Heterogeneous Systems
6.7.4.1 Ab-initio Emulsion Polymerization
Besides the kinetic considerations illustrated in Section 6.7.3, performing a LRP by ab-initio emulsion polymerization is still the objective of major research efforts by many groups, mainly because this process is generally simpler and more commer-cially established, albeit less versatile, than other emulsion processes. Nonetheless,this process turns out to be much more complicated than others (for example,miniemulsion) when applied to LRP, since the multiphase environment greatly complicates the global kinetics of the process. The need to have the RX species,which is generally very hydrophobic, inside the polymeric reaction locus demands fast material transport of this species out of the monomer droplets, where it is ini-tially stored, across the water phase to the polymer particles, where the reaction actually takes place. This transport must satisfy two fundamental requirements[74]: (1) RX must be readily available in the particles so that all the chains can start growing from the beginning of the process, and (2) RX must be uniformly dis-tributed among all the particles.Despite the initial rather unsuccessful efforts to conduct an NMP in emulsion,mainly due to stability problems, this process proved to give good results [76].Great care must be put into the choice of the nitroxide: the use of nitroxides that are too hydrophobic leads to uncontrolled reactions, while nitroxides that are too hydrophilic lead to long induction periods, mainly due to the significant duration of LRP in water [77]. It has also been observed that polymerization in the mono-mer droplets plays a fundamental role. Although this always happens to a small extent also in non-LRP, the impact of this process is very limited, due to the large difference in radical concentrations (and therefore reaction rates) between mono-mer droplets and polymer particles. On the contrary, in systems like NMP and ATRP, segregation is not effective at enhancing the radical concentration in the polymer particles, and thus polymerization in monomer droplets proceeds as in particles [78].Unfortunately ATRP, which is more versatile than NMP, can with difficulty be applied to ab-initio emulsion polymerization, mainly because its chemistry, which involves one more species (the metal–ligand complex), becomes even more com-plicated [79]. Earlier attempts failed just because anionic surfactants reacted with the metal [80]. However, switching to nonionic ones did not bring substantial improvements. Micron-sized particles were often observed, with the exception of n-butyl methacrylate (BMA), where submicron particles were obtained, although rather susceptible to coagulation [81].As shown earlier, the RAFT mechanism is better suited to emulsion polymeriza-tion, where it is possible to exploit the segregation typical of these systems. How-ever, even in this case, its application to ab-initio emulsion polymerization did not enjoy much success. The reasons for such behavior are always the same: (1) poor transport of very hydrophobic species from monomer droplets to particles, and (2)the possible occurrence of some polymerization in the monomer droplets that

6.7 Living Polymerization in Emulsion 285
makes these species even more insoluble [82]. A notable exception is again the polymerization of BMA [83]. As for NMLP, more water-soluble RAFT agents failed to solve the problem, since they mainly move the polymerization into the water phase. A simple solution to this problem could come from the use of surface-active RAFT agents. These are sufficiently water-soluble to diffuse fast across water, but their surface activity is high enough to keep them away from the aqueous-phase chemistry. The best example of such a species is represented by xanthates [84],even though they suffer from low activity in controlling the polymer growth. The poor results in terms of polydispersity are even worse because these RAFT agents remain anchored to the surface, although this can turn out to be an effective way to produce core–shell particles.
6.7.4.2 Miniemulsion Polymerization
Among different alternatives, a very effective way to operate an LRP in segregated systems is indeed miniemulsion. In this case, small monomer droplets are the pri-mary locus of reaction and all the difficulties from interphase transfer vanish, since monomer and all the other hydrophobic species required to run an LRP are already in the main reaction locus. However, further difficulties have been reported, such as incomplete droplet nucleation and colloidal stability problems [74, 82, 85]. More subtle is the evidence of instabilities in the miniemulsion due to the kinetics of LRP. In contrast to conventional systems, where long chains are created from the beginning, in LRP all the polymer chains are short initially. This might lead to superswelling states of the droplets and, eventually, to destabilization [86].Despite these difficulties, literature abounds with examples of styrene miniemul-sion polymerizations by NMP; all show good control of the MWD and the afore-mentioned problem of slow polymerization rates [76]. Due to the improvements in nitroxide efficiency, studies involving polymerization of acrylates have also ap-peared [87]. However, a fast buildup of nitroxides is often observed in this case,which depresses the polymerization rate. A possible solution is represented by the removal of the excess of nitroxides, which has also given good results in styrene polymerization [87].ATRP is also very effective when applied to miniemulsion systems. Still, care is needed in the choice of surfactant (nonionic) and ligand (not too water-soluble).Also, it proved effective to run a so-called ‘‘reverse ATRP’’, that is, starting from the metal in the oxidized form, since the original metal complex (such as Cu(I))is rather sensitive to oxidation during miniemulsion formation by sonication[88].Finally, RAFT also has been successfully applied to miniemulsion polymeriza-tion with several monomers [82]. Figure 6.10 shows a typical result of a block co-polymerization of MMA and styrene, which proved to give good control of polymer growth together with high polymerization rates. Still, some problems remain un-solved. Among them is the evidence of long induction times, generally attributed to high desorption rates as a consequence of the initial exchange reaction to very short species. This has been shown to be easily solved by using oligomeric RAFT agents [74].

286 6 Emulsion Polymerization Fig. 6.10. Example of application of RAFT to miniemulsion polymerization for the formation of poly(methyl methacrylate)-b-polystyrene [74]. Left: conversion profile; right: evolution of molecular weight distribution as measured by GPC.Emulsion Polymerization Reactors Emulsion polymers are ‘‘products-by-process’’ whose microstructure and proper-ties are determined during the polymerization. Therefore, the reactor type, the op-eration mode, and the control strategy play a key role in achieving an efficient, safe,and consistent production of high-quality emulsion polymers.Reactor Types and Processes In principle, both stirred-tank reactors and tubular reactors, and combinations thereof, may be employed.
6.8.1.1 Stirred-tank Reactors
The stirred-tank reactor is the reactor most commonly used in emulsion polymer-ization. This reactor may operate in batch, semibatch, or continuous mode.A batch reactor is a closed system in which time is the only independent vari-able. The batch operation can be used for small-scale production of homopolymers from monomers with a relatively low heat of polymerization. However, the draw-backs associated with this type of operation limit its industrial use. These draw-
backs are:

Control of the polymer properties is impracticable.
Productivity is low, considering the load, unload, and cleaning times.
Because all of the monomer is initially charged into the reactor, the heat genera-tion rate is high and the control of the reactor temperature is very difficult.
Batch-to-batch variations due to irreproducible particle nucleation may jeopardize product consistency. The use of seeded polymerization mitigates the problem.

6.8 Emulsion Polymerization Reactors 287
In semibatch operation, some fraction of the reactants (the initial charge) is charged into the reactor initially, and the rest of the formulation is fed in con-tinuously over a period of time. Most commercial emulsion products are manu-factured in semibatch reactors. The main characteristic of this process is its great flexibility. By varying the composition and amount of the initial charge,as well as the composition and flow rates of the feeds, both temperature and poly-mer quality can be controlled. A wide range of products are accessible using this technique, which allows any polymer property to be tailored, including copolymer composition, molecular weight distribution, polymer architecture, particle mor-phology, and particle size distribution. In addition, a large portfolio of products can be produced with a single reactor. The main drawback of this operation mode is the relatively low productivity, which is being compensated by using larger reactors.In continuous operation mode, both feed and effluent streams flow continu-ously. The main characteristic of a continuous stirred tank reactor (CSTR) is the broad residence time distribution (RTD), which is characterized by a decreasing ex-ponential function. The same behavior describes the age of the particles in the re-actor and hence the particle size distribution (PSD) at the exit. Therefore, it is not possible to obtain narrow monodisperse latexes using a single CSTR. In addition,CSTRs are liable to suffer intermittent nucleations [89, 90] that lead to multimodal PSDs. This may be alleviated by using a tubular reactor before the CSTR, in which polymer particles are formed in a smooth way [91]. On the other hand, the copoly-mer composition is quite constant, even though it is different from that of the feed.The broad RTD together with the problem of heat removal in large stirred tanks make it difficult to achieve high conversions in a single tank. An arrangement of multiple stirred tanks in series allows better heat removal and presents a narrower residence time distribution, which in turn leads to a narrower PSD. Moreover,copolymer composition and molecular weight can be controlled by intermediate feeds of monomer or chain-transfer agents. CSTRs in series are used for high-tonnage productions such as styrene–butadiene rubber (SBR), but are not well adapted to the production of specialties because of the difficulties associated with grade transitions.
6.8.1.2 Tubular Reactors
From a safety point of view, tubular reactors are advantageous because they have a large area/volume ratio and hence the heat removal capacity is higher than that of the CSTR.A continuous plug-flow reactor is somewhat similar to a batch stirred-tank reac-tor, where reaction time is equivalent to the space time (t) in the tube. For this re-actor type, grade transition between different polymer grades is instantaneous, but the control of polymer properties is almost impracticable. An important disadvan-tage of the tubular reactor is the inadequate mixing that can lead to phase separa-tion, reactor plugging, and wall fouling [92]. Several modifications have been per-formed to improve radial mixing and minimize the associated problems, but to date tubular reactors have not been widely utilized for industrial production. The

288 6 Emulsion Polymerization most important modified tubular reactors include loop reactors, pulsed-flow reac-tors, wicker-tube reactors, and Couette–Taylor flow reactors.The continuous-loop reactor is probably the only tubular reactor used in com-mercial production of emulsion polymers [93, 94] and its use is limited to produc-tion of vinyl acetate homopolymers and copolymers (with ethylene and Veova 10)[95–97]. A continuous-loop reactor consists of a tubular loop that connects the inlet and the outlet of a recycle pump. The macromixing of such reactors is between that of plug-flow and well-mixed reactors. Commonly, the loop recirculation rate is significantly greater than the feed rate. Under these circumstances, the resi-dence time distribution is very close to that of a CSTR [98]. Its high heat-transfer area/reactor volume ratio allows efficient heat removal and hence high conversions in short residence times can be achieved. This results in a substantial reduction in the reactor volume. The main disadvantage of this reactor is that highly mechani-cally stable formulations are required to prevent shear-induced coagulation at high recycling rates.Pulsed-flow reactors consist basically of a column in which a periodic external pulsation is provided. The goal of the pulsed flow regime is to achieve highly effec-tive mixing, minimizing phase segregation, wall fouling, and tube plugging [99,100]. The introduction of sieved plates, Raschig rings, or baffles is reported to im-prove mixing [101, 102]. However, these internal elements may be sources of coag-ulation, and cleaning coagulum from these reactors will be very difficult.The wicker-tube reactor consists of a coiled tube which meanders between solid,fixed, cylindrical supports. The heat removal capacity is high and it is claimed that the multiple changes in flow direction allow the production of a polymer disper-sion with a very low coagulum content [103, 104].The Couette–Taylor reactor consists of two concentric cylinders of which the outer one is fixed and jacketed, while the inner one rotates as a stirrer. In this way, the reaction takes place in the annulus formed between the two cylinders.For a specific configuration, if the inner cylinder exceeds a certain speed the fluid inside the gap develops counter-rotating toroidal vortices. The boundaries of the vortices represent a barrier for axial mixing, while radial mixing inside each vortex is good. Consequently the Couette–Taylor reactor may be modeled as a hydro-dynamically formed train of continuous stirred tank reactors. Good radial mixing,which reduces phase segregation and plugging, and high heat removal capacity are the main characteristics of these reactors [105–108].Reactor Equipment Despite the availability of different reactors, commercial emulsion polymerization is mostly carried out in stirred-tank reactors or in reactors that have a similar macromixing behavior, such as the loop reactor. The mixing problems associated with tubular reactors, together with a lack of flexibility in controlling product prop-erties, lowers the attractiveness of tubular reactors for commercial production.Therefore the discussion will be limited to stirred-tank reactors.

6.8 Emulsion Polymerization Reactors 289
The stirred-tank reactors used for the production of emulsion polymers have sizes ranging from 5 to 50 m 3 . Stainless steel vessels with a height/diameter ratio between 1.1 and 1.3:1 are usually employed. These reactors must be equipped for mixing and heat transfer.
6.8.2.1 Mixing
For agitator selection it is critical to define correctly the mixing requirements. In practice, the most common cause of agitator failure is not miscalculation of the power or rotational speed, but incorrect definition of the agitator main task. In emulsion polymerization, the mixing equipment must ensure that the tasks of emulsification and blending are performed well, and it must facilitate mass and heat transfer without causing coagulation [109].The power consumption of an impeller is the product of the pumping capacity(circulation flow rate) and the velocity head, which is directly related to shear rate and turbulence. Depending on the type and size of the impeller, either the flow or the turbulence can be favored [110]. Axial flow impellers usually produce a fluid motion that is downward at the central axis of the vessel and upward in the wall region. They are designed to produce a high flow/power ratio with little turbulent loss. The designs of axial-flow impellers are derived from three-blade propellers.Radial-flow turbines produce a radial fluid motion from the impeller to the wall,where the radial flow separates into an upper and a lower circulation loop. They are characterized by a relatively low flow/power ratio, with much of the energy dis-sipated by turbulence around the impeller. Radial-flow turbines have flat blades or a disk with flat blades.In emulsion polymerization, a high shear rate may cause coagulation. However,a certain amount of turbulence is required for emulsification and to avoid phase segregation. Moreover, high fluid circulation is needed in order to guarantee the macroscopic uniformity and to enhance mass and heat transfer. In this way,mixed-flow turbines with features of both radial and axial flow can be useful. The most common of these impellers is the 45 angled blade turbine. Multiple impel-lers on the same shaft can also be employed.In any case, the agitation requirements in emulsion polymerization are often re-lated to the operation mode. In batch operations, vigorous agitation is required in the initial stages to avoid monomer segregation and promote good phase disper-sion. Later on, a lower agitation intensity is needed to avoid shear-induced coagula-tion. In semicontinuous operations, multiple impellers are normally used to en-sure agitation as the level goes up. In addition, it is very important for mixing of the entering reactants to be instantaneous: this requires a high circulation flow rate. If neat monomer is fed in, a certain amount of turbulence is required to facil-itate dispersion. This requirement may be avoided by feeding a pre-emulsion. The location of the addition point of the reactants is important, to avoid monomer pools and other problems derived from the accumulation of initiators, which in-creases ionic strength and thus promotes coagulation, and the accumulation of emulsifiers, which leads to local nucleations and/or flocculations. Continuous op-eration requires mixing characteristics similar to those of the semibatch process.

290 6 Emulsion Polymerization Empirical correlations for the key aspects of mixing, such as power consump-tion, emulsification, liquid circulation, and mixing time, can be found elsewhere[111]. The advances in computational fluid dynamics (CFD) combined with new techniques of measuring the local flow pattern are likely to transform the whole
6.8.2.2 Heat Transfer
Polymerization reactions are highly exothermic and the heat generated must be removed in order to control reactor temperature. In commercial emulsion poly-merization, the heat removal rate from the reactor is often the factor limiting pro-ductivity. Both safety and product quality depend on the heat removal capacity of the reactor. For industrial reactors, it is often not sufficient to operate with a simple cooling jacket and other devices must be incorporated in the design [112]:
improved jacket design to increase turbulence, and hence the heat-transfer co-efficient (dimpled, half-pipe);
cooled baffles (the use of internal coils is not an option because they tend to in-crease fouling and coagulation);
external loop heat exchangers for the reaction medium – in this case, a mechan-ically stable formulation is required; external heat exchangers can be used also to cool the feedstreams in continuous and semicontinuous operations;
reflux condensers.Increasing the agitation speed to increase the internal heat-transfer coefficient is not an option because a high impeller speed can lead to coagulum formation.In any case, heat-transfer requirements are largely determined by the operation mode. Batch is the most critical operation because high polymerization rates are achieved due to the high monomer concentration. In semibatch mode, the heat generated can be easily controlled by the monomer feed rate. In these reactors, ex-tra cooling is provided by the cold feed. In continuous mode, the continuous cold feed facilitates the control of the reactor temperature, particularly when the reactor temperature is high.
6.9
Reaction Engineering Independently of the operation mode (batch, semibatch, or continuous), in well-mixed stirred-tank reactors the properties do not vary significantly with the posi-tion in the reactor, and time is the only independent variable. Therefore, the neces-sary balances for the reactor design may be made at macroscopic level – that is, for the reactor as a whole.

292 6 Emulsion Polymerization Tab. 6.3. Usual conversion definitions.[a]Conversion Batch Semicontinuous Continuous
Ð
Ni0  Ni Ni0 þ Fie dt  Ni Fie  Fis Fractional conversion of monomer i ÐNi0 Ni0 þ Fie dt Fie
P P Ð P
ðNi0  Ni Þ ðNi0 þ Fie dt  Ni Þ ðFie  Fis ÞOverall fractional conversion P P Ð P Ni0 ðNi0 þ Fie dtÞ Fie
P Ð
ðNi0 þ Fie dt  Ni ÞOverall global conversion P
[a] N
i0 ¼ number of moles of monomer i at time zero; NTi ¼ the total amount of monomer i [mol] to be fed in a semicontinuous operation.In order to calculate the gravimetric conversions, it is necessary to multiply each term by the molecular weight of the corresponding monomer.The calculation of the monomer conversion depends on the operation mode. Table
6.3 summarizes the expressions for the different reactors. The instantaneous co-
polymer composition refers to the composition of the copolymer that is being formed at a given time. Referred to monomer 1 this composition is given by Eq.
(60), where Rpi is the polymerization rate of monomer i.
Rp1
y1i ¼ X ð60Þ
Rpi
The cumulative composition is the average composition of the copolymer formed up to a given time, as stated in Eq. (61), where Npoli is the amount [mol] of mono-mer i polymerized.
Npol1
y1cum ¼ X ð61Þ
Npoli
In continuous operations under steady-state conditions, y1i ¼ y1cum during the whole process.
6.9.2
Heat Balance Assuming that the energy balance for the reacting systems is essentially an en-thalpy balance, this reduces to Eq. (62), where cpi and cpie [J mol1 K1 ] are the heat capacity of compound i in the reactor and under the entry conditions, respec-tively. Rpi [mol m3 s1 ] is the polymerization rate of monomer i, (DHri ) [J mol1 ]

6.9 Reaction Engineering 293
is the polymerization heat of monomer i under the reactor conditions, T [K] is the reactor temperature, Te [K] the temperature of the feeds, Q loss [J s1 ] the heat losses to the surroundings (for example, through the reactor lid), Q stirring [J s1 ]the heat produced by the agitator, and Q transfer [J s1 ] the heat removed through the heat removal devices (cooling jacket, cooling baffles, external heat exchanger and reflux condenser).
X dT X X
Ni cpi ¼ Rpi ðDHri ÞV  Fie cpie ðT  Te Þ
dt
þ Q transfer þ Q loss þ Q stirring ð62ÞFor heat removal through the cooling jacket Q transfer is given by Eq. (63), where U[J m2 s1 K1 ] is the overall heat-transfer coefficient, A [m2 ] the total heat-transfer area and DTml the logarithmic mean temperature difference between the cooling fluid and the reaction medium.Q transfer ¼ UADTml ð63ÞDTml is given by Eq. (64), where Twe and Tws [K] are the inlet and outlet tempera-tures of the cooling fluid (normally water) in the jacket. If Twe A Tws then DTml A ðT  Tw Þ.ðT  Twe Þ  ðT  Tws ÞDTml ¼ ð64Þ
ðT  Twe Þ
ðT  Tws Þ
The overall heat-transfer coefficient includes several resistances in series, but the internal resistance usually controls the heat-transfer rate (hi A U). The internal heat-transfer coefficient is a function of several factors such as the impeller type and dimensions, the impeller speed, the reactor diameter, and physical properties of the fluid. Empirical correlations based on dimensionless groups can be used.Equation (65) presents the usual form of these expressions [111], where Nu; Pr and Re are the Nusselt, Prandtl, and Reynolds numbers, j and jw the viscosity of the reaction medium at the reactor and wall temperatures respectively, and a; b; c,and d are constants.


j d
Nu ¼ a Re b Pr c ð65Þ
jw
Due to changes in the properties of the reaction media (for example, viscosity) the overall heat-transfer coefficient changes during the process. In semicontinuous op-eration, the heat-transfer area varies during the operation.

294 6 Emulsion Polymerization A practical method to determine the heat transferred in any cooling device is to measure the flow rate and the inlet and outlet temperatures of the cooling fluid,which are related by Eq. (66) where m_ w [kg s1 ] is the mass flow rate and cpw[J kg1 K1 ] the heat capacity.Q transfer ¼ m_ w cpw ðTwe  Tws Þ ð66ÞCombination of Eqs. (62) and (63) or (66) allows the estimation of the polymeriza-tion rate from temperature measurements. This method, which is called reaction calorimetry (see Section 6.10.1.6), is a powerful noninvasive on-line monitoring technique and it has been extensively applied to polymerization reactors [113, 114].Polymer Particle Population Balance (Particle Size Distribution)Particle size distribution strongly affects rheology [115], which in turn influences heat removal rate, mixing, mass transfer, and stability of the latex. On many occa-sions all of these aspects determine the scaleup and the feasibility of the operation.In addition, particle size distribution affects film formation and some application properties [116, 117].There are several ways in which the PSD can be represented, and often this de-pends on the method used to measure it. Thus, histograms are used when the PSD is determined by transmission electron microscopy. However, the mathemat-ical analysis is simpler if the PSD is defined in terms of the number density of polymer particles of unswollen volume v [mparticle ], nðvÞ. The units of nðvÞ are the number of particles per unit of unswollen volume of particle. From this definition,the number of particles with unswollen volumes between v1 and v2 is given by Eq.
(67), and the total number of particles by Eq. (68).
ð v2
Np ðv1 ! v2 Þ ¼ nðvÞ dv ð67Þ
v1
ðy
Np ¼ nðvÞ dv ð68ÞEquation (69) gives the macroscopic population balance for a CSTR, where the left-hand side accounts for the accumulation of particles in the reactor, the first term on the right-hand side accounts for the entry of particles into the reactor, the sec-ond for the exit of particles from the reactor, the third for the formation and loss of particles of unswollen volume v due to particle growth, the fourth for the loss of particles by coagulation with other particles and the fifth term accounts for the for-mation of particles of unswollen volume v by particle coagulation. In Eq. (69) nðvÞand ne ðvÞ are the reactor and inlet number density of polymer particles, Q s [m 3 s1 ]is the volumetric flow rate, V [m 3 ] the reactor volume, rv ðvÞ [mparticle s1 ] the volu-metric growth rate of each particle of volume v, kðv; v 0 Þ the coagulation rate con-

6.9 Reaction Engineering 295
stant for particles of volumes v and v 0 , and v0 the volume of particles formed by nucleation.
ð
qnðvÞ Q s Qs qrv ðvÞnðvÞ nðvÞ y¼ ne ðvÞ  nðvÞ   kðv; v 0 Þnðv 0 Þ dv 0 qt V V qv V v0
ð
1 vv0
þ kðv  v 0 ; v 0 Þnðv  v 0 Þnðv 0 Þ dv 0 ð69Þ
2V v0
This equation is a partial-differential-integral equation and the nucleation term is best incorporated as a boundary condition at volume v0 as in Eq. (70) [118], where R nuc is the nucleation rate given by Eqs. (20), (26), (31), and (33).
R nuc
nðv0 Þ ¼ V ð70Þ
rv ðv0 Þ
The first and second terms on the right-hand side of Eq. (69) should be removed for batch reactors, as well as for semicontinuous reactors to which no particles are fed. On the other hand, the coagulation terms may be neglected for stable formu-lations. Equations (69) and (70) are conveniently solved by using orthogonal collo-cation [119, 120].
6.9.4
Scaleup
The main objective of scaleup is to reproduce the laboratory results in commercial-scale reactors in such a way that the end-use polymer properties can be main-tained. In this way, a successful scaleup takes place when the latex is produced on a large scale at planned rates, at the projected manufacturing cost, and to the de-sired quality standards [121]. To achieve this goal, knowledge of the mechanisms that control the reactor behavior and the product properties is required [122].Then, the design variables that affect these controlling mechanisms must be main-tained constant from the smaller scale to the commercial equipment. In theory,this could be done by applying the principle of similarity by means of dimen-sionless groups characterizing the phenomena of interest. However, when several mechanisms are involved in the process, it is impossible to maintain all the dimen-sionless groups constant simultaneously. In this case, it is very important to know which similarities to keep and which to sacrifice.The flow into the vessel, together with heat and mass transfer, must be con-sidered in emulsion polymerization reactors because all of these mechanisms can have a great influence on polymer properties and reactor behavior. Unfortunately,the mixing and heat-transfer parameters do not scale equally. In Table 6.4 several scale factors are shown for a change from a 50 L pilot-scale vessel to a full-scale vessel of 50 m 3 . The calculated values assume geometric similarity with a constant impeller/tank diameter ratio and a constant height/tank diameter ratio. This table

296 6 Emulsion Polymerization Tab. 6.4. Several scale factors during scaleup.Parameter [a] Pilot scale Plant scale of 50 m 3 (scaleup procedure)P (power) 1 10 3 10 8 100 10 5 0.1 N (impeller rpm) 1 0.21 10 0.1 1 0.01 Q (impeller flow) 1 215 10 4 100 10 3 10 P=V (power/volume) 1 1 10 5 0.1 100 0.0001 q=V (heat transfer rate/volume) 1 0.08 1 0.05 0.21 0.01 ND (tip speed) 1 2.1 100 1 10 0.1 Q=Q S (impeller flow/feed rate) 1 0.21 10 0.1 1 0.01 Q=H (impeller flow/head) 1 46.8 1 100 10 10 3 Re (Reynolds number) 1 21.5 10 3 10 100 1[a] Units are as given in the Notation section.shows, in different columns the change in selected parameters when one of them is held constant.It is often essential to maintain the same rate of heat transfer in the large-scale unit. Nevertheless this scaleup criterion is impractical because it demands exces-sive impeller tip speeds and high power costs. For this reason, to compensate the negative effect in the heat-transfer rate when another scaleup criterion is selected,additional cooling devices are necessary in most cases [123].With respect to mixing characteristics, it is very important to identify the param-eters that have a major effect on the polymer properties. This task is difficult in emulsion polymerization, where mixing affects several polymer properties. Mono-mer droplet and particle size, mass transfer between the different phases, coagula-tion and kinetics may be affected by mixing parameters. The power per volume or the tip speed are the more usual scaleup criteria, but an adequate combination of several mixing parameters can be better to maintain product quality. In some cases, changes in the formulation can be utilized to maintain dynamic similarity.Changes in viscosity during the process affect mixing, and scaleup becomes more difficult. Intermediate pilot plant experiments are required in most cases to make a successful scaleup. Computational fluid dynamics will probably transform scaleup in the near future.
6.10
On-line Monitoring in Emulsion Polymerization Reactors In the production of dispersed polymers, the main objectives to be fulfilled are:
Safety: The reactor temperature must be kept under safe limits to avoid thermal runaways. In addition, violation of environmental regulations both in the plant environment and in the finished products must be avoided.

6.10 On-line Monitoring in Emulsion Polymerization Reactors 297

Production rate: The goal is to maximize the production rate of the available
Product quality: The required quality is given by the end-use properties such as viscosity, scrub resistance, tensile strength, flexibility, elasticity, toughness, and glosss.In order to implement the process conditions that lead to the required specifica-tions of safety, production rate, and product quality, it is necessary to develop suit-able control strategies. Reaction control strategies rely on both efficient monitoring techniques and state estimation and filtering techniques. In this section the main focus is on the instrumentation available for monitoring emulsion polymerization reactors. Detailed reviews for on-line monitoring techniques for polymerization re-actors are available [124–126].Polymer latexes are ‘‘products-by-process’’, and therefore the required structural and morphological properties of the polymer that yield the adequate end-use prop-erties are produced in the reactor. In polymer latexes the properties that one would like to control during the polymerization include: monomer(s) conversion, copoly-mer composition, MWD, PSD, particle morphology, branching and crosslinking,gel content, and particle size distribution. Not all of these properties can be moni-tored on-line, although some can be made observable by combining available measurements and mathematical models. Examples of observable properties are copolymer composition by means of calorimetric or density measurements [127–129] and instantaneous molecular weights from measurements of unreacted monomer and CTAs (gas chromatography [130] or reaction calorimetry [131] in linear polymers).Other properties, such as molecular weight distribution of nonlinear polymers and particle morphology, are currently neither measurable nor observable.
6.10.1
On-line Sensor Selection One of the issues when monitoring an emulsion polymerization reactor is selec-tion of the most appropriate technique [124, 126]. For instance, monomer conver-sion and copolymer composition can be monitored on-line by means of densim-etry, refractive index, gas chromatography, calorimetry, ultrasound, fluorescence,ultraviolet reflection, and other spectroscopic methods such as Raman, mid-range infrared, and near-infrared.Figure 6.11 can be used as a guide for the selection of a technique for monitor-ing monomer conversion and copolymer composition. The main idea behind this plot is the fact that the amount of information provided by the available techniques is different and furthermore, the implementation of these techniques involves dif-ferent degrees of difficulty (including here robustness in a harsh environment,maintenance, and know-how required). The ideal technique would be located in the upper left-hand corner, but unfortunately no such technique is currently available.

298 6 Emulsion Polymerization Fig. 6.11. Guideline chart for sensor selection.In the rest of Section 6.10.1, a brief description of the monitoring techniques shown in Figure 6.11 will be presented, emphasizing the advantages and disadvan-tages of each one, and finally examples of application for emulsion polymerization reactors will be given.
6.10.1.1 Latex Gas Chromatography
In this technique, a latex sample from the reactor is first diluted and then injected into the GC to analyze unreacted monomers and CTAs [132, 133]. Although suc-cessful implementation of this technique for latexes with a high solids content has been reported [134], the setup is liable to suffer clogging. Additional disadvan-tages of this monitoring technique include the time delays associated with the analysis itself when multicomponent copolymerizations are monitored. This draw-back can be partially alleviated by using capillary columns and modern gas chro-matography equipment with reduced analysis time.
6.10.1.2 Head-space Gas Chromatography
This GC-based monitoring technique circumvents the lack of robustness of the latex GC technique by analyzing the reactor head-space [135, 136]. However, the estimation of the monomer concentrations in the polymer particles requires that equilibrium between the latex and gas phase is attained. In addition, the values of the equilibrium parameters are required, which introduces an additional uncer-tainty to the measurements. Note also that formulation ingredients such as CTAs are not measurable by this means.
6.10.1.3 Densimetry
This technique is based on the density change that occurs when monomer is con-verted to polymer. This difference, which is obviously a maximum in bulk polymer-ization, allows emulsion polymerizations of relatively high solids content to be ac-curately monitored [137]. The main disadvantage of on-line densimetry is that, as for latex GC, a sample of the reaction medium must be introduced in the thermo-stated measurement cell and the system is liable to suffer clogging. A further dis-

6.10 On-line Monitoring in Emulsion Polymerization Reactors 299
advantage of densimetry is that it provides only overall conversion measurements for copolymerization, and hence partial conversions and copolymer compositions must be inferred using a model [129]. An advantage of densimetry in comparison to GC is that there is no delay in the measurement of density, which can be moni-tored almost continuously.
6.10.1.4 Ultrasound
The principle of ultrasound relies on the propagation of the ultrasonic wave pres-sure. The measurable properties are the ultrasonic velocity and the attenuation of the wave, which are a function of the medium where the wave propagates and hence of the density, viscosity, and compressibility. The attenuation measurements are not very reliable for on-line monitoring because the attenuation of the wave in dispersed systems is complex and depends on the dispersion medium, the viscous losses within the particles and at the interface between the particle and the contin-uous phase, thermal losses, sound scattering in dispersed media, and the dynamic relaxation of the polymeric material [138, 139]. Furthermore, there are technical problems in measuring the attenuation at high solids and the presence of gas bubbles can make the measurements difficult. Sound propagation velocity is better suited to monitor emulsion polymerization reactors and it has been proven in sys-tems with high conversion and high solids content [140–142]. In principle, the main advantages of this technique are that it is noninvasive in nature (although care should be taken with emitter–receiver transducers located inside the reactor),it is fast and cheap, and, in addition to monomer conversion, other latex properties such as particle size, monomer solubilities, and critical micellar concentration can also be measured [141].The main disadvantage is that to exploit the information on the sound veloc-ity and attenuation fully, calibration is required to obtain accurate predictions of monomer conversion or other properties such as particle size because the theo-retical models linking those characteristics to sound propagation velocity are not predictive.Nevertheless, on-line sound velocity is one of the more promising techniques for monitoring industrial-size polymerization reactors because of the fast and robust measurement and easy maintenance at a relatively low cost.
6.10.1.5 Spectroscopic Techniques
Since the mid-1990s, there has been plenty of activity regarding the use of spectro-scopic techniques for on-line evaluation of polymer properties [143–146]. This has been possible due to the recent development of fiber-optic probes, which allow in-situ measurements in remote and harsh environments (high temperatures, pres-sures, toxic environments, and so on). An additional advantage is that a fiber-optic probe can be installed in an existing reactor within a short time without expensive modifications. Fluorescent, ultraviolet (UV), infrared (IR), near-infrared (NIR),mid-infrared (MIR) and Raman spectroscopic techniques can be used for poly-merization reaction monitoring. These can be divided between absorption- and emission-based techniques. IR, NIR, and MIR are absorption-based.

300 6 Emulsion Polymerization IR is not well suited to monitor polymerizations in dispersed media because water gives a strong absorption, and hence important bands are overlapped or hidden by that of the water. In addition, transmission through fiber-optics is still relatively poor in the infrared region, which makes IR not as suitable as other tech-niques for remote monitoring.NIR spectroscopy corresponds to the spectral region 14000–4000 cm1 . The main advantages of NIR are that no sample preparation is required and that the NIR signal can be easily transmitted through fiber-optics made of silica, which have a low cost. The main disadvantage of NIR comes from the fact that in this re-gion, absorption bands are broad because they do correspond not to specific bonds or groups of molecules, but to combinations of them. This makes the accurate quantitative analysis difficult. Furthermore, the detection of the minor compounds in polymerization formulations is not easy. Although NIR has been used to moni-tor emulsion polymerization reactions [147], difficulties due both to broad bands associated with water absorption and to the effect of the particle size have been re-ported [141, 148, 149]. Another difficulty of NIR is the short penetration depth of the NIR signal. Furthermore, the presence of big monomer droplets might cause inaccuracies in the quantitative analysis due to inhomogeneous sampling of the reactor by the NIR probe.The MIR spectral region is from 4000 to 400 cm1 . This region is very rich in fundamental absorptions and hence the potential of this technique is high. How-ever, the use of MIR spectroscopy to monitor emulsion polymerization reactors is scarce, mainly because remote monitoring is not possible at low cost as currently only a few exotic materials are known to be able to transmit in this region. In addi-tion, water is absorbed in this region and it may hide bands that are important for further analysis.Raman spectroscopy is an emission-based technique. Although conventional dis-persive Raman spectroscopy (laser wavelengths between 500 and 700 nm) has not been successfully used to monitor polymerization reactions due to the tremendous effect of fluorescence on the spectra, FT-Raman (laser wavelength in the NIR re-gion, 1034 nm) or modern dispersive Raman equipments (laser wavelengths over 800 nm) overcome this difficulty. Currently, Raman spectroscopy can be considered as the spectroscopic technique with the greater potential to monitor polymerization reactors, and especially emulsion polymerization reactors, in situ. Raman spectros-copy presents several advantages over the absorption techniques (MIR and NIR).The most important ones are:
Water is a weak scatterer and hence Raman is well suited to monitor polymeriza-tion in dispersed media.
It is very sensitive to CbC bonds. This makes it possible to follow, with high ac-curacy, the disappearance of monomer by polymerization.
Low-cost fiber-optic technology can be used to monitor polymerization reactors remotely.
The penetration depth of the Raman signal is greater than for NIR and hence in emulsion polymerization interference by monomer droplets can be avoided.

6.10 On-line Monitoring in Emulsion Polymerization Reactors 301
0.006
0.005
Weight Fraction MMA
0.004
0.003
0.002
0.001
0 0.2 0.4 0.6 0.8 1
Conversion
0.05
Weight Fraction BA
0.04
0.03
0.02
0.01
0 0.2 0.4 0.6 0.8 1
Conversion
Fig. 6.12. On-line monitoring of a seeded semibatch emulsion copolymerization of MMA/BA. Evolution of (a) MMA; (b) BA.Figure 6.12 shows an example of monitoring a semibatch emulsion polymerization of MMA/BA of high solids content (55 wt.%) by means of on-line FT-Raman spec-troscopy [150]. It should be pointed out that this monomer system was challenging because the chemical structures of the monomers are very similar, and hence most of the bands overlap. Chemometric (partial least squares, PLS) analysis was neces-

302 6 Emulsion Polymerization 0 0.2 0.4 0.6 0.8 1
Conversion
Fig. 6.12. (c) solids content. g, gravimetry and gas chromatography; , FT-Raman.sary to quantitatively determine the concentrations of monomers and the solids content. Furthermore, in some cases nonlinear calibration techniques are neces-sary because PLS, a multivariate linear calibration, may not be sufficient if the degree of nonlinearity between the spectra and the properties of interest is sig-nificant.
6.10.1.6 Reaction Calorimetry
Reaction calorimetry is probably the cheapest, easiest, and most robust monitoring technique for polymerization reactors, due to the large enthalpy of polymerization of most monomers. The technique is noninvasive (basically, only temperature sen-sors are required), and it is industrially applicable [151, 152]. It yields continuous information on the heat released by polymerization and hence it is also very useful for safety issues. The main drawback is that only overall polymerization rates can be obtained. Consequently, the determination of the individual rates requires esti-mation techniques [114, 153–155].Reaction calorimetry is based on the energy balance in the reactor, given by Eq.
(71),
X dT
Ni cpi ¼ Q r þ Q feed þ Q transfer þ Q loss þ Q stirring ð71Þ
dt
where the term on the left-hand side is the heat accumulated in the reactor, Q r is the heat generation rate due to polymerization, Q feed is the sensible heat genera-

6.10 On-line Monitoring in Emulsion Polymerization Reactors 303
tion rate due to the feeding of reagents into the reactor, Q transfer is the heat flow across the reactor wall, and Q loss and Q stirring represent the rate of heat losses and of heating due to stirring, respectively. The generation rate of the heat of reaction,Q r , can be calculated from the other terms, provided that these can be calculated with sufficient accuracy. In emulsion polymerization reactors, the largest of these terms is Q transfer .In heat-flow calorimetry, Q transfer is calculated from the measurements of the re-actor (T) and jacket (Tw ) temperatures by applying Eq. (72), where U is the overall heat-transfer coefficient and A the heat-transfer area.Q transfer ¼ UAðT  Tw Þ ð72ÞThe implementation of heat-flow calorimetry requires knowledge of the evolution of U. This is the weakest point of this technique.In heat-balance calorimetry, eqs. (71) and (72) are coupled with the energy bal-ance in the jacket to produce Eq. (73), where m w is the mass of cooling fluid in the jacket, m_ w the mass flow rate of cooling fluid in the jacket, cpw its specific heat capacity, Twe the inlet jacket temperature and Tws the outlet jacket temperature.
dTws
m w cpw ¼ UAðTws  TÞ þ m_ w cpw ðTwe  Tws Þ ð73Þ
dt
Heat-balance calorimetry allows the simultaneous estimation of U and Q r , pro-vided that Twe  Tws could be accurately measured. Therefore, heat-balance calo-rimetry is best suited for large-scale industrial reactors because a significant differ-ence between the jacket inlet and outlet temperatures is necessary, and this is the case in industrial reactors. The advantage of this approach is that a-priori informa-tion of the overall heat-transfer coefficient is not necessary and hence it is more robust and reliable than heat-flow calorimetry.Oscillation calorimetry also allows simultaneous determination of the heat of re-action and the overall heat-transfer coefficient from temperature measurements of the reactor and the jacket. This is done by taking advantage of the different dynam-ics of heat transfer (fast) and heat of the reaction (slow) when an oscillation of either the reactor or jacket temperature is created artificially. The analysis of the os-cillatory temperature signals allows calculatation of the UA term and Q r simulta-neously [156, 157]. The oscillation of the jacket and reactor temperatures can be achieved in different ways, but perhaps the most practical one is by imposing a si-nusoidal oscillation in the set-point of the reactor temperature. This approach can only be applied to small reactors (< 20 L) with high flow rates of the cooling fluid,because the oscillating signal is strongly attenuated as the reactor size increases.The higher reactor time-constants make the estimation of Q r and UA very uncer-tain [158].The amount of free monomer and the copolymer composition in emulsion poly-merization reactors can be inferred from measurement of the heat of reaction, Q r

304 6 Emulsion Polymerization
Gravimetry
Free Monomer (g)Chromatography Calorimetry 0 50 100 150 200 250 300 350
Time (min)
1.0
Calorimetry Full Points GC Terpolymer Composition
0.8
Open Points NMR
0.6
BA
0.4
MMA
0.2
VAc
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Conversion
Fig. 6.13. On-line monitoring of emulsion polymerization reactor by means of calorimetric measurements: (a) free monomer in VAc/BA/AA semibatch emulsion polymerization;(b) terpolymer composition in the VAc/MMA/BA semibatch emulsion polymerization.[127, 128]. Figure 6.13(a) shows the evolution of the free monomer concentration as inferred from on-line reaction calorimetry compared with the off-line measure-ment (gravimetry and gas chromatography) for a VAc/BA/AA high solids content semibatch emulsion polymerization. In Figure 6.13(b), the estimation of the ter-polymer composition from calorimetric measurements is compared with NMR and GC measurements for a VAc/MMA/BA emulsion terpolymerization [159].

306 6 Emulsion Polymerization The closed-loop control strategy requires calculation of the set-point, that is, the trajectories or profiles of the state variables as a function of a measured variable(overall conversion). These profiles can be calculated by means of an optimization algorithm. In what follows, a brief description of the calculation of the optimal tra-jectories for copolymer composition and the MWD control is presented. The goal of the optimization algorithm is to calculate the set-point trajectories of the state(controlled) variables that ensure the production of an emulsion polymer of the de-sired copolymer composition and MWD in the minimum process time. To achieve this goal, the objective function to be minimized is as expressed in Eq. (74), where Rp is the polymerization rate and XT is the overall conversion.
ð 1 
Min dXT ð74Þ
½M p R
0 p
The minimization of Eq. (74) is subjected to the following constraints:Constraint 1: product composition and MWD The polymer produced must have the desired copolymer composition and the final MWD.The condition to produce a latex with a given copolymer composition is that the ratio of the monomer concentrations in the polymer particles must be kept at the value that ensures the production of the desired composition. This comono-mer ratio can be calculated from the Mayo–Lewis equation, Eq. (75), where r1 and r2 are the reactivity ratios and y1i is the instantaneous composition referred to monomer 1.½M1 p ð2y1i  1Þ G fð2y1i  1Þ 2  4r1 ð y1i  1Þ y1i r2 g 1=2
¼ ð75Þ
½M2 p 2r1 ð y1i  1ÞThe calculation of the condition to produce a latex with a given MWD is based on the fact that for linear polymers produced by free-radical polymerization, the poly-mer chains do not suffer any modification once they are formed. This opens the possibility of decomposing the desired final MWD in a series of instantaneous MWDs to be produced at different stages of the reaction [130]. When chain trans-fer to a CTA is the main termination event, each of those MWDs can be character-ized by the number-average chain length, X ni , according to Eq. (76).k p1 ½M1 p þ k p2 ½M2 p X ni ¼ ð76Þktr; CTA ½CTAp Therefore the problem reduces to calculating the sequence of the values of X ni that provide the desired final MWD. In order to calculate the X ni values that should be produced at each value of XT , the final MWD is discretized as in Eq. (77), where XTf is the final overall conversion; X nij is the instantaneous number-average chain length produced in the conversion increment j, Wj ðnÞ is the instantaneous MWD

6.11 Control of Emulsion Polymerization Reactors 307
produced in the conversion increment j, and k is the number of increments into which XTf is divided.
1 X k
DXT Xk
n n
Wc ðnÞ ¼ Wj ðnÞDXTj ¼ exp  ð77ÞXTf j¼1 XTf j¼1 X nij X nij Note that in the discretization the most probable distribution is used because, if chain transfer to CTAs is the main termination event, the instantaneously formed polymer obeys this distribution (polydispersity index ¼ 2).For a given number of conversion increments, the required values of X nij can be calculated by minimizing the Eq. (78), where Wcd ðnÞ and Wc ðnÞ are the desired and the calculated MWDs.
X
Min ½Wcd ðnÞ  Wc ðnÞ 2 ð78Þ
X ni n
This is a nonlinear optimization in which the number of values of n should be greater than the number of conversion increments. A priori any MWD, Wcd ðnÞ,with polydispersity index equal or greater than 2 can be prepared by this method.Strictly speaking, the maximum molecular weight achievable with this technique is that produced with the minimum amount of CTA that ensures the termination by chain transfer to CTAs is the main termination event. In practice, this is very close to the molecular weight obtained without CTA. On the other hand, MWDs contain-ing very low molecular weights may require the use of amounts of CTA that exceed the maximum allowable quantities (usually lower than 1 wt.% based on monomer)used in industrial practice. The minimization of Eq. (78) provides the values of X nij to be produced at different DXT [Eq. (79)].½M1 p þ ½M2 p X ni o ¼ f ðXT Þ ð79Þ
½CTAp
Equations (75) and (79) are used as the constraints in the optimization algorithm to produce a copolymer of constant instantaneous composition Y1i and a given MWD, Wcd ðnÞ.Constraint 2: Safety considerations Polymerizations are exothermic processes that can cause runaways, so the maximum amounts of the monomer that can be pres-ent in the reactor should be limited for safety reasons. Therefore, to design safe processes an analysis of the risk parameters must be made in order to obtain the limits in reaction conditions for safe operation: namely, the limits in monomer concentration and temperature that ensure that the pressure buildup in the reactor will not exceed the maximum pressure that the reactor can withstand. The risk parameters are the onset temperature, the adiabatic temperature increase, and the maximum temperature and pressure that may be reached during a polymerization

308 6 Emulsion Polymerization process under adiabatic conditions. To assess risk parameters, adiabatic calorime-ters of low thermal inertia (phi-factor, F, less than 15%) are used.The onset temperature of the polymerization reaction, Tonset , is defined as the temperature at which the self-heating rate is equal to a given arbitrary onset crite-rion. This criterion depends on the sensitivity of the equipment and is used as a reference to calculate the adiabatic temperature increase, DT, which depends on the total amount of monomer in the formulation, MiTOT , the heat of polymeriza-tion (DHri ), and the heat capacity of the reaction medium according to Eq. (80),where Ni and cpi are the amount and the heat capacity, respectively, of each reagent i present in the reaction medium.MiTOT ðDHri ÞDT ¼ X ð80Þ
Ni cpi
The adiabatic temperature increase, DT, is calculated for the experiments carried out in the adiabatic calorimeters by subtracting the Tonset from the maximum tem-perature achieved, Tmax .DT ¼ Tmax  Tonset ð81ÞThus, the dependence of the risk parameters on process variables such as the monomer concentration in the polymer particles, particle size, solids content and initiator/monomer ratio are of paramount importance to establish the safe regions of operation of an emulsion polymerization reactor, and furthermore to develop op-timal control strategies under safe conditions.A typical result obtained in an adiabatic calorimeter for an emulsion polymeriza-tion reaction is shown in Figure 6.15. From the evolution of the temperature, the onset of the adiabatic reaction and the maximum temperature achieved can be de-termined. Therefore, adiabatic temperature increases under these conditions can be easily calculated. The evolution of the pressure provides an indication of the maximum pressure reached.Figure 6.16 shows the evolution of the risk parameters as a function of the polymer/monomer ratio for the emulsion polymerization of VAc/BA/AA
(78.5:18.5:3) as obtained in a VSP2 reactor (Fauske & Associates). The particle size
of the seed and the initiator/monomer ratio were the same in all the experiments.The plot shows that the onset temperature decreases as the monomer concentra-tion in the polymer particles increases. Tmax ; DT, and pressure (the latter not shown) increase upon increasing the monomer content. The analysis can be extended for other process variables such as solids content, particle size, and initiator/monomer ratios.An example of the safety limits for the VAc/BA/AA emulsion polymerization system is shown in Figure 6.17. The plot, which is based on the data displayed in Figure 6.16, is constructed assuming that the polymerization will be carried out at

6.11 Control of Emulsion Polymerization Reactors 309
T (°C)
0 50 100 150 200 250 300
time (min)
3,5
P (atm)
2,5
1,5
0,5
0 50 100 150 200 250 300
time (min)
Fig. 6.15. Time evolution of (a) temperature; (b) pressure during emulsion polymerization of VAc/BA/AA (78.5:18.5:3)with initial monomer/polymer ratio ¼ 50:50; initiator/monomer ratio ¼ 0.002 wt.%; final solids content ¼ 50 wt.%; seed particle size ¼ 156 nm.80 C (Twork ) and that the maximum temperature allowed to run the process, Tlimit ,is 100 C. Note that this temperature must be calculated on the basis of the maxi-mum pressure that the reactor can withstand, and is also a function of the process variables.The graph shows that the region of polymer/monomer ratio below 3:1 (concen-tration of monomer in the polymer particles higher than ½Mp ¼ 2:90 mol L1 ) will not be safe. In other words, if higher monomer concentrations are used in the process and the cooling system fails, there is a risk of exceeding the Tlimit tempera-

310 6 Emulsion Polymerization
Tonset
Tmax
0 2 4 6 8 10 Polymer/Monomer Fig. 6.16. Evolution of the risk parameters as a function of the polymer/monomer ratio for a VAc/BA/AA emulsion polymerization of high solids content.
T + ∆T
120 work
limit
work
SAFE REGION
20 ∆T
0 2 4 6 8 10 DANGER Polym er / M onom er Fig. 6.17. Safety regions for a VAc/BA/AA emulsion polymerization. Polymerization temperature ¼ 80 C; maximum temperature allowed for the process ¼ 100 C.

6.11 Control of Emulsion Polymerization Reactors 311
ture (Twork þ DT > Tlimit ) and the runaway will take place. Above the limit polymer/mononor ¼ 3 (½Mp < 2:90 mol L1 ) the process can be operated safely because in none of the cases does the [Twork þ DT] operating line exceed the limit of the safe operation temperature.Constraint 3: Nonremoval of monomer and CTA The monomers and CTA already charged in the reactor cannot be removed (Eqs. (82)–(84), where the subscripts f and pol stand for free and polymerized amounts, respectively).d½M1f þ M1pol 
b0 ð82Þ
dXT
d½M2f þ M2pol 
b0 ð83Þ
dXT
d½CTA f þ CTA pol 
b0 ð84Þ
dXT
Constraint 4: Limitation on monomer and CTA addition The maximum amounts of the monomer and CTA that can be added to the reactor are the total amounts of these compounds in the formulation, M1TOT ; M2TOT , and CTATOT , respectively[Eqs. (85)–(87)].M1f þ M1pol a M1TOT ð85ÞM2f þ M2pol a M2TOT ð86ÞCTA f þ CTA pol a CTATOT ð87ÞThe optimization provides the amounts of monomers and CTAs in the reactor at any overall conversion. These profiles are independent of the kinetics of the pro-cess and can be regarded as master curves. Once the trajectories of the amounts of monomers and CTAs as a function of the conversion are calculated, the imple-mentation of the closed-loop strategy (Figure 6.14) reduces to tracking these pro-files. To do so, on-line measurements of the overall conversion and of the free amount of monomers and CTA are necessary. Reaction calorimetry plus state esti-mation is probably the easiest, cheapest, and most robust option from an industrial perspective.At each sampling time a nonlinear controller calculates the values of the manip-ulated variables u (flow rates of monomer and CTA) that must be added during the sampling interval to ensure tracking of the master trajectories and hence to pro-duce the desired polymer. A number of nonlinear controllers have been reported in the literature for this purpose: nonlinear model predictive controllers (NMPCs)[162], nonlinear geometric controllers (NGCs) [163], and internal model control-lers (IMCs) [164] being the ones that have gained more attention in the specialized literature.

312 6 Emulsion Polymerization Total amounts of monomers (mol/L)Butyl acrylate Total amount of CTA (mol/L)
Styrene
0.012
CTA
0.009
0.006
0.5
0.003
0 0.2 0.4 0.6 0.8 1 Overall Conversion Fig. 6.18.Optimal trajectories for the amount of monomer and CTA to produce S/n-BA ¼ 50:50 copolymer with a bimodal MWD.An example of the performance of an on-line control strategy like the one de-picted above is shown in Figures 6.18 and 6.19 [165]. A copolymer with constant composition, styrene/n-butyl acrylate (S/BA) ¼ 50:50 and a bimodal MWD with two peaks (50 wt.% of the polymer in each peak) of different polydispersities were sought: Mw1 ¼ 1:05  10 6 and PI1 ¼ 2:5 and Mw2 ¼ 1:15  10 5 and PI2 ¼ 3. Fig-ure 6.18 presents the optimal profiles of styrene, n-butyl acrylate, and tert-dodecyl mercaptan (TDM) to produce the copolymer with the properties mentioned. It can be seen that to maintain the comonomer ratio the required amount of n-BA was always greater than that of styrene. Moreover, during the production of the first 50% of the polymer, the amount of TDM required was low since the mode with the high molecular weight was prepared first; at 50% conversion, a sudden addi-tion of TDM was required to start producing the low molecular-weight mode. Fig-ure 6.19 presents a comparison between the desired and obtained copolymer com-position and MWD during the controlled experiment.
Notation
A total heat-transfer area [m 2 ]Ap total surface area of the polymer particles [m 2 ]as saturated surface of the polymer particles covered by 1 mol of surfactant[m 2 mol1 ]c overall termination rate coefficient in the polymer particles [Eq. (9)] [s1 ]cc rate coefficient for bimolecular termination by combination [s1 ]

Notation 313
1.0
Copolymer Composition
0.8
0.6
0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0
Overall Conversion
0.6
Desired
df/dlog(M )
w
X=0.18
0.5
X=0.47
0.4 X=0.60
X=0.77
0.3 X=0.96
0.2
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log (M )
w
Fig. 6.19.Experimental results of the on-line controlled emulsion polymerization of S/n-BA: (a) cumulative copolymer composition; (b) MWD produced at different conversions.cd rate coefficient for bimolecular termination by disproportionation [s1 ]CMC critical micellar concentration [mol m3 ]cpi heat capacity of compound i in the reactor [J mol1 K1 ]cpie heat capacity of compound i under the entry conditions [J mol1 K1 ]cpw heat capacity of the cooling fluid [J kg1 K1 ]½CTAp concentration of CTA in polymer particle [mol m3 ]CTA f free amount of CTA in the reactor [mol]

314 6 Emulsion Polymerization CTA pol amount of CTA polymerized [mol]CTATOT total amount of CTA in the formulation [mol]D impeller diameter [m]dd diameter of the monomer droplets [nm]dp diameter of the polymer particles [nm]f efficiency factor of the initiator radicals fac frequency of activation in living polymerization [s1 ]fc frequency of bimolecular termination [s1 ]fde frequency of deactivation in living polymerization [s1 ]Fie inlet molar flow rate of component i [mol s1 ]Fis outlet molar flow rate of component i [mol s1 ]H velocity head [J kg1 ]hi internal heat-transfer coefficient [J m2 s1 K1 ](DHri Þ polymerization heat of monomer i under the reactor conditions [J mol1 ]I amount of initiator [mol][I] concentration of initiator in the aqueous phase [mol m3 ]icrit critical length of the oligoradicals formed from desorbed radicals jcrit critical length of the oligoradicals formed from initiator Ki partition coefficient of monomer i between the phase j and the aqueous phase.kðv; v 0 Þ coagulation rate constant for particles of volumes v and v 0[m 3 particle1 s1 ]ka entry rate coefficient [m 3 mol1 s1 ]kac activation rate coefficient in NMP or ATRP [s1 or m 3 mol1 s1 ]kadd addition rate coefficient in RAFT [m 3 mol1 s1 ]kam rate coefficient for radical entry into the micelles [m 3 mol1 s1 ]kd rate coefficient for radical exit [s1 ]kd ðnÞ rate coefficient of radical exit from particles with n radicals [s1 ]kde deactivation rate coefficient in NMP or ATRP [m 3 mol1 s1 ]kex exchange rate coefficient in DT [m 3 mol1 s1 ]
ðÞ
kfrag fragmentation rate coefficient in RAFT, backward reaction [s1 ]
ðþÞ
kfrag fragmentation rate coefficient in RAFT, forward reaction [s1 ]kI rate coefficient for initiator decomposition [s1 ]kp propagation rate constant [m 3 mol1 s1 ]k pi average propagation rate constant of monomer i in copolymerization[m 3 mol1 s1 ]kpji propagation rate constant of radicals with terminal unit j with monomer i [m 3 mol1 s1 ]kt termination rate coefficient in the polymer particles [m 3 mol1 s1 ]kt reversible termination rate coefficient in living polymerization[m 3 mol1 s1 ]ktr; CTA average chain transfer to CTA rate constant in copolymerization[m 3 mol1 s1 ]k tr; M chain transfer to monomer rate coefficient [m 3 mol1 s1 ]k tw termination rate coefficient in the aqueous phase [m 3 mol1 s1 ]

Notation 315 m partition coefficient of small radicals between polymer particles and the aqueous phase [Eq. (11)]Micrit number of oligoradicals of critical length formed from desorbed radicals Mif amount of free monomer i in the reactor [mol]Mipol amount of monomer i polymerized [mol]½Mi p concentration of monomer i in the polymer particles [mol m3 ]MiTOT total amount of monomer i in the formulation [mol]Mm number of inactive chains of length m Mmi number of inactive chains of length m in particles with i radicals Mn cumulative number-average molecular weight Mni instantaneous number-average molecular weight½Mp monomer concentration in the polymer particles [mol/m 3 ]mw mass of the cooling fluid in the jacket [Kg]m_ w mass flow rate of the cooling fluid [Kg/s]Mw cumulative weight-average molecular weight Mwi instantaneous weight-average molecular weight½Mw monomer concentration in the aqueous phase [mol/m 3 ]N impeller rpm n number of radicals in a particle n average number of radicals per particle NA Avogadro’s number nðvÞ number of polymer particles per unit of unswollen volume of particle[particles m3
particle ]
ne ðvÞ inlet number density of polymer particles [particles m3
particle ]
Ni total amount of compound i in the reactor [mol]Ni0 moles of monomer i in the reactor at time zero [mol]nm aggregation number of surfactant [molecules/micelle]Nm number of micelles Nn number of polymer particles with n radicals Np number of polymer particles Npoli moles of monomer i polymerized [mol]Npr number of precursor particles NTi total amount of monomer i to be feed in a semicontinuous operation
[mol]
Nu Nusselt dimensionless number P impeller power consumption [J s1 ]Pj time-averaged probability of finding an active chain with ultimate unit of
type j
PM average molecular weight of the repeating unit in the polymer chain Pr Prandtl dimensionless number q heat transfer rate [J s1 ]Q impeller flow [m 3 s1 ]Q feed sensible heat generation rate due to feeding into the reactor [J s1 ]Q loss rate of heat losses to the surroundings [J s1 ]Qr heat generation rate by polymerization reaction [J s1 ]

316 6 Emulsion Polymerization Qs volumetric flow rate [m 3 s1 ]Q stirring rate of heat production by the agitator [J s1 ]Q transfer heat removal rate [J s1 ]Re Reynolds dimensionless number Ri net generation rate of component i [mol m3 s1 ]ri reactivity ratio of monomer i R jcrit number of oligoradicals of critical length formed from initiator Rm number of radicals of length m Rmi number of radicals of length m in particles with i radicals R nuc nucleation rate of polymer particles [particles m3 s1 ]Rp overall polymerization rate [mol m3 s1 ]Rpi polymerization rate of monomer i [mol m3 s1 ]Rpp polymerization rate per polymer particle [mol particles1 s1 ]RX amount of the ‘‘capping’’ species in controlled polymerization [mol]rv volumetric growth rate of one polymer particle [m 3 s1 ]rv ðvÞ volumetric growth rate of a particle of volume v [m 3 s1 ]Rnk generation rate of the kth moment of the distribution of inactive chains½Rw concentration of radicals in the aqueous phase [mol m3 ]ST total amount of surfactant in the reactor [mol]Sw amount of surfactant in the aqueous phase [mol]t reaction time [s]T reactor temperature [K]DT adiabatic temperature increase Te temperature of the feed [K]Tg glass transition temperature [K]Tlim maximum temperature achievable for a safe operation [K]Tmax maximum temperature obtained during an adiabatic polymerization reaction [K]DTml logarithmic mean temperature difference Tonset temperature at which the polymerization start under adiabatic condi-
tions [K]
Tw jacket temperature [K]Twe inlet temperature of the cooling fluid in the jacket [K]Tws outlet temperature of the cooling fluid in the jacket [K]U overall heat-transfer coefficient [J m2 s1 K1 ]V reactor volume [m 3 ]Vi volume of monomer i [m 3 ]Vd volume of the droplet phase [m 3 ]Vw volume of the aqueous phase [m 3 ]Vp volume of the polymer particles [m 3 ]vp volume of a swollen polymer particle [m 3 ]Vpol volume of polymer [m 3 ]W volume of water [m 3 ]Wc ðnÞ cumulative weight MWD Wcd ðnÞ desired cumulative MWD

References 317 Wj ðnÞ instantaneous weight MWD of the polymer formed at conversion incre-
ment j
[X] concentration of ‘‘living agent’’ in controlled polymerization [mol m3 ]Xi conversion of monomer i X ni instantaneous number-average chain length X nij instantaneous number-average chain length of the polymer formed in conversion increment j XT overall conversion XTf final overall conversion DXTj jth overall conversion increment y1cum cumulative copolymer composition referred to monomer 1 y1i instantaneous copolymer composition referred to monomer 1
Greek
a1 parameter of Eq. (29)a2 parameter of Eq. (30)g generation rate of small radicals by chain transfer [Eq. (11)] [mol m5 ]d critical length for entry of radicals generated from the initiator h consumption rate of small radicals generated by chain transfer [Eq. (11)]
[m2 ]
l overall mass-transfer rate coefficient [m 3 s1 ]mk kth-order moment of the distribution of active chains nk kth-order moment of the distribution of inactive chains r frequency of radical entry [s1 ]t space time [s]j viscosity of the reaction medium at the reactor temperature [kg m1 s1 ]F fraction of the heat of reaction used to heat the reactor walls fM volume fraction of monomer in the polymer particles fi volume fraction of monomer i in phase j fp volume fraction of polymer in the polymer particles fww volume fraction of water in the aqueous phase C parameter of Eq. (17)jw viscosity of the reaction medium at the wall temperature [kg m1 s1 ]
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158 S. Kraemer, R. Gesthuisen, S. 162 B. W. Bequette, Ind. Eng. Chem. Res.Engell, J. R. Leiza, J. M. Asua, Comp. 1991, 30, 1391–1413.Chem. Eng., in press, 2004. 163 C. Kravavis, M. Soroush, AIChE J.159 I. Sáenz de Buruaga, J. R. Leiza, 1990, 30, 249–264.J. M. Asua, Polym. React. Eng. 2000, 8, 164 C. G. Economou, M. Morari, B.39–75. Plasson, Ind. Eng. Chem. Res. 1986,160 O. Elizalde, M. Vicente, J. R. Leiza, 25, 403–411.J. M. Asua, Polym. React. Eng. 2002, 165 M. Vicente, J. R. Leiza, J. M. Asua,10, 265–283. AIChE J. 2001, 47, 1594–1606.

Klaus-Dieter Hungenberg This chapter will deal with ionic chain growth polymerization for monomers which are of some industrial importance. It will deal mainly with those polymer reaction engineering aspects which are relevant for designing processes and products. As it cannot cover the entire subject, the systems dealt with are chosen as examples of the most important features of ionic polymerization.
7.1
Introduction There are several recent monographs [1–7] on anionic and cationic polymerization covering various mechanistic, kinetic and preparative aspects, so here just some fundamental issues which are relevant for reaction engineering will be discussed.Like free-radical polymerization ionic polymerization is also a chain polyaddi-tion. After the formation of an active center (radical, anionic, or cationic species),monomer molecules are added to this active center to form long-chain molecules.However, from a kinetic point of view there are two major differences between rad-ical and ionic polymerization.The first difference is that free-radical polymerization is monomer-based, which means that the kinetics is (almost) exclusively determined by the monomer M.Once a radical R
is formed and added to the monomer molecule to build the growing chain R- - -M
, the reactivity of this growing chain in all reactions is deter-mined by the nature of the monomer irrespective of the nature of the initiating radical, which just forms the tail of that growing chain. So, from a practical point of view, in radical polymerization it is sufficient to determine the kinetic scheme and parameters and their dependences on the system variables temperature and pressure for a monomer system with one kind of radical initiator. Furthermore,all active radical centers from one monomer are identical, and they are hardly in-
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

fluenced by the nature of the surroundings, nor by the solvent nor by penultimate units of the chain; at least, the influences of penultimate units [8] or solvents [9]are quite small. In contrast, ionic polymerization is site-based: that is, the nature of the active center strongly depends on the nature of the initiator. Propagation is by monomer insertion at an ion pair [Eq. (1) or (2)], where the counter ion comes from the initiator system.@@Pnþ X þ M ! @@Pnþ1
þ
X ð1Þ
or
@@Pn Y þ þ M ! @@P
nþ1 Y
þ
ð2Þ
So the kinetic scheme and parameters not only depend on M, but they may differ considerably according to the initiator used. Furthermore, the active centers are not uniform as in free-radical polymerization, but because of their ionic nature there usually exists a complex chemical equilibrium between different species,even if the initiator has a unique structure. This equilibrium [Eq. (3)] between free ions, solvent-separated ion pairs, contact ion pairs, covalent polarized bonds,and between associated and non-associated species, as well as the concentration of these species strongly depend on the polarity or solvating power of the solvent sys-tem, the solvent itself, and the presence of other salts.X þ Y þ $ X ==Y þ $ X ; Y þ $ X d Y dþ $ 1=n ðXYÞn ð3ÞAll these species can in principle participate in all the reactions with different rate coefficients. These facts make ionic polymerizations difficult to access for kinetic modeling from a practical point of view in an industrial environment.There is another important difference between free-radical and ionic polymeriza-tion. In free-radical polymerization, there are system-immanent, unavoidable ter-mination reactions, the bimolecular disproportionation and combination between two radicals. Because these termination reactions are very fast (k t ¼ 10 7 –10 9 M1 s1 ) compared to propagation (k p ¼ 10 0 –10 3 M1 s1 ) and radical formation by initiator decomposition (kd ¼ 103 –101 s1 ) the pseudo-steady-state hypothesis can be applied.In ionic systems, in general, there are no such system-immanent termination re-actions between species carrying the kinetic chain and, if there are reactions which terminate the active species, they often occur on a similar time scale to initiation and propagation, or they are even slower.Examples of such reactions, which irreversibly terminate the kinetic chain, are elimination reactions where the eliminated species is not able to re-initiate the po-lymerization, such as in the elimination of LiH or NaH during anionic polymeriza-tion of styrene or butadiene [10–13]. In cationic polymerization, the collapse of ion pairs to covalent species, as in reaction (4) where the CaF bond is too strong for the ion pair to be reformed again, is a true termination reaction.

Tab. 7.1. Monomers with different initiation mechanisms.Monomers Type of initiation Radical Cationic Anionic Ethylene þ  þa-Olefins  þ 1,1-Dialkyl olefins  þ 1,3-Dienes þ þ þStyrene, a-methylstyrene þ þ þHalogenated olefins þ  Vinyl esters þ  Acrylates, methacrylates þ  þAcrylonitrile, methacrylonitrile þ  þAcrylamide, methacrylamide þ  þVinyl ethers  þ N-Vinylcarbazole þ þ N-Vinylpyrrolidone þ þ Aldehydes, ketones  þ þ16], styrenic polymers, and block copolymers of styrene and dienes [17–21] as well as the ring-opening polymerization of cyclic ethers, lactones, or lactams. Here es-pecially, polymers from ethylene oxide and propylene oxide are important as base materials for polyurethanes [22–24].
7.2.1
Anionic Polymerization of Hydrocarbon Monomers – Living Polymerization There are two main reasons why anionic polymerization of hydrocarbon mono-mers such as styrene, a-methylstyrene, butadiene, isoprene, and so on has gained scientific and industrial importance: These monomers may be polymerized under such conditions that termination and transfer reactions are absent, resulting in a so-called ‘‘living’’ polymerization. The microstructure of the polydienes so produced can be varied over a wide range.So the microstructure of dienes strongly depends on the counter ion (see Table
7.2.) and the solvent or the presence of additives, which are capable of shifting the
equilibrium in Eq. (3) from the right to the left (see Table 7.3.). An attempt to ra-tionalize the influence of polar additives on the microstructure is given in Ref. 25.Generally, the vinyl content increases with the fraction of free ions.
7.2.1.1 Association Behavior/Kinetics
There are numerous publications [1, 5, 6] dealing with the equilibrium of Eq. (3)between free ions as one extreme possibility for the structure of the initiating and/

7.2 Anionic Polymerization 327
Tab. 7.2. Microstructure of polybutadienes produced in hydrocarbon solvent with different counter ions (from [25a]).Alkali metal cis-1,4 [ %] trans-1,4 [ %] 1,2 [ %]Li 35 52 13 Na 10 25 65
K 15 40 45
Rb 7 31 62
Cs 6 35 59
or polymerizing species, and aggregation of more or less covalently bonded species as the other extreme. All these species may participate in the polymerization, and the rates for the same reaction may differ by several orders of magnitude depend-ing on the species involved. For example, the propagation rate coefficient for anionic polymerization of styrene in THF at 25 C is 8  10 4 M1 s1 for the free ion, 200 M1 s1 for the ion pair [26], and values of 0.0155 and 0.024 M0:5 s1 are reported [27, 28] for the overall propagation rate coefficient k p  K 1=2 of the equilibrium system from monomeric and dimeric polystyryllithium in benzene and cyclohexane.In particular, the question of association in lithium-based polymerization in non-polar solvents, one of the most important industrial systems, and the determina-tion of the kinetic scheme and parameters, are still under debate. In general, the association behavior is formulated as in Eq. (9), where n gives the association num-ber, which generally is between 2 and 6, depending on the structure of P (mono-mer or initiator), solvent, and temperature (see Table 7.4).
K ass
ðPaLiÞn ! nPaLi ð9ÞTab. 7.3. Vinyl content of polybutadienes produced in hydrocarbon solvent with butyllithium as initiator with different additives (from [28a]).Additive Molar ratio additive/lithium 1,2 [ %]Diethyl ether 10:1 16 Diethyl ether 5:1 10
THF 6:1 43
THF 3:1 25
THF 1:1 17
Diglyme 4:1 87 Diglyme 2:1 85 Diglyme 1:1 78

Tab. 7.4. Association number for different Li organyls.[a]Li alkyl Solvent n n-BuLi benzene 6–6.3 cyclohexane 6
THF 2–2.8
sec-BuLi cyclohexane 4
benzene 4
THF 1.1
t-BuLi benzene 4
hexane 4
Menthyl-Li cyclohexane 2 Benzyl-Li benzene 2
THF 1
Poly(styryl)-Li benzene 2 cyclohexane 2
hexane 2
Poly(isoprenyl)-Li benzene 2
hexane 2
cyclohexane 2–4 Poly(butadienyl)-Li cyclohexane 2–4.3 benzene 2–3.7
hexane 2
[a] Compilation of data given in Ref. 6, pp. 16, 20, 138.With the assumption that association is high and the concentration of the mono-meric species is low, the concentration of the monomeric species is given by Eq.
(10), where PaLi is either the initiating lithium alkyl or the propagating species.

1=n
K ass
½PaLi ¼ ½PaLi0 ð10ÞAssuming that the associated species are not reactive, or at least they are much less reactive than the non-associated one, the reaction rates for initiation and propaga-tion are given by Eq. (11), where k is either ki or k p.


K ass 1=n 1=n 1=n r ¼ k½PaLi½M ¼ k ½PaLi0 ½M ¼ kobs ½PaLi0 ½M ð11ÞThis correlation of the association number with the reciprocal of the reaction order may be too simple. Extensive discussions on this subject are summarized in Refs.1, 5, 6, and 29–33. Here, just an overview of the various interpretations is given without trying to judge which is the right one, but in every case of ionic polymer-ization, when one is trying to set up a realistic mechanistic process model, similar questions must be answered. Therefore the anionic polymerization of hydrocarbon monomers in hydrocarbon solvents, which is one of the best-investigated anionic

7.2 Anionic Polymerization 329
systems, is used here as an example to point out which aspects may become impor-tant for reactor and process layout.One reason for the debate is the high sensitivity of ionic systems to impurities,which may cause experimental errors leading to various interpretations of the data.However, from a mechanistic point of view also there are arguments that this sim-ple correlation does not seem to be a general rule. For styrene with t-butyllithium initiation is reported to be independent of monomer concentration [34]; for dienes various orders are reported with differences between fractional reaction order and degree of aggregation (see Table 7.1 in Ref. 6). Furthermore, cross-aggregation be-tween initiating and propagating species is likely to occur and must be consid-ered, as well as intermediate equilibria between hexamers, tetramers, dimers, and monomers. In Ref. 35 this problem is addressed for the cross-association of styryl-and butadienyllithium. There are a number of recent publications reporting much higher association numbers of up to 100 and more [36–39], which are under dis-cussion [40, 41]. Bywater discussed [42, 43] the difficulties in separating K ass and k p . Nevertheless, there are some reports on the separate determination of K ass ,[44–47], which however are contradictory. With the exception of those in Ref. 47,most of the data are from investigations at rather low temperatures (< 30 C) com-pared to industrial conditions, which are at considerably higher temperatures (60–100 C). So, extrapolating the association behavior from low to higher temperatures may cause errors.In addition it must be stated that Eq. (10) is only valid if the concentration of non-associated species is low. Otherwise the equilibrium of Eq. (9) must be consid-ered a priori: for example, for n ¼ 2 in the case of styrene, the concentration of non-associated PaLi is given by Eq. (12).
s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K ass K ass 2 K ass½PaLi ¼  þ þ ½PaLi0 ð12ÞFigure 7.1 gives the temperature dependence of K ass and k p for styrene from Ref.
44. When the fraction a of non-associated polystyryllithium is calculated with the
10000000 1000
10000
-1
kp / M s
K ass / M
-1
0.01
0.1
0.00001
0.00000001 0.001
0.0025 0.003 0.0035 0.004
1/T / 1/K
Fig. 7.1. Association equilibrium constant K ass and propagation rate constant k p as f ðTÞ [44].

0.1
α
0.01
0.001
0.000001 0.0001 0.01 1
Kass / M
Fig. 7.2. Fraction a of non-associated species as a function of K ass for dimerization for various polystyryllithium concentrations (b, 0.0001 M; , 0.001 M; þ, 0.01 M). Broken lines: exact solution according to Eq. (12); full lines:approximate solution according to Eq. (10).simplification in Eq. (10) and the correct solution in Eq. (12), Figure 7.2 shows that there may be considerable deviations for values of K ass ¼ 105 M and higher. So,for industrial temperatures the fraction of associated species may be rather low and a first-order kinetic law with respect to lithium concentration may result [10,Table 7.5 gives some examples of the temperature dependence of the propaga-tion rate coefficient for Li-initiated styrene polymerization according to Eq. (11)with n ¼ 2. From Figure 7.3 it can be seen that there are deviations of one order of magnitude between the results of different authors, even when considering the same solvent and a rather limited temperature range of 4–60 C. In Ref. 48 data for styrene and butadiene are given up to 70 C assuming n ¼ 2 for styrene and n ¼ 2 or 4 for butadiene. More difficulties will arise when extrapolating to higher temper-atures and tackling the problem of temperature dependence of the association be-havior. In Ref. 10 an attempt is made to describe the propagation over a wider tem-Tab. 7.5. Literature values for propagation rate coefficient kp0 ¼ k p ðK=2Þ 0:5 for Li-initiated styrene polymerization in hydrocarbon solvents.kOp; 0 [MC0:5 sC1 ] Ea [kJ molC1 ] Solvent ˚Temperature [ C] Ref.4:5  10 8 60.3 none 20–50 492:6  10 9 64.0 none 4–21 502:9  10 11 78.6 cyclohexane 30–50 515:2  10 8 60.3 benzene 10–30 275:1  10 7 56.9 toluene 20–50 526:8  10 8 63.6 toluene 30–60 53

7.2 Anionic Polymerization 331
acc. to [49]acc. to [50]acc. to [51]
0.1 acc. to [27]
acc. to [53]acc. to [52]
-1 -1
kp K / M s
0.01
1/2
0.001
0.0001
0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037
-1
1/T / K
Fig. 7.3. Comparison of literature values for propagation rate coefficient kp0 ¼ k p ðK=2Þ 0:5 for lithum-initiated styrene polymerization in hydrocarbon solvents.perature range, from 10 C up to 100 C, with k p ¼ 1  10 11 e7900=T M1 s1 and K ¼ 3:2  10 38 e27600=T M according to Eq. (12), which collapses to a simple first-order kinetic with respect to initiator above 40 C.There is no judgment on the quality of the various results, but this overview can serve as an example for the difficulties in ionic polymerization kinetics in general,and careful checking of literature kinetic data is strongly recommended before they are used for reactor layout.This somewhat extensive discussion on the association behavior of lithium-initiated polymerization, the best-investigated ionic system, shows the difficulties in kinetic modeling of ionic polymerization. Contrary to free-radical polymeriza-tion, the existence of all possible different species must be considered for every sys-tem under investigation.
7.2.1.2 Molecular Weight Distribution of Living Polymers
The living nature of the anionic polymerization of hydrocarbon monomers has been revealed by Szwarc [54, 55]. There is still an ongoing debate on the exact def-inition of living polymerization [5, 6, 55–57]. For the scope of this chapter we will refer pragmatically to living polymers if the active end groups of the individual chains ‘‘retain the propensity of growth for at least as long a period as needed for the completion of the intended synthesis’’ [5]: that is, initiation and propagation[(Eqs. (13) and (14)] are the only reactions, irreversible termination and transfer re-actions being absent.

I þ M ! P1 initiation ð13ÞP1 þ M ! Piþ1

propagation ð14ÞIn the sense of this definition, the associated species discussed above can also be called living, even if they do not participate in a propagation reaction, but they are in equilibrium with the active non-associated species on a much shorter time scale than the overall reaction. For simplicity, in the following discussion the association is not considered explicitly, but if it is known quantitatively it can be considered by making the rate coefficients a function of [PaLi].From the reactions in Eqs. (13) and (14), the set of differential equations (15)–
(20) can be derived.
d½I 
 ¼ k i ½I ½M ð15Þ
dt
d½M Xy
 ¼ k i ½I ½M þ k p ½M ½Pi  ¼ RP ð16Þ
dt i¼1
d½P1 
 ¼ k i ½I ½M þ k p ½P1 ½M ð17Þ
dt
d½Pi 
 ¼ k p ½Pi1 ½M þ k p ½Pi ½M ð18Þ
dt
X
½Pi  ¼ ½I 0  ½I  ð19Þ
i¼1
X
i½Pi  ¼ ½M0  ½M ð20Þ
i¼1
For fast initiation (k i g k p ), all initiator is transferred to growing chains immedi-ately, so monomer conversion and the kinetic chain length are given by Eqs. (21)–
(23).
½M ¼ ½M0 ek p ½I 0 t ð21Þ½M0  ½M ½M0  ½M0 ekP ½I 0 t ½M0 n¼ ¼ ¼ xM ð22Þ½I 0 ½I 0 ½I 0
with
X
½Pi  ¼ ½I 0 ð23Þ
i¼1

7.2 Anionic Polymerization 333
The resulting frequency distribution hðiÞ of chain lengths i can be derived by solv-ing the system of differential equations sequentially starting with ½P1 t¼0 ¼ ½I 0 and the definition of the kinetic chain length given above [Eq. (24)].½Pi  ½Pi  nði1Þ en hðiÞ ¼ X ¼ ¼ ð24Þ½Pi  ½I 0 ði  1Þ!The weight distribution is given by Eq. (25).
nði1Þ en
wðiÞ ¼ ð25Þði  1Þ!ðn þ 1ÞThe average number- and weight-average degrees of polymerization and the poly-dispersity for this Poisson distribution are given by Eqs. (26)–(28).
½M0  ½M
Pn ¼ ¼1þn ð26Þ
½I 0
Pw ¼ 1 þ n þ A 1 þ Pn ð27Þ
1þn
Pw
D¼ A1 ð28Þ
Pn
If, however, k i g k p does not hold, meaning if the initiation rate coefficient is in the order of k p or even less, there is no immediate conversion of initiator to grow-ing chains and the simplification of Eq. (23) becomes invalid and must be replaced by Eq. (19). The solution for this general case is given in Refs. 58 and 59. From Eqs. (15) and (16), the variation of monomer conversion as a function of initiator conversion is given by Eq. (29) with r ¼ k p =k i > 1.
½I 
½M  ½M0 ¼ ð1  rÞð½I   ½I 0 Þ þ r½I 0 ln ð29Þ
½I 0
The variation of M and I with time cannot be solved analytically in this case, but must be found numerically. The frequency and weight distributions of the so-called Gold distribution are given by Eqs. (30) and (31), with R ¼ r  1, q ¼ r=R and u ¼ ðr  1Þ lnð½I 0 =½I Þ. The averages of this distribution are given by Eqs.
(32) and (33).

i
r qu Xy
uj
½Ni  R j¼i hðiÞ ¼ y ¼ ð30ÞX rð1  eu=R Þ
½Ni 
i¼1

  
i½Ni  r i ru=ðr1Þ Xy
uj ru
wðiÞ ¼ y ¼ e ð1  rÞð1  eu=R Þ þ ð31Þ
X r1 j! R
j¼i
i½Ni 
i¼1
ru
ð1  rÞð1  eu=ðr1Þ Þ þPn ¼ r1 ð32Þð1  eu=ðr1Þ Þ
ru 1  ru
þ 2ru þ ð2r  1Þðr  1Þð1  eu=ðr1Þ ÞPw ¼ r  1 r  1 ð33Þ
ru
ð1  rÞð1  eu=ðr1Þ Þ þ
r1
A Poisson distribution will also result when using bifunctional initiators [60], with the peak maximum at 2½M0 =½I 0 . Figure 7.4 gives a comparison of the Poisson dis-tribution for n ¼ 50 (k i > k p ), the Gold distribution for (k i < k p , r ¼ k p =k i ¼ 100),and the Schulz–Flory or most probable distribution resulting from step growth po-lymerization or free-radical polymerization with termination by disproportionation.This most probable distribution is given by Eq. (34), where p is the conversion of end groups in step growth polymerizations or polycondensation or the probability of propagation in chain growth polymerization.hðiÞ ¼ p i1 ð1  pÞ (34)The width of the distributions reflects the sharpness of the initiation reaction; with k i > k p all chains start at the same time and the distribution is very narrow, just reflecting some statistics of monomer addition. If chain initiation is delayed for some chains (k i < k p ), the distribution is skewed and will become narrower if reac-
0.06
0.05
0.04 Poisson
h(i)
0.03
0.02 Gold Schulz-Flory
0.01
0 50 100 150 200 Fig. 7.4. Comparison of various distributions for Pn ¼ 51:Schulz–Flory distribution with p ¼ 0:98, Pw ¼ 102; Gold distribution with r ¼ k p =k i ¼ 100, u ¼ 86:513, Pw ¼ 64; and Poisson distribution for n ¼ 50. The corresponding weight-averages are 102, 64, and 52.

7.2 Anionic Polymerization 335
1.4 1
0.8
1.3
0.6
xM
1.2
0.4
1.1
0.2
0 0.2 0.4 0.6 0.8 1
xI
Fig. 7.5. Polydispersity D and monomer conversion xM as a function of initiator conversion xI for r ¼ 10 (full lines) and r ¼ 100 (broken lines) according to Eqs. (29), (32), and (33).½I0 ¼ 0:101 M, ½M0 ¼ 3 M. Number- and weight-averages at xM ¼ 1 are 31 and 34 for r ¼ 10, and 51 and 64 for r ¼ 100.tion proceeds and more chains will be initiated and propagate. The broadest distri-bution is the most probable distribution if, as in step growth polymerization, there is no initiation but only chain propagation.Slow initiation has some important consequences, which are shown in Figure
7.5. The monomer may be consumed before initiator conversion is complete. This
may be important when using functionalization techniques or synthesizing block copolymers. Moreover, the polydispersity may be considerably higher than 1, the value which is usually assumed to be typical for living polymerization and used as a criterion for ‘‘livingness’’.Besides the reactivity ratio r, the equilibrium between different species –associated and non-associated ones or free ions and ion pairs, active and dormant species – which may have different reactivities in propagation reactions, can also have an impact on broadening the molecular weight distribution [61–73].The MWD discussion up to now has concerned batch reactors, but there are also a number of publications dealing with other reactors. In ideal plug flow reactors under steady-state conditions, polymers with the same characteristics of the MWD as in batch reactors are built – the time axis is transformed to the length axis of the plug flow reactor [74, 75], and for fast initiation the MWD is a Poisson distribution.For laminar plug flow reactors [75, 76] some broadening of the distribution is ob-served, depending on ½M0 =½I 0 and conversion. The long-term stability of laminarflow tubular reactors is questioned [76], because of the possibility of very long chains growing near the reactor walls, where the residence time approaches infin-ity, if radial diffusion of the monomer occurs from the inner tube to the walls.The MWD resulting from semi-batch operations of a stirred tank reactor with monomer feed under various conditions is treated in Refs. 77–82. In a homoge-neous continuous stirred tank reactor (HCSTR), the steady-state concentrations of monomer and initiator can be derived from the monomer and initiator mass bal-

ance, neglecting volume changes, by Eqs. (35) and (36), which must be solved si-multaneously. ½Mf and ½I f are the feed concentrations, V is the reactor volume,and vin; I ; vin; M are the volume feed flows of initiator and monomer.
vin; M
½Mf t
½M ¼ V ð35Þk i ½I  þ kP k i ½M 2 ½I t 2 þ ½M
vin; I
½I f t
½I  ¼ V ð36Þk i ½Mt þ 1 In a CSTR, the narrow Poisson distribution resulting from the chemistry is super-imposed by the broad residence time distribution of the HCSTR – chains can leave the reactor after seconds of growth as very short chains, but there are also chains which reside in the reactor for very long time, growing to very long molecules. This results in a Schulz–Flory distribution [83] [Eqs. (37)–(40)], where t is the mean residence time and [M] is the steady state-monomer concentration.

j1
1=t kP ½M
hð jÞ ¼ ð37ÞkP ½Mt þ 1=t kP ½M þ 1=t
2
j1
1=t kP ½M
wð jÞ ¼ j ð38ÞkP ½Mt þ 1=t kP ½M þ 1=t Pn ¼ 1 þ kP ½Mt ð39ÞPw ¼ 1 þ 2kP ½Mt ð40ÞEquation (37) is equivalent to Eq. (34) with the propagation probability given by Eq.
(41).
kP ½M
p¼ ð41Þ
kP ½M þ 1=t In a segregated stirred tank reactor (SCSTR) [74] the width of the distribution ranges from that of an HCSTR for rather low conversion or mean residence time,up to a value approaching that of a batch reactor for high conversions or long mean residence times. The influence of termination, chain transfer, and long-chain branching in batch and CSTR reactors is described in Refs. 84 and 85.
7.2.1.3 Side Reactions
Pure living polymerization of hydrocarbon monomers usually occurs at low tem-peratures, where side reactions are not important. At higher temperatures, how-ever, two side reactions do become important, elimination of LiH and transfer to compounds with an acidic hydrogen [5, 6, 31, 86, 87].

7.2 Anionic Polymerization 337
There have been some investigations on the stability of alkyllithium deriva-tives and of growing chains. Generally, linear alkyllithiums are more stable than branched ones, with first-order decomposition half-lives of 1–6 h at 87–98 C[88, 89]. From Ref. 89, k tt ¼ 1:0  10 10 expð11798=TÞ s1 for s-BuLi and k tt ¼9:5  10 10 expð13407=TÞ s1 for n-BuLi can be evaluated. As with all rate coeffi-cients of ionic polymerization, this elimination also depends on the reaction sys-tem. The lithium hydride (LiH) elimination from butyllithium in the presence of polar additives such as lithium butoxide is reported to be several times faster than from pure lithium alkyl [90, 91], and in ethereal solvents proton abstraction or ether cleavage may occur [92].Growing chains such as polydienyl- and polystyryllithium are less stable. For polystyryllithium, the LiH elimination is followed by proton transfer to another polystyryllithium [93], resulting in an allylic anion, which is unable to propa-gate (see Scheme 7.1). The same holds for polystyrylsodium [94]. Quantitative data for LiH elimination are given in Refs. 13 (k tt ¼ 295  e5280=T s1 ) and 10(k tt ¼ 3:92  10 6 eð8700=TÞ ). In Ref. 95 the influence of THF was investigated, but rather similar values were obtained (k tt ¼ 2:4  10 6 eð8459=TÞ ). It must be noted that these termination reactions are much slower than propagation, that is to say the half-life for the active chains is much higher (hours) than the half-life for prop-agation (seconds) at temperatures of 80 C and higher.
Bu Li Bu
k tt + LiH
Bu Bu -
+ PsLi + PsH Scheme 7.1. Termination in anionic polymerization of styrene.Polybutadienyllithium is somewhat more stable than polystyryllithium (k tt ¼6:7  105 s1 versus 1:9  104 s1 at 93 C [13]), but contrary to styrene, the side reactions here may cause coupled and branched polymers [95–98].The other class of important side reactions is transfer to compounds with acidic hydrogen atoms, such as toluene, ethyl benzene and so on (Scheme 7.2).Some quantitative data have been given for chain transfer from polystyryllithium to aromatic solvents [10, 33, 47, 99–101], and from polydienyllithium to alkenes[102, 103] and toluene, and for the influence of polar additives [104]. There is usu-ally no effect on the polymerization rate, because the number of active chain ends

Bu Li Bu Li
k tr
Li Li
kp
Scheme 7.2. Transfer to ethylbenzene in anionic polymerization of styrene.is maintained, but the chain length of the macromolecules, that is, the molecular weight, is reduced. The presence of transfer reactions limits the production of block copolymers by sequential addition of monomers to the living chains. How-ever, in Refs. 33 and 99 it is pointed out that the presence of transfer reactions may be advantageous, for example, for homopolymerzation of styrene in a CSTR,because they reduce the amount of initiator necessary to get the desired molecular weight.The effect of side reactions, such as termination by monomer, impurities, or spontaneous termination and transfer to monomer or impurities, on the molecular weight distribution are dealt with in Refs. 66 and 105–119; they generally result in some broadening of the distribution.
7.2.1.4 Copolymerization
The most important copolymers are those from styrene and butadiene, either as statistical copolymers or block copolymers. From a kinetic point of view the as-sociation behavior discussed above becomes even more complex because of the possible cross-association between the different growing chain ends. This issue has seldom been addressed [35, 120].But in spite of this complex association behavior, in most cases the simple Mayo terminal model [Eq. (42)] with two copolymerization parameters, which was devel-oped originally for free-radical polymerization, is in many cases sufficient to de-scribe anionic copolymerization also.r1 f12 þ f1 f2
F1 ¼ ð42Þ
r1 f1 þ 2f1 f2 þ r2 f22 However, there is one important difference between free-radical and living anionic polymerization, and this is the lifetime of the growing chain. This difference be-comes important when considering the dependence of copolymer composition on conversion. Equation (42) gives the copolymer composition or mole fraction F1 of monomer M1 in the polymer as a function of mole fraction f1 in the monomer

7.2 Anionic Polymerization 339
feed for incremental conversion. Because of the different reactivity of the mono-mers, usually f1 and consequently F1 change with increasing conversion.In free-radical polymerization, where the lifetime of the growing polymer chain is in the order of seconds or less, compared to hours for the overall polymerization reaction, and where new chains are initiated throughout the overall reaction time by decomposition of radical initiators such as peroxides or azo compounds, those chains which are initiated at different times or levels of conversion will differ in their composition, but those which are initiated at the same time will have the same composition. This difference in composition from chain to chain is calledfirst-order chemical heterogeneity.If, however, as in living polymerization, all chains start at the same time and live throughout the polymerization, every chain will see all the changes in monomer feed composition, and consequently the composition will change along the chain and not from chain to chain. This is known as second-order chemical heteroge-neity. The differences between these two kinds of heterogeneities, one between different chains, the other within different fractions of one chain, are shown in Scheme 7.3.Increasing conversion First-order chemical heterogeneity
I-BBBBBSBB
I-BBBBBSBB
I-BBBBBSBB I-SBSBSSBS
I-SBSBSSBS
I-SSSSBSSS
Second-order chemical heterogeneity R-BBBBBSBB-Li R-BBBBBSBBSBSBSSBS-Li R-BBBBBSBBSBSBSSBSSSSSSSSS-Li Scheme 7.3. Chemical heterogeneity with increasing conversion for free-radical (first-order) and living polymerization (second-order).The severity of the chemical heterogeneity strongly depends on the copolymer-ization parameters. In free-radical polymerization there is just one pair of parame-ters, which may depend somewhat on temperature, for one pair of monomers;whereas in ionic polymerization these parameters for every pair of monomers strongly depend on the counter ion and solvent polarity (see Table 7.6).These extreme values have a very drastic effect on the shift not only in composi-tion but also in rates. Taking the absolute rate coefficients given by Ohlinger[35] for Li-initiated copolymerization of butadiene and styrene at 20 C in toluene(kSS ¼ 0:45, kSB ¼ 110, kBB ¼ 0:084, kBS ¼ 0:0066 M1 s1 ), the resulting overall

Tab. 7.6. Copolymerization parameters for Li-initiated copolymerization of styrene (¼ M1 ) and butadiene (¼ M2 ).[a]T [ C] ˚ Solvent r1 r225 bulk 0.04 11.220 toluene 0.004 12.930 benzene 0.035 1050 cyclohexane 0.025 15.130 heptane 0.1 778 THF 11 0.040 THF 0.2 5.325 THF 0.3 425 diethyl ether 0.4 1.7[a] Examples taken from Ref. 6, p. 247ff.conversion versus time curves are given in Figures 7.6 and 7.7. The interesting fea-ture is that even though the homopolymerization of styrene is faster than that of butadiene, during copolymerization nearly all the butadiene is consumed at a lower rate than in homopolymerization before styrene is incorporated to any appre-ciable extent, and after the butadiene is consumed the overall polymerization rate becomes that of pure styrene.
xS = 1
0.8
xS = 0
xS = 0.35
0.6
conversion
xS = 0.5
0.4
xS = 0.65
0.2
0 5000 10000 15000 20000 25000 30000
t/s
Fig. 7.6. Batch copolymerization of styrene and butadiene with n-butyllithium (1 g) at 20 C in toluene (10 kg) with various styrene feed fractions xS (20 mol monomer overall). Simulation of overall conversion with data from Ref. 35.

7.2 Anionic Polymerization 341
0.8
x S = 0.65
0.6
x S = 0.5
xS inpolymer
0.4 x S = 0.35
0.2
0 5000 10000 15000 20000 25000 30000
t/s
Fig. 7.7. Batch copolymerization of styrene and butadiene with n-butyllithium (1 g) at 20 C in toluene (10 kg) with various styrene feed fractions xS (20 moles monomer overall).Simulation of mole fraction of styrene in copolymer with data from Ref. 35.The reason is that kSB has a high value and kBS has a low value. This combina-tion is responsible for nearly all chain ends existing as butadienyllithium ends,which predominantly propagate by addition of butadiene (kBB > kBS ). If one of the occasional propagation reactions by addition of styrene gives a styryllithium chain end, there will be a very fast addition of butadiene to regenerate butadienyllithium ends. Thus, in a batch copolymerization, very few styrene units are incorporated into the chains during the first part of the reaction and they exist as isolated units.Incorporation of styrene will increase only when nearly all the butadiene is con-sumed. Overall, therefore, there is a block of nearly pure butadiene with some iso-lated styrene units, followed by a rather short block where styrene and butadiene are incorporated at almost the same rates – a so-called tapered block, followed by a final block of nearly pure styrene (see Scheme 7.3). From Table 7.6 one can see that a similar but reverse situation is valid in polar solvents; here styrene is con-sumed first and then butadiene, but again a copolymer with nearly pure blocks of homopolymer will result.
7.2.1.5 Tailor-made Polymers by Living Polymerization – Optimization
Anionic living polymerization offers a unique opportunity for tailor-made polymers because all the chains and their living ends are accessible throughout the course of the reaction. In general, there are two important tasks in tailoring polymers: one concerns their chemical distribution and the other the molecular weight distribu-

The structure of copolymers in terms of composition or sequence length distri-bution may vary between two extremes – copolymers with a very high level of chemical heterogeneity within each chain, and homogeneous copolymers with the comonomer units randomly distributed along the chain.The various pathways to block copolymers are described in a number of reviews and monographs [6, 121–127]. In principle, block copolymers are accessible by se-quential addition of monomers to either monofunctional or bifunctional [128, 129]initiators (see Scheme 7.4, where the route from a monofunctional initiator to a three-block copolymer is shown as an example). The amount of undesired by-products – homopolymers or di-block copolymers – depends on the reactivity ratios relative to propagation given in the scheme. These are obviously the ratios for transfer and termination at higher temperatures, but also the ratios for initiation and the cross-propagation reactions. From Figure 7.5 and Eq. (29) it can be seen that the monomer for forming one block may be consumed before the initiator or the ends of the previously formed block are completely transferred to the ends of the new block, and so the yield of tri-block copolymers is reduced.R-Li Reactivity ratios Dead chains for side reactions
ki
+nS
k p,S
R-So-Li + R-Li SSSSSS
k S Bu
+ m Bu k tt
k p , Bu
k p ,i
R-So-Bup-Li + R-Buq-Li + R-Li SSSSBBBBB + BBBBB
k tr
k p ,i
k Bu S
k p ,S
R-So-Bup -Sr –Li + R-Buq-Ss-Li + R-St-Li SSSSBBB +SSSSBBBSSSS Scheme 7.4. Pathway to SBS tri-block copolymer and possible side products.The optimal production of tri-block copolymers in a series of CSTRs, where the reactors are operated in an isokinetic way, is described in Ref. 130.To produce a copolymer with a more random structure in spite of the extreme values of the copolymerization parameters, there are several possibilities. One is

7.2 Anionic Polymerization 343
to use small amounts of polar additives such as THF, ethers, alkoxides, and so on as modifiers [131, 132], to bring the extreme r-values nearer to unity. Another obvi-ous method is to polymerize in a continuous stirred tank reactor, where there is no shift in composition. For the steady state, the copolymerization equation can be written as a function of feed (index 0) and reactor concentrations [Eq. (43)].
½M1 
r1 þ1
½M1 0  ½M1  ½M2 
¼ ð43Þ
½M2 0  ½M2  ½M2 
r2 þ1
½M1 
A third possibility is model-based feed control [133], where butadiene is fed to a mixture of styrene and butadiene in such a way that a certain monomer ratio is maintained.The other important task is to tailor the molecular weight distribution of the polymer, and especially for this task the ‘‘livingness’’ in anionic polymerization is advantageous [134]. Optimized reactor operations for broadened or bimodal distri-butions in a tubular reactor are described in Refs. 135 and 136. Semi-batch opera-tion with a programmed initiator feed [137, 138] and oscillating feeds to homoge-neous CSTRs offer the possibility of a wide range of MWDs [139–144].
7.2.1.6 Industrial Aspects – Production of Living Polymers
In general, the requirements for purity of the reagents, solvents, and monomers are much higher in anionic polymerization than in free-radical polymerization.Necessary distillations and/or adsorption towers, often with activated aluminum,increase the costs for feed preparation compared to free-radical processes.Anionic polymerization of styrene has attracted much attention during recent years and several publications and patents have been published on this issue. The main reason for this interest is the high rate of polymerization which can be achieved compared to free-radical polymerization, and the low content of residual styrene and oligomers in the final polymer [145–147].There are several concepts for an anionic process, which in many cases are oper-ated in an adiabatic or at least non-isothermal mode. Some are using recirculated loop reactors [148–151] or single-pass tubular reactors [75, 152, 153], which may be segmented [154]. A spray tower [155] has a similar residence time distribution,but heat removal is by a countercurrent nitrogen stream. Boiling CSTRs are de-scribed in Refs. 33 and 156. One problem seems to be fouling or gel formation in regions of the reactor where the flow of living chains is limited [76].Polybutadiene rubbers are produced either with transition metal catalysts to give high-cis-butadiene rubber or with lithium alkyls to give medium-cis-butadiene rubber (see Table 7.7). The latter is mainly used as the rubber component in the production of HIPS (high impact polystyrene).These rubbers are generally produced in aliphatic solvents [6, 157–159] in a CSTR or a series of CSTRs. The vinyl content, which is usually about 10% in ali-phatic solvents, can be varied by the addition of polar compounds (see Table 7.3)over a wide range.

Tab. 7.7. Microstructure of polybutadienes produced in hydrocarbon solvent with different catalyst systems.Catalyst cis-1,4 [ %] trans-1,4 [ %] 1,2 [ %]
Nd 98 1 1
Co 96 2 2
Ni 96 3 1
Ti 93 3 4
Li 36 52 12 Copolymers from styrene and butadiene (SBRs) with a more or less random co-monomer distribution are produced either in an emulsion polymerization process(cold or hot E-SBR), a free-radical polymerization, which however gives crosslinked polymers, or in a solution process using lithium initiators (S-SBR). There are con-tinuous processes [157, 158, 160–166] as well as (semi-) batch processes (167, 168)with a controlled feed of butadiene [133, 169] to maintain a constant monomer ra-tio. Randomizers are often used [131, 170] to control the comonomer distribution.Block copolymers and also star-shaped [171] block copolymers from styrene and dienes are generally produced batch-wise with the monomers added in appropriate sequences. These sequences depend on the functionality of the initiator (mono- or bifunctional initiators) [172–175], on the use of coupling agents [176, 177], and on the sharpness of the transition between the various blocks. Another reason for ap-portioning the monomer feed is the limited heat removal capacity of the reactor.
7.2.2
Anionic Polymerization of Vinyl Monomers Containing Heteroatoms The possibilities of living polymerization described above have always attracted re-searchers toward extending this technique to other vinyl monomers such as acryl-ates (i), methacrylates (ii), cyanoacrylates (iii), nitriles (iv), and vinyl aldehydes or ketones (v) (see Scheme 7.5). However, almost no industrial applications have H H H CH3 H CN H COOR H COOR H COOR
H H H H
H CN H COR
iv v
Scheme 7.5. Monomers with heteroatoms for anionic polymerization.

7.2 Anionic Polymerization 345
been developed up to now for this group of monomers. Some overviews can be found in Refs. 5 and 7. An interesting example of the application of these mono-mers are the cyanoacrylates, which can be polymerized by very weak bases such as water or skin proteins and which are used as superglues.The reason for this deficiency in application is the presence of side reactions,which can only be suppressed by proper selection of the structure of the monomer itself and the reaction conditions, especially the polymerization temperature. The propagating center in the anionic polymerization of (meth)acrylates is the ester enolate anion, which is formed according to Eq. (44).H CH3 A CH2 O
A + ð44Þ
H COOMe
H3C OMe
The most important side reaction is the attack at the ester group, which can be inter- or intramolecular [Eqs. (45) and (46)].H CH3 H CH3
Me O ð45Þ
H COOMe H COR CH3 CH3 CH3 CH3 CH2 CH2 OMe
OMe
O O + HC O
MeO O 3
H3C H3C O
OMe OMe ð46ÞAnother side reaction is the enolization of acrylate polymer chains [Eq. (47)].
H
A + CH2 C CH2 C + AH ð47Þ
COOR COOR
Whether these side reactions are termination or transfer reactions depends on the ability of the anions to re-initiate the polymerization and on the fate of the vinyl ketone formed in Eq. (45). Generally, this vinyl ketone is rapidly incorporated into the chain, and a relatively unreactive dormant end group is produced. The alkoxide from reactions such as Eq. (45) is usually not reactive enough to initiate another chain.These side reactions can be suppressed to some extent at temperatures below 0 C, which make them hardly accessible in industrial applications. The t-butyl group is reported to prevent side reactions. In general, the kinetic formalism for the heteroatom-containing monomers is the same as that described in Section

7.2.2, but side reactions are more important. Some recent developments in the po-
lymerization of these monomers have been described in Refs. 178 and 179, and these papers also cover the relevant mechanistic literature.However, the anionic polymerization of (meth)acrylate monomers has not yet gained industrial importance.
7.2.3
Anionic Polymerization of Monomers Containing Hetero Double Bonds Besides monomers containing carbon–carbon double bonds, monomers with het-eroatoms at the double bond can also be polymerized via an anionic mechanism.The only monomer with some industrial importance is formaldehyde. It is poly-merized [180] via its carbon–oxygen double bond to give polyacetal homopolymer or polyoxymethylene [Eq. (48)].nCH2 O ! ðaCH2 aOaÞn ð48ÞThe polymerization is a precipitation polymerization from gaseous formaldehyde in an inert solvent such as cyclohexane at fairly low temperatures to prevent depoly-merization reactions. Amines such as tri-n-butyl amine are used as initiators. The polymer precipitates in the dispersing agent as powder. After completion of the polymerization, it is necessary to stabilize the hydroxyl end groups, for example by esterification with acetic anhydride to prevent the unzipping reaction of Eq.
(49).
aðaCH2 aOaÞn CH2 aOH ! aðaCH2 aOaÞn1 CH2 aOH þ CH2 O ð49Þ
7.2.4
Anionic Polymerization via Ring Opening There are a number of heterocyclic monomers, for example, epoxides, cyclic sul-fides, lactones and lactides, lactams, cyclic carbonates, and cyclosiloxanes, which can be polymerized by ring-opening reactions; many of them can be polymerized by an anionic as well as by a cationic mechanism. They cannot all be covered here,but there are a number of monographs and reviews on this subject [181–185].In polymerization via double bonds the driving force for chain growth is the en-ergy difference between one double and two single bonds, which in most cases is high enough to overcome the loss in entropy when going from monomers to poly-mers. In ring-opening polymerization the number and nature of the bonds remain the same, and there is just the release of ring strain to make chain growth of cyclic monomers an exothermic reaction. The Gibbs free energy of polymerization DGP is given in Eq. (50).DGP ¼ DHP  TDSP (50)

7.2 Anionic Polymerization 347
The equilibrium constant KP for the chain growth step Pi þ M ! Piþ1 is defined in Eq. (51) and the equilibrium monomer concentration ½Me in Eq. (52).kp ½Piþ1  1 KP ¼ ¼ A ð51Þkd ½Pi ½Me ½Me ln½Me ¼ ðDHP  TDSP Þ=RT ð52ÞFor monomer concentrations below this equilibrium concentration, no polymeriza-tion occurs. As for most polymers, the changes for entropy and enthalpy are both negative (see Table 7.8), there exists a limiting temperature TC above which poly-merization is thermodynamically forbidden. TC is given by Eq. (53), where the standard state refers to unit concentration.
DHP DHP0
TC ¼ ¼ 0
ð53Þ
DSP DSP þ R ln½MFrom Table 7.8 it can be seen that for most of the cyclic monomers, the reversibil-ity of the propagation must be taken seriously into account. It has been shown[186] that, even for a living polymerization without any termination or transfer re-Tab. 7.8. Standard thermodynamic parameters for some cyclic monomers for anionic polymerization [182].Monomer Atoms in ring States [a] DHp0 [kJ molC1 ] DSp0 [ J molC1 KC1 ] [M]e [M]O 3 gc 140 174 4  1016 O 4 lc 82.4 74 3  1011
O O
5 ss 14 13.5 1:6  102
O OMe
6 lc 7.1 27.6 1.58
N O
H
O O 6 ss 22.9 41.1 1:16  102
O O
7 lc 13.8 4.6 2:2  103
H
[a] State of monomer (first letter) and polymer (second letter): g – gaseous, l – liquid, s – solution, c –condensed.

action but where the propagation step is reversible, the limiting molecular weight distribution is not a Poisson distribution but a most probable distribution with D ¼ 2.As the cyclic monomer and the linear polymer chain consist of the same bonds,there are two further side reactions which are inherent to the system – inter- and intramolecular chain transfer to polymer (Scheme 7.6).
X X X X X
X X X X X X X X X X
+ X +
X
X X X X X X X X X X
kinter
X
X X X X X
kintra X X
+ X X X
+ X
kp
X X X X X X Scheme 7.6. Chain growth, inter- and intramolecular transfer in ring-opening polymerization.By intermolecular chain transfer there is no change in Pn , because the number of chains remains the same, but a scrambling of the molecular weight distribution will occur, leading to the most probable MWD as a limiting distribution. Intra-molecular chain transfer will lead to macrocycles, which also can participate in propagation reactions. The equilibrium concentration according to the Jacobsen–Stockmayer theory [187] of these non-strained macrocycles of size n is given by Eq. (54).
kp ðnÞ
½Mn e ¼ A n5=2 ð54Þ
k p ðnÞ
The anionic polymerization of lactams [188, 189] is usually via a so-called activated anionic mechanism with a two-component catalyst system from a strong base and an N-acyllactam. It is interesting in the sense that the active center of the growing chain is an N-acyllactam end group to which a lactamate anion, formed from the monomer, is added (Scheme 7.7). First, there is H-abstraction from the monomer to give the lactamate anion. In principle, this anion could add to another lactam molecule, but this step is rather slow. Instead it will add much faster to the N-acyllactam activator. A subsequent proton transfer gives the active N-acyllactam end group and the lactamate anion is regenerated. The chain propagation consists of the same sequence of elementary reactions.This activated anionic polymerization is living in the sense that the number of active centers is preserved; but the inevitable intermolecular chain transfer to poly-

7.2 Anionic Polymerization 349
initiation
H O O
N N
O O O O O O R N N R N N O O O H O O H O O O R N N N R N N N propagation
O H O O O
N O H O O O
* N N
+ * N N N
O H O O
O H O O O H O O O N N N N N NH N N
n n
Scheme 7.7. Activated anionic mechanism for ring-opening polymerization of lactams.mer, here the transamidation, together with the propagation–depropagation of the cyclic monomer and macrocycles, will broaden the molecular weight distribution[189].This polymerization has attracted much attention because it is very fast com-pared to the hydrolytic polymerization of e-caprolactam, which takes several hours,whereas the anionic polymerization is complete within minutes, even below the melting point of the polymer. However, no large-scale process is known [190,191]. The anionic polymerization of caprolactam is described in melt in extruders[192, 193], in direct polymerization to fibers [194] and in casting and RIM pro-cesses [195]. The latter processes, particularly, are performed below the melting temperature, so the equilibrium monomer (and oligomer) concentration is much lower (1–3%) compared to the hydrolytic polymerization (10–12%), where a monomer extraction step is necessary.The polymerization of ethylene oxide was one of the first living polymerizations[196]. Polymers from oxiranes, polyoxyalkylenes, are used in a wide range of mo-

lecular weights and applications [22–24]. The most important ones are rather low molecular weight polymers, which are used either as nonionic surfactants [197] or as raw materials for polyurethanes [23, 24].Scheme 7.8 is a general reaction scheme for the copolymerization of ethylene oxide and propylene oxide with multifunctional alcohols as initiators.
OH OH
HO OH + KOH HO O K + H2O+ initiatior formation
OH O K
HO O K HO OH
OK
OK O
OK
OK O copolymerization
EO / PO
OK
OK O
+ + OK
OK O O
R R
+ + H2O + H2 O Termination
OK OH
R R R R
O O O O
n OH +
nO K
+ + mO K
mOH
chain transfer Scheme 7.8. Anionic ring-opening copolymerization of ethylene and propylene oxide.Usually potassium alkoxides, often from multifunctional alcohols, are used as initiators. They are formed in a separate step starting from the respective alcohol and potassium hydroxide. The resulting water must be removed to ensure maxi-mum conversion to the potassium alkoxide. Remaining water will terminate grow-ing chains, forming hydroxyl end groups. Chains with alkoxide end groups can undergo chain transfer reaction with hydroxyl-terminated chains. This proton ex-change, which may also happen with multifunctional initiator molecules, is the fastest reaction in this scheme. The acidity of the participating species is rather similar; the differences are generally those between primary and secondary hy-droxyl groups, so the equilibrium constant for the proton exchange is around unity.

onic polymerization follows the electron-donating ability of the substituents at the double bond. Electron-donating substituents increase the nucleophilicity of the double bond in the addition reaction and stabilize the resulting carbenium ion.The general order of reactivity in cationic polymerization is given in Eq. (55).
Me Me
H2C CH H2C CH H2C CH H2C H2C CH H2C
N Me
OR
OMe ð55Þ
Initiation is by either strong protonic acids (perchloric acid, triflic acid, and so on)or Lewis acids together with a co-initiator such as BF3/H2 O, AlX3/RX. As discussed for anionic polymerization, for cationic polymerization also several different spe-cies may be involved in propagation, but it has been pointed out (Ref. 2, pp. 190,
205) that the reactivity differences between the various species – carbenium ions
and ion pairs – are not very great. Covalent species are not active in propagation.In general, propagation rate coefficients in cationic polymerization are very large.A very comprehensive review is given in Ref. 214. For isobutene, values of 1:5  10 8 M1 s1 are reported at 0 and 78 C [215]. For such high rate coeffi-cients the polymerization may become diffusion-controlled, which has been ad-dressed [216] for a tubular reactor.In cationic polymerization of alkenes, there is an equilibrium between active ions or ion pairs and inactive covalent species, where the ionization constant is rather low (105 M1 in the isobutene/BCl3 system). The dynamics of this equilib-rium strongly affects the width of the molecular weight distribution. For the se-quence of reactions [Eq. (56)] the polydispersity of the distribution is given by Eq.
(57) [217], if there are no other side reactions such as transfer or irreversible termi-
nation.kp k ass þM H3 CaCHþ A !  H3 CaCHA ! ð56Þ
þM k ion
R R
D ¼ Pw =Pn ¼ 1 þ ½I 0 k p =kass ð57ÞBimodal distributions may result, if free ions and ion pairs have the same reactivity but different lifetimes [218].However, transfer reactions are nearly unavoidable side reactions in cationic polymerization and may involve b-proton transfer even to rather weak bases like the monomer itself, to transfer agents and solvent as well as Friedel–Crafts alkyla-tion of aromatic rings, if monomers like styrene are polymerized. Proton transfer yields unsaturated chains [Eq. (58)], with the double bond in either the endo or exo position. These may lead to branched polymers, if the double bonds are accessible to homo- or copolymerization.

7.3 Cationic Polymerization 353
* C A + C
A + + ð58Þ
As proton transfer usually has the higher activation energy, the molecular weight of the polymer decreases with temperature. Tabulated transfer coefficients may be taken from Refs. 219 and 220.The problem of living or non-living cationic polymerization of alkenes is ad-dressed in Refs. 220 and 221. It is shown that, depending on the molecular weight,even for transfer constants up to 5  104 the system exhibits the behavior of a liv-ing system, such as a linear increase of molecular weight with conversion, rather narrow distributions, and so on.Copolymerization parameters for cationic polymerization can be found in Refs.3, 222, and 223. Note that the scattering of data in higher and so the reliability of copolymerization parameters for cationic polymerization are much less than for radical polymerization.Commercial polymers [2, p. 683ff, 157, 224] from isobutene involve isobutene homopolymers and butene rubbers. Polyisobutene homopolymers comprise a wide range of polymers with molecular weights from a few thousand grams per mole up to 10 6 g mol1 , and butene rubbers are copolymers of isobutene with 1–3 mol% of isoprene. High molecular weight homopolymers are produced in the continuous BASF belt process at 100 C in boiling ethylene to remove the heat of polymerization. The ethylene evaporates during polymerization and the polymer is scraped from the belt. Volatiles are removed in twin-screw extruders. Homopoly-mers and rubbers are also produced in the Exxon slurry process in methyl chloride at 95 C. The polymerization rate is controlled by initiator feed and cooling is with liquid ethylene and/or propylene. The polymer slurry is withdrawn from the top of the reactor and worked up downstream.Medium and low molecular weight polymers are produced in solution processes in light hydrocarbons in CSTR reactors or in recirculating tubular (loop) reactors.The molecular weight is controlled by monomer purity, initiator, water content,and temperature.
7.3.2
Cationic Ring-opening Polymerization Monographs and reviews of this subject can be found in Refs. 181–185 and 225–
227. The most important polymers from cationic ring-opening polymerization are
polyoxyalkylenes with one or four CH2 groups produced by the polymerization of trioxane and tetrahydrofuran. The smaller epoxides, ethylene and propylene oxide,though able to polymerize by a cationic and anionic mechanism, are not polymer-ized cationically, because of the existence of side reactions leading to dioxane or

dioxolane [228]. In contrast to these smaller homologs, THF and trioxane can only be polymerized by a cationic mechanism [23, 229].Initiation can be by strong protonic acids such as perchloric acid and fluorosul-fonic acid, oxonium salts like triethyloxonium (Et3 Oþ , A ¼ BF4 or SbCl6 ), and Lewis acids, for example BX3 , PF5 , SBF5 , and so on. Lewis acids usually need a co-initiator, which in many cases is water, which is present to some extent in the monomer even after purification. Hetero polyacids of the general formula Hn Xm Mtz Oy with X ¼ P or Si and Mt ¼ Mo; W or V, have gained more and more importance, especially for THF. The active chain end is usually an oxiranium ion
(2).
R O
Ions and ion pairs are both active in propagation, but their respective contribution usually cannot be distinguished because of the high exchange rates between the two. Covalent species may reversibly result from association with counter ions such as CF3 SO 3 or ClO4 . They may participate in propagation reactions, but their reactivity is much less than that of the ionic species (see Table 7.9). A detailed dis-cussion on the kinetics of propagation for THF is given in Ref. 182.Because of the low energy gain from ring strain, the equilibrium monomer con-centration for the five- and six-membered rings such as THF and trioxane during polymerization is rather high (see Figure 7.8).As already discussed (see Section 7.2.4), inter- and intramolecular transfer reac-tions to polymer are of great importance with regard to the molecular weight dis-tribution (Scheme 7.6). In complete thermodynamic equilibrium, the conversion should be given by the equilibrium monomer concentration and the molecular weight distribution should be the most probable one with D ¼ 2, because of the intermolecular chain transfer, which is responsible for scrambling of the molecular Tab. 7.9. Propagation rate coefficients for some monomers for ionic and covalent species [2].Monomer Counter ion Solvent ˚T [ C] Propagating species k p [MC1 sC1 ]THF CF3 SO3 CCl4 25 ionic 4  102 covalent 6  105 CH2 Cl2 25 ionic 3:1  102 covalent 1:7  104 CH3 NO2 25 ionic 2:4  102 covalent 5  104 Oxepane CF3 SO3 H3 NO2 25 covalent 1:4  104 ionic 4  104 Oxazoline CH3 C6 H4 O CD3 CN 40 covalent 1:9  103 ionic 1:8  105

7.3 Cationic Polymerization 355
2.5
1.5
ln[M]
0.5
0.0028 0.003 0.0032 0.0034 0.0036
1/T / 1/K
Fig. 7.8. Equilibrium THF concentration during polymerization according to Ref. 252.weight distribution. Intramolecular chain transfer will give macrocycles, the con-centration of which should be proportional to n 5=2 [Eq. (54)], if ring formation is reversible and all equilibrium conditions are fulfilled. Examples of this ideal equi-librium between linear and cyclic polymers are found for 1,3-dioxolane and sil-oxanes [230, 231].However, for THF it has been shown [232] that, even after monomer conversion has reached the equilibrium concentration, there is still a buildup of cyclic mono-mers [Eq. (59)]: that is, the system is not yet in equilibrium with respect to macro-cycles and transfer reactions may still proceed.
O O
+ O + O
ð59Þ
So, in spite of these transfer reactions, linear polymers with a narrow molecular weight distribution can be formed by kinetic control, if, as for THF, the transfer rate coefficient to polymer is rather small. Chain growth, and consequently conver-sion and kinetic chain length, may have reached their equilibrium faster than the other equilibrium reactions. Depending on reactivity ratios in propagation and transfer, the polymerization can be stopped at low conversions, yielding the desired linear and narrow product [233].In contrast, transfer rates for trioxane to polymer are extremely fast. This can be seen from the observation that during copolymerization of trioxane with 1,3-dioxolane, ethylene oxide, and similar comonomers, the comonomer is nearly com-pletely consumed at an early stage of reaction [229, 234–236]. One would therefore expect longer sequences of these comonomers. However, it has been shown, from

NMR and stability studies, that the comonomers are randomly distributed along the chains [237, 238]. This is attributed to the fact that intermolecular chain trans-fer (transacetalization), contrary to THF polymerization, proceeds on a time scale similar to propagation, and the same holds for the intramolecular chain transfer leading to cyclic polymer [239, 240]. Cationic polymerization of trioxane is usually a precipitation polymerization leading to crystalline polymers, and the size of the macrocycles is determined by the size of the crystalline lamellae [240].The polymerization–depolymerization equilibrium is not between the growing chain and the originally propagating monomer, trioxane, but to formaldehyde [Eq.
(60)] [241, 242]. Thus formaldehyde should be considered as a comonomer [236,
243], which will be accumulated until its equilibrium concentration and pressure is reached.RaOCH2 aOCH2 aOCH2 aOaCHþ þ2 T RaOCH2 aOCH2 aOaCH2 þ CH2 O ð60ÞThis unzipping depolymerization occurs during polymerization, but it may also oc-cur under thermal stress with the neutralized polymer; unzipping then starts from neutral but unstable end groups such as aOH or aCHO or from statistical chain scission. Unzipping will stop at comonomer units such as those from ethylene ox-ide, dioxepane, and similar monomers, which are not able to depolymerize, and a then stable copolymer will result. Homopolymers, which are usually polymerized anionically from formaldehyde (see Section 7.2.3) will not be stable unless unstable end groups are transformed to stable ones.There are several industrial processes for poly-THF and polyoxymethylene.Cationic polymerization of poly-THF may be batch-wise or continuous [23, 244]using strong protonic acids in a temperature range of 30 to 60 C, depending on the desired molecular weight. Newer processes use solid catalysts like zirco-nium oxide together with acetic anhydride to form poly-THF with two acetate end groups, which later are hydrolyzed to give the dihydroxy polymer. Other acidic cat-alysts such as zeolites or acid montmorillonite clay are also described [245–247].The cationic copolymerization of trioxane with ethylene oxide, 1,3-dioxolane, and suchlike is initiated either with strong protonic acids or Lewis acids, for example BF3 . Molecular weight is controlled by the catalyst concentration and monomer purity, and also by chain transfer agents such as methylal [248], which may lead to more stable end groups. Most processes are run below the melting temperature of the polymer (164–167 C) in precipitating agents or in bulk, and are carried out in kneaders or double-screw reactors [249, 250], but there are also some descrip-tions of melt processes [251].
7.4
Conclusion
Ionic polymerization offers several advantages. In many cases, it is much faster than free-radical polymerization or polycondensation. This is either because of

Notation 357 very high rate coefficients for propagation, especially in the case of cationic poly-merization, or because of the high stationary concentration of active centers, for example in living anionic polymerization compared to free-radical polymerization.Furthermore, because of its site-based nature, it offers more control over the struc-ture of the resulting polymer by changing the reactivity of the active center when using different counter ions, solvents, complexing agents, or salts as additives. So a wide range of structural isomers can be obtained, and copolymerization parame-ters may be varied over a wide range, resulting in structures ranging from block copolymers to random copolymers from the same pair of monomers, which is not possible by free-radical polymerization.However, from a kinetic and modeling point of view, this site-based nature of ionic polymerization also has some disadvantages. Because the reactivity of the active center strongly depends on the nature of the initiator and on all the other factors in the polymerizing system, the kinetic scheme and parameters of every system are different, and so in general they must be determined again for every change in the system.The high sensitivity to impurities necessitates much more effort to purify mono-mers and solvents. This may prevent the wider employment of ionic polymeriza-tion. Generally, ionic polymerization is used for monomers which cannot be poly-merized by free radicals, or for the production of polymer structures which are otherwise not accessible.
Notation
D polydispersity index fi mole fraction of monomer i in the monomer mixture Fi mole fraction of monomer i in polymer chains being formed hðiÞ frequency distribution of chain length i degree of polymerization½I f initiator feed concentration kt rate coefficient for termination [M1 s1 ]k tt rate coefficient for thermal termination, for example, LiH elimination
[s1 ]
kAB rate coefficient for chain propagation [M1 s1 ] for monomer B added to chain ending with A kass rate coefficient of association k ion rate coefficient of ionization, dissociation kp rate coefficient for chain propagation [M1 s1 ]kd rate coefficient for decomposition [s1 ]ki rate coefficient for chain initiation [M1 s1 ]K equilibrium constant, general K ass equilibrium constant of association[M] monomer concentration [M]½Meq equilibrium monomer concentration

½Mf monomer feed concentration Mn number-average molecular weight [g mol1 ]Mw weight-average molecular weight [g mol1 ]½Ni  concentration of polymer chains with degree of polymerization i n association number or ring size Pn number average degree of polymerization Pw weight average degree of polymerization r1 ; r2 copolymerization parameters according to the terminal model r ratio of k p =k i for living polymerization Rp rate of propagation [M s1 ]R gas constant T temperature TC ceiling temperature v_ in; I initiator feed flow [L s1 ]v_ in; M monomer feed flow [Ls1 ]wðiÞ mass distribution of chain length xM monomer conversion xI initiator conversion
Greek
DGp Gibbs free energy of polymerization DHp enthalpy of polymerization DSp entropy of polymerization t residence time n average kinetic chain length
Acronyms
s- (t-, n-)Bu sec.- (tert.-, n-)butyl CSTR continuous stirred tank reactor HCSTR homogeneous continuous stirred tank reactor M, M
monomer molecule (
, with active center)T, T
transfer agent (
, with active center)THF tetrahydrofuran PaLi growing polymer chain with lithium counter ion PsH polystyrene molecule ending with hydrogen MWD molecular weight distribution Pi polymer chain with degree of polymerization i HIPS high-impact polystyrene NMR nuclear magnetic resonance RIM reaction injection molding SBR styrene–butadiene rubber SCSTR segregated continuous stirred tank reactor

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177 Sumitomo, US 4 845 173, 1989. Soc., 1952, 74, 3152.178 M. van Beylen, S. Bywater, G. 200 E. Santacesaria, M. DiSerio, R.Smets, M. Szwarc, D. J. Worsfolg, Garaffa, G. Addino, Ind. Eng. Chem.Adv. Polym. Sci., 1988, 86, 89. Res., 1992, 31, 2413.179 D. Baskaran, Prog. Polym. Sci., 2003, 201 A. J. Lowe, B. Weibull, J. Polym. Sci.,28, 521. 1954, 12, 493.180 Polyacetal, PERP report Nexant, Chem 202 W. Hreczuch, G. Bekierz, Tenside Systems, New York, 2002. Surf. Deterg., 1995, 32, 55.181 J. E. McGrath (ed.), Ring-Opening 203 F. Heatley, G. Yu, C. Booth, T. G.Polymerization – Kinetics, Mechanisms, Blease, Eur. Polym. J., 1991, 27, 573.and Synthesis, ACS Symp. Ser. 286, 204 M. Adal, P. Flodin, Tenside Surf.American Chemical Society, Deterg., 1994, 31, 1.Washington DC, 1985. 205 M. DiSerio, G. Vairo, P. Iengo, F.182 D. J. Brunelle (ed.), Ring-Opening Felippone, E. Santacesaria, Ind. Eng.Polymerization – Mechanisms, Catalysis, Chem. Res., 1996, 35, 3848.Structure, Utility, Hanser Publishers, 206 E. Santacesaria, M. DiSerio, P.Münich, 1993. Iengo, in Reaction Kinetics and the 183 K. J. Ivin, T. Saegusa (eds.), Ring- Development of Catalytic Processes, G. F.Opening Polymerization, Vols. 1–3, Froment, K. C. Waugh, (eds.),Elsevier, New York, 1984. Elsevier, New York, 1999.184 J. S. Riffle, K. Matyjeszwski (eds.), 207 M. DiSerio, R. Tesser, A. Dimiccoli,Makromol. Chemie, Macromol. Symp. E. Santacesaria, Ind. Eng. Chem. Res.,42/43, 1991. 2002, 41, 5196.185 F. Franta, P. Rempp (eds.), Makromol. 208 M. DiSerio, S. Di Martino, E.Chemie, Macromol. Symp. 32, 1990. Santacesaria, Ind. Eng. Chem. Res.,186 A. Miyake, W. H. Stockmayer, 1994, 33, 509.Makromol. Chem., 1965, 88, 90. 209 A. de Lucas, L. Rodriguez, M.187 H. Jacobsen, W. H. Stockmayer, J. Perez-Collado, Polym. Int., 2002, 51,Chem. Phys., 1950, 18, 1600. 1066.188 F. Fahnler, in Kunststoff-Handbuch, 210 L. E. Pierre, C. C. Price, J. Am.Vol. 3/4, Polyamide, p. 48ff, Hanser Chem. Soc., 1956, 78, 3432.Publishers, Münich, 1998. 211 D. M. Simons, J. J. Verbanc, J. Polym.189 K. Hashimoto, Prog. Polym. Sci., Sci., 1960, 44, 303.2000, 25, 1411. 212 S. D. Gagnon, Polyethers (Propylene 190 M. J. Kohan, Polyamides, in Oxide Polymers), in Kirk-Othmer Ullmann’s Encyclopedia of Industrial Encyclopedia of Chemical Technology,Chemistry, Vol. 31, 6th ed., Wiley-VCH, Vol. 19, 4th. ed., John Wiley, New Weinheim, 2003. York, 1996.191 Atofina, EP 1 165 661, 2000. 213 D. M. Bach, E. M. Clark, R.192 S. K. Ha, J. L. White, Int. Polym. Ramachandran, Polyethers (Ethylene Processing, 1998, 13, 136. Oxide Polymers), in Kirk-Othmer 193 B. J. Kim, J. L. White, Int. Polym. Encyclopedia of Chemical Technology,Processing, 2002, 17, 33. Vol. 19, 4th. ed., John Wiley, New 194 BASF, US 6 441 109, 2002. York, 1996.

214 P. H. Plesch, Prog. Reaction Kinetics, 234 V. Jaacks, ACS Adv. Chem. Series,1993, 18, 1. 1969, 91, 371.215 R. B. Taylor, F. Williams, J. Am. 235 Y. Yamashita, T. Tusda, M. Okada,Chem. Soc., 1969, 91, 3728. S. Iwatsuki, J. Polym. Sci., A-1, 1966,216 A. A. Berlin, S. A. Vol’fson, Polym. 4, 2121.Sci. USSR (Engl. Transl.), 1994, 36, 236 G. Opitz, Plaste Kautschuk, 1973, 20,
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219 P. H. Plesch, The Chemistry of Sci., 1974, 18, 1927.Cationic Polymerization, Pergamon 240 M. Hasegawa, K. Yamamoto, T.Press, Oxford, 1963. Shiwaku, T. Hashimoto,220 P. Sigwalt, Makromol. Chem. Macromolecules, 1990, 23, 2629.Macromol. Symp., 1991, 47, 179. 241 W. Kern, V. Jaacks, J. Polym. Sci.,221 K. Matyjaszewski, P. Sigwalt, Polym. 1960, 48, 399.Int., 1994, 35, 1. 242 W. K. Busfield, D. Merigold,222 J. P. Kennedy, T. Kelen, F. Tüdos, J. Makromol. Chem., 1970, 138, 65.Polym. Sci., Polym. Chem. Ed., 1975, 243 G. L. Collins, R. K. Greene, F. M.13, 2277. Berardinelli, W. H. Ray, J. Polym.223 T. Kelen, F. Tüdos, B. Turcsnayi, Sci., Polym. Chem. Ed., 1981, 19,J. P. Kennedy, J. Polym. Sci., Polym. 1597.Chem. Ed., 1977, 15, 3047. 244 A. Echte, Handbuch der Technischen 224 K. S. Whiteley, T. G. Higgs, H. Polymerchemie, VCH, Weinheim,Koch, R. L. Mayer, W. Immel, 1993.Polyolefins, in Ullmann’s Encyclopedia 245 DuPont, PCT Int. Appl. 92/14 773,of Industrial Chemistry, Vol. 28, 6th ed., 1992.Wiley-VCH, Weinheim, 2003. 246 G. Pruckmayr, P. Dreyfuss,225 S. Penczek, P. Kubisa, K. Matyjas- Polyethers, Tetrahydrofuran, in Kirk-zewski, Adv. Polym. Sci., 1985, 68/69, Othmer Encyclopedia of Chemical
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226 S. Penczek, P. Kubisa, K. Matyjas- New York, 1996.zewski, Adv. Polym. Sci., 1980, 37, 1. 247 P. Dreyfuss, M. P. Dreyfuss, G.227 T. Saegusa, S. Kobayashi, Prog. Pruckmayr, Tetrahydrofuran Polym. Sci., 1973, 6, 107. Polymers, in Encyclopedia of Polymer 228 S. Kobayashi, K. Morikawa, T. Science, Vol. 16, 2nd ed., Wiley-Saegusa, Polym. J., 1979, 11, 405. Interscience, New York, 1989.229 K. Weißermel, E. Fischer, K. 248 H. Nagahara, K. Hamanaka, K.Gutweiler, D. D. Hermann, Yoshida, T. Iwaisako, J. Masamoto,Kunststoffe, 1964, 54, 410. Chem. Lett., 2000, 2.230 J. M. Andrews, J. A. Semlyen, 249 G. Sextro, Polyoxymethylenes, in Polymer, 1972, 13, 142. Ullmann’s Encyclopedia of Industrial 231 J. A. Semlyen, P. V. Wright, Polymer, Chemistry, Vol. 28, 6th ed., Wiley-VCH,1965, 10, 643. Weinheim, 2003.232 J. M. McKenna, T. K. Wu, G. 250 J. Masamoto, Prog. Polym. Sci., 1993,Pruckmayr, Macromolecules, 1977, 10, 18, 1.
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João B. P. Soares and Leonardo C. Simon
8.1
Polyolefin Properties and Applications
8.1.1
Introduction Polyolefins are among the most important commodity plastics today due to their low production costs, reduced environmental impact, and a very wide range of ap-plications. They are found in products as diverse as prosthetic implants, gas pipe-lines, automobile parts and accessories, synthetic fabrics, films, containers, and toys.It may seem surprising that polymers composed only of carbon and hydrogen atoms can be so flexible, but the versatility of polyolefins can be easily explained by the way the monomer molecules – ethylene, propylene, and higher a-olefins –are connected to form the polymer chains. It is true for any polymer, but dramati-cally so for polyolefins, that chain microstructure is the key to understanding their physical properties.Polyolefins are produced in practically all types of reactor configurations –autoclaves, tubular reactors, loop reactors, fluidized-bed reactors – making them a prime choice for polymer reaction engineering studies. Polymerization may take place in either gas or liquid phases. For liquid-phase reactors, the monomers can be either liquid (as in the case of propylene and higher a-olefins) or dissolved in an inert diluent. Industrial catalysts for olefin polymerization are mainly heteroge-neous, but some processes also use soluble catalysts. There are many different types of catalysts for olefin polymerization and they can be used to synthesize poly-mer chains with very different microstructures and properties.Despite the fact that polyolefins use comparatively simple monomers and have been around for many years, it can be said with confidence that the degree of sophistication of polyolefin manufacturing and characterization techniques has
1) The symbols used in this chapter are listed at
the end of the text, under ‘‘Notation’’.Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

366 8 Coordination Polymerization no equal among other synthetic polymers. In this chapter, we will introduce the reader to this fascinating field of polymer reaction engineering by giving an over-view of the different scientific and technological aspects related to olefin polymer-ization with coordination catalysts.Types of Polyolefins and Their Properties Polyethylenes are the commercial polyolefins with the highest tonnage. Figure 8.1 compares the molecular structures of some polyethylene resins made by coordina-tion and free-radical polymerization. (Free-radical polyolefins are not the main topic of this chapter and will be described only in contrast with polyolefins made with coordination catalysts.)The oldest type of commercial polyolefin is low-density polyethylene (LDPE)made by free-radical polymerization. Low-density polyethylene is made using supercritical ethylene under severe polymerization conditions in autoclaves(1500–2000 atm, 180–290
C) or tubular reactors (1500–3500 atm, 140–180
C). Be-cause of backbiting and chain transfer to polymer reactions, LDPE has both short-and long-chain branches. These branches decrease the crystallinity and density of the polymer – typically in the range 0.915–0.935 g cm3 – and affect several other mechanical and rheological properties. Low-density polyethylene is used predomi-nantly for making films because of its limp feel, transparency, and toughness.Additionally, the long-chain branches give excellent processability and high melt tension to LDPE.High-density polyethylene (HDPE) was first made with Ziegler–Natta catalysts.When produced as a homopolymer, HDPE has no short-chain branches, but a H H H H H H H H H H H H H H C C C C C C C C C C C C C C H H H H H H H H H H H H H H LDPE LLDPE / VLDPE HDPE
0.915-0.935 g/cm3 0.915-0.94 g/cm3 / 0.88 - 0.912 g/cm3 0.96-0.97 g/cm3
Fig. 8.1. Polyethylene microstructures.

8.1 Polyolefin Properties and Applications 367
very small amount of a-olefin comonomer can be copolymerized with ethylene to form short-chain branches that decrease its density. Typically, the density of HDPE resins is in the range from 0.96 to 0.97 g cm3 . Since HDPE has few or no short-chain branches, it has much greater rigidity than LDPE and can be used in structural applications.Linear low-density polyethylene (LLDPE) is a copolymer of ethylene and a-olefins(generally 1-butene, 1-hexene, or 1-octene) with densities in the range 0.915–0.94 g cm3 . Products with even lower densities, down to 0.88 g cm3 , are sometimes called very low-density polyethylene (VLDPE) but are chemically identical to LLDPE. Copolymerization of ethylene with increasing amounts of a-olefins dis-rupts the order of the linear polyethylene chains by introducing short-chain branches. As a consequence, the density, crystallinity, and rigidity of LLDPE are lower than for HDPE. Linear low-density polyethylene is used predominantly infilms, and shares the market with LDPE.Polypropylene is the second most important commercial polyolefin. While free-radical polymerization can only produce atactic polypropylene, Ziegler–Natta cata-lysts can make highly isotactic polypropylene. The chain regularity of isotactic polypropylene is responsible for its high melting temperature and crystallinity,making it ideal for injection molding and extrusion applications due to its excellent rigidity, toughness, and temperature resistance. Metallocene catalysts can produce several types of polypropylenes with other stereosequences, as illustrated in Figure
8.2 for some representative catalysts. Isotactic polypropylene is by far the leader
among the commercial stereoisomers of polypropylene, but syndiotactic polypropy-lene and some stereoblock elastomeric polypropylenes made with metallocene cat-alysts have attracted some interest lately [1–3].Olefins are commonly polymerized in a single reactor or in two or more reactors in series. Single reactors are used when a polyolefin with more uniform composi-tion is required, while reactors in series are employed for the commercial produc-atactic Cp2ZrCl2 isotactic Et(Ind)2ZrCl2 syndiotactic iPr(Flu)(Cp)HfCl2
isotactic-
stereoblock (NMCp)2ZrCl2
isotactic-
atactic- Et(Me4Cp)(Ind)TiCl2 stereoblock hemi-isotactic iPr(Cp)(Ind)ZrCl2 Fig. 8.2. Polypropylene microstructures and some metallocene catalysts used in their synthesis.

368 8 Coordination Polymerization propene + ethene
PC PC
propene
catalyst +
diluent
product
TC TC
c.w. c.w.homopolymer matrix Fig. 8.3. Process for the production of high-impact polypropylene.tion of homopolymers and copolymers with more elaborate microstructural distri-butions. A series of reactors allows for significant flexibility during polymerization,since each reactor can be kept in different polymerization conditions to make a polymer that can be considered a blend of two or more components (reactor blends). High-impact propylene/ethylene copolymers are the most typical example of this polymerization technique. Propylene is fed to the first reactor, producing isotactic polypropylene particles. A mixture of propylene and ethylene is fed to the second reactor to make a propylene–ethylene amorphous copolymer phase which is dispersed within the isotactic polypropylene particles already formed. The amor-phous copolymer fraction is responsible for increasing the impact strength of the homopolymer matrix formed in the first reactor, as illustrated in Figure 8.3.Another important example of polyolefins produced with reactors in series com-prises HDPE resins for pipe applications. It will be shown later that when ethylene and a-olefins are copolymerized with a heterogeneous Ziegler–Natta catalyst in a single reactor, the low molecular weight chains are also the ones that contain the highest a-olefin fraction. For pipe applications this is undesirable because it de-creases the environmental stress cracking resistance of the polymer [4, 5]. One way of overcoming this problem is to produce a high molecular weight copolymer in the first reactor in the absence of chain-transfer agent and a lower molecular weight homopolymer in the second reactor in the presence of chain-transfer agent.Coordination catalysts are also used to make elastomers, notably ethylene–propylene–diene (EPDM) rubbers [6]. In this case, homogeneous catalysts are pre-ferred to heterogeneous ones because they generally produce polymers with a more uniform distribution of crystallinity. The unreacted double bonds of the dienes are used during rubber crosslinking reactions. Figure 8.4 shows some typi-cal examples of dienes used for the manufacture of EPDM rubbers.

8.1 Polyolefin Properties and Applications 369
5-ethylidene-2-norbornene CH - CH31,4-hexadiene CH2 = CH - CH2 - CH = CH - CH3 Fig. 8.4. Different dienes used for the production of EPDM.
8.1.3
The Importance of Proper Microstructural Determination and Control in Polyolefins Since polyolefins are composed of only carbon and hydrogen atoms, the way these atoms are connected – in other words, the microstructure of a polyolefin –determines their properties. At the most elementary level, the microstructure of a polyolefin is defined by its distributions of molecular weight, chemical composi-tion, long-chain branching, and stereoregularity. Therefore, determining the micro-structure of polyolefins can be considered the first and most important step toward the understanding of their properties.High-temperature gel permeation chromatography (GPC) is the standard tech-nique for measuring the molecular weight distribution of polyolefins. Since gel permeation chromatography is a very well-established technique it will not be dis-cussed here. Several publications provide additional discussions on the GPC tech-nique [7–9].The chemical composition distribution of polyolefins is measured (indirectly)by either temperature rising elution fractionation (Tref ) or crystallization analysis fractionation (Crystaf ). These two techniques provide similar information on the chemical composition distribution of polyolefins and can be used interchangeably in the vast majority of cases. Both methods are based on the fact that the crys-tallizability of HDPE and LLDPE depends strongly on the fraction of a-olefin como-nomer incorporated into the polymer chains, that is, chains with an increased a-olefin fraction have a decreased crystallizability. A similar statement can be made for polypropylene and other polyolefin resins that are made with prochiral mono-mers: resins with high stereoregularity and regioregularity have higher crystalliz-abilities than atactic resins.In Crystaf, polyolefin chains are crystallized from a dilute solution by slowly lowering the solution temperature. Chains with fewer a-olefin units crystallize at higher temperatures, while chains with a higher a-olefin fraction crystallize at lower temperatures. This information is used to generate a Crystaf profile relating crystallization temperatures to the fraction of polymer that crystallizes at those

370 8 Coordination Polymerization Crystaf vessel Cumulative Differential Crystaf Profile Crystaf Profile
Tc Tc
Chemical Composition Distribution Calibration Curve
w Tc
α-olefin fraction α-olefin fraction Fig. 8.5. Estimation of the chemical composition distribution of a polyolefin using a Crystaf profile and a calibration curve.temperatures. The Crystaf profile is then converted into the chemical composition distribution by means of a calibration curve relating the fraction of a-olefin in the copolymer to the crystallization temperature, as illustrated in Figure 8.5. Tempera-ture rising elution fractionation operates in a similar way to Crystaf, but involves one additional step, when the chains crystallized during the crystallization period described above are eluted from the Tref column to generate the Tref profile. A Tref calibration curve, similar to the Crystaf calibration curve, is used to transform the Tref curve into the chemical composition distribution. More details of the Tref and Crystaf analytical procedures can be obtained from many recent publications[7, 10, 11].For our purposes in this chapter, it suffices to say that the chemical composition distribution of LLDPE made with heterogeneous Ziegler–Natta catalysts is gener-ally bimodal or multimodal, as illustrated in Figure 8.6. This clearly indicates that heterogeneous Ziegler–Natta catalysts have at least two – but probably more –distinct active-site types with different reactivity ratios for a-olefin incorporation:one site type produces polymer with a low a-olefin content, while the other site type makes polymer with a much higher a-olefin content. The presence of two or more active sites on heterogeneous Ziegler–Natta catalysts is also confirmed by the fact that they produce polyolefins with broad molecular weight distributions and polydispersities much higher than the theoretical value of 2 that would be expected for polymers made with coordination catalysts containing a single site type, as will

8.1 Polyolefin Properties and Applications 371
Easier to crystallize Weight fraction More difficult Mol % α-olefin comonomer Crystallization temperature Fig. 8.6.Chemical composition distribution of a polyolefin made with a heterogeneous Ziegler–Natta catalyst.be discussed in more detail below. In addition, the chains with the lower a-olefin fractions are also the ones with the higher molecular weight averages, as depicted in Figure 8.7 [12]. In this way, the distributions of molecular weight and chemical composition of polyolefins made with heterogeneous Ziegler–Natta catalysts are al-ways coupled and are, in fact, considered the ‘‘fingerprint’’ of these resins.The molecular weight distribution of polypropylene made with heterogeneous Ziegler–Natta catalysts is also broad, with polydispersities higher than 2. In addi-tion, some catalysts will also produce polypropylene with a distribution of stereo-regularities that can be traced back to the presence of multiple active-site types with distinct stereochemical control characteristics [13].On the other hand, several homogeneous Ziegler–Natta and metallocene cata-lysts make HDPE, LLDPE, and polypropylene with narrow distribution of chemical composition and molecular weight with the theoretical polydispersity of 2. The av-erage a-olefin fraction for these polymers is constant and independent of molecular weight, which confirms that they are made by a single-site catalyst [14]. In fact, the ability to make polyolefins with narrow microstructural distributions, and con-sequently with a much better control of polymer microstructures and properties,is one of the main reasons why metallocene catalysts have had such an impact on the polyolefin manufacturing industry since the 1980s.An equally important contribution of metallocene catalysts to polyolefin syn-thesis is the ability to produce polyethylene and polypropylene with long-chain branches. These narrow-MWD resins have excellent processability and melt strength due to the presence of few long-chain branches (often less than 1 per 1000 carbon atoms) and have had a marked impact on the polyolefin industry[15]. The only absolute way of measuring long-chain branches in polyolefins is by carbon-13 nuclear magnetic resonance ( 13 C NMR) but, due to the low levels of long-chain branches generally present in these resins, these measurements are

372 8 Coordination Polymerization
2x105
Normalized response (a.u.) 0.04
Mw
8x104
0.02
6x104
0.00 4x104
0 2 4 6 8 10 x, mole % butene Fig. 8.7. Average weight-average molecular weight of ethylene–1-butene copolymers made with a heterogeneous Ziegler–Natta catalyst as a function of average 1-butene content [11a].very often subject to significant experimental error. Nonetheless, very good evi-dence of the presence of long-chain branches can be found by studying the rheo-logical response of these resins. Long-chain branch formation with polyolefins will be discussed in detail below in Section 8.3.4.
8.2
Catalysts for Olefin Polymerization The class of catalysts we currently call Ziegler–Natta was first used by Ziegler in 1953 to polymerize ethylene at low pressure, and further developed by Natta in 1954 to produce isotactic polypropylene. Both Ziegler and Natta were awarded the Nobel Prize in chemistry in 1963; since then, this field has grown incessantly, with the development of improved catalysts and new industrial processes.Many other catalysts capable of polymerizing olefins have become available since the original Ziegler–Natta catalyst based on crystalline titanium chloride was intro-duced. More recently, the discovery of soluble metallocene/aluminoxane catalysts opened the doors to a new revolution in the production of polyolefins. These cata-lyst systems are able to make polyolefins in very high yields and with a degree of microstructural control not possible to achieve using conventional Ziegler–Natta catalysts.Most industrial processes today still use heterogeneous Ziegler–Natta catalysts,although the market share of metallocene resins is increasing due to the enhanced

8.2 Catalysts for Olefin Polymerization 373
properties of polyolefins made with these catalysts and the fact that polymerization processes that were originally designed to use Ziegler–Natta catalysts can be con-verted, with minimal changes, to operate with metallocenes – the so called ‘‘drop-Although traditional heterogeneous and homogeneous Ziegler–Natta catalysts are commonly used as the standard example of coordination polymerization, co-ordination catalysts include any complex of transition metals and organic ligands:Phillips catalysts are heterogeneous, chromium-based complexes that are not clas-sified as Ziegler–Natta catalysts; metallocenes are complexes of a transition metal –in most cases an early transition metal – and cyclopentadienyl or cyclopentadienyl-derivative ligands; late transition metal catalysts may have a variety of ligands con-taining heteroatoms, such as phosphorus, nitrogen, or oxygen, directly bonded to the transition metal.The active site in coordination catalysts for olefin polymerization is, therefore, a transition metal surrounded by ligands. The catalytic properties depend on the fine tuning between the transition metal and the ligands in terms of geometry and elec-tronic character. In most cases the active site is produced by the activation of a complex called a pre-catalyst, or catalyst precursor. The pre-catalyst is commonly stable and easy to handle; sometimes it is stable even to moisture and oxygen.The creation of the active site by reaction of the pre-catalyst with an activator or cocatalyst is made just prior to its injection in the polymerization reactor or inside the polymerization reactor itself. The activator alkylates the pre-catalyst complex to form the active sites and stabilizes the resulting cationic active site. Common acti-vators are based on organoaluminum or organoborane compounds [16]. Because the activator works as a Lewis acid (electron acceptor), it is also used to scavenge polar impurities from the reactor. These impurities are electron donors, such as oxygen, sulfur and nitrogen compounds, and moisture, that poison the cationic active site. Figure 8.8 shows a simplified chemical equation for the activation mechanism and the corresponding equation used for modeling in the Section 8.3.
L X L R
A AlR3 A AlR2X2
L X L
C Al C*
A = transition metal center (Ti, Zr, Ni, …)L = ligands X = halogen (Cl, Br)AlR3 = alkylaluminum cocatalyst R = alkyl group (methyl, ethyl)Fig. 8.8. Catalyst activation by reaction of a pre-catalyst and a cocatalyst.

374 8 Coordination Polymerization
L R L R L
A A A
L L L
C* M P*1
Pol
L L ( )
A n A n
L L
P*1 nM P*1+n Fig. 8.9. Monomer coordination and insertion.Polymerization with coordination catalysts proceeds via two main steps: mono-mer coordination to the active site, and monomer insertion into the growing poly-mer chain, as illustrated in Figure 8.9. Before insertion, the double bond in the olefin monomer coordinates to the coordination vacancy of the transition metal.After the olefin is inserted into the growing polymer chain, another olefin mono-mer can coordinate to the vacant site; thus the process of insertion is repeated to increase the size of the polymer chain by one monomer unit at a time until chain transfer takes place [17]. In the case of copolymerization, there is a competition be-tween the comonomers to coordinate to the active sites and to be inserted into the growing polymer chains. Different rates of coordination and insertion of comono-mers determine the final chemical composition of the copolymer chain.Insertion of a prochiral a-olefin such as propylene creates a chiral carbon bonded to the active site. Differently from free-radical polymerization, coordination poly-merization can be regio- and stereoselective, depending on the design of the li-gands bonded to the transition metal. Regioselectivity determines the sequence of 1–2 or 2–1 insertions while stereoselectivity determines whether the polymer is isotactic, syndiotactic, or atactic (Figure 8.10).Several chain-transfer mechanisms are operative in coordination polymerization:transfer by b-hydride elimination; transfer by b-methyl elimination; transfer to monomer; transfer to cocatalyst; and transfer to chain-transfer agent – commonly hydrogen – or other small molecules. The type of termination reaction determines the chemical group bound to the active site and the terminal chemical group in the polymer chain. The first three types produce unsaturated chain ends, while the last two types produce saturated chain ends. Figure 8.11 illustrates these five transfer mechanisms.Reaction of the active site with polar impurities deactivates the catalyst. Due to the cationic nature of the active sites, nucleophilic groups with a lone pair of elec-

8.2 Catalysts for Olefin Polymerization 375
Regioselectivity 1,2-insertion
Pol L
L L 2
1 Pol
2,1-insertion
Pol L
L L
A 2 1
Pol
Stereoselectivity
Pol L
L L
Pol
Pol L
L L
Pol
Fig. 8.10. Regio- and stereoselectivity in coordination polymerization.trons (generally substances containing oxygen, nitrogen, or sulfur) can coordinate irreversibly with the active site, causing irreversible catalyst deactivation. Bimo-lecular catalyst deactivation happens when two active sites form a stable complex that is inactive for monomer polymerization. This type of bimolecular intermediate is favored at high catalyst concentrations and is reversible [18]. The term ‘‘latent state’’ can be used to describe certain catalytic intermediates that reduce the cata-lyst activity. The formation of latent states is difficult to probe and the mechanisms are generally not well understood. Figure 8.12 shows chemical equations for this catalyst deactivation mechanism.

376 8 Coordination Polymerization chain transfer by β-hydride elimination
A A A Pol
L Pol L L
Pol
P*r P*H Dr
chain transfer by β-methyl elimination L L CH3 L CH3
A A A Pol
L Pol L L
Pol
P*r P* Dr
chain transfer to hydrogen A H2 A 2 A Pol L Pol L Pol L P*r H2 P*H Dr chain transfer to monomer L Pol L Pol L
A A A Pol
L L L
P*r M P *r Dr chain transfer to cocatalyst
L Pol L
A AlR3 A Pol AlR2
L L R
P*r Al P*A D Al Fig. 8.11. Chain termination mechanisms.
L R L R L
A A A
L L R L
2 C* 2 Cd
Fig. 8.12. Catalyst deactivation by bimolecular reactions.

8.2 Catalysts for Olefin Polymerization 377
Dr
H2
Dr
Dr
AlR3 M
P*H
C C* P*r P*H P*Me
M
M
M
M
Fig. 8.13. Catalytic cycle for polymerization.The catalytic cycle is a convenient graphical way to describe the central role played by the active site in the mechanism of polymerization. Changes in the na-ture of the active site will affect the catalytic mechanism and consequently the activity and the selectivity of the polymerization. Changes in the polymerization re-actor conditions, such as temperature and monomer concentration, play a vital role in the catalyst mechanism because they affect the rate constants of each of these steps. Figure 8.13 shows a catalyst cycle for olefin polymerization with coordina-tion catalysts.Catalytic activity is a measurement of how fast the monomer can be poly-merized. Unfortunately, it may be expressed in several ways in the literature,which very often makes it difficult to compare experimental results from different research groups. A common, but not recommended, way is to express catalyst activity as the mass of polymer made by one mole of catalyst per unit of time, for instance kgpolymer (mol catalyst h)1 . Since polymerization rate is proportional to monomer concentration, this value of the catalyst activity cannot be used for differ-ent monomer concentrations. A better way is to normalize the catalyst activity with respect to the moles of monomer present in the reactor, that is, to express the cat-alyst activity in units of kgpolymer (mol catalyst h molmonomer )1 . Since the measured polymerization rate does not necessarily have a first-order dependence on mono-mer concentration, this form may also be oversimplified, but it is certainly pref-erable to the first one. In addition, since coordination catalysts generally deactivate during polymerization, one should be very careful when extrapolating these aver-age catalyst activities to different polymerization times. In general, it is advisable that the complete plot of catalyst activity as a function of polymerization time be available for the proper quantification of polymerization kinetics with coordination catalysts. Such profiles are illustrated in Figure 8.14 for characteristic polymeriza-tion cases. Another commonly used parameter when accessing catalyst activity is the turnover frequency, in time1 units, which stands for the time necessary for one monomer insertion to take place.

378 8 Coordination Polymerization polymerization rate
Decay type
Build-up type polymerization time Fig. 8.14.Polymerization kinetics with coordination catalysts showing different deactivation rates.
8.2.1
Ziegler–Natta, Phillips, and Vanadium Catalysts The low-pressure ethylene polymerization catalyst introduced by Ziegler was TiCl4 /AlR3 , where R is commonly an alkyl group such as methyl or ethyl. A cata-lyst for stereospecific polymerization of propylene, TiCl3 /AlR3 , was disclosed by Natta shortly after Ziegler’s discovery. The active sites for polymerization are located at the surface and edges of the crystalline structure of titanium chloride.Differences of atomic arrangements around the titanium atoms on the titanium chloride surface affect the chemical properties and consequently the activity and the selectivity of the polymerization catalyst. First-generation Ziegler–Natta cata-lysts consisted of particulate titanium chloride crystals. Two major advances in het-erogeneous Ziegler–Natta catalysts were the use of supports and internal donors.Inert supports such as magnesium chloride can improve the morphology of thefinal polymer particle in the reactor, increase catalytic activity by many times, and consequently decrease the load of transition metal in the final polymer. Internal donors help to control the selectivity of the catalyst by blocking certain sites and therefore are necessary to control polymer structure. The evolution of heteroge-neous Ziegler–Natta catalysts is a truly fascinating story that has been reported in detail in many references [19, 20].Phillips catalysts are based on Cr(IV) supported on silica and alumina. The true structure of the Phillips catalyst is not well understood. A mixture of chromium oxide and silicon oxide (CrO3/SiO2 ) is used to create the active sites. The catalyst does not require addition of chemical activators before the polymerization, since the active site is produced prior to the polymerization by thermal activation at high temperatures (600
C, for instance) [21, 22]. Phillips catalysts are used in both gas-phase and slurry processes. Polyethylene made with Phillips catalysts

8.2 Catalysts for Olefin Polymerization 379
has a very broad molecular weight distribution, with polydispersities ranging from 12 to 24. Interestingly, hydrogen is not an effective chain-transfer agent and gener-ally decreases catalyst activity.Vanadium catalysts are used to manufacture ethylene–propylene–diene rubbers.The catalyst is produced by activating a mixture of VCl3 and VCl4 with AlR3 (alky-laluminum) compounds. Vanadium catalysts are very sensitive to reaction condi-tions [23]. Changes in the composition of the solvent, polymerization time, and temperature can affect catalyst activity and selectivity.Metallocene Catalysts Metallocene catalysts have structures similar to ferrocene (Cp2 Fe), which is a widely known complex with a ‘‘sandwich’’ structure, where the transition metal lies between two cyclopentadienyl (Cp) rings. Cp2 ZrCl2 (Figure 8.15) is a well studied example of a metallocene catalyst for olefin polymerization. Metallocenes are soluble and therefore can be used as homogeneous catalysts, but they can also be supported on inert carriers such as silica, alumina, and magnesium chloride,among other inorganic and organic supports, and used as heterogeneous catalysts[24].The cocatalysts commonly used with Ziegler–Natta catalysts (AlEt2 Cl, AlEt3 ) are not able to activate metallocene catalysts very well. Even though metallocenes can be activated with these cocatalysts, they generally deactivate very fast, and consequently have no commercial interest. The use of bulky activators, such as methylaluminoxane (MAO), is necessary to produce highly active metallocene cata-lysts. In fact, it can be said that it was the discovery of MAO as a good cocatalyst for metallocenes that made possible the metallocene revolution that we are cur-rently experiencing in polyolefin manufacture [25]. Other bulky Lewis acids such as trispentafluorophenylborane (B(C6 F5 )3 ) are also very good cocatalysts for metal-locenes.A large variety of metallocene catalysts can be obtained by altering the simple structure of Cp2 ZrCl2 . The nature of the transition metal and the structure of the ligand have a large effect on catalyst behavior. The shape, geometry, and chemical structure of the ligand can affect the activity and selectivity of the catalyst. The symmetry imposed by ligands around the active site determines the geometry for monomer coordination and insertion, and consequently the relative orientation of catalyst and growing polymer chain. The possibility of variations in the chemical
Zr
Fig. 8.15. Structure of a typical metallocene catalyst, Cp2 ZrCl2 .

380 8 Coordination Polymerization C2v Cs (meso) C2 (racemic)(atactic PP) (atactic PP) (isotactic PP)Ex: Cp2ZrCl2 Et(Ind)2ZrCl2 Ex: Et(Ind)2ZrCl2
Cs C1
(syndiotactic PP) (no unequivocal prediction)Ex: iPr(Flu)(Cp)HfCl2 Ex: (Flu)(MeCp)ZrCl2 Fig. 8.16. Symmetry requirements for the production of polypropylenes with different stereoregularities.substitutions and/or presence of bridging groups on the cyclopentadienyl rings creates an endless set of possible structures, especially if substitutions of ligands with heteroatoms are considered. The size and the type of the bridge connecting the cyclopentadienyl rings affect the opening on the opposite side of the transition metal, and consequently the relative rates of the elementary steps in the polymer-ization mechanism. This effect is called the ‘‘biting angle’’. The literature on metal-locene catalysts is very rich – one may even say opulent – and thousands of metal-locene catalyst structures have been documented since the early 1980s [1, 2, 26].The appropriate selection of ligand structure has a major effect on the selectivity of the polymerization mechanism. A practical and very interesting example is the control of tacticity in polypropylene. There are two mechanisms to explain control of stereoregularity: chain-end or enantiomorphic site control. In the former, the ori-entation of the growing polymer chain determines the orientation of the monomer during insertion. In the latter, this orientation is determined by the active site, as determined by the arrangements between the transition metal and its ligands [27].Since the stereochemical control of most metallocenes is of the enantiomorphic type, it is possible to classify them according to the type of polypropylene chain they will produce based merely on the symmetry of the ligands around the active site, as elegantly illustrated in Figure 8.16.Furthermore, certain ligands can change the orientation during the growth of a polymer chain, for instance oscillating between configurations that produce atactic and isotactic polypropylene blocks. By the proper choice of polymerization condi-tions it is possible to produce polypropylenes that have chains varying from mainly atactic through block atactic–isotactic to mainly isotactic configurations [28].

8.2 Catalysts for Olefin Polymerization 381
CH3
H3C
CH3
H3C
CH3
H3C
CH3
H3C N
H3C CH3
CH3
Fig. 8.17. A half-sandwich metallocene.Another important class of metallocene catalysts, the ‘‘half-sandwich’’ metallo-cenes (Figure 8.17) have a more open structure because only one cyclopentadienyl ring is coordinated to the transition metal. Catalysts with open structures have very high reactivity ratios toward the incorporation of longer a-olefins and are used for the production of ethylene/1-octene copolymers [29]. More importantly, these cata-lysts can form polyolefins containing long-chain branches via a mechanism that is analogous to the copolymerization of long a-olefins, as will be described in Sec-tion 8.3.4.
8.2.3
Late Transition Metal Catalysts This family of coordination catalysts is characterized by having a transition metal from groups 8, 9, or 10. A great advantage of some polymerization catalysts based on late transition metals is that the active sites are more tolerant to polar comono-mers (and impurities) than the early transition metals used with metallocenes. The tolerance toward polar comonomers is attributed to the electronic configuration of the central metal and its interaction with its ligands. This feature is very attractive for modifying the chemical composition of polyolefins by copolymerization with vinyl alcohols, acrylates, or other vinyl polar comonomers. A few polar functional groups can significantly increase the hydrophilicity of polyolefins and enhance sev-eral of their properties, such as dyeability, adhesion, and compatibility with other polar polymers.Late transition metal catalysts are very active and produce high molecular weight polyethylene. The catalyst can be used in homogeneous solution or supported in inert carriers such as silica. The active site can be generated by activation of a metal halide complex such as (diimine)NiCl2 with organoaluminum compounds such as(CH3 )3 Al or MAO. As for metallocenes, the active species are cationic. These cata-lysts have a well defined structure and in homogeneous systems can behave as single-site catalysts.

382 8 Coordination Polymerization Forward Forward Chain Chain Walking Walking A
P P P
Backward
A Chain
Walking
Chain Walking followed by Insertion
Monomer
A Insertion
P A P
Ethyl Branch
Insertion
A = metal active center P = polymer chain Fig. 8.18. Chain walking mechanism.A class of late transition metal catalysts based on Ni and Pd with bulky diimine ligands has an interesting feature called the ‘‘chain walking mechanism’’ [30].With the appropriate choice of ligands and reactor conditions, it is possible to pro-duce branched polyethylene from pure ethylene without addition of comonomers.The chain walking mechanism involves simultaneous isomerization and polymer-ization steps, as indicated in Figure 8.18. Isomerization reactions cause the active site to move along the growing polymer chain without terminating chain growth.A short-chain branch is created when monomer insertion takes place after the active site has moved away from the chain end. The number of carbons in this short branch equals the number of carbons by which the active site has moved away from the chain end. Branches from methyl to hexyl or longer are produced during ethylene polymerization, although methyl branches are predominant [31].The number of branches produced by the chain walking mechanism can be controlled by varying the polymerization temperature and monomer concentration.Increasing the polymerization temperature or decreasing the monomer concen-tration favors chain walking and increases the number of short-chain branches.By choice of appropriate polymerization conditions, ethylene can be used to make polyethylene with properties varying from very low density (highly branched) to high density (few or no branches).The molecular weight of polyethylene made with some late transition metal catalysts based on iron can be very high (> 1 000 000 g mol1 ) [30]. At low tem-peratures polymerization of a-olefins with certain diimine–NiCl2 catalysts behave as a living polymerization: that is, the number-average molecular weight increases linearly with polymerization time and the polydispersity index (PDI) is approxi-mately 1.

384 8 Coordination Polymerization Free-radical polymerization is monomer-based:RM* + M RMM*R(M)n* + M R(M)n+1*Coordination polymerization is site-based:Ti CH2 - CH2
CH2 CH2
Fig. 8.19. Comparison of the monomer propagation step in free-radical and coordination polymerization.polymerization can be divided into five main classes of reaction: catalyst activation with the cocatalyst; catalyst initiation with monomer; chain propagation; chain transfer; and poisoning and deactivation [32].No bimolecular termination reactions – termination by combination or disproportionation – as observed in free-radical polymerization take place with co-ordination catalysts. Some catalysts, under certain polymerization conditions, may polymerize dead polymer chains containing terminal vinyl unsaturations, leading to the formation of chains with long-chain branches. We will discuss the mecha-nism of long-chain branch formation with coordination catalysts in Section 8.3.4.Coordination catalysts must be activated via reaction with a cocatalyst, which is generally present in great excess. (In some references, the complex that is reacted with the cocatalyst is called the pre-catalyst and the product of the reaction between the pre-catalyst and the cocatalyst is named the catalyst. Even though this termi-nology is more rigorous, we will adopt the looser terminology by which the transi-tion metal compound is called the catalyst, even before reaction with the cocata-lyst.) The cocatalyst/catalyst molar ratio is commonly 5:1–20:1 for heterogeneous Ziegler–Natta catalysts, but it can be as high as 1000:1 or more for metallocene catalysts. Little is known, from a quantitative point of view, about the kinetics of catalyst activation for coordination polymerization. Qualitatively, it has been shown that the cocatalyst is required to reduce and alkylate the active site. This reaction is generally assumed to proceed very quickly and to be first order with respect to cat-alyst, C, and cocatalyst or activator, Al, to form the active site C  [Eq. (1)].

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 385
C þ Al ! C  ð1ÞAfter the catalyst is activated by reaction with the cocatalyst, it is ready to poly-merize the monomer, M, forming a living polymer chain P1 of length 1 [Eq. (2)].C  þ M ! P1 ð2ÞA subsequent monomer propagation reaction with a growing polymer of chain length r, Pr , increases its length to r þ 1 [Eq. (3)].
kp
Pr þ M ! Prþ1 ð3ÞIn general, the first monomer insertion step is considered to have a different reac-tion rate constant, k i , from the subsequent monomer propagation steps, k p . Since in practice these constants are very difficult (if not impossible) to estimate in-dependently, most kinetic models make the reasonable simplifying assumption that k i G k p .The most important transfer reactions in coordination polymerization are: (1) b-hydride elimination; (2) transfer to chain-transfer agent; (3) transfer to monomer;and (4) transfer to cocatalyst.b-Hydride elimination is a first-order reaction in which the hydrogen atom at-tached to the b-carbon in the living chain is abstracted by the active center, forming a metal hydride center, CH , and a dead polymer chain containing a terminal vinyl unsaturation, D¼ r [Eq. (4)].
k tb
Pr ! CH þ D¼
r ð4Þ
The metal hydride center is available for polymerization and will undergo an initi-ation reaction in the presence of monomer [Eq. (5)], similarly to that of [Eq. (2)].
k iH
CH þ M
! P1 ð5ÞThe independent estimation of k i and k iH can be very involved; often the simplify-ing assumption k i G k iH is adopted.For polypropylene polymerization, b-hydride transfer will produce a dead poly-mer chain with a vinylidine chain end. On the other hand, vinyl-terminated chains are produced when b-methyl-transfer reactions take place [Eq. (6)].
k tbCH3

Pr


! CCH 3
þ D¼
r ð6Þ
Chain-transfer agents are commonly used to control the molecular weight of the polymer. By far the most common chain-transfer agent used in olefin polymeriza-tion is hydrogen [Eq. (7)].
k tH
Pr þ H2
! CH þ Dr ð7Þ

386 8 Coordination Polymerization Chain transfer to hydrogen leads to the production of a dead chain with a saturated chain end, Dr , and a metal hydride active site that can be initiated with monomer according to Eq. (5).Transfer to monomer also occurs during olefin polymerization, leading to a vinyl-terminated chain for the case of polyethylene (and a vinylidine-terminated chain for the case of polypropylene) [Eq. (8)].
k tM
Pr þ M
! P1 þ D¼
r ð8Þ
Finally, the cocatalyst may also act as a chain-transfer agent [Eq. (9)]. If the cocata-lyst is trimethylaluminum, (CH3 )3 Al, this elementary step produces a methylated-active center, which can also be initialized by the monomer in a subsequent step:
k tAl
Pr þ Al
! CCH3 þ Dr; Al ð9ÞCoordination catalysts are very sensitive to polar impurities and may also be deac-tivated due to mono- or bimolecular mechanisms. Even though these elementary reactions are not very well understood from a quantitative point of view, we will show next some simple reaction mechanisms proposed to describe them.Reactions with polar impurities, I, leading to the formation of a deactivated site,Cd , and a dead polymer chain have been modeled with the very simple equations[Eqs. (10) and (11)].
kdI
Pr þ I ! Cd þ Dr ð10Þ
kdI
C  þ I ! Cd ð11ÞLikewise, monomolecular and bimolecular deactivation reactions have been de-scribed with the simple kinetic schemes described by Eqs. (12)–(14).
kd
Pr ! Cd þ Dr ð12Þ
 kd
C ! Cd ð13Þ
 kd
2C ! 2Cd ð14ÞThese deactivation equations are not easy to prove, but have been successful in de-scribing several olefin polymerization processes with Ziegler–Natta and metallo-cene catalysts.The term ‘‘single-site catalyst’’ has a very precise meaning in coordination poly-merization. From the point of view of catalyst structure, single-site catalysts are those where all active sites are represented by the same chemical species and have the same polymerization kinetic parameters. In other words, single-site catalysts can be represented with a single set of the elementary reactions described by Eqs.
(1)–(14) or any other equivalent set of polymerization mechanism equations. From
a polymer microstructure point of view, single-site catalysts will produce linear

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 387
polymers that have a theoretical polydispersity of 2 instantaneously. These two con-ditions are not independent. In fact, the second condition is the consequence of The qualification ‘‘instantaneously’’ used above should not be taken lightly, since this may lead to significant misunderstanding of catalyst behavior. An instanta-neous property refers to the property of the polymer made at a given instant in time under a set of invariant polymerization conditions. If these conditions change throughout the polymerization, so will the instantaneous property under examina-tion. Therefore, even though polymer made with a single-site coordination catalyst has a polydispersity of 2 instantaneously, the polydispersity of the accumulated polymer may be much higher (but never lower) than 2 if reaction conditions are allowed to vary during the polymerization, as indicated in Figure 8.20 for the case of decreasing chain-transfer agent concentration as a function of polymerization time. It is only when the polymerization is operated at steady state that the instan-taneous properties of the polymer will be equal the accumulated ones.Polymers made by a mechanism where the chain can either grow by propagation or terminate by transfer reactions, as quantified by Eqs. (1)–(14), follow Flory’s most probable chain length distribution [33] [Eq. (15)].
 
r r
wðrÞ ¼ exp  ð15Þ
rn2 rn
1.4
time
chain transfer agent
1.2
0.8
time
w (r )
0.6
0.4
0.2
1 1.5 2 2.5 3 3.5 4 4.5
log r
Fig. 8.20. Instantaneous chain length distributions of a polymer made with a single-site catalyst when the concentration of chain transfer agent the reactor decreases with increasing polymerization time.

388 8 Coordination Polymerization In Eq. (15), wðrÞ is the weight distribution of chains of length r and rn is the number-average chain length of the polymer population. Therefore, from a poly-mer microstructure point of view, a coordination catalyst is considered to have only one site type if it produces polymer with a chain length distribution that fol-
8.3.2.2 Copolymerization
The polymerization model most commonly adopted for olefin copolymerization is the terminal model, particularly for studies of polymerization kinetics. In the ter-minal model, only the last monomer molecule added to the chain end influences polymerization and transfer rates. Besides the fact that it is logically expected, there is also significant experimental evidence supporting the terminal model for olefin polymerization. Since monomer propagation and chain-transfer reactions take place by insertion between the chemical bond formed by the metal in the active site and the polymer chain end, it is certainly reasonable to assume that both the nature of the active site and the type of monomer last added to the chain will affect these reactions. On the other hand, higher-order models such as the penultimate and pen-penultimate models have not found widespread use in coordination polymerization.Copolymerization models are similar to homopolymerization models, with the added complexity that more polymerization rate constants are required. An ac-cepted form of the terminal model for the binary copolymerization of olefins is shown in Table 8.1. Notice that, except for the fact that the polymerization rate con-stants now depend on the monomer and the chain end type, the mechanism is es-sentially the same as the one described in Section 8.3.2.1 for homopolymerization.For the case of linear binary copolymers, the instantaneous bivariate chain length and chemical composition distribution, wðr; yÞ, is described by Stockmayer’s distri-bution [Eqs. (16)–(19)] [34]. rffiffiffiffiffiffiffi  
r r r ry 2
wðr; yÞ ¼ exp  exp  ð16Þrn2 rn 2pk 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik ¼ FA ð1  FA Þ 1  4FA ð1  FA Þð1  rA rB Þ ð17Þk p; AA k p; BB rA ¼ ; rB ¼ ð18Þk p; AB k p; BA y ¼ FA  FA ð19ÞAs indicated in Eq. (19), y is the deviation from the average molar fraction of monomer type A in the copolymer, F A . As usual, the instantaneous value of F A can be calculated using the molar fraction of monomer A in the reactor, fA , and the reactivity ratios rA and rB by the Mayo–Lewis equation, Eq. (20) [35].ðrA  1Þ fA2 þ fA
FA ¼ ð20Þ
ðrA þ rB  2Þ fA2 þ 2ð1  rB Þ fA þ rB

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 389
Tab. 8.1. Terminal model for binary copolymerization of olefins.[a]
ka
Activation C þ A ! C
k i; A
Initiation C  þ A
! P1; A
k i; B
C  þ B
! P1; B
k p; AA
Propagation Pr; A þ A

! Prþ1; A
k p; AB
Pr; A þ B

! Prþ1; B
k p; BA
Pr; B þ A

! Prþ1; A
k p; BB
Pr; B þ B

! Prþ1; B
k tb; A
b-Hydride elimination Pr; A

! CH þ D¼
r; A
k tb; B
Pr; B

! CH þ D¼
r; B
k tH; A
Transfer to hydrogen Pr; A þ H2

! CH þ Dr; A
k tH; B
Pr; B þ H2

! CH þ Dr; B
k t; AA
Transfer to monomer Pr; A þ A

! P1; A þ D¼
r; A
k t; AB
Pr; A þ B

! P1; B þ D¼
r; A
k t; BA
Pr; B þ A

! P1; A þ D¼
r; B
k t; BB
Pr; B þ B

! P1; B þ D¼
r; B
kAl; A

Transfer to cocatalyst Pr; A þ Al

! CCH3 þ Dr; Al
kAl; B

Pr; B þ Al

! CCH3 þ Dr; Al
kdI; A
Deactivation with impurities Pr; A þ I

! Cd þ Dr; A
kdI; B
Pr; B þ I

! Cd þ Dr; B
kd; A
Monomolecular deactivation Pr; A

! Cd þ Dr; A
kd; B
Pr; B

! Cd þ Dr; B[a] A and B represent the two monomer types. All other symbols are the same as those used for homopolymerization, but the reaction rate constants indicate the type of chain end and reacting monomer with the generic nomenclature k i; j where i is the chain end type and j represents the monomer type.In the same way that Flory’s distribution is the necessary outcome of adopting the homopolymerization model described by Eqs. (1)–(14), Stockmayer’s distribution is the consequence of adopting the binary copolymerization mechanism described in Table 8.1.The attentive reader may have noticed that Stockmayer’s distribution is an exten-sion of Flory’s distribution for the case of copolymers. In fact, upon integration in the interval y a y a y, Stockmayer’s distribution reduces to Flory’s distribu-tion as demonstrated by Eq. (21).ðy  rffiffiffiffiffiffiffi    r r r ry 2 r r
wðrÞ ¼ 2
exp  exp  dy ¼ 2 exp  ð21Þy rn rn 2pk 2k rn rn Therefore, the chain length distributions of linear binary (or multicomponent) co-polymers also follow Flory’s most probable distribution and have, instantaneously,a polydispersity of 2.

390 8 Coordination Polymerization Similarly, an expression for the chemical composition distribution alone can be obtained by integrating Eq. (16) over all chain lengths, that is, in the interval

ðy rffiffiffiffiffiffiffi  r r r ry 2 3
wð yÞ ¼ 2
exp  exp  dr ¼ rffiffiffiffiffi 5=2 ð22Þ0 n rn 2pk 2k 2k y 2 rn
4 1þ
rn 2k
It is difficult to overestimate the explanatory power contained in these few simple expressions. Equation (16) teaches us that longer polymer chains also have narrow chemical composition distributions, as illustrated in Figure 8.21. This result sim-ply reflects the fact that statistical deviations from the average copolymer composi-tion become less likely as the chain increases in length, as would be expected since chains with infinite length should all have exactly the same average copolymer composition. Equation (16) also shows that the tendency to form comonomer blocks (rA rB ! y) will necessarily broaden the chemical composition distribution of copolymers, while the tendency toward monomer alternation (rA rB ! 1) will narrow the chemical composition distribution and in the limit make all chains have the composition F A ¼ 0:5, as illustrated in Figure 8.22. In fact, the chemical composition distribution is best apprehended by plotting Eq. (22) as a function of the lumped parameter rn =k. Figure 8.23 shows that the chemical composition dis-
0.01
0.009 r = 250
0.008
0.007
0.006
w (r ,y )
0.005
0.004
0.003
0.002
0.001
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Fig. 8.21. Chemical composition distribution for several chain lengths, as described by Stockmayer’s bivariate distribution.Distribution parameters: rn ¼ 1000, FA ¼ 0:5, rA rB ¼ 1.

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 391
r1r2 = 0.01
0.03 r1r2 = 1.0
r1r2 = 5
0.025
0.02
w (r , y )
0.015
0.01
0.005
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
y
Fig. 8.22. Chemical composition distribution for the same chain length and varying rA rB , as described by Stockmayer’s bivariate distribution. Distribution parameters: rn ¼ 1000,FA ¼ 0:5, r ¼ 1000.rn /κ = 1000 rn /κ = 2000 rn /κ = 4000
w( y)
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
y
Fig. 8.23.Chemical composition distribution as a function of the lumped parameter rn =k in Eq. (22).

392 8 Coordination Polymerization tribution becomes broader as the value of rn =k decreases, reflecting the fact that polymers with shorter number-average chain lengths and a tendency to form blocks (see Eq. (17): k increases when rA rB increases) have broader chemical com-position distributions. In conclusion, the width of the instantaneous chemical composition distribution of binary copolymers made by coordination polymeriza-tion is a function of a single lumped parameter rn =k and, for a given copolymer composition, depends only on the number-average chain length and reactivity ratio
8.3.3
Polymerization Kinetics with Multiple-site Catalysts All heterogeneous Ziegler–Natta and Phillips catalysts have two or more active-site types and many soluble Ziegler–Natta and metallocene catalysts may also show multiple-site behavior [36, 37]. In addition, several metallocene catalysts, when supported on organic and inorganic carriers, may behave like multiple-site cata-lysts even if they behaved as single-site catalysts in solution polymerization. There-fore, several of the catalysts used industrially for polyolefin manufacturing have in fact two or more active-site types.The kinetics of polymerization with multiple-site catalysts is generally consid-ered to be the same as with single-site catalysts, as described by Eqs. (1)–(14) for homopolymerization and in Table 8.1 for copolymerization, with different polymer-ization kinetic parameters assigned to each site type. In some cases, the polymer-ization mechanism may be extended to include site transformation steps, where sites of one type may change into sites of another type, such as the one described with the reversible reaction in Eq. (23), where D could be a catalyst modifier such as an electron donor, for instance.C1 þ D $ C2 ð23ÞSince these additional site transformation steps may affect the relative ratio of site types present in the reactor but do not influence the general polymerization behavior with multiple-site catalysts, for the sake of simplicity they will not be fur-ther considered in this chapter.It is usually straightforward to detect the presence of multiple-site types on a co-ordination catalyst because these catalysts will produce polymer with polydispersity higher than 2 even under invariant polymerization conditions. The simplest way to visualize this phenomenon is to assume that every different site type on a multiple-site catalyst produces polymers that follow a distinct Flory’s distribution: that is,those with a distinct number-average chain length, rn; i [38]. In this way, the chain length distribution for the whole polymer is a combination of distinct Flory’s dis-tributions weighted by the mass fraction of polymer made on each site type, m i[Eq. (24)].

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 393
0.9
0.8
0.7
0.6
w (r )
0.5
0.4
0.3
0.2
0.1
1 1.5 2 2.5 3 3.5 4 4.5
log r
Fig. 8.24. Chain length distribution of a polymer made with a coordination catalyst containing three different active-site types.See Figure 8.25 for the corresponding chemical composition distribution. Distribution parameters: rn; 1 ¼ 500, rn; 2 ¼ 1000,rn; 3 ¼ 2000, m1 ¼ 0:3, m2 ¼ 0:3, m3 ¼ 0:4.
X
n  
r r
wðrÞ ¼ m i 2 exp  ð24Þ
i¼1
rn; i rn; i The graphical representation of Eq. (24) for a coordination catalyst having three distinct site types is shown in Figure 8.24. It is important to remember that the use of Eq. (24) is a direct consequence of assuming that the mechanism of poly-merization for multiple-site catalysts is described with the same set of equations,Eqs. (1)–(14), used to describe single-site catalysts. In other words, Flory’s distribu-tion is the logical consequence of the mechanism adopted for coordination poly-merization.For polymers made with multiple-site catalysts, the instantaneous number- and weight-average chain lengths are given by Eqs. (25) and (26).
!1
X
mi
rn ¼ ð25Þ
i¼1 n; i
X
rw ¼ 2 m i rn; i ð26Þ
i¼1

394 8 Coordination Polymerization
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
fraction of ethylene Fig. 8.25. Chemical composition distribution length distribution. Distribution parameters:of polymer made with a coordination catalyst ðrn =kÞ1 ¼ 2000, ðrn =kÞ2 ¼ 4000,containing three different active-site types. ðrn =kÞ3 ¼ 8000, FA; 1 ¼ 0:88, FA; 2 ¼ 0:90,See Figure 8.24 for the corresponding chain FA; 3 ¼ 0:93, m1 ¼ 0:3, m2 ¼ 0:3, m3 ¼ 0:4.Therefore, the polydispersity index of a polymer made with a catalyst having n site types is given by Eq. (27), which is always greater than or equal to 2.
rw Xn X
mi
PDI ¼ ¼ 2 m i rn; i ð27Þ
rn i¼1
i¼1 n; i
Similarly, the bivariate molecular weight and chemical composition distributions of binary copolymers made with multiple-site catalysts can be described as a weighted superposition of Stockmayer’s distributions [39]. If we consider only the chemical composition component of the distribution, as described by Eq. (22), the distribution of polymer made with a multiple-site catalyst becomes Eq. (28).
X
wð yÞ ¼ m i sffiffiffiffiffiffiffiffiffiffiffiffiffi    5=2 ð28Þi¼1 k y 2 rn
4 2 1þ i
rn i 2 k i
Figure 8.25 shows the chemical composition distribution obtained with Eq. (28) of a polyolefin made with a catalyst having three distinct active-site types, whose chain length distribution has already been presented in Figure 8.24. Notice the characteristic bimodal distribution of ethylene/a-olefin copolymers made with het-

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 395
Fig. 8.26. Bivariate distribution of chain Distribution parameters: rn; 1 ¼ 500,length and chemical composition of a rn; 2 ¼ 1000, rn; 3 ¼ 2000, FA; 1 ¼ 0:88,polymer made with a coordination catalyst FA; 2 ¼ 0:90, FA; 3 ¼ 0:93, m1 ¼ 0:3, m2 ¼ 0:3,containing three different active-site types. m3 ¼ 0:4, ðr1 r2 Þ1 ¼ ðr1 r2 Þ2 ¼ ðr1 r2 Þ3 ¼ 1.erogeneous Ziegler–Natta catalysts. These two figures illustrate a very important aspect of olefin polymerization with heterogeneous Ziegler–Natta catalysts, the fact that the sites that make polymer chains with higher number-average chain lengths also have a lower reactivity ratio toward comonomer incorporation. This seems to be a universal property of heterogeneous Ziegler–Natta catalysts and is dramatically illustrated in Figure 8.26 for the bivariate distribution of chain length and chemical composition of a model polymer. Much research has been done to overcome this phenomenon, which can very often be seen as a limitation of heter-ogeneous Ziegler–Natta catalysts, since low molecular weight chains with high a-olefin fractions are easily extracted from the polymer particles and may cause reac-tor fouling, a high extractable content in films – a serious limitation for food and medical applications – and odor problems during processing.The multiplicity of active sites on heterogeneous Ziegler–Natta and Phillips cata-lysts is a truly complex phenomenon and other explanations proposed to describe the observed broad chain length and chemical composition distributions have also been proposed, but most rely on similar approaches to the one described herein.The references at the end of the chapter give more information on this topic [40,41].
8.3.4
Long-chain Branch Formation The mechanism of long-chain branch (LCB) formation with coordination polymer-ization catalysts is terminal branching. In this mechanism, a dead polymer chain containing a terminal unsaturation – generally a vinyl group – is copolymerized with a growing polymer chain to form an LCB (Figure 8.27) [15]. We have already seen that dead polyethylene chains with terminal unsaturations will be formed by

396 8 Coordination Polymerization
Pr,2


B Zr
CH
B Zr
2 =
CH
Ds,1=
Fig. 8.27. Mechanism of long-chain branch formation with coordination catalysts.b-hydride elimination and transfer to monomer reactions, as described by Eqs. (4)and (8), respectively. Since these chains can be reincorporated into the growing polymer chains via LCB-formation reactions, they are frequently called macro-monomers. Macromonomers can be made in situ via b-hydride elimination and transfer to monomer reactions or they can be added, as an additional comonomer type, to the polymerization reactor.In this way, long-chain branch formation can be seen as a copolymerization reac-tion with a very long a-olefin and, as expected, catalysts that have a high reactivity ratio toward a-olefin incorporation can also form polymers with high LCB con-tents. Several metallocene catalysts and some homogeneous Ziegler–Natta cata-lysts are effective in forming polymer chains with LCBs. In contrast, heteroge-neous Ziegler–Natta catalysts make only linear polyolefins.The polymerization mechanism described by Eqs. (1)–(14) for homopolymeriza-tion needs to be augmented by only one additional equation, Eq. (29), to include long-chain branch formation.
kLCB
Pr; i þ D¼
s; j
! Prþs; iþjþ1 ð29ÞIn Eq. (29), the subscripts i and j indicate the number of LCBs per chain, while the subscripts r and s are their respective chain lengths. Therefore, when a linear grow-ing chain (i ¼ 0) reacts with a linear macromonomer ( j ¼ 0), a chain with one LCB is formed (i þ j þ 1 ¼ 1). Similar equations are easily developed for copolymeriza-tion.The weight distribution of chain length for polymer populations containing i LCBs per chain, wðr; iÞ is given by Eq. (30) [42], in which the parameter t is given by Eq. (31).wðr; iÞ ¼ r 2iþ1 t 2iþ2 expðtrÞ ð30Þð2i þ 1Þ!rate of chain transfer þ rate of LCB formation
t¼ ð31Þ
rate of propagation

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 397
0.0004
Linear
0.00035
0.0003
0.00025
1 LCB
w (r , y )
0.0002
2 LCB
3 LCB
0.00015 4 LCB
5 LCB
0.0001
0.00005
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Fig. 8.28. Chain length distributions of the several polymer populations of branched polyolefins made with a single-site coordination catalyst (1=t ¼ 1000).Therefore, in the absence of LCB formation, t ¼ 1=rn , and Eq. (30) reduces to Flory’s most probable distribution, Eq. (15). Figure 8.28 shows the chain length distribution for several polymer populations with increasing numbers of LCBs per chain. As expected, the chain length average increases with an increasing number of LCBs per chain.An analytical solution for the instantaneous chain length distribution for the whole polymer produced in a CSTR is also available [Eq. (32)] [15]. The function I1 is the modified Bessel function of the first kind of order 1, given by Eq. (33).Bessel functions are easily found in mathematical tables and are readily available is most scientific software applications [43, 44].
 pffiffi 
ð1  aÞt expðrtÞ rt a wðrÞ ¼ pffiffi I1 2 ð32Þð1 þ aÞ a 1þa
X
y
ðx=2Þ 12k
I1 ðxÞ ¼ ð33Þ
k¼0
k!Gðk þ 2Þ
The parameter a is defined by Eq. (34), where f ¼ is the molar fraction of macro-monomer in the reactor, measured with respect to the total concentration of poly-mer; s is the reciprocal of the average reactor residence time; kLCB is the rate con-stant for LCB formation; and Y is the concentration of growing polymer chains in the reactor.

398 8 Coordination Polymerization
0.00035
0.0003
α = 0.1
0.00025
0.0002
w (r )
0.00015
0.0001
α = 0.3
0.00005
α = 0.5
0 5000 10000 15000 20000 25000 30000 35000 40000 Fig. 8.29. Overall chain length distribution for branched polyolefins made with a single-site coordination catalyst(1=t ¼ 1000).
f¼
a¼ ð34Þ
1 þ s=ðkLCB YÞFigure 8.29 shows how the chain length distribution of the whole polymer is af-fected by the value of the parameter a. First, notice that a belongs to the interval[0, 1]. All chains are linear when a ¼ 0 since this implies that either f ¼ ¼ 0 (with-out macromonomers, no LCBs can be formed) or s=ðkLCB YÞ ! y. The latter con-dition is obeyed when either s ! y (that is, the reactor residence time tends to zero) or kLCB Y ¼ 0. Both cases imply that no macromonomers are accumulated in the reactor. On the other hand, LCB formation is maximal when a ¼ 1, a condi-tion obeyed only when all dead polymer chains in the reactor contain terminal un-saturations, f ¼ ¼ 1, and the residence time in the reactor is infinite, s ! 0, or the rate of LCB formation is infinite, kLCB Y ! y. Therefore, Eq. (32) captures, in a very elegant way, all the factors determining LCB formation with a coordination catalyst.The parameter a can also be related to the number of LCBs per chain for the whole polymer, B, by Eq. (35).
B¼ ð35Þ
1a

8.4 Single Particle Models – Mass- and Heat-transfer Resistances 399
Notice that the average number of LCBs per chain can vary from 0 when a ¼ 0 to infinity when a ¼ 1. Since most long-chain-branched polyolefins made with coordi-nation catalysts are only sparsely branched, with values of B rarely exceeding unity,the upper limit of the paramter a should be considered only as a theoretical possi-bility never to be reached in practical situations.Chain length averages and polydispersity index for the whole polymer can also be related to the parameters a or B via Eqs. (36)–(38).r n ¼ ð1 þ 2BÞ ð36Þr w ¼ ð1 þ 2BÞð1 þ BÞ ð37ÞPDI ¼ 2ð1 þ BÞ ð38ÞThese equations demonstrate that the polydispersity index of long-chain-branched polyolefins is always greater than 2 and that the chain length averages increase with an increasing number of LCBs per chain.It is also interesting to calculate the mass fraction of polymer populations con-taining i LCBs per chain, m i , by Eq. (39).ð2iÞ! a i ð1  aÞð2i þ 1Þ
mi ¼ ð39Þ
i!ði þ 1Þ! ð1 þ aÞ 2iþ2 Notice that, for sparsely branched polymers, most of the chains are linear, but the number of more highly branched species increases as a ! 1, as illustrated in Fig-ure 8.30.Similarly, an extension [Eq. (40)] of Stockmayer’s distribution can be derived for binary copolymers containing LCBs formed by terminal branching [45].rffiffiffiffiffiffiffi  
1 r ry 2
wðr; y; iÞ ¼ r 2iþ1 t 2iþ2 expðrtÞ exp  ð40Þð2i þ 1Þ! 2pk 2k These equations give a very accurate portrait of the chain microstructure of these polyolefins. They are, in fact, a window into their chain architecture and can be very useful in understanding the constitution of these complex polymers.
8.4
Single Particle Models – Mass- and Heat-transfer Resistances Most commercial processes for the manufacture of polyolefins use solid catalysts,such as heterogeneous Ziegler–Natta and Phillips catalysts. Many metallocene cat-alysts have also been supported on inorganic carriers, typically silica, for industrial

400 8 Coordination Polymerization
α = 0.05
0.9 α = 0.1
α = 0.2
0.8
α = 0.4
α = 0.8
0.7
0.6
mi
0.5
0.4
0.3
0.2
0.1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 i (LCB/chain)Fig. 8.30. Mass fraction of chains with i LCB per chain, as a function of the parameter a.use [24]. As in any solid-catalyzed reaction, interparticle and intraparticle mass-and heat-transfer resistances may become a limiting step in heterogeneous olefin polymerization processes.Olefin polymerization with heterogeneous catalysts has several very distinct char-acteristics that should be discussed from a qualitative point of view before a more quantitative treatment is presented. Effective heterogeneous catalysts for olefin polymerization are highly porous, as is usual with most heterogeneous catalysts.When a fresh catalyst particle is first fed to the reactor it is evidently free of poly-mer, but its pores quickly become filled with polymer molecules formed as mono-mer diffuses from the bulk phase in the reactor to the surface of the active sites on the catalyst pores. At this point, the three alternatives illustrated in Figure 8.31 are possible: (1) the polymer phase clogs the catalyst pores and inhibits any additional polymerization from happening; (2) the structure of the catalyst is not strong enough to resist the expansion forces of the fast-forming polymer chains and ‘‘ex-plodes’’ in several smaller particles, generating fines; (3) the growing polymer chains are capable of fragmenting the catalyst particle in an ordered way and act as a binder between the catalyst fragments, forming an expanding polymer particle composed of polymer chains surrounding catalyst fragments.The last alternative is the only one that has industrial interest; much catalyst re-search is behind the design of catalyst particles that can be properly fragmented to form uniform polymer particles. This leads to the so-called ‘‘replication phenome-non’’, whereby the size distribution of the catalyst particles is neatly replicated by the size distribution of the polymer particles exiting the reactor, as illustrated in

8.4 Single Particle Models – Mass- and Heat-transfer Resistances 401
catalyst particle with pores clogged with polymer catalyst fragmentation and formation of fines desired expanding catalyst-polymer Fig. 8.31. Particle growth and fragmentation for polymerization with heterogeneous coordination catalysts.Figure 8.32. Proper replication of the catalyst particles is essential for stable reactor operation and also for the handling of the polymer particles in post-reactor pro-cesses. Reactor residence time distribution in CSTRs may have an important effect on the replication phenomenon; the references at the end of the chapter provide more details on this subject [46–50].This picture of particle fragmentation and growth has been captured in its most important details by the multigrain model [36, 51–60] which was originally devel-oped to describe the crystalline structures of TiCl3 and TiCl4 /MgCl2 Ziegler–Natta catalysts, but has also been used extensively to describe metallocene and late tran-sition metals catalysts supported on inorganic carriers. In the multigrain model,the polymer particle is divided into two levels of mass-transfer resistances: the mi-croparticles or primary particles, and the macroparticle or secondary particle. The secondary particle is the porous catalyst particle itself that is fed to the reactor. It is considered to be formed by the agglomeration of several nonporous primary particles having polymerization active sites on their surfaces. As polymer grows around the primary particles, the secondary particle expands as a function of poly-merization time and activity. The multigrain model is illustrated in Figure 8.33.Notice that the multigrain model does not deal directly with the initial seconds of particle fragmentation, when the catalyst pores are being filled with polymer chains that start to fragment the catalyst particles. Since many industrial catalysts are in fact pre-polymerized in milder conditions in a separate reactor before being fed to the polymerization reactor, this should not be seen as a limitation of the multigrain model for most industrial applications. Catalyst pre-polymerization in

402 8 Coordination Polymerization Polymerization Catalyst Polymer Cumulative %Particle size Fig. 8.32.Replication phenomenon for polymerization with heterogeneous coordination catalysts.milder conditions is generally recommended to avoid the formation of intraparticle hot spots and the improper fragmentation of the catalyst particles when subjected to the more severe polymerization conditions existing in industrial polymeriza-tion reactors. Several older and some more recent simulation studies deal with these initial instants of polymerization, but they are beyond the scope of this chapter. These initial particle-fragmentation models are more interesting for pre-polymerization studies, since the amount of polymer made during this very short initial stage is unlikely to have a marked influence on the overall properties of the Secondary Particle Primary Particle
or or
Macroparticle Microparticle
Growing
Catalyst polymer fragment shell Fig. 8.33. The multigrain model.

8.4 Single Particle Models – Mass- and Heat-transfer Resistances 403
polymer produced in the reactor. A 2001 review covers some of these alternative The multigrain model equation for spherical secondary particles is the classic diffusion-reaction equation in a sphere, Eq. (41), where Ms is the monomer con-centration in the secondary particle as a function of polymerization time, t, and radial position, rs .
 
qMs 1 q qMs¼ 2 Deff rs2  RVp ð41Þqt rs qrs qrs The effective diffusivity in the secondary particle, Deff , can be estimated using the conventional expression for effective diffusivity in porous heterogeneous catalysts,Eq. (42), where Db is the monomer bulk diffusivity in the reaction medium, and e and t are the void fraction and tortuosity of the polymer particle, respectively. The fact that both e and t are likely to vary as a function of the degree of fragmentation and expansion of the secondary particle is certainly one of the difficulties in getting a good estimate for Deff .
eDb
Deff ¼ ð42Þ
Finally, RpV is the average volumetric rate of polymerization in the secondary particle at a given radial position. Since, in the multigrain model, the polymeriza-tion is assumed to take place only at the surface of the primary particles, this term couples the models for the primary and secondary particles.Equation (41) is subjected to the following boundary conditions given by Eqs. (43)and (44), where Rs is the radius of the secondary particle, ks is the mass-transfer coefficient in the external film surrounding the secondary particle, and Mb is the monomer concentration in the bulk phase. Equations (43) and (44) are the classic boundary conditions for heterogeneously catalyzed chemical reactions, namely symmetry at the center of the particle and stationary convective mass transfer through the mass-transfer boundary layer surrounding the particle, respectively.Finally, the initial condition is given by Eq. (45).
qMs
ðrs ¼ 0; tÞ ¼ 0 ð43Þ
qrs
qMs
Deff ðrs ¼ Rs ; tÞ ¼ ks ðMb  Ms Þ ð44Þ
qrs
Ms ðrs ; t ¼ 0Þ ¼ Ms0 ð45ÞThe initial concentration in the secondary particle, Ms0 , may be set to zero for a monomer-free catalyst condition, but this generally leads to stiff differential equa-tions that may be very hard to solve. It is also common to assume a pseudo-steady-

404 8 Coordination Polymerization state concentration at t ¼ 0 to obtain the initial condition for Eq. (41). Unless one is interested in the monomer profiles for the very first seconds of polymerization,this approximation generally leads to a system of partial differential equations that Spherical primary particles are modeled with a similar equation [Eq. (46)].
 
qMp 1 q 2 qMp¼ 2 Dp rp ð46Þqt rp qrp qrp In Eq. (46), the subscript p refers to values in the primary particles. Equation (47)has been suggested to estimate the effective diffusivity of monomer in the primary particle, where Da is the diffusivity of monomer in amorphous polymer and w and i are correction factors to account for the decrease in diffusivity due to chain crystal-linity and immobilization of the polymer amorphous phase due to the crystallites.As can be very well imagined, these parameters are also hard to determine and Dp is generally used as an adjustable parameter in the model.
Da
Dp ¼ ð47Þ
wi
Notice that Eq. (46) does not contain a polymerization reaction term. Because the multigrain model assumes that polymerization takes place at the surface of the cat-alyst fragment embedded within the primary particle, the reaction term appears as one of the two required boundary conditions [Eq. (48)]. Equation (48) states that the rate of monomer diffusion at the surface of the catalyst fragment, R c , equals the rate of monomer consumption due to polymerization at rate of Rpc , and Eq.
(49) imposes the condition that the concentration at the surface of the primary par-
ticle equals the equilibrium concentration of monomer absorbed onto the polymer phase, Meq .
qMp 4
4pR 2c Dp ðrp ¼ R c ; tÞ ¼ pR 3c R pc ð48Þ
qrp 3
Mp ðrp ¼ R p ; tÞ ¼ Meq a Ms ð49ÞThe equilibrium concentration of monomer in polymer can be related by Eq. (50)to the monomer concentration in the secondary particle at a given radial position and time if a partition coefficient, KMP , between the two phases is known.
Ms
Meq ¼ ð50Þ
KMP
The polymerization rate at the surface of the catalyst fragment is given by Eq. (51),where C  is the concentration of active sites at the surface of the catalyst fragment.

8.4 Single Particle Models – Mass- and Heat-transfer Resistances 405
Tab. 8.2. Temperature profiles in the primary and secondary particles according to the Secondary particle Primary article
   
qTs 1 q qTs qTp 1 q qTp rp Cp ¼ 2 ke rs2 þ ðDHp ÞRpV rp Cp ¼ 2 ke rp2 qt rs qrs qrs qt rp qrp qrp qTs 2 qTp 4ðrs ¼ 0; tÞ ¼ 0 4pRc ke ðrp ¼ R p ; tÞ ¼ ðDHp Þ pRc3
qrs qrp 3
qTs
ke ðrs ¼ Rs ; tÞ ¼ hðTb  Ts Þ Tp ðrp ¼ R p ; tÞ ¼ Ts
qrs
Ts ðrs ; t ¼ 0Þ ¼ Ts0 Tp ðrp ; t ¼ 0Þ ¼ Tp0 R pc ¼ k p C  Mðrp ¼ R c ; tÞ ð51ÞFinally, the initial condition for the primary particle is stated in Eq. (52).Mp ðrp ; t ¼ 0Þ ¼ Mp0 ð52ÞOnce again, a pseudo-steady-state approximation may be adopted to reduce the stiffness of the system of differential equations for short polymerization times.The multigrain model also includes a set of equations to describe the temper-ature profiles in the primary and secondary particles. These equations are sum-marized in Table 8.2. Mathematical models for solving this system of partial differ-ential equations with moving boundaries are involved and have been discussed in the literature [36, 51–60].This versatile model has been used extensively to describe polymerization with heterogeneous Ziegler–Natta catalysts. Although it is difficult to make general statements for such complex systems, it can be said with confidence that most of the mass- and heat-transfer resistances will take place at the beginning of the polymerization when the concentration of active sites on the secondary particles is at its maximum. As polymer is formed, it pushes apart the active sites in what can be visualized as a dilution effect. In this way, as the secondary particle grows,the catalyst concentration decreases, and naturally intraparticle mass- and heat-transfer effects become less prevalent. For the same reason, particle hot spots are more likely to be observed with highly active catalyst particles at the beginning of the polymerization. This effect is highly undesirable since it may lead to softening of the polymer phase and result in particle agglomeration and severe reactor foul-ing, especially in gas-phase reactors.The equations presented so far for the multigrain model are mass- and energy-balance equations in a spherical catalyst particle used for conventional hetero-geneously catalyzed reactions subjected to a moving boundary due to polymer formation. To predict polymer properties such as chain length and chemical com-position, these monomer and temperature profiles must be coupled with an addi-tional set of equations that describes polymerization and termination mechanisms

406 8 Coordination Polymerization taking place on the surface of the catalyst. The method of moments is generally the preferred technique used in conjunction with the multigrain model, but its discus-sion will be deferred to Section 8.5. Instead, we will use Stockmayer’s bivariate dis-tribution, Eq. (16), to illustrate how polymer properties can be conveniently pre-dicted from the multigrain model.First, it should be remembered that Stockmayer’s distribution is an instanta-neous distribution; that is, it describes the distributions of chain length and chem-ical composition for polymer made at a given instant in time at a given spatial lo-cation in the reactor. Now, consider the polymerization of ethylene and an a-olefin,1-hexene for instance, taking place in a spherical porous catalyst particle. Hydro-gen is used as the chain-transfer agent. Assume also that the catalyst has only one type of active site, as would be observed when supporting a metallocene on a po-rous silica particle, for instance. The primary and secondary particles at a given in-stant in time are subject to mass-and heat-transfer resistances that result in radial monomer concentration and temperature profiles. Assuming that ethylene propa-gates at a much higher rate than 1-hexene and that both have comparable diffu-sivities, the radial profile for ethylene will be much steeper than for 1-hexene. Sim-ilarly, the radial profile for hydrogen can be assumed to be very flat, since the hydrogen diffusivity is high and its reaction rate is low. (Notice that a low reaction rate of the chain-transfer agent as compared to the polymerization rate for the monomers is a requirement for the production of high molecular weight poly-mers.) Given that the monomer concentration and temperature affect the parame-ters of the Stockmayer distribution, rn =k, each radial position i in the secondary particle is associated with a unique Stockmayer’s distribution, wi ðr; yÞ, as depicted in Figure 8.34. Polymer richer in the slow reacting comonomer, 1-hexene, is pro-duced near the center of the particle because the molar ratio of 1-hexene/ethylene increases from the surface to the center of the particle. Likewise, chains with lower molecular weight are produced at the center of the particle because the molar ratio of hydrogen/(ethylene þ 1-hexene) increases from the surface to the center of the particle. These concentration gradients will, therefore, broaden the distributions of molecular weight and chemical composition of polymer made with a heteroge-neous catalyst, even a single-site catalyst, due to the spatial variations of concentra-tions within the particle.The summation of all these distributions over the polymer particle, weighted by the amount of polymer made at each radial position, gives the distribution for the whole particle at a given instant in time wp ðr; yÞ, as described by Eq. (53), where Rp; i is the rate of polymerization at radial position i and m is the number of radial positions used in the discretization of the macroparticle.
X
Rp; i wi ðr; yÞwp ðr; yÞ ¼ i¼1 m ð53Þ
X
Rp; i
i¼1

8.4 Single Particle Models – Mass- and Heat-transfer Resistances 407
w (fraction of ethylene)
20 4 0.8
w (r )
3 0.6
0.4 3
1 0.2
0.7 0.72 0.74 0.76 0.78 0.8 0.82
1 2 3 4 5
fraction of ethylene
log r
H2
C6H12
C2H4
Fig. 8.34. Using Stockmayer’s distribution with the multigrain model to predict the distribution of chain length and chemical composition of polyolefins.These instantaneous distributions can then be integrated in time to obtain the cumulative distribution in the reactor per polymer particle, Wp ðr; yÞ [Eq. (54)].
Xm ð
Rp; i wi ðr; yÞ dt Wp ðr; yÞ ¼ i¼1 m ð ð54Þ
X
Rp; i dt
i¼1
For the case of catalysts containing multiple-site types, such as heterogeneous Ziegler–Natta catalysts, a similar approach applies, by defining one Stockmayer’s distribution for each active-site type. In this case, the overall distribution of chain length and chemical composition in the particle at a given instant equals the sum-mation of the distributions over all site types and all radial positions in the particle[Eq. (55)], where the subscript j indicates the site type of a catalyst containing n site types.

408 8 Coordination Polymerization Rp; i; j wi; j ðr; yÞ
i¼1 j¼1
wp ðr; yÞ ¼ ð55Þ
Rp; i; j
i¼1 j¼1
Many other single-particle model formulations exist with lower or higher levels of sophistication. Most of these models generate qualitative results that are similar to the ones described in this section. As usual with any modeling effort, the degree of model sophistication must be justified by the quality of the experimental data avail-able to support the model assumptions. Our recent (2001) review gives coverage of these alternative models [36].
8.5
Macroscopic Reactor Modeling – Population Balances and the Method of Moments Population balances coupled with the method of moments can be considered the method of choice in most olefin polymerization simulation models. Population balances are molar balances (steady-state or dynamic) of all the important chemical species present in the reactor: living and dead polymer chains, catalyst sites, mono-mers, and chain-transfer agents. Solving the dynamic population balances for liv-ing and dead chains of length r allows the recovery of the complete chain length distribution as a function of polymerization time. Alternatively, instead of solving the whole population balance, one may use the method of moments to solve for only a few moments of the chain length distribution, a technique that requires much less computational effort. In this case, it is possible to model how chain length averages vary as a function of polymerization time, but the information about the complete distribution is irretrievably lost for more complex cases.In this section, we will first illustrate how to use the method of moments for homopolymerization and then show how these equations can be easily adapted to copolymerization using the method of pseudo-kinetic constants.
8.5.1
Homopolymerization In the following model development, we will use the polymerization kinetics mechanism described by Eqs. (1)–(14) to derive the population balances and mo-ment equations for homopolymerization with a catalyst containing only one site type. Catalysts containing two or more site types are handled similarly by defining a set of equations with distinct polymerization kinetic constants for each different site type.For living polymer chains of length r b 2 made in a CSTR, the dynamic popula-tion balance can be derived as Eq. (56).

8.5 Macroscopic Reactor Modeling – Population Balances and the Method of Moments 409
¼ Prin þ k p MðPr1  Pr Þ  ðk tb þ k tH H2 þ k tAl Al þ k tM M þ kdI I þ kd þ sÞPr
dt
ð56Þ
In Eq. (56) and in the subsequent ones, the notation x in represents the feed flow rate of a given chemical species to the reactor, and s is the reciprocal of the average residence time in the CSTR. Notice that Eq. (56) is simply the molar balance for chains of length r: chains are generated by propagation of a chain of length r  1 or by transfer from a feed stream (coming from a previous reactor in a series of reactors, for instance), and are consumed by either propagation to chains of length r þ 1, by transfer and deactivation reactions leading to dead polymer chains, Dr , or by exiting the reactor in an outlet stream.Similarly, for chains of unit length, Eq. (57) applies.
dP1
¼ P1in þ ðk i C  þ k iH CHi ÞM  k p P1 M
dt
 ðk tb þ k tH H2 þ k tAl Al þ k tM M þ kdI I þ kd þ sÞP1 ð57ÞThe population balances for C  and CH are needed to solve Eqs. (56) and (57).They are given in Eqs. (58) and (59) respectively, where Y 0 ¼ y r ¼1 Pr .
dC 
¼ C in þ ka CAl þ ðk tAl Al þ k tM MÞY 0  ðk i M þ kdI I þ kd þ sÞC  ð58Þ
dt
dCH in
¼ CH þ ðk tb þ k tH HÞY 0  ðk iH M þ kdI I þ kd þ sÞCH ð59Þ
dt
Often, the reaction between catalyst and cocatalyst is considered instantaneous.Consequently, the term ka CAl can be dropped from Eq. (58) and it is assumed that C  ðt 0 Þ ¼ Cðt 0 Þ.The concentration of deactivated catalyst sites at any time is obtained via a molar balance [Eq. (60)].Cd ðtÞ ¼ C  ðt 0 Þ  C  ðtÞ  CH ðtÞ  Y 0 ðtÞ ð60ÞPopulation balances for dead polymer chains are also easily derived from the poly-merization mechanism equation, Eq. (61).dD^ r dDr dD¼ dDr; Al
¼ þ r þ
dt dt dt dt^ in þ ½ðk tH H2 þ kdI I þ kd Þ þ ðk tb þ k tM MÞ þ k tAl AlPr  sD
¼D ^r ð61Þ

410 8 Coordination Polymerization In Eq. (61), dead chains with all types of chain ends (Dr ; D¼ r , and Dr; Al ) have been combined in a single variable D ^ r for simplicity. Separate population balances for dead chains with different chain end types could also be kept, if necessary.Molar balances for monomer, chain-transfer agent, impurities, catalysts, and co-catalyst [Eqs. (62)–(66)] are also required to solve the system of ordinary differen-tial equations defined by Eqs. (56)–(61).¼ M in  ðk p Y 0 þ k tM Y 0 þ k i C  þ k iH CH þ sÞM G M in  ðk p Y 0 þ sÞM ð62Þ
dt
dH2
¼ H2in  ðk tH Y 0 þ sÞH2 ð63Þ
dt
¼ I in  ½kdI ðY 0 þ C  þ CH Þ þ sI ð64Þ
dt
dC
¼ C in  ðka Al þ sÞC ð65Þ
dt
dAl
¼ Al in  ðka C þ k tAl Y 0 þ sÞAl ð66Þ
dt
Because of the long-chain approximation, the simplification k p Y 0 g k tM Y 0 þ k i C þ k iH CH in Eq. (62) is very often applied.Elegant ways of solving the population balances defined by Eqs. (56)–(66) have been developed to model how the complete chain length distribution of polyolefins varies as a function of polymerization time in batch, semibatch, or continuous re-actors [61, 62].When only chain length averages are required, the method of moments is the most adequate technique for solving this problem. The ith moment, mðiÞ , of a given distribution, f ðxÞ, is defined by Eq. (67).
X
mðiÞ ¼ x i f ðxÞ ð67Þ
x
Therefore, the zeroth moment is simply the total number (or concentration) of liv-ing polymer chains [Eq. (68)].
X
y X
y
Y ð0Þ ¼ Pr ¼ P1 þ Pr ð68Þ
r ¼1 r ¼2
Notice that, for coordination polymerization, Y ð0Þ will be approximately equal to the number of active sites at a given time in the reactor, since initiation reactions tend to be very fast in these systems.

8.5 Macroscopic Reactor Modeling – Population Balances and the Method of Moments 411
Taking the first derivative of Eq. (68), one obtains Eq. (69).
dPr
¼ þ ð69Þ
dt dt r ¼2
dt
Substituting Eqs. (56) and (57) into Eq. (69) generates the final expression for the zeroth moment of living polymer chains in a CSTR. After some algebraic manipu-lation, Eq. (70) is obtained.¼ Y ð0Þin þ K i M  ðK t þ K d þ sÞY ð0Þ ð70Þ
dt
Here the several kinetic parameters were lumped into the constants K t ; K d , and K i according to Eqs. (71)–(73), to allow for a more concise notation.K t ¼ kbt þ k tH H2 þ k tM M þ k tAl Al ð71ÞK d ¼ kd þ kdI I ð72ÞK i ¼ k i C  þ k iH CH ð73ÞSimilarly, the first moment of the living chains corresponds to the weight of these chains [Eqs. (74) and (75)].
y X
Y ð1Þ ¼ rPr ¼ 1  P1 þ rPr ð74Þ
r ¼1 r ¼2
dY ð1Þ dP1 X y
dPr
¼ þ r ð75Þ
dt dt r ¼2
dt
Once again, substituting Eqs. (56) and (57) into Eq. (75) leads to the final expres-sion for the first moment of living polymer chains in a CSTR, Eq. (76).
dY ð1Þ
¼ Y ð1Þin þ K i M  ðK t þ K d þ sÞY ð1Þ þ k p MY ð0Þ ð76Þ
dt
The equations for the second moment of living polymer, Eqs. (77)–(79), are derived in an analogous manner.
y X
Y ð2Þ ¼ r 2 Pr ¼ 1 2  P1 þ r 2 Pr ð77Þ
r ¼1 r ¼2

412 8 Coordination Polymerization
dPr
¼ þ r2 ð78Þ
dt dt r ¼2
dt
¼ Y ð2Þin þ K i M  ðK t þ K d þ sÞY ð2Þ þ k p Mð2Y ð1Þ þ Y ð0Þ Þ ð79Þ
dt
A similar procedure leads to the moment equations for the chain length distribu-tion of dead polymer molecules [Eqs. (80) and (81)].X ðiÞ ¼ r i Pr ð80Þ
r ¼1
dX ðiÞ X y
dXr
¼ ri ð81Þ
dt r ¼2
dt
Substituting Eq. (61) into Eq. (81) produces the general expression for the ith mo-ment of the dead polymer chains, Eq. (82).
dX ðiÞ
¼ X ðiÞin þ ðK t þ K d ÞðY ðiÞ  P1 Þ  sX ðiÞ ð82Þ
dt
Notice that P1 is subtracted from Y ðiÞ for exactness since ‘‘dead chains’’ of length 1 are simply monomer units and should not be counted as dead polymer chains.This correction, however, is negligible for high polymers and can be omitted for most modeling applications.Equation (70), (76), (79), and (82) can be solved with the molar balances for the reactants, Eqs. (62) to (66), to calculate the leading moments of the chain length distribution. The values of the moments, as a function of polymerization time,can then be used to compute the chain length averages with the expressions de-scribed below.The number-average chain length, rn , is easily related to the zeroth and first mo-ments of the distributions of chain length of living and dead polymer by Eq. (83).rðDr þ Pr ÞX ð1Þ þ Y ð1Þ X ð1Þrn ¼ r ¼1 ¼ G ð0Þ ð83Þ
Xy ð0Þ
X þY ð0Þ X
ðDr þ Pr Þ
r ¼1
Since, for most coordination polymerizations, the amount of dead polymer far ex-ceeds the amount of living polymer in the reactor, the approximation indicated in Eq. (83) is very commonly used.

8.5 Macroscopic Reactor Modeling – Population Balances and the Method of Moments 413
The weight-average chain length, rw , is likewise obtained from the ratio of the second to the first moments of living and dead chains, obtained from Eq. (84).r 2 ðDr þ Pr ÞX ð2Þ þ Y ð2Þ X ð2Þrw ¼ r ¼1 ¼ G ð1Þ ð84Þ
Xy ð1Þ
X þY ð1Þ X
rðDr þ Pr Þ
r ¼1
Finally, the polydispersity index, PDI, is given by Eq. (85).rw ðX ð2Þ þ Y ð2Þ ÞðX ð0Þ þ Y ð0Þ Þ X ð2Þ X ð0ÞPDI ¼ ¼ 2 G ð85Þrn ðX ð1Þ þ Y ð1Þ Þ ðX ð1Þ Þ 2 For the case of multiple-site catalysts, population balances are derived for each cat-alyst site type, and the chain length-averages for the whole polymer are found by averaging the values calculated for each site type [Eqs. (86) and (87)].
ð1Þ ð1Þ
ð1Þ
ðXj þ Yj Þ Xj
j¼1 j¼1
rn ¼ n G n ð86Þ
X ð0Þ ð0Þ
X ð0Þ
ðXj þ Yj Þ Xj
j¼1 j¼1
ð2Þ ð2Þ
ðXj þ Yj Þ Xj
j¼1 j¼1
rw ¼ n G n ð87Þ
X ð1Þ ð1Þ
X ð1Þ
ðXj þ Yj Þ Xj
j¼1 j¼1
Population balances and the method of moments can also be combined with the multigrain model and other polymer particle growth models. In this case, the pop-ulation balances are defined for each position in the particle to obtain the radial profiles of chain length averages [36, 51–60].
8.5.2
Copolymerization Population balances for copolymerization can be developed using the polymeriza-tion kinetics presented in Table 8.1. This approach generates equations that are similar to the ones obtained for homopolymerization but contain more terms to account for the effect of chain ends on the kinetics of propagation and termination.

414 8 Coordination Polymerization We will first show how to derive such population balances for living polymer chains, but instead of applying the same approach to all other species we will intro-duce the concept of pseudo-kinetic rate constants [63, 64]. When pseudo-kinetic rate constants are defined, the equations derived for homopolymerization can also be used for copolymerization with only one minor modification, thus considerably simplifying the time required for model development.The population balance for living polymer chains terminating in monomer type A with r b 2 is given by Eq. (88).¼ Pr;inA þ k p; AA ðPr1; A  Pr; A ÞA þ k p; BA Pr1; B A  k p; AB Pr; A B
dt
 ðk tb; A þ k tH; A H2 þ k tAl; A Al þ k t; AA A þ k t; AB B þ kdI; A I þ kd; A þ sÞPr; A
ð88Þ
Similarly, for living polymer chains terminating in monomer type B with r b 2,Eq. (89) applies.
dPr; B
¼ Pr;inB þ k p; BB ðPr1; B  Pr; B ÞB þ k p; AB Pr1; A B  k p; BA Pr; B A
dt
 ðk tb; B þ k tH; B H2 þ k tAl; B Al þ k t; BA A þ k t; BB B þ kdI; B I þ kd; B þ sÞPr; B
ð89Þ
Equations (88) and (89) can be added to obtain the differential equation for Pr ¼Pr; A þ Pr; B , Eq. (90).¼ Pr;inA þ Pr;inA þ ðk p; AA Pr1; A A þ k p; AB Pr1; A B þ k p; BA Pr1; B A
dt
þ k p; BB Pr1; B BÞ  ðk p; AA Pr; A A þ k p; AB Pr; A B þ k p; BA Pr; B A þ k p; BB Pr; B BÞ ðk tb; A Pr; A þ k tb; B Pr; B Þ  ðk tH; A Pr; A þ k tH; B Pr; B ÞH2 ðk tAl; A Pr; A þ k tAl; B Pr; B ÞAl ðk t; AA Pr; AA A þ k t; AB Pr; A B þ k t; BA Pr; B A þ k t; BB Pr; B BÞ ðkdI; A Pr; A þ kdI; A Pr; B ÞI  ðkd; A Pr; A þ kd; B Pr; B Þ  sPr ð90ÞApplying the definitions in Eqs. (90)–(93) to Eq. (90), one obtains Eq. (94).fA ¼ ; fB ¼ 1  fA ð91ÞPr; A þ Pr; B fA ¼ ; fB ¼ 1  fA ð92Þ
AþB
M ¼AþB ð93Þ

8.5 Macroscopic Reactor Modeling – Population Balances and the Method of Moments 415
¼ Prin þ ðk p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB ÞPr1 M
dt
 ðk p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB ÞPr M ðk tb; A fA þ k tb; B fB ÞPr  ðk tH; A fA þ k tH; B fB ÞPr H2 ðk tAl; A fA þ k tAl; B fB ÞPr Al ðk t; AA fA fA þ k t; AB fA fB þ k t; BA fB fA þ k t; BB fB fB ÞPr M ðkdI; A fA þ kdI; A fB ÞPr I  ðkd; A fA þ kd; B fB ÞPr  sPr ð94ÞEquation (94) can be expressed in the more compact form of Eq. (95), where the pseudo-kinetic constants are defined by Eqs. (96)–(102).¼ Prin þ k^p MðPr1  Pr Þ  ðk^tb þ k^tH H2 þ k^tAl Al þ k^tM M þ k^dI I þ k^d þ sÞPr
dt
ð95Þ
k^p ¼ k p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB ð96Þk^tb ¼ k tb; A fA þ k tb; B fB ð97Þk^tH ¼ k tH; A fA þ k tH; B fB ð98Þk^tAl ¼ k tAl; A fA þ k tAl; B fB ð99Þk^tM ¼ k t; AA fA fA þ k t; AB fA fB þ k t; BA fB fA þ k t; BB fB fB ð100Þk^dI ¼ kdI; A fA þ kdI; A fB ð101Þk^d ¼ kd; A fA þ kd; B fB ð102ÞNotice that Eqs. (56) and (95) are equivalent, with the only difference that Eq. (95)uses pseudo-kinetic constants in place of the actual kinetic constants found in Eq.
(56). The beauty of this modeling approach is that it is not necessary to develop
new equations for copolymerization (binary or higher): the equations developed for homopolymerization, including the moment equations, are equally applicable to copolymerization, provided that pseudo-kinetic constants are used to replace the actual polymerization kinetic constants.To calculate the pseudo-kinetic constants one must know the values of fA and fA at each polymerization time. Values for fA are easily calculated from the molar bal-ance for the monomers, Eq. (103).
dA
¼ Ain  ðk i C  þ k i; H CH ÞA  ðk p; AA fA þ k p; BA fB ÞAY 0
dt
 ðk t; AA fA þ k t; BA fB ÞAY 0  sA ð103ÞSince most of the monomer is consumed in polymerization reactions, Eq. (103) is commonly reduced to the simpler form of Eq. (104).

416 8 Coordination Polymerization¼ Ain  ðk p; AA fA þ k p; BA fB ÞAY 0  sA ð104Þ
dt
The analogous equation for monomer B is Eq. (105).
dB
¼ B in  ðk p; BB fB þ k p; AB fA ÞBY 0  sB ð105Þ
dt
The long-chain approximation can be used to calculate the values of fA via Eqs.
(106)–(108).
k p; AB Pr; A B ¼ k p; BA Pr; B A ð106Þand therefore:k p; AB fA fB ¼ k p; BA fB fA ¼ k p; BA ð1  fA Þ fA ð107Þ
k p; BA fA
fA ¼ ð108Þ
k p; BA fA þ k p; AB fB The use of Eq. (108) to calculate fA assumes that this value is not a function of chain length; compare Eq. (91). It has been shown that this hypothesis is valid for high polymers [63, 64].Finally, the average copolymer composition [Eq. (109)] is easily obtained from Eqs. (104) and (105).FA ¼ ; FB ¼ 1  FA ð109Þ
AþB
8.6
Types of Industrial Reactors The polymerization of olefins with coordination catalysts is performed in a large variety of polymerization processes and reactor configurations that can be classified broadly into solution, gas-phase, or slurry processes. In solution processes, both the catalyst and the polymer are soluble in the reaction medium. These processes are used to produce most of the commercial EPDM rubbers and some polyethylene resins. Solution processes are performed in autoclave, tubular, and loop reactors.In slurry and gas-phase processes, the polymer is formed around heterogeneous catalyst particles in the way described by the multigrain model. Slurry processes can be subdivided into slurry–diluent and slurry–bulk. In slurry–diluent pro-cesses, an inert diluent is used to suspend the polymer particles while gaseous(ethylene and propylene) and liquid (higher a-olefins) monomers are fed into the reactor. On the other hand, only liquid monomer is used in the slurry–bulk pro-

8.6 Types of Industrial Reactors 417
Fig. 8.35. Reactor configurations used with olefin polymerization: (a) Autoclave; (b) Loop; (c) Fluidized-bed;(d) Vertical gas-phase; (e) Horizontal gas-phase.cess. Polyethylene and polypropylene can be produced in slurry–diluent reactors,while slurry–bulk reactors are restricted to polypropylene and its copolymers.Slurry processes involve the use of autoclaves or loop reactors. Gas-phase reactors are also used to polymerize ethylene, propylene, and higher a-olefins. They can be classified into fluidized-bed and stirred-bed reactors. A gaseous stream of mono-mer and nitrogen fluidizes the polymer particles in fluidized-bed reactors, while mechanical stirring is responsible for suspending the polymer particles in gas-phase stirred-bed reactors. Gas-phase stirred-bed reactors are further subdivided into horizontal and vertical reactors. These different reactor configurations are il-lustrated in Figure 8.35.Several polymerization processes use only one reactor, but two or more reactors can also be operated in series (tandem reactor technology) to produce polyolefins with more complex microstructures [5]. Each reactor in the series is maintained under different operating conditions to produce products that are sometimes called‘‘reactor blends’’. Although, in principle, the post-reactor blending of different resins could lead to the same product, in reactor blends the chains are mixed on the molecular scale, permitting better contact between the polymer chains made in different reactors at a lower energy cost.The oldest example of this procedure is the manufacture of high-impact poly-propylene, as already described (see Section 8.1.2). Other applications have become more popular lately, especially for the production of bimodal resins. Figure 8.36 illustrates a tandem process using two gas-phase vertical stirred-tank reactors. Sev-eral other reactor combinations are used industrially [65]. For heterogeneous pro-cesses, the first reactor(s) in the series can be either slurry or gas-phase, but com-monly the second reactor (or set of reactors) is a gas-phase reactor. This is especially important when the production of polymers with lower crystallinity

418 8 Coordination Polymerization
unreacted
monomer
catalyst
cocatalyst
offgas
N2
Reactor 1 Reactor 2
powder
silo
polymer
pellets
propylene propylene
N2
ethylene ethylene
Extruder
hydrogen hydrogen Fig. 8.36. Example of a gas-phase tandem reactor process.occurs in the second set of reactors, because this fraction is more easily extracted in slurry reactors, leading to fouling problems. Of course, these processes should be operated in such a way as to avoid particle agglomeration caused by the formation of sticky polymers of lower crystallinity.For heterogeneous catalysts, tandem reactor technology also relies on the fact that each polymer particle is in fact a microreactor operated in semibatch mode,into which monomers and chain-transfer agents are fed continually, while the poly-mer formed never leaves the microreactor. In this way, polymer populations with different average properties are produced in each reactor and accumulate in the polymer particle microreactor, as illustrated in Figure 8.37. In theory, an optimal balance does exist between the fractions of these different populations to meet certain performance criteria. This creates a truly fascinating reactor and product design problem because the fractions of the different polymer populations per par-ticle will be a function of the residence time distribution in the individual reactors in the reactor train.Consider first the case of two tubular reactors in series, making high-impact polypropylene. Reactor 1 produces isotactic polypropylene, while random ethylene–propylene copolymer is made in Reactor 2. Assuming that both reactors are ideal plug-flow reactors, the residence time of all the polymer particles in each reactor is exactly the same. Consequently, if the distribution of active sites in the

8.6 Types of Industrial Reactors 419
Product from Product from reactor 1 reactors 1 and 2(reactor blend)catalyst and
cocatalyst
monomers and chain transfer
agents
monomers and chain transfer
agents
Fig. 8.37. Production of reactor blends in tandem reactors.catalyst particles is uniform, the fractional polypropylene content in the ethylene–propylene copolymer is exactly the same in each polymer particle exiting Reactor 2.Figure 8.38 illustrates this situation.A very different picture emerges when using two CSTRs in series. Because the residence time distribution of an ideal CSTR, EðtÞ, with average residence time tr ,is given by the usual exponential decay equation [Eq. (110)], then some particles will leave Reactor 1 after a short time while others will only leave after spending a considerably longer time in the reactor.
 
1 t
EðtÞ ¼ exp  ð110Þ
tr tr
Since the same will happen in Reactor 2, in the end the ratio of polypropylene to ethylene–propylene copolymer per particle exiting Reactor 2 will also vary widely,which may be undesirable in some applications. Some of the reactor configura-tions shown in Figure 8.35 can reduce this phenomenon, particularly the configu-ration adopted for the gas-phase horizontal reactor, because the residence time dis-tribution of this reactor is the equivalent to about three to four CSTRs in series.(Remember that the residence time of an infinite series of ideal CSTRs is that of a plug-flow reactor.) A more recent solution for this problem, in fact a completely new alternative to tandem reactor technology, is the multizone reactor that will be described in more detail below (see Section 8.6.4).

420 8 Coordination Polymerization Fig. 8.38. Reactor blends produced in two plug-flow reactors and two CSTRs in series.These polymerization processes were originally designed to polymerize olefins with heterogeneous Ziegler–Natta catalysts, with the exception of some solution processes that were designed to work with homogeneous Ziegler–Natta catalysts for the production of EPDM rubbers or some types of polyethylene resins. How-ever, heterogeneous Ziegler–Natta processes can be adapted to the use of metallo-cene catalysts with minimal changes if the metallocenes are supported on an inert carrier such as silica. Although some differences in particle morphology have been noticed when heterogeneous Ziegler–Natta catalysts are replaced by supported metallocenes, these Ziegler–Natta processes are also currently being used with metallocene catalysts on an industrial scale. In fact, the possibility of using metal-locenes in conventional Ziegler–Natta reactors is one of the main reasons for the fast adoption of metallocenes by the polyolefin manufacturing industry.
8.6.1
Gas-phase Reactors Gas-phase reactors for olefin polymerization are divided into two classes: fluidized-bed reactors and stirred-bed reactors. The stirred-bed reactors can be further clas-sified into vertical and horizontal.

8.6 Types of Industrial Reactors 421
Gas-phase reactors, especially fluidized-bed reactors, are the most common con-figuration for the polymerization of ethylene to produce HDPE and LLDPE. They are also a very common choice for the second reactor in the production of high-Fluidized-bed reactors were developed by Union Carbide (Unipol Process) –currently Univation – and British Petroleum (BP). Some details in their config-uration may vary, but their main characteristics are the same. In fluidized-bed reactors, gaseous monomers, chain-transfer agent, inerts, and catalysts are fed con-tinuously into the reactor. The polymerization temperature should be kept well below the melting point of the polymer to avoid particle agglomeration, loss of flu-idization, and bed collapse. Since polymer particles are formed in the gas phase in absence of diluent, there is no need for further separation steps when the product is exiting the reactor (except for removal of unconverted monomer), which is a clear advantage of gas-phase processes over slurry and solution processes.The heat of polymerization can be removed by heat exchangers placed on an ex-ternal recirculation loop. However, low boiling point hydrocarbons and a-olefin co-monomers can be introduced into the reactor in the liquid phase to absorb the heat of polymerization by their latent heat of evaporation in an operation procedure called condensed mode. Since most polymerization reactors are limited by their heat removal capability, this technique permits a substantial increase in reactor productivity.There are two principal designs for stirred-bed gas-phase reactors: vertical (origi-nally developed by BASF/Targor) and horizontal (originally developed by Amoco–Chisso). These reactors have several of the advantages of fluidized-bed reactors but the polymer bed is suspended by mechanical agitation. Therefore, impeller design is of the utmost importance in these reactors to avoid reactor fouling and particle agglomeration. Stirred-bed reactors are generally smaller than fluidized-bed reac-tors, thus permitting grade transition with less production of off-specification ma-terial. Both vertical and horizontal designs can be operated in condensed mode to increase productivity. As mentioned above, the narrower residence time distribu-tion of horizontal gas-phase stirred-bed reactors is advantageous for the production of reactor blends with a narrow distribution of their different components. Narrow residence time distributions are also useful if frequent grade transitions are re-quired.Gas-phase reactors have several advantages, notably: Good temperature control due to high turbulence and heat-transfer coefficients,and heat removal by the latent heat of vaporization of inerts and monomers. Lower operational costs due to lack of diluent recovery operations and, in the case of fluidized-bed reactors, due to the absence of moving parts. Grade flexibility for molecular weight control, since hydrogen concentration is regulated by varying the partial pressure of hydrogen in the reactor. In slurry and solution reactors, the low solubility of hydrogen in the diluent may reduce the range of possible molecular weights for a given catalyst. Higher comonomer incorporation (that is, production of copolymer with lower

422 8 Coordination Polymerization crystallinity). This is possible since there is no risk of copolymer dissolution in the reaction medium. However, care should be taken not to form sticky polymer particles that can lead to particle agglomeration or reactor fouling. The narrower residence time distribution of horizontal stirred-bed reactors leads to higher yields per pass, formation of less off-specification material, and more uniform impact copolymers and reactors blends.Some disadvantages of gas-phase reactors are: For the particular case of fluidized-bed reactors, fluidization is not a trivial pro-cess and therefore demands better process control and the design of catalyst par-ticles that fluidize well. Severe fouling and polymer particle agglomeration can occur, with occasional re-actor shutdown. Fluidized-bed reactors, in particular, are prone to the formation of polymer sheets on the walls (‘‘sheeting’’) or polymer ‘‘chunks’’ that may lead periodic interruption of reactor operation. Fluidized-bed reactors are generally very large, which makes grade transition more time-consuming and might lead to the production of significant amounts of off-specification products.
8.6.2
Slurry Reactors Slurry processes for olefin polymerization are performed in autoclave or loop reac-tors. Both reactor configurations are rather old and date from the beginning of commercial olefin polymerization. Most first-generation olefin polymerization pro-cesses used autoclave reactors, while the Phillips process employed a loop reactor.Slurry–diluent processes use an inert diluent to suspend the polymer particles.Although the diluent does not directly affect the polymerization, it has been shown that different diluents might change catalyst behavior, probably due to electronic interaction with the active sites. Gaseous monomers and hydrogen are continu-ously bubbled through the diluent. Liquid a-olefin comonomers, diluent, catalysts,and cocatalyst are continuously fed into the reactor. Alternatively, liquefied propy-lene can be fed into the reactor (slurry–bulk process). Except from this difference,all other conditions are similar to the slurry–diluent process.Both autoclave and loop reactors have a residence time distribution of CSTRs(loop reactors are operated at very high recirculation ratios), so they share several of the same characteristics. There is a tendency nowadays to favor the loop reactor configuration for the production of polypropylene in slurry–bulk mode.Some advantages of slurry reactors are: Their simplicity of design and low cost make them a common choice for laboratory-scale studies for screening catalysts and investigation of polymeriza-tion kinetics, particularly autoclave configurations. The large amount of diluent used (or liquid monomer) acts as a heat sink, per-

8.6 Types of Industrial Reactors 423
mits very good temperature control, and minimizes the risk of runaway polymer- In loop reactors, the high recirculation rate leads to high turbulence and high heat-transfer coefficients. Additionally, the high heat-transfer area available in these reactors permits efficient removal of heat of polymerization and therefore high polymer yields.However, slurry processes also have several disadvantages, such as: It is necessary to remove the diluent from the polymer formed and recycle it back to the polymerization reactor after purification, thus increasing operational costs and environmental hazards. With Ziegler–Natta catalysts, molecular weight is generally controlled by transfer to hydrogen. Since the solubility of hydrogen in the diluent is not very high,there is less flexibility for controlling molecular weight with this type of reactor.This is not as important for Phillips catalysts, where molecular weight control is achieved via support treatment, but can become a limiting factor with Ziegler–Natta catalysts. Metallocenes are generally very sensitive to the presence of hydro-gen and therefore less influenced by this reduced solubility that Ziegler–Natta catalysts. Less crystalline copolymer chains can dissolve in the diluent – particularly the ethylene–propylene copolymer fraction in high-impact polypropylene –causing fouling and increasing the viscosity of the diluent. Therefore, certain low-crystallinity grades cannot be produced with these reactors.It has been speculated that, for the case of metallocene catalysts, some of the limi-tations encountered with slurry reactors (both CSTR and loop) when producing copolymers of lower crystallinity can be minimized or completely eliminated. As discussed above, heterogeneous Ziegler–Natta catalysts produce LLDPE with very broad chemical composition distributions containing low-crystallinity tails. Such low-crystallinity tails are absent in most polyolefins made with metallocene cata-lysts, thus minimizing the risk of copolymer dissolution during polymerization in slurry reactors.
8.6.3
Solution Reactors Solution processes use autoclave, tubular, or loop reactors. As compared to slurry and gas-phase polymerization, solution processes are commonly operated at a much higher temperature to keep the polymer dissolved in the reaction medium,and at much lower average residence times (5–20 min, as opposed to 1–4 h). Since polymerization conditions are more uniform in solutions reactors – there are no inter- and intraparticle heat- and mass-transfer resistances, for instance – this configuration is commonly used for the production of EPDM rubbers with soluble Ziegler–Natta vanadium-based catalysts. Composition homogeneity is a require-

424 8 Coordination Polymerization ment of most EPDM rubbers since the formation of populations with higher crys-tallinity is generally not acceptable in the rubber industry.Solution reactors can be also used to produce polyethylene resins with soluble Ziegler–Natta or metallocene catalysts.The short residence time used in solution reactors because of their high opera-tion temperature is often an advantage during grade transition. The fact that the polymer is in solution is also beneficial for process control, since solution viscosity can be used as a measure of polymer molecular weight, for instance. However,high solution viscosities are also a limiting factor for these reactors, reducing the achievable polymer concentration in solution, especially for high molecular weight resins. Solution reactors can also operate in a wider range of temperatures than slurry and gas-phase reactors, for which the temperature should be high enough
Internal
gas/solid separator
RISER
upward
pneumatic DOWNCOMER transport packed bed moving downward
Gas
fan
Product
discharge
Catalyst
inlet
Heat exchanger Fig. 8.39. Schematic of a multizone circulating reactor [50].

Notation 425 to permit high polymerization rates but not so high as to soften the polymer and cause particle agglomeration and reactor fouling. This wider temperature range al-lows for more flexibility in terms of catalyst types and polymer structural control.In addition, solution reactors can be used to produce polymer from very high to very low crystallinity, since there are no problems with reactor fouling caused by sticky low-crystallinity polymers.
8.6.4
Multizone Reactors The multizone circulating reactor (see Figure 8.39) is a novel concept for propylene polymerization, developed by Basell. This reactor combines a fast fluidization reac-tor with a moving packed-bed reactor and can produce reactor blends in a single reactor instead of in a series of reactors [65]. The reactor operates in a cycle: poly-mer is transferred from the bottom of the fixed-bed section (downcomer) to the bottom of the fluidized-bed section (riser). A gas stream, containing monomers and inerts, conveys the polymer particles to the top of the riser and a centrifugal separator settles them at the top of the downer. Finally, the particles flow by gravity from the top to the bottom of the downer, where the cycle is repeated again.The polymer particles can pass through these two sections of the reactor several times before exiting the reactor. If these two (or more) zones are kept in difference polymerization conditions, a multimodal reactor blend polymer can be produced.It is claimed that because the polymer particles can be made to circulate between the different reactor zones several times before exiting the reactor, a more uniform distribution of blend components will result than in an equivalent resin made on two reactors in series. Of course, this is what would be expected from a reactor blend made in several reactors in series.
Notation
A monomer type A Al cocatalyst or activator B number of LCBs per chain for the whole polymer; monomer type B
C catalyst
C active center

CCH 3
methylated active center Cd deactivated site CH metal hydride center D catalyst modifier (electron donor)Da diffusivity of monomer in amorphous polymer [cm 2 s1 ]Db monomer bulk diffusivity in the reaction medium [cm 2 s1 ]Deff effective diffusivity in the secondary particle, [cm 2 s1 ]Dp effective diffusivity in the primary particle, [cm 2 s1 ]Dr dead chain with a saturated chain end

426 8 Coordination Polymerization D¼r dead polymer chain containing a terminal vinyl unsaturation D dead polymer chains with all possible types of chain ends Dr; Al dead polymer chain formed via a transfer to cocatalyst reaction EðtÞ reactor residence time distribution f¼ molar fraction of macromonomer in the reactor, measured with respect to the total concentration of polymer fA molar fraction of monomer A in the reactor FA average fraction of comonomer A in the copolymer H2 hydrogen i number of long-chain branches per polymer chain I polar impurity I1 modified Bessel function of the first kind of order 1 ka rate constant for catalyst activation [s1 ]kd rate constant for monomolecular or bimolecular deactivation [s1 ]Kd lumped deactivation constant, defined in Eq. (72)kdI rate constant for deactivation by impurity [L mol1 s1 ]ki initiation rate constant [L mol1 s1 ]Ki lumped initiation constant, defined in Eq. (73)k iH initiation rate constant for a metal-hydride center [L mol1 s1 ]kLCB rate constant for LCB formation [L mol1 s1 ]KMP monomer partition coefficient between bulk and polymer phases kp propagation rate constant [L mol1 s1 ]k p; ij propagation rate constant for monomer type i and chain end type j[L mol1 s1 ]ks mass-transfer coefficient in the external film surrounding the secondary particle [cm s1 ]Kt lumped chain-transfer constant, defined in Eq. (71)k tb b-hydride elimination rate constant [s1 ]k tb-CH3 b-methyl elimination rate constant [s1 ]k tAl rate constant for transfer to cocatalysts [L mol1 s1 ]k tH rate constant for transfer to hydrogen [L mol1 s1 ]k tM rate constant for transfer to monomer [L mol1 s1 ]m number of radial positions used in the discretization of the macro-
particle
mi mass fraction of polymer made on site type i, mass fraction of polymer with i long-chain branches per chain
M monomer
Mb monomer concentration in the bulk phase [mol L1 ]Meq equilibrium concentration of monomer absorbed onto the polymer phase [mol L1 ]Mp monomer concentration in the primary particle [mol L1 ]Mp0 initial monomer concentration in the primary particle [mol L1 ]Ms monomer concentration in the secondary particle [mol L1 ]Ms0 initial concentration of monomer in the secondary particle [mol L1 ]n number of active-site types in a multiple-site catalyst

Notation 427 PDI polydispersity index PDI polydispersity index for branched polymers Pr growing polymer of chain length r r chain length rA ; rB reactivity ratios Rc radius of the primary particle [cm]rn number-average chain length rn number-average chain length for branched polymers Rp; i rate of polymerization at radial position i [mol L1 s1 ]Rpc polymerization rate at surface of catalyst fragment [mol L1 s1 ]RpV average volumetric rate of polymerization in the secondary particle at a given radial position [mol L1 s1 ]rp radial coordinate in the primary particle [cm]rs radial coordinate in the secondary particle [cm]Rs radius of the secondary particle [cm]rw weight-average chain length rw weight-average chain length for branched polymers s reciprocal of the average reactor residence time [s1 ]t polymerization time [s]tr average reactor residence time [s]wðrÞ weight distribution of chains of length r (Flory’s most probable chain length distribution)wðr; iÞ weight distribution of chains of length r containing i long-chain
branches
wðr; yÞ weight distribution of chains of length r and chemical composition y(Stockmayer’s bivariate distribution)wðr; y; iÞ weight distribution of chains of length r, chemical composition y, and i long-chain branches wð yÞ weight distribution of chains with chemical composition y wðrÞ instantaneous chain length distribution for the whole polymer produced in a CSTR in the presence of branching reactions wp ðr; yÞ instantaneous distribution of chain length r and chemical composition y in the polymer particle Wp ðr; yÞ cumulative distribution of chain length r and chemical composition y in the polymer particle X ðiÞ ith moment of the dead polymer chains y deviation from the average molar fraction of monomer type A in the
copolymer
Y concentration of growing polymer chains in the reactor Y ðiÞ ith moment of the living polymer chains
Greek
a defined in Eq. (34)e void fraction of the polymer particle

428 8 Coordination Polymerization i correction factor to account for the decrease in diffusivity due to chain immobilization of the polymer amorphous phase due to the crystallites mðiÞ ith moment of a chain length distribution t defined in Eq. (31); tortuosity of the polymer particle fI fraction of chains terminated in monomer of type i w correction factor to account for the decrease in diffusivity due to chain crystallinity
Subscripts
i; j site type; number of long-chain branches r; s chain length Superscripts in feed flow rate of a given chemical species to the reactor [mol L1 s1 ]^ pseudo-kinetic constant
Acronyms
Cp cyclopentadienyl Crystaf crystallization analysis fractionation CSTR continuous stirred-tank reactor EPDM ethylene–propylene–diene rubber GPC gel permeation chromatography HDPE high-density polyethylene LCB long-chain branch LDPE low-density polyethylene LLDPE linear low-density polyethylene MAO methylaluminoxane Tref temperature rising elution fractionation
References
1 G. W. Coates, Chem. Rev., 2000, 100, Kim, J. Polym. Sci.: Part B: Polym.
1223. Phys., 2000, 38, 1267.
2 L. Resconi, L. Cavallo, A. Fait, F. 5 J. Scheirs, L. L. Böhm, J. C. Boot,Piemontesi, Chem. Rev., 2000, 100, P. S. Leevers, Trends Polym. Sci., 1996,
1253. 4, 408.
3 K. Angermund, G. Fink, V. R. 6 G. Ver Strate, in Encyclopedia of Jensen, R. Kleinschmidt, Chem. Rev., Polymer Science and Engineering, Vol. 6,2000, 100, 1457. J. I. Kroschwitz (Ed.), John Wiley &4 J. B. P. Soares, R. F. Abbott, J. D. Sons, New York, 1986, p. 522.

References 4297 J. B. P. Soares. Fractionation, in 23 S. Gambarotta, Coord. Chem. Rev.,Encyclopedia of Polymer Science and 2003, 237, 229.Technology, 3rd Edition, Wiley-VCH, 24 G. G. Hlatky, Chem. Rev., 2000, 100,Weinheim, 2004, pp. 75–131. 1347.8 D. Campbell, R. A. Pethrick, J. R. 25 H. Sinn, I. Schimmel, M. Ott, N.White, Polymer Characterization, 2nd von Thienen, A. Harder, W.edition, Stanley Thornes Publishers, Hagendorf, B. Heitmann, E. Haupt,Cheltenham, 2000. in Metalorganic Catalsyts for Synthesis 9 W. W. Yau, J. J. Kirkland, D. D. Bly, and Polymerization, W. Kaminsky Modern Size-Exclusion Liquid Chroma- (Ed.), Springer, Berlin, 1999, pp. 105–tography: Practice of Gel Permeation and 122.Gel Filtration Chromatography, John 26 H. Butenschön, Chem. Rev., 2000,Wiley & Sons, New York, 1979. 100, 1527.10 S. Anantawaraskul, J. B. P. Soares, 27 S. A. Miller, R. Waymouth, in P. M. Wood-Adams, Fractionation Ziegler Catalysts, G. Fink, R.of semi-crystalline polymers by Mulhaupt, H. H. Brintzinger crystallization analysis fractionation (Eds.), Springer, Berlin, 1995, pp.(Crystaf ) and temperature rising 441–454.elution fractionation (Tref ). Adv. 28 H. G. Alt, A. Köppl, Chem. Rev.,Polym. Sci., in press. 2000, 100, 1205.11 J. B. P. Soares, A. E. Hamielec, in 29 S. Chum, W. J. Kruper, M. J. Guest,Experimental Methods in Polymer Adv. Mater., 2000, 12, 1759.Characterization, R. A. Pethrick (Ed.), 30 S. D. Ittel, L. K. Johnson, Chem.John Wiley & Sons, Chichester, 1999, Rev., 2000, 100, 1169–1203.p. 15. 31 L. C. Simon, C. P. Williams, J. B. P.12 A. Faldi, J. B. P. Soares, Polymer, Soares, R. F. de Souza, Chem. Eng.2001, 42, 3057. Sci., 2001, 56, 4181.13 J. B. P. Soares, A. E. Hamielec, 32 J. B. P. Soares, Chem. Eng. Sci., 2001,Polymer, 1996, 37, 4606. 56, 4131.14 A. E. Hamielec, J. B. P. Soares, Prog. 33 P. J. Flory, Principles of Polymer Polym. Sci., 1996, 21, 651. Chemistry, Cornell University Press,15 J. B. P. Soares, Macromol. Mater. Eng., Ithaca, 1953.2004, 289, 70. 34 W. H. Stockmayer, J. Chem. Phys.,16 E. Y.-X. Chen, T. J. Marks, Chem. 1945, 13, 199.Rev., 2000, 100, 1391. 35 P. C. Hiemenz, Polymer Chemistry,17 G. Henrici-Olive, S. Olive, Angew. Marcel Dekker, New York, 1984.Chem. Int. Ed. Engl., 1971, 10(2), 10. 36 T. F. McKenna, J. B. P. Soares,18 H. H. Britzinger, D. Fischer, R. Chem. Eng. Sci., 2001, 56, 3931.Mülhaupt, B. Rieger, R. M. 37 J. B. P. Soares, A. E. Hamielec,Waymouth, Angew. Chem. Int. Ed. Polym. React. Eng., 1995, 3, 261.Engl., 1995, 34, 1143. 38 J. B. P. Soares, A. E. Hamielec,19 K. Soga, T. Shiono, Prog. Polym. Sci., Polymer, 1995, 36, 2257.1997, 22, 1503. 39 J. B. P. Soares, A. E. Hamielec,20 R. Mulhaupt, Macromol. Chem. Phys., Macromol. Theory. Simul., 1995, 4,2003, 204, 289. 305.21 H. L. Krauss, in Transition Metals and 40 J. B. P. Soares, Polym. React. Eng.,Organometallics as Catalysts for Olefin 1998, 6, 225.Polymerization, W. Kaminsky, H. 41 T. Keii, Macromol. Theory Simul.,Sinn (Eds.), Springer-Verlag, Berlin, 1995, 4, 947.1988, pp. 163–168. 42 J. B. P. Soares, A. E. Hamielec,22 P. C. Thune, J. Loos, A. M. de Jonga, Macromol. Theory Simul., 1996, 5, 547.P. J. Lemstrab, J. W. 43 M. R. Spiegel, J. Liu, Mathematical Niemantsverdriet, Topics in Catalysis, Handbook of Formulas and Tables,2000, 13, 67. McGraw-Hill, New York, 1999.

430 8 Coordination Polymerization 44 D. A. McQuarrie, Mathematical Ray, J. Appl. Polym. Sci., 1986, 32,Methods for Scientists and Engineers, 5451.University Science Books, Sausalito, 55 S. Floyd, T. Heiskanen, T. W.California, 2003. Taylor, G. E. Mann, W. H. Ray, J.45 J. B. P. Soares, A. E. Hamielec, Appl. Polym. Sci., 1987, 33, 1021.Macromol. Theory Simul., 1997, 6, 591. 56 R. A. Hutchinson, C. M. Chen, W.46 J. B. P. Soares, A. E. Hamielec, H. Ray, J. Appl. Polym. Sci., 1992, 44,Macromol. Theory Simul., 1995, 1085. 1389.47 A. Prasetya, L. Liu, J. Litster, F. 57 R. A. Hutchinson, W. H. Ray, J.Watanabe, K. Mitsutani, G. H. Ko, Appl. Polym. Sci., 1990, 41, 51.Chem. Eng. Sci., 1999, 3263. 58 R. A. Hutchinson, W. H. Ray, J.48 J. J. Zacca, J. A. Debling, W. H. Ray, Appl. Polym. Sci., 1991, 43, 1271.Chem. Eng. Sci., 1996, 51, 4859. 59 R. A. Hutchinson, W. H. Ray, J.49 J. J. Zacca, J. A. Debling, W. H. Ray, Appl. Polym. Sci., 1991, 43, 1387.Chem. Eng. Sci., 1997, 52, 1941. 60 R. A. Hutchinson, W. H. Ray, J.50 J. A. Debling, J. J. Zacca, W. H. Ray, Appl. Polym. Sci., 1987, 34, 657.Chem. Eng. Sci., 1997, 52, 1969. 61 P. Canu, W. H. Ray, Comp. Chem.51 J. A. Debling, W. H. Ray, Ind. Eng. Eng., 1991, 15, 549.Chem. Res., 1995, 34, 3466. 62 M. Wulkov, Macromol. Theory Simul.,52 S. Floyd, K. Y. Choi, T. W. Taylor, 1996, 5, 393.W. H. Ray, J. Appl. Polym. Sci., 1986, 63 T. Xie, A. E. Hamielec, Makromol.32, 2935. Chem., Theory Simul., 1993, 2, 421.53 S. Floyd, K. Y. Choi, T. W. Taylor, 64 T. Xie, A. E. Hamielec, Makromol.W. H. Ray, J. Appl. Polym. Sci., 1986, Chem., Theory Simul., 1993, 2, 455.31, 2231. 65 M. Covezzi, G. Mei, Chem. Eng. Sci.,54 S. Floyd, R. A. Hutchinson, W. H. 2001, 56, 4059.

P. D. Iedema and N. H. Kolhapure
9.1
Introduction In this chapter some mathematical methods to solve kinetic modeling problems are explained. A very sound basis for this was already laid many years ago by Flory[1]. Here, we want to present modern mathematical tools that have recently been developed through the use of computers. The focus is on the link between kinetic rate data and reactor type on one hand, and distributive properties – in one or more dimensions – on the other. These distributive or microstructural properties are concerned not only with countable quantities, such as the number of monomer units in a polymer molecule, but also with structure in the case of branched poly-mer molecules. With structure, we discuss the connectivity of branch points and the lengths of the segments between them. In the greater part of the text, reactors are treated in a simplified manner. We consider continuous and batch reactors, but all of them ideally mixed. The effect of incomplete mixing (segregation, macro-and micromixing) is addressed in classical textbooks such as Biesenberger [2] and Dotson et al. [3]. Some recent attempts to include the impact of micromixing on distributions are available in the literature [57–59], but this field is still in its in-fancy. Nevertheless, to still provide a sound basis for issues of mixing, we devote one section to the use of computational fluid dynamics in polymer reaction engi-neering problems.The microstructural properties that we address are chain length, number of monomer units of one kind (copolymer), number of branch points, number of un-saturated bonds, and number of reactive monomer units (end groups in poly-condensation). Problems may require solution of one or more of these properties simultaneously. Here, we will denote this as the dimensionality of the problem at hand. For instance, growth in addition polymerization can be described by a simple 1D (chain length) reaction equation and population balance.
kp
R n þ M ! R nþ1 ð1ÞHandbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

dR n
¼ k p MðR n
1
R n Þ ð2Þ
dt
In contrast, growth in polycondensation of a trifunctional monomer A with a bi-functional monomer B requires a 3D description. The 3D distribution R n; i; k , where subscripts denote chain length, number of A end groups, number of B end groups,respectively, obeys:
kc
R n; i; k þ R m; j; l ! R nþm; iþj
1; kþl
1 ð3ÞdR n; i; k X
n
1 Xi X k
¼ kc jðk
l þ 1Þlði
j þ 1ÞR m; j; l R n
m; i
jþ1; k
lþ1 ð4Þdt m¼1 j¼0 l ¼0 Note that this describes a reaction between single end groups of two different molecules; end group combinations within one (longer) molecule require a similar approach.These examples illustrate the importance of the dimensionality of the problem at hand. In general, low dimensionality can be dealt with using analytical or differen-tial methods, while higher dimensionality soon requires a Monte Carlo sampling approach. Note that all of these methods, except MC, start with a population bal-ance. Here we will mainly discuss ways to solve such balances of lower or higher dimensionality. This chapter will start with the description of lower-dimensional problems and show to what extent these can be successful. This often involves the reduction of the problem to lower dimensionality, inevitably leading to averaging over one or more dimensions. The most well-known is the method of moments,which, however, does not solve full distributions. This is followed by the introduc-tion of a fairly recent mathematical method based on differential equations that does solve full distributions: the Galerkin h–p finite element method (FEM). Sub-sequently, we will present advanced applications of the Galerkin-FEM (G-FEM)method, being classes and pseudo-distribution modeling. A completely different approach – originating from polymer network modeling – is then discussed: prob-ability generating functions. To conclude the methods for countable properties, we give an overview of full Monte Carlo simulation methods as introduced by Tobita[11–15, 48–51], mostly for cases where analytical or differential equation ap-proaches fail. A separate section then is spent on very recently developed condi-tional Monte Carlo methods to synthesize branched architectures. Finally, a section is devoted on applications of computational fluid dynamics in polymer reaction engineering.
9.2
Discrete Galerkin h–p Finite Element Method The discrete Galerkin h–p finite element method (FEM) is a powerful numerical method to solve chain length distributions for a wide set of polymerization prob-

9.2 Discrete Galerkin h–p Finite Element Method 433
lems. It has been implemented in the commercially available package PREDICI.A great proportion of the problems discussed in this chapter are solved with this approach. A detailed description of the mathematics of the method is given by Wulkow [4]. Below the main mathematical features are given following the de-scription in Ref. 4.Any set of population balance equations (see, for example, Table 9.2, below) can be written as a set of countable ordinary differential equations [Eqs. (5)].us0 ðtÞ ¼ fs fu1 ðtÞ; . . . ; us tot ðtÞg s ¼ 1; . . . ; stot ð5ÞHere, the us ðtÞ are the concentrations of all the macromolecules with length n at time t, represented by vector us ðtÞ; stot is the dimension (chain length) of the sys-tem, for polymer systems typically very large, up to 10 6 . For an approximation u1 of the solution uðt þ tÞ after a time step from t to t þ t a semi-(linear)-implicit Euler scheme is applied [Eq. (6), where j ¼ uðtÞ, A is the derivative f u ðjÞ, and I the identity matrix].ðI
tAÞDu ¼ tf ðjÞ ð6Þu1 ¼ j þ Du The solution of this equation is approximated by a finite element Galerkin method.This is realized by a multilevel algorithm, according to which a subdivision of the s-axis is constructed (see Figure 9.1). On each interval h a local expansion us of us is used, where j is the level number and l is the interval number:
X j j
usj jh j ¼ a kl t k; l ðsÞ ð7Þ
k¼0
( h1j , p1j ) ( hmj , pmj )Fig. 9.1. Division of chain length axis into intervals of length h, where distribution is approximated by Chebyshev polynomials of order p.

(h1,p1) (h2,p2) (h3,p3) (h4,p4) (h5,p5)
1 smax
(h1,p1+1) (h2,p2) (h3L,1) (h3R,1) (h4,p4+1) (h5,p5)Fig. 9.2. Refinement strategy. Orders of intervals (h1 ; p1 ) and(h4 ; p4 ) increased by 1. Interval (h3 ; p3 ) is split into two intervals with order 1 (order may be higher according to optimization strategy). Other intervals remain unchanged.The polynomials t k; l are discrete Chebyshev polynomials of degree k. The number of expansion coefficients pl may differ from interval to interval, such that fitting it to a concentration distribution can be solved by varying grid and order. Note that this feature gives the name to the method: Galerkin h (varying intervals) – p (vary-ing order). The node-order-distribution on the final grid is chosen in such a way that the work necessary to compute the whole distribution is minimal:DF ¼ fðhl ; pl Þ . . . ðhm ; pm Þg ð8ÞThe construction is started with an initial grid D 0 on the interval [0; s max ], where s max may be very large. It proceeds from level to level by refinements or increases of the order, using information from the previous level. An example is shown in Figure 9.2.The Galerkin h–p method is very flexible, being able to solve simple distributions with few broad intervals, but also complicated, including multimodal distributions with steep flanks requiring intensive local adaptation. Such adaptations are man-aged automatically with the method. The algorithm as implemented in PREDICI utilizes chain length truncation. A truncation index is calculated, being the maxi-mum chain length up to which a distribution is calculated. From the index for a
old
calculated distribution, s max , a new index is calculated from the moments of the distribution:sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2
new m 1 m2 m1 s max ¼ þ k
k ¼ 10 ð9Þ
m0 m0 m0
When higher accuracy is required in tail calculations weight- or wðlog sÞ-based truncation indices can be calculated, where moments m 0 ; m1 , and m 2 are replaced by mk ; mkþ1, and mkþ2 , with k ¼ 1 or 2, respectively. For example, a Flory distribution with average n n ¼ 100 is represented until s max ¼ 1100. A first demonstration of the capabilities of the Galerkin-FEM method will be given in Section 9.4 in a com-parison with the method of moments. Further and extensive use is made of the

Tab. 9.2. (Population) balances for radical polymerization with transfer to polymer and random scission by H-abstraction.
dM Xy
t ¼ M0
M
k p tM R n ¼ M0
M
k p tMl 0 (a)
dt n¼1
dI2
t ¼ I20
I2
kd tI2 (b)
dt
t ¼
I þ 2kd tI2
k i tIM (c)
dt
t ¼ S0
S
ks tSl 0 (d)
dt
dR1
t ¼
R1 þ k i tIM þ ks tSl 0
k p tR1 M þ k rs tl 0 m 0 (e)
dt
dR n Xy
¼
k p MR n þ k p MR n
1
k tp R n m1 þ k tp l 0 Pn þ k rs l 0 Pk
k rs l 0 m1 þ (f )
dt k¼nþ1

ðk tc þ k td Þl 0 R n
ks SR n
k m MR n
R n =t
dPn Xy
¼
k tp l 0 nPn þ k tp m1 R n þ k rs l 0 Pk þ k rs m1 R n
k rs l 0 ðn
1ÞPn (g)
dt k¼nþ1
1 X n
1
þ k td l 0 R n þ k tc R m
n R m þ ks SPn þ k m MPn
Pn =t
2 m ¼1
rate coefficient for combination between chains with terminal units p and s. The full population balance reads as [5] (the ‘‘þ¼’’ duet means that this is one out of possibly more contributions to the population balance of Pn; i ):dPn; i X 2 X 2 X
n
1 X i
þ¼ kc; ps R m; j R sn
m; i
j ð11Þdt r ¼1 q¼1 m¼1 j¼0 The moments of distributions R and P are defined as:
y X
X y
y X
X y
lab ¼ na i b R n; i mab ¼ na i b Pn; i ; p ¼ 1; 2 ð12Þn¼1 j¼1 n¼1 j¼1 The population balance of Eq. (11) is expressed in moments by performing the summations as in Eq. (12) and collecting all the terms. In its most general form the result can be expressed as [5]:
b   
dmab 1X 2 X 2 Xa X
a b r r
þ¼ kc; ps l l ð13Þdt 2 p¼1 s¼1 g¼0 d¼0 g d g; d a
g; b
d Usually, when number- and weight-average molecular weights are to be calculated,

9.3 Method of Moments 437
Tab. 9.3. Moment equations for radical polymerization with transfer to polymer and random scission by H-abstraction.¼ k p MR1
ðk tc þ k td Þl 20
ks Sl 0
k m Ml 0
l 0 (a)
dt t
dl1 1
¼ k p Ml 0
k tp m1 l1 þ k tp m 2 l 0 þ k rs l 0 ðm 2
m1 Þ
k rs l 0 ðm1
m 0 Þ (b)
dt 2

ðk tc þ k td Þl 0 l1
ks Sl1 1
k m Ml1
l1
 
dl 2 1 1 1
¼ 2k p Ml1
k tp m1 l 2 þ k tp m3 l 0 þ k rs l 0 m3
m 2 þ m1 þ
k rs l 2 ðm1
m 0 Þ (c)
dt 3 2 6

ðk tc þ k td Þl 0 l 2
ks Sl 2
k m Ml 2
l 2
 
dl3 1 1 1
¼ 3k p Ml 2
k tp m1 l3 þ k tp m4 l 0 þ k rs l 0 m4
m3 þ m 2 þ
k rs l3 ðm1
m 0 Þ (v)
dt 4 2 4

ðk tc þ k td Þl 0 l3
ks Sl3
k m Ml3
l3
 
dl4 1 1 1 1¼ 4k p Ml3
k tp m1 l4 þ k tp m5 l 0 þ k rs l 0 m5
m4 þ m3
m1 (e)dt 5 2 3 30þ
k rs l4 ðm1
m 0 Þ
ðk tc þ k td Þl 0 l4
ks Sl4
k m Ml4
l4
 
dm 0 1 1
¼ k rs l 0 ðm1
2m 0 Þ þ k tc þ k td l 20 þ ks Sl 0 þ k m Ml 0
m 0 (f )
dt 2 t
 
dm1 1 1
¼ k rs l 0 m 2
m1
m 0 þ k rs l1 ðm1
m 0 Þ
k rs l 0 ðm 2
m1 Þ þ k tp l1 m1 (g)
dt 2 2

k tp l 0 m 2 þ ðk tc þ k td Þl 0 l1 þ ks Sl1 þ k m Ml1
m1
 
dm 2 1 1 1
¼ k rs l 0 m3
m 2 þ m1
m 0 þ k rs l 2 ðm1
m 0 Þ
k rs l 0 ðm3
m 2 Þ þ k tp l 2 m1 (h)
dt 3 2 6

k tp l 0 m3 þ k tc ðl 0 l 2 þ l12 Þ þ k td l 0 l 2 þ ks Sl 2 þ k m Ml 2
m 2
 
dm3 1 1 1
¼ k rs l 0 m4
m3 þ m 2
m 0 þ k rs l3 ðm1
m 0 Þ
k rs l 0 ðm4
m3 Þ þ k tp l3 m1 (i)
dt 4 2 4

k tp l 0 m4 þ k t ðl 0 l3 þ 3l1 l 2 Þ þ k td l 0 l3 þ ks Sl3 þ k m Ml3
m3
 
dm4 1 1 1 1¼ k rs l 0 m5
m4 þ m3
m1
m 0 þ k rs l4 ðm1
m 0 Þ
k rs l 0 ðm5
m4 Þ ( j)dt 5 2 3 30þ k tp l4 m1
k tp l 0 m5 þ k t ðl 0 l4 þ 4l1 l3 þ 3l 22 Þ þ k td l 0 l4 þ ks Sl4 þ k m Ml4
m4 it is sufficient to have a; b ¼ 0; 1; 2. Although these forms seem to be complex, they can be derived straightforwardly and solved without any additional assumption or model for linear polymerization – that is, when rate factors are independent of chain length, comonomer content, and so on. In fact, contributions to the popula-tion balances from all other kinetic mechanisms – propagation, transfer to solvent or monomer, disproportionation – can be treated in similar but simpler ways. For linear polymerization the method of moments provides exact solutions for number

and weight averages of molecular weight and copolymer composition. Most well known are the averages for homopolymerization expressed as:
m1 m2
Mn ¼ m 0 Mw ¼ m 0 ð14Þ
m0 m1
9.3.3
Nonlinear Polymerization In nonlinear polymerization rate factors depend on chain length or other micro-structural properties, which leads to special problems when utilizing the method of moments [6]. We here address transfer to polymer in radical polymerization leading to long-chain branching (LCB) and random scission. The reaction equation for transfer to polymer in a 1D formulation with the rate factor proportional to chain length, k tp m is Eq. (15).
k tp m
R n þ Pm ! Pn þ R m ð15ÞThe population balance contribution for dead chains is given by:
dPn Xy Xy
þ¼ k tp ðm1 R n
l 0 nPn Þ m1 ¼ nPn ; l 0 ¼ Rn ð16Þ
dt n¼1 n¼1
Py
Defining moments as in Eq. (12), ma ¼ na Pn , and constructing the moment
n¼1
equations from Eq. (16) yields:
dma
þ¼ k tp ðm1 la
l 0 maþ1 Þ ð17Þ
dt
This illustrates a typical closure problem. In linear polymerization, differential equations of the ath moment contain RHS terms with ath or lower moments,which allows direct solution of the set. In contrast, as shown by Eq. (17), nonlinear polymerization leads to higher moments on the RHS. This implies that the set cannot readily be solved without additional assumptions. Another example is ran-dom scission of linear dead chains into two living chains (macroradicals):
k rs n
Pn ! R m þ R n
m ð18ÞAgain, the rate factor is proportional to chain length. The 1D population balance
reads:
dR n X
y
þ¼ k rs nPn ð19Þ
dt m¼nþ1

9.3 Method of Moments 439
Constructing equations for 0th through 4th moments by summing over chain length leads to:
þ¼ 0; ð20Þ
dt
dl1 1
þ¼ k rs ðm 2
m1 Þ; ð21Þ
dt 2
 
dl 2 1 1 1
þ¼ k rs m3
m 2 þ m1 ; ð22Þ
dt 3 2 6
 
dl3 1 1 1
þ¼ k rs m4
m3 þ m 2 ; ð23Þ
dt 4 2 4
 
dl4 1 1 1 1þ¼ k rs m5
m4 þ m3
m1 : ð24Þdt 5 2 3 30 Hulburt and Katz [7] have developed a method to obtain estimates of the higher moments in terms of lower ones using a distribution approximation method.Most previous work has been based on the 3rd-moment closure. A general expres-sion is constructed for moment mi , where l is the highest order of the series of Laguerre polynomials in the approximation, while a and g are parameters to be
specified:
ði þ g
1Þ! m 0
mi ¼
ðg
1Þ! ðg=aÞ i
X l
km X m
j m!ðm þ g
1Þ!ðm þ i þ g
1
jÞ!
þ i
ð
1Þ ð25Þ
m¼3 ðg=aÞ j¼0 j!ðm
jÞ!ðm þ g
1
jÞ!The series is truncated at l ¼ i
1, so that the computation of mi requires the coef-ficient k i
1 . This implies that ml and higher moments equal zero. Coefficients k i follow in their turn from the lower moments m0...i
1 by:
X
ðg
1Þ! ðg=aÞ i
j ki ¼ ð
1Þ j m ; ð26Þ
j¼0
j!ði þ g
1
jÞ! ði
jÞ! i
j
m a2
so k 0 ¼ m 0 . It is further assumed that: a ¼ 1 ; g ¼ , which leads to m0 m 2 =m 0
a 2 k1 ¼ k2 ¼ 0.Thus, closure expressions have been obtained for m3 ; m4, and m5 :
m2
m3 ¼ ð2m 2 m 0
m 21 Þ ð27Þ
m 0 m1

ð2m 21
3m 2 m 0 Þð3m 21 m 2
6m22 m 0 þ 4m 0 m1 m3 Þ
m4 ¼ ð28Þ
m02 m12

ð12m14 m 2
42m12 m 22 m 0 þ 36m 20 m 32 þ 20m13 m 0 m3
30m1 m 20 m 2 m3 þ 5m12 m 20 m4 Þð3m12
4m 2 m 0 Þ
m5 ¼
m 20 m13
ð29Þ
9.4
Comparison of Galerkin-FEM and Method of Moments Certain features of both methods can nicely be illustrated in a simple nonlinear problem: radical polymerization with transfer to polymer in a continuous stirred-tank reactor (CSTR) with termination by either disproportionation or recombina-tion. Data and results are shown in Figures 9.3–9.5. Typical for the moment method is the occurrence of a point beyond which no stable solution exists. In this case this happens for k tp ¼ 1000 m 3 (kmol s)
1 for the case with dispropor-tionation and k tp ¼ 160 m 3 (kmol s)
1 when recombination is by the termination mechanism. In both cases the second moment of dead chains, m 2 , goes to infinity at that point. Such instabilities are not found by the G-FEM method. At higher k tp we observe further clear discrepancies, especially for the living chains having much higher Mw according to G-FEM. However, the living chain Mw values are not calcu-lated correctly in all cases by the G-FEM method. This is best demonstrated by comparing the chain length distributions (CLDs) of living and dead chains for the recombination case shown in Figures 9.1 and 9.2. One observes that the CLD tail of the living chains extends over a much wider range than that of the dead chains.This is due to the existence of computation limits in the G-FEM PREDICI package:concentrations Pn below a specified minimum are put equal to zero. For dead chains this limit is located at chain lengths of around 10 7 , where concentrations are more than 10 orders of magnitude smaller than the highest dead chain concen-trations. Living chain concentrations drop less rapidly with length; in terms of n 2 R n they even rise at high n, and consequently they are calculated up to a higher limit: n ¼ 10 9 . Now, some of the living chains are produced from dead ones of the same length through transfer to the polymer. The computation limit of dead chains is therefore visible as a discontinuity in the living CLD, beyond which it is no longer calculated correctly. This also results in an error in the total living chain concentration, l 0 , which ultimately would affect the overall population balance, in-cluding the dead chain CLD [6]. Checking the monomer balance with the dead polymer balance (m1 ), one learns that for the case shown the result is still reliable.For larger k tp we see that gradually more of the monomer units polymerized be-come contained in living chains, of which the CLD is not computed correctly, so that the overall solution is no longer valid. Estimation procedures for monomer units contained in tails beyond the computation limit are described by Iedema et al. [6]. In the case of disproportionation this problem automatically resolves itself by increasing k tp (>1800 m 3 /(kmol s)
1 ). This is because living chain concen-

9.4 Comparison of Galerkin-FEM and Method of Moments 441
x 10-5 kmole/m3
1.8
1.6
n2Rn
1.4
Concentration
1.2
0.8
0.6
0.4 Computation
limit P n
0.2
0 0 2 4 6 8 10 1210 10 10 10 10 10 10 Chain Length
kmole/m3
1.8
1.6
n2Pn
1.4
Concentration
1.2
Computation
limit
0.8
0.6
0.4
0.2
0 0 2 4 6 810 10 10 10 10 Chain Length Fig. 9.3. Radical polymerization in CSTR with k p ¼ 1:4  10 5 m 3 (kmol s)
1 ; k tc ¼ 5  10 10 transfer to polymer. Reactor and kinetic data: m 3 (kmol s)
1 ; k tp ¼ 180 m 3 (kmol s)
1 ;initiator feed: I2; f ¼ 5  10
3 kmol m
3 ; conversion x ¼ 0:249. The discontinuity in the monomer feed Mf ¼ 16:75 kmol m
3 ; living CLD is due to a vanishing dead chain residence time: t ¼ 30 s; kd ¼ 0:5 1 s
1 ; concentration.trations increase more rapidly within the computation limit of 10 9 and reach the same magnitude as the dead chain tail concentrations; see Figure 9.3. Thus, living and dead CLDs nicely overlap over the whole range and are correctly calculated,even while a considerable proportion of the monomer units are now polymerized in living chains.

105 kg/mole Living chains,4 recombination termination only
Mw
Galerkin-FEM Moments method(closure µ 3)1010 20 40 60 80 100 120 140 160 180 ktp ( m3/(kmole.s) )
kg/mole
Moments method(closure µ 3)
Mw100
Galerkin-FEM 20 Dead chains,recombination termination only 0 20 40 60 80 100 120 140 160 180 ktp (m3/(kmole.s))Fig. 9.4. Radical polymerization; for data, see Figure 9.3.Comparison of Galerkin-FEM and method of moments. The deviation is strongest for living chains.We conclude that the method of moments yields accurate solutions for all moments (hence Mn ; Mw ; Mz ) in the case of linear polymerization. In those cases where Flory distributions form the solution of instantaneous or steady-state poly-merizations, full CLDs can be constructed as simple combinations of Flory dis-tributions. This is not possible when distributions are resulting from combina-tion reactions between other distributions, such as recombination termination. In nonlinear polymerization the method of moments requires estimation of higher moments, by which errors are introduced unavoidably. In those situations the

9.4 Comparison of Galerkin-FEM and Method of Moments 443
6 Living chains 5 Galerkin-FEM Mw 10 Moments method 4 (closure µ 3)Dead chains
3 Living
chains
Dead chains Disproportionation termination only 0 500 1000 1500 2000 2500 3000 ktp (m3/(kmole.s))
kmole/m3
0.9 0.01
0.8 0.008 Dead
0.7 0.006
Concentration
0.004
0.6
0.002 Living
0.5
0 5 6 7 8 910 10 10 10 10
0.4
0.3 Close-up
0.2
0.1
0 0 2 4 6 8 1010 10 10 10 10 10 Chain Length Fig. 9.5. Radical polymerization; see Figure 9.3. Termination is through disproportionation only. The living chain concentration is the same order of magnitude as the dead chain concentra-tion at high chain length.

Galerkin-FEM method is preferred, although care must be taken when extremely long chain lengths have to be calculated.
9.5
Classes Approach
9.5.1
Introduction The classes approach is applicable to multidimensional problems where the range of the second and higher dimensions is restricted to a small range. In the examples of this approach we will discuss presently, these second dimensions are the num-bers of terminal double bonds (TDBs) on a chain in the case of poly(vinyl acetate)(PVAc) and the numbers of radical sites on a chain in the case of low-density poly-ethylene (LDPE). The idea simply is to solve the problem rigorously in the first dimension, chain length, for separate classes with a fixed value for the second di-mension. Thus, a 2D problem is reduced to a set of 1D problems, where the size of the set is determined by the number of values of the second variable – the number of classes – for which the solution is desired. Obviously, this is only feasible when the number of classes is limited, since the equations have to be implemented sep-arately for each class. A different classes approach is followed by Pladis and Kippar-issides [8], who combined it with a method of moments.
9.5.2
Computing the CLD of Poly(vinyl acetate) for a Maximum of One TDB per Chain The most general set of reaction equations is given in Table 9.4 in terms of all three concentration distribution variables – chain length, number of TDBs, num-Tab. 9.4. Reactions for radical polymerization of vinyl acetate.
kd
Initiator dissociation I2 ! 2I
ki
Initiation I þ M ! R1; 0; 0
kp
Propagation R n; i; k þ M ! R nþ1; i; k
k td
Termination by disproportionation R n; i; k þ R m; j; l ! Pn; i; k þ Pm; j; l(without TDB creation)
k td
Termination by disproportionation (with TDB creation) R n; i; k þ R m; j; l ! Pn; iþ1; k þ Pm; j; l
k tc
Termination by recombination R n; i; k þ R m; j; l ! Pnþm; iþ j; kþl
km
Transfer to monomer R n; i; k þ M ! Pn; i; k þ R1; 1; 0
k tp m
Transfer to polymer R n; i; k þ Pm; j; l ! Pn; i; k þ R m; j; lþ1
kdb j
Terminal double bond propagation R n; i; k þ Pm; j; l ! R nþm; iþ j
1; kþlþ1 Notation: m; n: chain length; i; j: number of terminal double bonds(TDB) per chain; k; l: number of branches per chain.

Tab. 9.5. Full 3D set of population balance equations for radical polymerization of vinyl acetate in living and dead chain concentration variables R n; i; k and Pn; i; k . (for indices, see Table 9.4); summation over the branching index k, yielding a 2D formulation of exactly the same form, provides the basis for the TDB classes model.
dI2 I2
Initiator dissociation ¼
kd I2

dt t
dI I dR1; 0; 0 Initiation ¼ 2kd I2
k i MI
; þ¼ k i MI
dt t dt
dR n; i; k
Propagation þ¼ k p MðR n
1; i; k
R n; i; k Þ
dt
Termination by [a]dR n; i; k dPn; i; k 1 disproportionation þ¼
k td l 0 R n; i; k ; þ¼ k td l 0 ðR n; i
1; k þ R n; i; k Þ
dt dt 2
(TDB creation)Termination by dR n; i; k dPn; i; k 1 X n
1 X
i Xk
þ¼
k tc l 0 R n; i; k ; þ¼ k tc R m; j; l R n
m; i
j; k
l recombination dt dt 2 m ¼1 j¼0 l ¼0 dR n; i; k dR1; 1; 0 dPn; i; k Transfer to monomer þ¼
k m MR n; i; k ; þ¼ k m l 0 M; þ¼ k m MR n; i; k
dt dt dt
dR n; i; k dPn; i; k Transfer to polymer þ¼ k tp ðl 0 nPn; i; k
1
m1 R n; i; k Þ; þ¼ k tp ð
l 0 nPn; i; k þ m1 R n; i; k Þ
dt dt
dR n; i; k Xy X y X y n
1 X
X iþ1 X k
Terminal double bond þ¼ kdb
R n; i; k iPn; i; k þ jPm; j; l R n
m; i
jþ1; k
l
1 dt n¼1 i¼0 k¼0 m ¼1 j¼0 l ¼0 propagation
dPn; i; k
þ¼
kdb il 0 Pn; i; k
dt
[a] The first term between brackets equals zero in the case in which no account is taken of TDB creation by disproportionation.disproportionation does not lead to reactive TDBs and recombination is absent,chains may carry one TDB as a maximum, which is the case being generally dealt with in modeling studies of PVAc [10–18]. The specific situation of a maximum of one TDB per chain here serves as an example of the classes approach, while the more general case of more than one TDB will be addressed in the context of pseudo-distribution modeling (Section 9.6). Only two TDB classes are required to fully describe the problem of one TDB as a maximum: 0 and 1 TDB per chain.Consequently, population balances for only four distributions have to be solved –R n ; Pn ; R¼ ¼n ; Pn – and this can be done in an exact way without additional assump-tions. The equations are shown in Table 9.6. The resulting CLDs for a realistic set of kinetic data are shown in Figure 9.6 for the total dead chain concentrations, with and without TDB.
9.5.3
Multiradicals in Radical Polymerization In most models of radical polymerization, living chains are explicitly taken into account, except for instance in the Monte Carlo simulation approach by Tobita[11–15]. Usually, macroradicals are assumed to possess only one radical site

9.5 Classes Approach 447
Tab. 9.6. Full set of population balance equations for the PVAc problem in the case of a maximum of one terminal double bond per chain.
dR n dR¼n
Propagation þ¼ k p MðR n
1
R n Þ þ¼ k p MðR¼n
1
R¼n Þ
dt dt
dR n dPn
Termination by þ¼
k td l 0 R n ; þ¼ k td l 0 R n
dt dt
disproportionation ¼ ¼
dR n dPn
þ¼
k td l 0 R¼n ; þ¼ k td l 0 R¼n
dt dt
dR¼ dR¼n dPn¼Transfer to monomer 1þ¼ k m l 0 M; þ¼
k m MR¼n ; þ¼ k m MR¼n
dt dt dt
dR n dPn
þ¼
k m MR n þ¼ k m MR n
dt dt
dR n dPn
Transfer to polymer þ¼ k tp ðl 0 nPn
m1 R n Þ; þ¼ k tp ð
l 0 nPn þ m1 R n Þ
dt dt
¼ ¼
dR n dPn
þ¼ k tp ðl 0 nPn¼
m1 R¼n Þ; þ¼ k tp ð
l 0 nPn¼ þ m1 R¼n Þ
dt dt
dR n X y X
n
1
¼ ¼
Terminal double þ¼ kdb
R n Pn þ Pm R n
m dt n¼1 m ¼1 bond propagation
dPn¼
þ¼
kdb l 0 Pn¼
dt
kmole/m3
1.2
n 2 ( Pn + Pn= )
kdb = 0
kdb = 2500 m3/(kmole.s)
0.8
kdb = 6200 m3/(kmole.s)
0.6
0.4 kdb = 31000 m3/(kmole.s)
0.2
Fig. 9.6. Radical polymerization of vinyl Mf ¼ 9 kmol m
3 ; residence time: t ¼ 4000 s;acetate in a CSTR: sensitivity of chain length kd ¼ 9  10
6 s
1 ; k p ¼ 9500 m 3 (kmol s)
1 ;distribution to rate of TDB propagation. k m ¼ 2:38 m 3 (kmol s)
1 ; k td ¼ 10 8 m 3 Reactor and kinetic data: initiator feed: (kmole s)
1 ; k tp ¼ 5 m 3 (kmol s)
1 ;I2; f ¼ 3:85  10
3 kmol m
3 ; monomer feed conversion x ¼ 0:5.

Tab. 9.7. Reaction equations for radical polymerization of ethylene: 2D chain length–multiradical approach.
kd
Initiator dissociation I2 ! 2I
ki
Initiation I þ M ! R1; 1
kpi
Propagation R n; i þ M ! R nþ1; i
k td ij
Termination by disproportionation R n; i þ R m; j ! R n; i
1 þ R m; j
1
k tc ij
Termination by recombination R n; i þ R m; j ! R nþm; iþj
2
km i
Transfer to monomer R n; i þ M ! R n; i
1 þ R1; 1
km i
Transfer to solvent R n; i þ S ! R n; i
1 þ R1; 1
k tp jm
Transfer to polymer R n; i þ R m; j ! R n; i
1 þ R m; jþ1 Notation: m; n: chain length; i; j: number of radical sites per chain.(monoradical assumption). In principle, this is only correct for linear polymeriza-tion. In nonlinear polymerization, for example, transfer to polymer or TDB propa-gation, multiradicals can easily be created. For instance, transfer to polymer is fre-quently assumed to occur only in dead chains, but obviously it also takes place in living chains. In order to investigate the effects of taking multiradicals into ac-count, we apply the classes approach. This anticipates the expectation that transi-tion regimes exist, where the monoradical assumption no longer holds, while the numbers of radical sites per chain is still low.We take the example of radical polymerization with transfer to polymer; reaction and population balance equations are listed in Tables 9.7 and 9.8. Note that the sec-Tab. 9.8. (Population) balance equations for radical polymerization of ethylene: 2D chain length–multiradical approach.
dI2 I2
Initiator dissociation ¼
kd I2

dt t
dI I dR1; 1 Initiation ¼ 2kd I2
k i MI
; þ¼ k i MI
dt t dt
dR n; i
Propagation þ¼ k p iMðR n
1; i
R n; i Þ
dt
Termination by dR n; iþ¼ k td l 0 fði þ 1ÞR n; iþ1
iR n; i g disproportionation dt dR n; i 1 P P
n
1 iþ1
Termination by recombination þ¼ k tc jði
j þ 2ÞR m; j R n
m; i
jþ2 dt 2 m ¼1 j¼1 dR n; i dR1; 1 Transfer to monomer þ¼ k m Mfði þ 1ÞR n; iþ1
iR n; i g; þ¼ k m l 0 M
dt dt
dR n; i dR1; 1 Transfer to solvent þ¼ ks Sfði þ 1ÞR n; iþ1
iR n; i g; þ¼ k m l 0 S
dt dt
dR n; i
Transfer to polymer þ¼ k tp ½l 0 nfR n; i
1
R n; i g þ l1 f
ði
1ÞR n; i
1 þ iR n; i g
dt
y X
X y
la ¼ na R n; i
n¼1 i¼0

9.6 Pseudo-distribution Approach 449
ond subscript of R n; i denotes the number of radical sites per chain; hence chains with i ¼ 0 represent what is usually called ‘‘dead’’ chains. The transfer-to-polymer reaction is obviously responsible for the creation of multiradicals, as appears from the term R n; i
1 on the RHS. Two classes models are formulated, one with two classes (the usual living and dead chains) and one with five classes (up to four radical sites per chain). Note that the implementation of the termination reactions requires the implementation of 4 2 pairs of reaction equations of the form:k tc IJ IþJ
2 k td IJ
R nI þ R m
! R nþm ; R nI þ R m! R nI
1 þ R m
J
1
ð32Þ
where RnI denotes the 1D distribution of the class with I radical sites per chain.Thus we see that many more classes rapidly lead to a huge implementation task and the classes method is not appropriate for such problems.We calculated CLDs with the two classes models for various transfer-to-polymer rates. Up to k tp ¼ 500 m 3 (kmol s)
1 results are identical, proving the validity of the monoradical-assumption in this range. For k tp ¼ 5000 m 3 (kmol s)
1 , on the contrary, we see large differences, as shown in Figure 9.7. According to the five-classes model, chains are much shorter and also conversion is lower. This is due to differences in an increase in the reactivity of multiradical chains in different reactions. From the reaction equations we see that propagation increases linearly with the number of radical sites, but termination with the product of radical sites.Hence, accounting for multiradicals leads to an increased termination/propagation ratio and thus to shorter chains and lower conversion under the conditions simu-lated. Note that the relative heights of the distributions in Figure 9.7 suggest that for k tp ¼ 5000 m 3 (kmol s)
1 even more than five classes are required for a proper description.We conclude, however, that the classes approach in this case produces useful information on the multiradical issue, even when employing a limited number of classes.
9.6
Pseudo-distribution Approach
9.6.1
Introduction This strategy is based on the Galerkin-FEM method. The simultaneous description of CLD and DBD leads to a two-dimensional problem with the property coordi-nates chain length and number of branches. In order to avoid the computationally expensive 2D problem, a mathematical reduction technique is applied to the origi-nal model, which allows the computation of average branching degrees per chain length (and the respective variances). For that reason branching moment distribu-tions are introduced. The population balances for these distributions can be solved

450 9 Mathematical Methods 10-3 kmole/m32-classes model
0.3 5-classes model
Concentration
0.2
n2Rn,0
n2Rn,0
0.1
n2Rn,1
n2Rn,1
0 0 2 4 6 8 10 1210 10 10 10 10 10 10 Chain Length n 10-3 kmole/m3 n2Rn,1 n2Rn,42-classes model 5-classes model Concentration
n2Rn,3
n2Rn,2
n2Rn,1
0 4 6 8 10
10 10 10 10 Chain Length n Fig. 9.7. Radical polymerization with transfer conversion and shorter chains. Kinetic data:to polymer: multiradical problem. CLDs of kd ¼ 0:57 s
1 ; k p ¼ 1:38  10 5 m 3 (kmol s)
1 ;chains with 0 . . . 4 radical sites per chain. k m ¼ 138 m 3 (kmol s)
1 ; ks ¼ 790 m 3 Comparison of two- and five-classes model. (kmol s)
1 ; k tc ¼ k td ¼ 2:82  10 8 m 3 Accounting for multiradicals leads to lower (kmol s)
1 ; k tp ¼ 5000 m 3 (kmol s)
1 .

9.6 Pseudo-distribution Approach 451
as a 1D problem. Thus the original 2D problem is replaced by a series of 1D prob-lems, getting exact information on averages concerning the second (number of branches) distribution. The distributions of the original kinetic model will be called‘‘real distributions’’ whereas the additional branching moment distributions are named ‘‘pseudo-distributions’’.We will introduce this approach in the case of the 2D CLD/DBD computation for mixed-metallocene polymerization of ethylene. Subsequently, we present applica-tions of the approach to the 3D problems of radical polymerization of vinyl acetate(CLD/DBD/number of terminal double bonds distribution), AB radical copolymer-ization (CLD/comonomer composition distribution/sequence length distribution),and finally the 2D problem of radical polymerization of polyethylene, where ran-dom scission is a complicating factor.
9.6.2
CLD/DBD for Mixed-metallocene Polymerization of Ethylene
9.6.2.1 Formulation of Pseudo-distribution Problem
Long branches are created in this system by incorporation of chains with a termi-nal double bond (TDB) at the catalyst site (constrained-geometry catalyst, CGC)[19–30]. The reaction equations are listed in Table 9.9 and the definitions of the growing and dead chain branching moment distributions are elucidated in Table
9.10. Note that the 0th branching moment distributions represent the original nor-
mal chain length distributions. From the branching moment distributions F and C certain averages may be derived. For example, the average number of branches of chains of length n; Bn , as well as the branching density, rn , can be computed from the ratio (for each n) of the 1st and 0th branching moment distribution:Tab. 9.9. Reaction mechanisms and rate coefficients.Reaction Reaction equation Rate coefficient

Branching cat. activation and initiation[a] CCGC ! CCGC ; ka; CGC

CCGC þ M ! R1;b 0 k i; CGC

Linear cat. activation and initiation[a] Clin ! Clin ; ka; lin

Clin þ M ! R1;l 0 k i; lin
b b
Branching cat. propagation R n; i þ M ! R nþ1; i k p; CGC Linear cat. propagation R n; i þ M ! R nþ1; i k p; lin
b 
Branching cat. b-hydride elimination R n; i ! C CGC þ Pn;b i kw; CGC
l  l¼ [b]
Linear cat. transfer to monomer R n; i þ M ! Clin þ Pn; i km; lin
b ¼ b [b]
branching cat. terminal double bond propagation R n; i þ Pm; j ! R nþm; iþjþ1 k p; TDB[a] Catalyst activation and initiation are taken as one step.[b] Note that P ¼ b¼ l¼n; i ¼ Pn; i þ Pn; i .

Tab. 9.10. Notation for mixed-metallocene systems.R n; i polymer growing on branching catalyst R nl polymer growing on linear catalyst (0 branches)Pn;b¼i ; Pn;l¼0 ; Pn;¼ i dead polymer with terminal double bond from branching/linear cat.and their sum, resp.Pn;b i ; Pn;l 0 ; Pn; i dead polymer without terminal double bond from branching/linear cat and their sum, resp.
y X
X y Xy X y
m¼0 ¼ Pn;¼ i m0 ¼ Pn; i n¼1 i¼0 n¼1 i¼0
y X
X y X
y
l 0b ¼ R n; i l 0l ¼ R nl n¼1 i¼0 n¼1
X
y
R nb ¼ R n; i
i¼0
X
y X
y
Pnb¼ ¼ Pn;b¼i Pnl¼ ¼ Pn;l¼0 Pn¼ ¼ Pn;¼ i
i¼0 i¼0
X
y X
y
Pnb ¼ Pn;b i Pnl ¼ Pn;l 0 Pn ¼ Pn; i
i¼0 i¼0
X
y
Fnb1 ¼ iR n; i
i¼0
X
y X
y
Cn¼1 ¼ iPn;¼ i Cn1 ¼ iPn; i
i¼0 i¼0
X
y
Fnb2 ¼ i 2 R n;
i¼0
X
y X
y
Cn¼2 ¼ i 2 Pn;¼ i Cn2 ¼ i 2 Pn; i
i¼0 i¼0
Bn ¼ Cn1 =Pn ; rn ¼ Cn1 =ðnPn Þ ð33ÞIn fact, F and C represent the moments of the branching distributions of living and dead chains at the given chain length, n. Thus, the branching polydispersity Dn follows from the second branching moment according to:
Cn2 =Cn1
Dn ¼ ð34Þ
Cn1 =Pn
In order to derive the balance equations for the branching moment distributions,the following steps have to be performed [31]: For each reaction step of the kinetic model, the two-dimensional population balance is derived.

9.6 Pseudo-distribution Approach 453
 The two-dimensional balance is reduced to a one-dimensional balance by apply-ing the above summations over the branching index (in the same type of opera-tion as obtaining moments from distributions). The resulting terms have to be expressed in terms of the branching moment distributions and are added to the system of equations.All equations are solved simultaneously. This procedure can in principle be applied to any polyreaction model where additional properties have to be considered. The key is how a certain reaction step changes the additional property.The total set of balance equations for all the reaction mechanisms involved in the mixed-metallocene problem is:
dM  
¼
fk i; lin Clin þ k i; CGC CCGC þ ðk p; lin þ km; lin Þl 0b þ ðk p; CGC þ km; CGC Þl 0l gM
dt
M0
M
þ ð35Þ
dR n; 0  l l¼ k i; lin MClin dðn
1Þ þ k p; lin ð½R n
1; 0
R n; 0 ÞM
dt
R n; 0

ðkb; lin þ km; lin MÞR n; 0
ð36ÞdR n; i  b¼ k i; CGC MCCGC dðn
1Þ þ k p; CGC ðR n
1
R nb ÞM
dt
þ
ðkb; CGC þ km; CGC MÞR n; i
n
1 X
X i b
R n; i
þ k p; TDB
m¼0 R n;
iþ
¼
Pm; b
j R n
m; i
j
1
ð37Þ
m¼1 j¼0
dPn;l¼0 l l¼ b
Pn;l¼0
¼ ðkm; lin M þ kb; lin ÞR n; 0
k p; TDB Pn; 0 l 0
ð38Þ
dt t
dPn;b¼i b b¼ b
Pn;b¼i
¼ ðkm; CGC M þ kb; CGC ÞR n; i
k p; TDB Pn; i l 0
ð39Þ
dt t
dB B
¼ k p; TDB m¼0 l 0b
ð40Þ
dt t
Here dðiÞ is the Kronecker’s delta function which has value of 1 for i ¼ 0 and 0 for i 0 0. Note that growing chains at the linear catalyst do not carry branches; hence the second index is always zero. Now, we will derive the equations for the pseudo-distributions in a stepwise manner. Reactions not affecting the number of branches per chain, that is, those describing propagation, transfer to monomer,

and b-hydride elimination, will be presented firstly. The reaction creating branch-ing, TDB propagation, is more complex and will be discussed subsequently.Propagation step In this case the degree of branching is not altered, so this is a simple step from this point of view. The contribution to the balance of ath branch-ing moment distributions of living chains in reduced or pseudo-distribution nota-tion follows, by multiplication with ia and taking the summation over the branch-ing index i:
 i
dFan X y
dR n
¼ ia þ¼ k p Mð
Fan
1 þ Fan Þ ð41Þ
dt i¼0
dt
Transfer to monomer, b-hydride elimination Again, the degree of branching is not affected, hence the contributions to the pseudo-distribution balances of the living and dead chains are given by:
dFan dCan
þ¼
k m MFan ; þ¼ k m MFan ; ð42Þ
dt dt
dFan dCan
þ¼
kb Fan ; þ¼ kb Fan ; ð43Þ
dt dt
TDB propagation The one-dimensional contributions for this mechanism describ-ing chain length are obtained from the TDB propagation terms in the 2D equa-tions by summing over the number of branches, i:
dR nb
þ¼
k p; TDB R nb m¼0 ð44Þ
dt
dPn¼
þ¼
k p; TDB Pn¼ l 0b ð45Þ
dt
dR nb X
n
1
¼
þ¼ k p; TDB R nb Pn
m ð46Þ
dt m¼1
The first branching moment distribution follows by multiplication of Eqs. (44)–(46)with the number of branches i, and subsequent summation:
dFnb1
þ¼
k p; TDB Fnb1 m¼0 ð47Þ
dt
dCn¼1
þ¼
k p; TDB Cn¼1 l 0b ð48Þ
dt

9.6 Pseudo-distribution Approach 455
The production term requires some more extensive algebraic manipulation:
X i
1
b ¼
þ¼ k p; TDB i R m; j Pn
m; i
j
1 dt i¼0 m ¼1 j¼0
y Xy
b @ ¼ A
¼ k p; TDB R m; j iPn
m; i
j
1 m¼1 j¼0 i¼ j
y X
b ¼
¼ k p; TDB R m; j ði þ j þ 1ÞPn
m; i m¼1 j¼0 i¼0
y X
y X
y X
b ¼ b ¼
¼ k p; TDB jR m; j Pn
m; iþ R m; j ði þ 1ÞPn
m; i m¼1 j¼0 i¼0 j¼0 i¼0
¼ ¼1 b ¼
¼ k p; TDB ðFmb1 Pn
m bþ Rm Cn
m þ Rm Pn
m Þ ð49Þ
m¼1
The second branching moment distribution follows by multiplication of the TDB propagation terms in the 2D equations with the squared number of branches i 2 and subsequent summation:
dFnb2
þ¼
k p; TDB Fnb2 m¼0 ð50Þ
dt
dCn¼2
þ¼
k p; TDB Cn¼2 l 0b ð51Þ
dt
The production term again requires some more extensive algebraic manipulation:
dFnb2 Xy X
¼
þ¼ k p; TDB i2 b R m; j Pn
m; i
j
1 dt i¼0 m¼1 j¼0
y Xy
¼ k p; TDB b R m; @ i2P ¼ A j n
m; i
j
1 m¼1 j¼0 i¼ j
y X
¼ k p; TDB b R m; j ði þ j þ 1Þ 2 Pn
m;
¼
m¼1 j¼0 i¼0

0 y 1
X Xy Xy Xy
2 b ¼ b 2 ¼B j R m; j Pn
m; i þ R m; j i Pn
m; i C
B j¼0 C
B i¼0 j¼0 i¼0 C
B C
n
1 B
X Xy Xy Xy Xy C
B þ2 b
jR m; j ¼
iPn
m; i þ 2 b jR m; j Pn
m; i C
¼
¼ k p; TDB B C
m¼1B
B j¼0 i¼0 j¼0 i¼0 C
B C
B Xy Xy Xy Xy C
@ þ2 b
R m; j ¼
iPn
m; i þ b
R m; j ¼
Pn
m; i A
j¼0 i¼0 j¼0 i¼0
X
n
1
Fmb2 Pn
m
¼ b
þ Rm ¼2
Cn
m ¼1
þ 2Fmb1 Cn
m¼ k p; TDB b1 ¼1 b ¼1 b ¼
ð52Þ
m¼1 þ 2Fm Pn
m þ 2R m Cn
m þ R m Pn
m The principles of the pseudo-distribution model are applicable to any reactor. Re-sults are presented for a semi-batch reactor using kinetic data from a realistic mixed-metallocene system in Figure 9.8.
9.6.2.2 Construction of the Full 2D Distribution
The pseudo-distribution approach thus allows us to calculate the 1D CLD and the average number of branches versus chain length. The question arises of how to obtain the full bivariate chain length–number of branches distribution from the pseudo-distribution model. The issue is that construction of branching distri-butions at each chain length requires an assumption to be made about the shape of the branching distribution, or more precisely for chains originating from the branching catalyst (Pnb¼ ). For instance, if we suppose that each monomer unit in a chain of given length has equal probability of being branched, then the branching distribution can be described by a binomial distribution. However, from the branch-ing polydispersity as calculated from Eq. (34) we observe that the distribution for dead chains from the branching catalyst is narrower than a binomial distribution.An alternative approximation method is required, generating branching distribu-tions at a given chain length with correct values for both the average number of branch points N n and the branching polydispersity Dn. A three-parameter binomial distribution modified by raising of it to a power an (the third parameter) varying with chain length, the variable power binomial distribution [VPBD] can satisfy these requirements:
  an
pn; N ¼ ðrn Þ N ð1
rn Þðn
NÞ ð53ÞNote that in the VPBD the parameter a permits the distribution width to be changed: a > 1 leads to a narrower distribution. The fitting procedure is described in more detail by Iedema et al. [20]. The VPBD method turns out to yield identical solutions to the classes (Section 9.5) and pgf (Section 9.7) methods as applied to the mixed-metallocene problem. The resulting full bivariate CLD/DBD for the case of a CSTR is shown in Figure 9.9, together with the VPBD fit.

9.6 Pseudo-distribution Approach 457
1.4
Semibatch, rCGC = 0.2
1.2
dw/d{log(MW)}
0.8
0.6
0.4 600 s
0.2
t = 100 s
0 3 4 5 6 710 10 10 10 10 Molecular Weight
x 10-3
600 s
500 s
Branching density ρb
0.8 400 s
Batch time
300 s
0.6 200 s
0.4 100 s
0.2
Semibatch, rCGC = 0.2
0 2 3 4 5
10 10 10 10 Chain Length Fig. 9.8. Mixed-metallocene polymerization of flow: 8  10
7 kmol s
1 ; batch time: 600 s;ethylene in a semibatch reactor; branching k i; CGC =k p; CGC ¼ k i; lin =k p; lin ¼ 1; k p; CGC ½M ¼(constrained geometry) catalyst: CGC-Ti; linear k p; lin ½M ¼ 500 s
1 ; kb; CGC ¼ 0:3 s
1 ; kb; lin ¼catalyst: Et[Ind]2 ZrCl2 . Reactor and kinetic 0:7 s
1 ; km; CGC =k p; CGC ¼ 0; km; lin =k p; lin ¼data: initial concentration CGC-Ti: 8  10
7 0:0014; kH; CGC ¼ 250 m 3 kmol
1 s
1 ;kmol m
3 ; initial concentration Et[Ind]2 ZrCl2 : kH; lin ¼ 0; k p; TDB ¼ 1750 m 3 kmol
1 s
1 .3:2  10
6 kmol m
3 ; monomer molar feed We conclude that the pseudo-distribution approach can be applied successfully,provided a good approximation can be made for the branching distribution at given chain length from the branching moments. The method is valid for batch reactors as well, in contrast to, for example, the pgf–cascade method (Section 9.7), which is restricted to steady state reactors. It is to be preferred over classes methods in cases, like the metallocene one, where the second distribution dimension may as-sume high values as well.

458 9 Mathematical Methods
3 an
10-2 2
VPBD power
10-4 n4
Fraction
10 1 10 2 10 3 10 10 5
10-6
CSTR
10-8 rCGC = 0.2[H2] = 1.13 10-3 kmole/m3
10-10 101
102 n
10-12 103 e ng
0 L
10 n
20 104 ai
Number of
Branche s N
105 Ch
Fig. 9.9. Mixed-metallocene polymerization of the Galerkin-FEM model. Assumption: variable ethylene in a CSTR. Kinetic data are the same power binomial distribution of number of as in Figure 9.8. Residence time CSTR: 300 s; branch points at each chain length of dead feed concentrations identical to initial chains from CGC-Ti with power an as shown in concentrations in Figure 9.8. Bivariate chain the insert. The discontinuity at N ¼ 1 indicates length/number of branches distribution. the existence of a ‘‘ridge’’ at N ¼ 0, due to the Hydrogen present. Based on molecular weight contribution of branchless chains from distribution and branching distribution from Et[Ind]2 ZrCl2 .
9.6.3
CLD/Number of Terminal Double Bonds (TDB) Distribution for Poly(vinyl acetate) –More than one TDB per Chain
9.6.3.1 General Case
This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6.Here, we address the more general problem of more than one TDB per chain [9].This occurs as a consequence of insertion of TDB chains created by disproportion-ation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a 1D formulation by developing the TDB and branching mo-ment expressions. The (N; M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by:
X
y X
y X
y X
y
N; M N M N; M N M Fn;:;: ¼ i k R n; i; k Cn;:;: ¼ i k Pn; i; k ð54Þi¼0 k¼0 i¼0 k¼0

9.6 Pseudo-distribution Approach 459
Tab. 9.11. The general (N; M)th double moment formulation of the population balance equation set of Table
9.10, obtained by multiplying by the TDB number and branching number indices, i N and k M , and subsequent
summation over these indices.
X
y X
y
d iN k M R n; i; k dFnN; M i¼0 k¼0 N; M Propagation ¼ þ¼ k p MðFn
1
FnN; M Þ
dt dt
N; M
dFn
Termination by dispropor- þ¼
k td l 0 FnN; M
dt
tionation !dCnN; M 1 Xy Xyþ¼ k td l 0 iN k M R n; i
1; k þ FnN; M [a]dt 2 i¼0 k¼0
dFnN; M
Termination by recombination þ¼
k tc l 0 FnN; M
dt
dCnN; M 1 X y Xy X n
1 X i X kþ¼ k tc iN kM R m; j; l R n
m; i
j; k
l dt 2 i¼0 k¼0 m ¼1 j¼0 l ¼0 dFnN; M dF1N; M Transfer to monomer þ¼
k m MFnN; M ; þ¼ k m l 0 M
dt dt
dCnN; M
þ¼ k m MFnN; M
dt
dFnN; M Xy Xy Transfer to polymer þ¼ k tp l 0 n iN k M Pn; i; k
1
m1 FnN; M
dt i¼0 k¼0
dCnN; M Xy Xyþ¼ k tp
l 0 n iN k M Pn; i; k
1 þ m1 FnN; M
dt i¼0 k¼0
0Xy X y X y 1 B
R n; i; k iPn; j; l C dFnN; M Xy Xy B n¼1 j¼0 l ¼0 C
N MB C
Terminal double bond þ¼ kdb i k B C dt B Xn
1 X
iþ1 X k C
propagation i¼0 k¼0 @ Aþ jPm; j; l R n
m; i
jþ1; k
l
1 m ¼1 j¼0 l ¼0
dCnN; M
þ¼
kdb l 0 CnðNþ1Þ; M
dt
[a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.Performing the corresponding summations on the equations in Table 9.6, one ob-tains the (N; M)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N; M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M ¼ 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N ¼ 0; 1, and 2.Solving the model thus essentially means solving the population balances of the real concentration distributions R n and Pn and the pseudo-distributions FnN; M and CnN; M .

Tab. 9.12. The (0; 0)th moment distribution formulation of the population balance equation set of Table 9.11 (taking N ¼ 0 and M ¼ 0); this set is solved by the TDB moment distribution model.
dR n
Propagation þ¼ k p MðR n
1
R n Þ
dt
Termination by dR n dPnþ¼
k td l 0 R n ; þ¼ k td l 0 R n disproportionation dt dt Termination by dR n dPn 1 X n
1þ¼
k tc l 0 R n ; þ¼ k tc R m R n
m recombination dt dt 2 m ¼1 dR n dR1 dPn Transfer to monomer þ¼
k m MR n ; þ¼ k m l 0 M þ¼ k m MR n
dt dt dt
dR n dPn
Transfer to polymer þ¼ k tp ðl 0 nPn
m1 R n Þ; þ¼ k tp ð
l 0 nPn þ m1 R n Þ
dt dt
Terminal double bond dR n Xy X
n
1
dPn
þ¼ kdb
R n Cn1; 0 þ Cm1; 0 R n
m ; þ¼
kdb l 0 Cn1; 0 propagation dt n¼1 m ¼1
dt
The resulting equations for N ¼ 0 and M ¼ 0 are listed in Table 9.12. Here, the(0; 0)th moments are the usual 1D chain length distribution variables, defined by:
y X
X y y X
X y
Rn ¼ R n; i; k ð¼ Fn0; 0 Þ Pn ¼ Pn; i; k ð¼ Cn0; 0 Þ ð55Þi¼0 k¼0 i¼0 k¼0 All of the derivations are straightforward, but the TDB propagation deserves closer examination. From Table 9.11 we have the general (N; M) formulation in:dFnN; M Xy Xyþ¼ kdb iN kM
dt i¼0 k¼0
X
y X
y X
y X
n
1 X
iþ1 X
k

R n; i; k iPn; j; l þ jPm; j; l R n
m; i
jþ1; k
l
1 n¼1 j¼0 l¼0 m¼1 j¼0 l ¼0
ð56Þ
Taking M ¼ 0 and N ¼ 0, the first term between the brackets can be rewritten as:
X
y X
y X
y X
y X
y
R n; i; k iPn; i; k i¼0 k¼0 n¼1 i¼0 k¼0
y X
X y X
y X
y y X
X y X
y
¼ ðR n; i; k Cn1; 0 Þ ¼ Cn1; 0 R n; i; k ¼ R n Cn1; 0 ð57Þi¼0 k¼0 n¼1 n¼1 i¼0 k¼0 n¼1

9.6 Pseudo-distribution Approach 461
The second term between the brackets can be rewritten as:
X
y X
y X
n
1 X
iþ1 X
k
jPm; j; l R n
m; i
jþ1; k
l
1 i¼0 k¼0 m¼1 j¼0 l ¼0
n
1 <X
X y Xk X y Xy = Xn
1¼ @ jPm; j; l R n
m; i
jþ1; k
l
1 A ¼ Cm1; 0 R n
m ð58Þm¼1 k¼0 l¼0 j¼0 i¼j
1
; m¼1
Thus, we find the TDB propagation term for the living chains to be given by:
dR n Xy X
n
1
þ¼ kdb
R n; i; k Cn1; 0 þ 1; 0 Cm R n
m ð59Þ
dt n¼1 m¼1
Applying M ¼ 0 and N ¼ 0 to the dead chains formulation yields:
dPn
þ¼
kdb l 0 Cn1; 0 ð60Þ
dt
The RHS expressions of Eqs. (59) and (50) contain higher TDB moment (1; 0) dis-tributions (Cn1; 0 ) than the (0; 0) moment distributions, R n and Pn , they describe.Tab. 9.13. The (1; 0)th moment distribution formulation of the population balance equation set of Table 9.11(taking N ¼ 1 and M ¼ 0); this set is solved by the TDB moment distribution model.dFn1; 0 1; 0 Propagation þ¼ k p MðFn
1
Fn1; 0 Þ
dt
Termination by dispropor- dFn1; 0þ¼
k td l 0 Fn1; 0 ;tionation dt
!  
dCn1; 0 1 Xyþ¼ k td l 0 iR n; i
1 þ Fn1; 0 ¼ k td l 0 Fn1; 0 þ R n [a]
dt 2 i¼0
Termination by recom- dFn1; 0 dCn1; 0 X
n
1
þ¼
k tc l 0 Fn1; 0 ; þ¼ k tc Fm1; 0 R n
m bination dt dt m ¼1 dFn1; 0 dF11; 0 dCn1; 0 Transfer to monomer þ¼
k m MFn1; 0 ; þ¼ k m l 0 M; þ¼ k m MFn1; 0
dt dt dt
dFn1; 0 dCn1; 0 Transfer to polymer þ¼ k tp ðl 0 nCn1; 0
m1 Fn1; 0 Þ; þ¼ k tp ð
l 0 nCn1; 0 þ m1 Fn1; 0 Þ
dt dt
Terminal double bond dFn1; 0 Xy X
n
1
1; 0 1; 0 2; 0 1; 0 1; 0 1; 0þ¼ kdb
Fn Cn þ ðCm R n
m þ Cm Fn
m
Cm R n
m Þpropagation dt n¼1 m ¼1
dCn1; 0
þ¼
kdb l 0 Cn2; 0
dt
[a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.

This is a direct consequence of the fact that the reactivity of the dead chains de-pends on the number of TDBs they carry. Hence, at this point we are already con-fronted with the closure problem present in the TDB moment model, anticipating that solving these higher moments leads to even higher moments in the equations.Next, we will derive the higher moment equations. For N ¼ 1 and M ¼ 0 we ob-tain the set of population balance equations for the pseudo-distributions of order(1; 0) as listed in Table 9.13.The termination by recombination equation is derived by developing the sum-mation term as:
1X y n
1 X
X i
i R m; j R n
m; i
j 2 i¼0 m¼1 j¼01Xn
1 Xy X y¼ R m; j @ iR n
m; i
j A 2 m¼1 j¼0 i¼j 1Xn
1 Xy Xy¼ R m; j ði þ jÞR n
m; i 2 m¼1 j¼0 i¼01Xn
1 Xy Xy Xy Xy¼ jR m; j R n
m; i þ R m; j iR n
m; i 2 m¼1 j¼0 i¼0 j¼0 i¼0
1Xn
1 X
n
1
¼ ðFm1; 0 R n
m þ R m Fn
m
1; 0
Þ¼ Fm1; 0 R n
m ð61Þ
2 m¼1 m¼1
This yields for the recombination contribution to the population balance:
dCn1; 0 X
n
1
þ¼ k tc Fm1; 0 R n
m ð62Þ
dt m¼1
With respect to TDB propagation, with M ¼ 0 and N ¼ 1 the first term between brackets becomes that in Eq. (64)
y X
X y X
y X
y X
y
iR n; i; k iPn; i; k i¼0 k¼0 n¼1 i¼0 k¼0
X
y X
y X
y X
y X
y X
y X
y
¼ iR n; i; k Cn1; 0 ¼ Cn1; 0 iR n; i; k ¼ Fn1; 0 Cn1; 0 i¼0 k¼0 n¼1 n¼1 i¼0 k¼0 n¼1
ð63Þ
and the second term, somewhat more complicated, becomes

9.6 Pseudo-distribution Approach 463
n
1 X
iþ1 X
k
i jPm; j; l R n
m; i
jþ1; k
l
1 i¼0 k¼0 m¼1 j¼0 l¼0
y n
1 X
¼ i ð jPm; j R n
m; i
jþ1 Þi¼0 m¼1 j¼0<Xn
1 Xy Xy =¼ @ jPm; j iR n
m; i
jþ1 A:m ¼1 j¼0 i¼ jþ1
n
1 X
¼ jPm; j ði þ j
1ÞR n
m; i m¼1 j¼0 i¼0
n
1 X
¼ j Pm; j R n
m; i þ jPm; j ði
1ÞR n
m; i m¼1 j¼0 i¼0 j¼0 i¼0
n
1
2; 0
¼ ðCm R n
m þ Cm1; 0 Fn
m
1; 0

Cm1; 0 R n
m Þ ð64Þ
m¼1
Thus, we find the TDB propagation terms for the living and dead chains to be:dFn1; 0 Xy X
n
1
1; 0 1; 0 2; 0 1; 0 1; 0 1; 0þ¼ kdb
Fn Cn þ ðCm R n
m þ Cm Fn
m
Cm R n
m Þ ð65Þ
dt n¼1 m¼1
dCn1; 0
þ¼
kdb l 0 Cn2; 0 ð66Þ
dt
The higher moments observed earlier on the RHS are seen again.We finally present the result for one higher moment distribution, N ¼ 2, M ¼ 0,for which we obtain the set of population balance equations for the pseudo-distributions of order (2; 0) as listed in Table 9.14.First, the termination by recombination term is developed:
1X y X
n
1 Xi
i2 R m; j R n
m; i
j 2 i¼0 m¼1 j¼00 1 !
1Xn
1 Xy
2 A 1Xn
1 Xy Xy¼ R m; j i R n
m; i
j ¼ R m; j ði þ jÞ R n
m; i 2 m¼1 j¼0 i¼ j 2 m¼1 j¼0 i¼0

Tab. 9.14. The (2; 0)th moment distribution formulation of the population balance equation set of Table 9.11(taking N ¼ 2 and M ¼ 0); this set is solved by the highest TDB moment version of the TDB moment distribution model.dFn2; 0 2; 0 Propagation þ¼ k p MðFn
1
Fn2; 0 Þ
dt
Termination by dFn2; 0þ¼
k td l 0 Fn2; 0 ;disproportionation dt
!  
dCn2; 0 1 Xyþ¼ k td l 0 i 2 R n; i
1 þ Fn2; 0 ¼ k td l 0 Fn2; 0 þ Fn1; 0 þ R n [a]
dt 2 i¼0
Termination by dFn2; 0 dCn2; 0 X
n
1
þ¼
k tc l 0 Fn2; 0 ; þ¼ k tc ðFm2 R n
m þ Fm1; 0 Fn
m
1; 0
Þ
recombination dt dt m ¼1 dFn2; 0 dF12; 0 Transfer to monomer þ¼
k m MFn2; 0 ; þ¼ k m l 0 M
dt dt
2; 0
dCn
þ¼ k m MFn2; 0
dt
dFn2; 0 dCn2; 0 Transfer to polymer þ¼ k tp ðl 0 nCn2; 0
m1 Fn2; 0 Þ; þ¼ k tp ð
l 0 nCn2; 0 þ m1 Fn2; 0 Þ
dt dt
8 Xy 9
>
Fn2; 0 Cn1; 0 >dFn2; 0 < n¼1 Terminal double bond þ¼ kdb  >;dt > Xn
1  3; 0 propagation >> Cm R n
m þ Cm1; 0 Fn
m 2; 0þ 2Cm2; 0 Fn
m
1; 0 >
: þ >
2; 0 1; 0 1; 0 m ¼1
2C m R n
m
2C m Fn
m þ Cm1; 0 R n
m
dCn2; 0
þ¼
kdb l 0 Cn3; 0
dt
[a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.1Xn
1 Xy Xy Xy Xy Xy Xy¼ j 2 R m; j R n
m; i þ R m; j i 2 R n
m; i þ jR m; j iR n
m; i 2 m¼1 j¼0 i¼0 j¼0 i¼0 j¼0 i¼0
1Xn
1
2; 0 2; 0
X
n
1
¼ ðFm R n
m þ R m Fn
m þ 2Fm1; 0 Fn
m
1; 0
Þ¼ ðFm2 R n
m þ Fm1; 0 Fn
m
1; 0
Þ
2 m¼1 m¼1
ð67Þ
This yields for the recombination contribution to the population balance:
dCn2; 0 X
n
1
þ¼ k tc ðFm R n
m þ Fm1; 0 Fn
m
1; 0
Þ ð68Þ
dt m¼1
The first term in the equation describing TDB propagation is derived in a similar way to Eq. (62), while the second term follows as:

9.6 Pseudo-distribution Approach 465
Xy X y   X y Xy X
n
1 X
iþ1 X
k
dR n; i; k
i ¼ i2 jPm; j; l R n
m; i
jþ1; k
l
1
i¼0 k¼0
dt i¼0 k¼0 m¼1 j¼0 l¼0
n
1 X
¼ i ð jPm; j R n
m; i
jþ1 Þi¼0 m¼1 j¼0
<Xn
1 X
¼ @ jPm; j i 2 R n
m; i
jþ1 A:m¼1 j¼0 i¼ jþ1
n
1 X
¼ jPm; j ði þ j
1Þ 2 R n
m; i m¼1 j¼0 i¼0
n
1
¼ ðCm3; 0 R n
m þ Cm1; 0 Fn
m
2; 0
þ 2Cm2; 0 Fn
m
1; 0 2; 0

2Cm R n
m
m ¼1

2Cm1; 0 Fn
m
1; 0
þ Cm1; 0 R n
m Þ ð69ÞThe total contribution of TDB propagation to the population balance is thus given
by:
dFn2; 0 Xy
þ¼ kdb
Fn2; 0 Cn1; 0
dt n¼1
n
1  3; 0
X 
Cm R n
m þ Cm1; 0 Fn
m2; 0þ 2Cm2; 0 Fn
m
1; 0 2; 0

2Cm R n
m
þ
m¼1
2Cm1; 0 Fn
m
1; 0
þ Cm1; 0 R n
m
ð70Þ
For the dead chains we have:
dCn2; 0
þ¼
kdb l 0 Cn3; 0 ð71Þ
dt
It is obvious that even higher TDB moment distribution balances can be con-structed, but we will restrict ourselves to the ones developed for up to the second moments. When applying the TDB moment model, in principle two solution strat-egies are possible: one using the zeroth and first TDB moments only and the sec-ond with zeroth, first, and second TDB moments. In the first case we have to find a closure relationship for the second moment, and in the second case, one for the third moment. Below, we show that the system becomes simpler in the case of a maximum of one TDB per chain.

Closure relations We have adopted a simple form for these relationships:Cn1; 0 1; 0 Cn2; 0 ¼ Dn C ð72Þ
Pn n
Cn2; 0 2; 0 Cn3; 0 ¼ Dn0 C ð73Þ
Cn1; 0 n
Here the functions Dn and Dn0 are in fact polydispersities of the branching moment distributions and in principle are to be determined as functions of chain length n.Inserting these closure relationships in Eqs. (65), (66), and (70), (71), reduces them
to:
( )
Xy n
1 
X  
dFn1; 0 1; 0 1; 0 1; 0 1; 0 Cm1; 0 1; 0þ¼ kdb
Fn Cn þ Cm Fn
m þ Dm
1 Cm R n
m
dt n¼1 m¼1
Pm
ð65aÞ
dCn1; 0 Cm1; 0þ¼
kdb l 0 Dn Cn1; 0 ð66aÞ
dt Pn
dFn2; 0 Xy
þ¼ kdb
Fn2; 0 Cn1; 0
dt n¼1
2   39
Cm2; 0 >
n
1 C 1; 0 F 2; 0 þ 2C 2; 0 F 1; 0 þ D 0 X 6 m n
m m n
m m 1; 0
1 Cm R n
m 7=
2; 0
þ 4 Cm 5
m ¼1
þ
2Cm1; 0 Fn
m
1; 0
þ Cm1; 0 R n
m
ð70aÞ
dCn2; 0 C 2; 0þ¼
kdb l 0 Dn0 n1; 0 Cn1; 0 ð71aÞ
dt Cn
An exact determination of Dn and Dn0 is not possible, but they can be estimated from results of other methods, for instance a TDB classes model in regions with few TDBs per chain.
9.6.3.2 TDB Pseudo-distribution Approach for a Maximum of one TDB per Chain
This case allows a few simplifications. Firstly, only the first TDB moment distribu-tions, Fn1; 0 and Cn1; 0 , have to be solved in addition to the real concentration distri-butions, since all higher TDB moment distributions are identical to these. Second,the TDB propagation contributions to the population balances become greatly sim-plified and their closure problem vanishes, since with Cn2; 0 ¼ Cn1; 0 Eqs. (65) and
(66) become:

9.6 Pseudo-distribution Approach 467
n
1
þ¼ kdb
Fn1; 0 Cn1; 0 þ Cm1; 0 Fn
m
ð74Þ
dt n¼1 m¼1
dCn1; 0
þ¼
kdb l 0 Cn1; 0 ð75Þ
dt
This implies that under the condition of a maximum of one TDB per chain, the set of population balance equations of the TDB branching moment variant of the model is solvable without requiring any additional closure assumption. The results obtained with the pseudo-distribution model are identical to those obtained with the classes model shown before (see Figure 9.6).
9.6.3.3 TDB Pseudo-distribution Approach for More than one TDB per Chain
It is interesting here to compare results for the case of insertion of disproportiona-tion-produced TDBs – leading to more than one TDB per chain – to the case of a maximum of one TDB per chain. The chain length distributions for the two cases and high TDB propagation rates are depicted in Figure 9.10 (left-hand side), which reveals that the CLDs are quite different. Most interestingly, for the case of a max-imum of one TDB per chain, the CLD features a shoulder that becomes higher with increasing kdb . This shoulder is absent in the other case, which can be ex-plained by realizing that here the longer chains become more reactive with length because of the higher number of TDBs on longer chains (see Figure 9.10, right-hand side). Since these chains are more reactive they will be consumed by the TDB propagation reaction more intensively, thereby producing more living chains,which is consistent with our findings (see Figure 9.10, left-hand side). When ex-tending the trend of increasing TDB propagation rate, we would expect a transition to a situation with vanishing dead chain concentrations, while retaining very few but extremely long living chains. Thus, we observe the effect of dead chains with many TDBs acting as crosslinkers between living chains. It should be noted that under such conditions our model is no longer fully representative. The model should at least be extended by allowing living chains with TDBs to be subject to TDB propagation and, consequently, the existence of multiradicals.The graph on the right in Figure 9.10 shows the numbers of TDBs per chain as a function of chain length for both cases of TDB production. Here, the contrast is very sharp. While for the maximum of one TDB per chain case the average num-ber of TDBs per chain decreases with chain length, in the case of more than one TDB it increases linearly with n in the double-logarithmic plot. The decrease in the former case is caused by the simultaneous transfer-to-polymer reaction [9].The latter implies a reduction to a constant TDB ‘‘density’’, the number of TDBs per monomer unit in a dead chain. As noted before, this results into a linear increase in dead chain reactivity with chain length, similarly to the transfer-to-polymer reaction.

Fig. 9.10. Radical polymerization of vinyl The part of the living chain CLD from the TDB acetate in a CSTR. The reactor and kinetic data moment model beyond 10 10 is estimated.are the same as in Figure 9.6. Left: chain Right: numbers of TDBs per chain for various length distributions for dead and living chains. kdb .

9.6 Pseudo-distribution Approach 469
Tab. 9.15. Reaction mechanisms for LDPE (first subscript: chain length; second subscript:number of branch points).Mechanism Reaction equation Rate factor Initiation I2 ! 2I I þ M ! R1; 0 kd ; k i f Propagation R n; i þ M ! R nþ1; i kp Disproportionation termination R n; i þ R m; j ! Pn; i þ Pm; j k td Recombination termination R n; i þ R m; j ! Pnþm; iþj k tc Transfer to monomer R n; i þ M ! Pn; i þ R1; 0 km Transfer to chain-transfer agent S R n; i þ S ! Pn; i þ R1; 0 kS Transfer to polymer R n; i þ Pm; j ! Pn; i þ R m; jþ1 þ LCB k trp Pre-scission (formation of secondary R n; i þ Pm; j ! Pn; i þ Rm; j k rs macroradicals)Scission of macroradicals Rn; i ! R n
m; i
j þ Pm; j ksec
9.6.4
Radical Polymerization of Ethylene to Low-density Polyethylene (LDPE)
9.6.4.1 Introduction
This problem has received considerable attention for a long time. Modeling is com-plicated, since branching is involved, and the importance of random scission for LDPE has now been recognized [31, 32, 54]. We address the 2D problem of CLD/DBD calculation here. The full set of reaction equations is given in Table 9.15, no-tation in Table 9.16, and population balance equations in Table 9.17. For merely Tab. 9.16. Notation for LDPE system.R n; i Primary radical chains Rn; i Secondary radical chains Pn; i Dead polymer chains
X
y X
y X
y X
y X
y X
y
m0 ¼ Pn; i l0 ¼ R n; i l0 ¼ Rn; i n¼1 i¼0 n¼1 i¼0 n¼1 i¼0
X
y X
y X
y
Rn ¼ R n; i Rn ¼ Rn; i Pn ¼ Pn; i i¼0 i¼0 i¼0
X
y X
y X
y
Fn1 ¼ iR n; i F1 n ¼ iRn; i Cn1 ¼ iPn; i i¼0 i¼0 i¼0
X
y X
y X
y
Fn2 ¼ i 2 R n; i F2 n ¼ i 2 Rn; i Cn2 ¼ i 2 Pn; i i¼0 i¼0 i¼0 Bn ¼ Cn =Pn number of branches per chain rn ¼ Cn =ðPn nÞ. branching density

Tab. 9.17. Population balance equations for LDPE.2D population balance equations
dR n; i
¼ k p MðR n
1; i
R n; i Þ
ðk td þ k tc Þl 0 R n; i þ k trp ðnl 0 Pn; i
1
m1 R n; i Þ (a)
dt
þ
ðk m M þ8 ks SÞR n; i
k rs m1 R n; i 9
Xy < X y =

þ ksec gðm; nÞ ½bðm; j; njiÞR m; j 
m ¼nþ1
: j¼i
þ ðk m M þ ks SÞl 0 dðn
1ÞdðiÞ þ k i f MIdðn
1ÞdðiÞdPn; i 1 X n
1 X¼ k td l 0 R n; i þ k tc ðR m; j R n
m; i
j Þ þ k trp ð
nl 0 Pn; i þ m1 R n; i Þ (b)dt 2 m ¼1 j¼0þ ðk m M þ k8s SÞR n; i þ k rs ð
nl 0 Pn; i þ m1 9R n; i Þ
Xy < Xy =

þ ksec gðl; nÞ ½bðm; j; njiÞR m; j
l ¼nþ1
: j¼i

dR n; i 
¼ nk rs l 0 Pn; i
ksec R n; i (c)
dt
1D concentration distribution equations
dR n
¼ k p MðR n
1
R n Þ
ðk td þ k tc Þl 0 R n þ k trp ðnl 0 Pn
m1 R n Þ
ðk m M þ ks SÞR n (d)
dt
X
y

k rs m1 R n þ ksec gðm; nÞRl þ ðk m M þ ks SÞl 0 dðn
1Þ þ k i f MIdðn
1Þ
m ¼nþ1
dPn 1 X n
1¼ k td l 0 R n þ k tc R m R n
m þ ðk trp þ k rs Þðm1 R n
nl 0 Pn Þ þ ðk m M þ ks SÞR n (e)
dt 2 m ¼1
X
y

þ ksec gðm; nÞR m
m ¼nþ1
dR n
¼ nk rs l 0 Pn
ksec R n (f )
dt
1D pseudo-distribution equations, 1st branching moment
y  
dFn1 X dR n; i 1¼ i k p Mð
Fn
1 þ Fn1 Þ
ðkM M þ kS SÞFn1
ðk td þ k tc Þl 0 Fn1 (g)
dt i¼0
dt
þ k trp f
m1 Fn1 þ l 0 nðCn1 þ Pn Þg
k rs m1 Fn1
Xy Xy X

þ ksec gðm; nÞ R m; j ibðm; j; njiÞ þ ðkM M þ kS SÞl 0 dðn
1Þm ¼nþ1 j¼0 i¼0þ k i f Mdðn
1Þ
y  
dCn X
dPn; N X
n
1
¼ i ¼ ðkM M þ kS SÞFn1 þ k td l 0 Fn1 þ k tc Fm1 R n
m (h)
dt i¼0
dt m ¼1
X
y Xy X

þ ðk trp þ k rs Þðm1 Fn1
l 0 nCn1 Þ þ ksec gðn; mÞ R m; j ibðm; j; njiÞn¼mþ1 j¼0 i¼0
Xy dR 
dF1 n; i
¼ i ¼ nk rs l 0 Cn1
ksec F1
dt i¼0
dt

9.6 Pseudo-distribution Approach 471
1D pseudo-distribution equations, 2nd branching moment
 
dFn2 X y
dR n; i
¼ i2 2
¼ k p Mð
Fn
1 þ Fn2 Þ
ðkM M þ kS SÞFn2
ðk td þ k tc Þl 0 Fn2 ( j)
dt i¼0
dt
þ k trp f
m1 Fn2 þ l 0 nðCn2 þ 2Cn1 þ Pn Þg
Xy Xy X

þ ksec gðm; nÞ R m; j i 2 bðm; j; njiÞm ¼nþ1 j¼0 i¼0
k rs m1 Fn2 þ ðkM M þ kS SÞl 0 dðn
1Þ þ k i f Mdðn
1Þ
 
dCn2 X y
dPn; i X
n
1
¼ i2 ¼ ðkM M þ kS SÞFn2 þ k td l 0 Fn2 þ k tc ðFm2 R n
m þ Fm1 Fn
m
Þ (k)
dt i¼0
dt m ¼1
X
y Xy X
2 2  2
þ ðk trp þ k rs Þðm1 Fn
l 0 nCn Þ þ ksec gðn; mÞ R m; j i bðm; j; njiÞn¼mþ1 j¼0 i¼0
Xy   
dF2 dR n; i¼ i2 ¼ nk rs l 0 Cn2
ksec F2
dt i¼0
dt
computational reasons it is assumed that the scission reaction proceeds in two steps, a pre-scission reaction yielding a secondary macroradical Rn; i , and a subse-quent breakage of this radical:
k rs ðn
1Þ
Pn; i þ R m; j ! Rn; i þ Pm; j ð76Þ
ksec
Rn; i ! R n
m; i
j þ Pm; j ð77ÞThe term (n
1) in the pre-scission equation [Eq. (76)] is due to the fact that we are dealing with random (pre-)scission here: that is, every CaC bond has equal probability of being attacked to form a secondary radical. The population balance contribution from this reaction is similar to that from a transfer-to-polymer reac-tion (see Table 9.17). The 2D population balance contribution from the actual scis-sion step reads as (Table 9.17):
y <
X Xy =
dR n; i 
þ¼ ksec gðm; nÞ ½bðm; j; njiÞRm; j  ð78Þ
dt m¼nþ1
: j¼i
Here, the functions g and b are the most general form of probability functions, ex-pressing the way in which the fragment lengths are distributed (g) and how the branch points are redistributed on these fragments (b). The function gðm; j; nÞ de-scribes the probability that a chain of length n with j branch points breaks into a fragment with length m, while bðm; j; nj jÞ is the conditional probability that a frag-ment of length m created by scission of a chain n=i carries exactly j branch points.The fragment length distribution function reflects the scission mechanism acting.

The Galerkin-FEM approach allows us to choose any model describing such mech-anisms, either chemical or mechanical, if they are expressed in terms of fragment lengths as functions of overall chain length or numbers of branch points [54]. An overview of such models is given elsewhere [32]. In the most realistic models the effect of branching and architectures is accounted for; this gives rise to predomi-nantly long and short fragments. The fundamental problem here, to start with, is that the second dimension of the 2D population balance problem, the branching distribution, has to be solved to calculate the first dimension, CLD. Moreover, not only the branching distribution, but also the character of the branched architec-tures, determine the scission function. One way of addressing this problem is employing an empirical approximation relationship for the fragment length func-tion gðn; mÞ found for ‘‘topological scission’’ of LDPE architectures [33]:
þ
ð2s þ mÞ 2 ð2s þ n
mÞ 2 gðn; mÞ ¼ n
1 ( ) ð79Þ
X 1 1
2þ
m¼1 ð2s þ mÞ ð2s þ n
mÞ 2 with s the average segment length, according to:
nn
s¼ ð80Þ
1 þ 2  LCB=m 0 In Eq. (80), LCB is the overall concentration of long chain branches, following from a balance coupled with the transfer-to-polymer reaction (see Table 9.17). The fragment length distribution gðn; mÞ for topological scission is a function of chain and fragment length only, and independent of the number i of branch points on the original chain. Note that this is an approximation method that fits to the differ-ential equation approach. Obviously, it accounts for branching architectures in an averaged manner. Fully accounting for architectures is not possible with the differ-ential equation method, since it does not describe connectivity in molecules. Doing this in more rigorous ways, full [11–15] or conditional [33–35] Monte Carlo (MC)simulations can be used (see Section 9.8). We stress here that the strength of the differential method discussed here is its ability to implement any function de-scribing scission in overall molecular dimensions, such as chain length. This is not readily possible in MC simulations.In order to solve the one-dimensional chain length distribution problem, the scission contribution of Eq. (78) is summed over the number of branch points on fragments, j. By definition bðn; i; mj jÞ ¼ 1, so this population balance assumes a one-dimensional form: j¼0
dR m Xy
þ¼ ksec gðn; mÞR n ð81Þ
dt n¼mþ1

9.6 Pseudo-distribution Approach 473
The 1D population balance for the pseudo-distribution or first branching moment distribution Fn follows by taking the first branching moment of Eq. (78):
X
y
j¼0 dFm Xy Xy X i

¼ þ¼ ksec gðn; mÞ R n; i jbðn; i; mj jÞ ð82Þdt dt n¼mþ1 i¼0 j¼0 Depending on the branch point redistribution function, the term in square brackets(in fact the expectation value of the number of branches on a fragment) can be evaluated. Similarly, we find for the second branching moment expression:
X
y
d j 2 R m; j ( " #)j¼0 dFm2 Xy Xy Xi
 2
¼ þ¼ ksec gðn; mÞ R n; i j bðn; i; mj jÞ ð83Þdt dt n¼mþ1 i¼0 j¼0 The task is now to find empirical expressions for the first and second branching moment summation terms between brackets in Eqs. (82) and (83) similar to Eq.
(79) for the fragment lengths. The shape of these is strongly dependent on the scis-
sion mechanism [32]. Note that finding solutions for this problem is important for calculation of the branching density and the branching polydispersity as functions of chain length. From this the shape of the branching distribution at constant chain length can be estimated, which then produces an estimation of the full CLD/DBD.Finally, in Figure 9.11 we show MWDs for two different scission models. The‘‘linear scission’’ case assumes scission of unbranched chains. The topological scis-sion case employs the fragment length function of Eq. (79). A marked difference is observed.
9.6.5
Radical Copolymerization
9.6.5.1 Introduction
We now address the problem of finding the 3D distribution of chain lengths,copolymer composition, and sequence length distribution in radical copolymer-ization. We have several options here. One could be explicitly solving the 3D con-
seq
centration variable at the sequence level, Pn; i; s , denoting the concentration of se-quences of length s on chains with a total number of monomer units n (usually called chain length), and number of monomer units of one kind i. Note that i=n is then the fractional copolymer composition. We will not do this, but instead choose a simpler option and solve a different 3D concentration variable at the
seq
chain level, Pn; i; s . Subscripts n and i have the same meaning as in Pn; i; s , but s here denotes the number of monomer sequences of one kind. Solving this prob-

474 9 Mathematical Methods
0.6
0.6
0.5
0.5
0.4
dW/d{log(n)}
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0 0
10 101 102 103 104 105 106 107 1081e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 Chain Length Fig. 9.11.Radical polymerization of ethylene in a CSTR. Linear and topological scission with the same polydispersity of 26 as experimental MWD from SEC-MALLS [35]. Solid line,experimental MWD; dash-dot line, topological scission result;dash-dot-dot line, linear scission result.lem in the full three dimensions would yield the average sequence length on chains Pn; i; s , simply following as i=s. We will show next how to solve the problem using pseudo-distributions and thus find the average copolymer composition and sequence length as a function of chain length, n. Reaction equations are listed in Table 9.18, notation and definitions in Table 9.19. Note that macroradicals are specified according to terminal unit, with an upper index A or B.
9.6.5.2 Balance Equations
All the reactions listed in Table 9.18 have contributions to the set of balance equa-tions. We will give a few examples of the generation of these contributions from some of the reaction equations.Tab. 9.18. Reaction equations for radical copolymerization.Initiation Propagation Termination kd A kAA A A A kc I2 ! 2I R n; i; s þ A
! R nþ1; iþ1; s R n; i; s þ R m; j; t ! Pnþm; iþj; sþt
ki kBB kc
I þ A ! R1;A 1; 0 B
R n; B
i; s þ B
! R nþ1; i; s
R n; B
i; s þ R m; j; t ! Pnþm; iþj; sþt
ki kAB kc
I þ B ! R1;B 0; 0 A
R n; B
i; s þ B
! R nþ1; i; s
R n; B
i; s þ R m; j; s ! Pnþm; iþj; sþt
B kBA A
R n; i; s þ A
! R nþ1; iþ1; sþ1

9.6 Pseudo-distribution Approach 475
Tab. 9.19. Notation and definitions.
y X
X y X
y y X
X y X
y y X
X y X
y
A B
Overall moments l 0A ¼ R n; i; s l 0B ¼ R n; i; s m0 ¼ Pn; i; s n¼1 s¼1 i¼1 n¼1 s¼1 i¼1 n¼1 s¼1 i¼1
X
y X
y y X
X y
2D concentrations Pn; i ¼ Pn; i; s Pn; s ¼ Pn; i; s R nA ¼ R n; i; s s¼1 i¼1 s¼1 i¼1
y X
X y y X
X y y X
X y
A B
1D concentrations R nA ¼ R n; i; s R nB ¼ R n; i; s Pn ¼ Pn; i; s s¼1 i¼1 s¼1 i¼1 s¼1 i¼11st moment distributions
y X
X y y X
X y y X
X y
A B
GnA ¼ iR n; i; s GnB ¼ iR n; i; s Ln ¼ iPn; i; s of number of A units s¼1 i¼1 s¼1 i¼1 s¼1 i¼11st moment distributions X
y X y X
y X y X
y X y
A B
FnA ¼ s R n; i; s FnB ¼ s R n; i; s Cn ¼ s Pn; i; s number of A sequences s¼1 i¼1 s¼1 i¼1 s¼1 i¼1 Number of A units per chain as a function of n nnA ¼ Ln =Pn Number of A sequences per chain as a function of n snA ¼ Cn =Pn Average sequence length of A as a function of n lnA ¼ nnA =snA Average sequence length of B as a function of n lnB ¼ ðn
nnA Þ=snA ¼ nnB =snA
kBA
BxA-propagation: Rn,B i, s B A m RnB1,iB1, sB1 In this reaction all the indices in-crease by 1. The contribution to the balance equations from which the length distri-bution is calculated simply follows by taking the double sum over i and s:
y X
X y
d Rn;A i; s
s¼1 i¼1
y X
X y
B B
þ¼ kBA A Rn
1; i
1; s
1 ¼ kBA AR n
1 ð84Þ
dt s¼2 i¼2
For the first moment distribution of the number of A units we have:
y X
X y
d iR n; i; s s¼1 i¼1 dGnA Xy X y Xy
B B
¼ þ¼ kBA A iR n
1; i
1; s
1 ¼ kBA A iR n
1; i
1 dt dt s¼2 i¼2 i¼2
ð85Þ
The last term here contains the two-dimensional distribution multiplied by i:i.R n
1; i
1 , which upon summation over i, due to its index (i
1), produces the
B B
sum of two one-dimensional distributions: R n
1 and Gn
1 . Hence, we get:
dGnA B B
þ¼ kBA AðR n
1 þ Gn
1 Þ ð86Þ
dt

Obviously, in this case the problem in obtaining the first moment distribution of the number of A sequences is exactly identical to that for the A units, so we have:
dFnA B B
þ¼ kBA AðR n
1 þ Fn
1 Þ ð87Þ
dt
k AA
AxA-propagation: Rn,A i, s B A m RnB1,iB1, s Here, only two of the three indices in-crease by 1. The length distribution is again calculated simply by taking the double sum over i and s which is identical to the length distribution term found in the pre-vious propagation case, Eq. (84):
y X
X y
d R n; i; s
s¼1 i¼1
y X
X y
A A
þ¼ kAA A R n
1; i
1; s ¼ kAA AR n
1 ; ð88Þ
dt s¼2 i¼2
For the first moment distribution of the number of A units, for this propagation step we have:
y X
X y
d iRn;A i; s s¼1 i¼1 dGnA Xy X y Xy
A A
¼ þ¼ kAA A iRn
1; i
1; s ¼ kAA A iRn
1; i
1 ; ð89Þdt dt s¼2 i¼2 i¼2 which again is equal to the one found before, Eq. (85). Hence we end up with an expression containing two one-dimensional distributions on the right-hand side:
dGnA A A
þ¼ kAA AðR n
1 þ Gn
1 Þ ð90Þ
dt
For this propagation step the situation for the first moment distribution of A se-quences is different, since we now have
y X
X y
d sR n; i; s i¼1 s¼1 dFnA Xy X y Xy
A A
¼ þ¼ kAA A sR n
1; i
1; s ¼ kAA A sR n
1; s; ð91Þdt dt i¼2 s¼2 s¼2 where in the last term the summation over s proceeds with a two-dimensional dis-tribution Rn
1; s simply having s as the index. This yields:
dFnA A
þ¼ kAA AFn
1 ; ð92Þ
dt
containing only one one-dimensional distribution.

9.6 Pseudo-distribution Approach 477
BxB propagation Rn,B i, s m RnB1,i, s Only the chain length index is increased. Using a similar argument to previously, we can say that the right-hand side expressions for both the number of A units and the number of A sequences moment distribu-tions will produce two-dimensional distributions as intermediate results having i and s as indices. Summation of these will finally produce expressions like Eq. (92):
dGnB B
dt
dFnB B
dt
kc
Termination by combination: Rn,A i, s B Rm,j, t m P nBm, iBj, sBt This is the reaction equation for termination between macroradicals with identical terminal groups.The balance equation describing the production of dead chains Pn; i; s is:dPn; i; s 1 X
n
1 X
i
1 X
s
1
A A
¼ k tAA ðR m; j; t R n
m; i
j; s
t Þ ð95Þdt 2 m¼1 j¼1 t¼1 The first moment distribution of the number of A units leads to:
y X
X y
d iPn; i; s ( )s¼1 i¼1 dLn 1 X n
1 Xy i
1 X
X y Xs
1
A A
¼ ¼ k tAA i ðR m; j; t R n
m; i
j; s
t Þdt dt 2 m¼1 i¼1 j¼1 s¼1 t¼1
1 X
n
1 Xy X
i
1
A A
¼ k tAA i ðR m; j R n
m; i
j Þ ð96Þ2 m¼1 i¼1 j¼1 The last term can be rearranged to give for the first moment of the number of A
units:
( ) 0 1
dLn 1 X
n
1 Xy X
i
1
1 X
n
1 Xy Xy
¼ k tAA A
ðR m; A @R A A A i j R n
m; i
j Þ ¼ k tAA m; j iR n
m; i
j dt 2 m¼1 i¼1 j¼12 m¼1 j¼1 i¼ j
1 X
n
1 Xy Xy
A A
¼ k tAA R m; j ði þ jÞR n
m; i 2 m¼1 j¼1 i¼0
1 X
n
1 Xy Xy Xy Xy
A A A A
¼ k tAA jR m; j R n
m; iþ R m; j iR n
m; i 2 m¼1 j¼1 i¼1 j¼1 i¼1
1 X
n
1 X
n
1
¼ k tAA ðGmA R n
m
A A A
þ Rm Gn
m Þ ¼ k tAA GmA R n
m
ð97Þ
2 m¼1 m¼1

An expression for the contribution from the termination reaction to the first mo-ment distribution of the number of A sequences is obtained in an identical
manner:
dCn X
n
1
¼ k tAA CmA R n
m
ð98Þ
dt m¼1
For the termination between B-terminated macroradicals the expressions obtained are exactly the same as Eqs. (97) and (98). For the reaction between A- and B-terminated macroradicals we find:
dLn X
n
1
þ¼ k tAB ðGmA R n
m
B A B
þ Rm Gn
m Þ ð99Þ
dt m¼1
dCn X
n
1
þ¼ k tAB ðCmA R n
m
B A B
þ Rm Cn
m Þ ð100Þ
dt m¼1
The consumption terms for the macroradicals corresponding to the termination reactions have simpler forms like the balance equation in three dimensions:dRn;A i; s X
n
1 X
i
1 X
s
1 X
n
1 X
i
1 X
s
1
þ¼
k tAA Rn;A i; s A
Rm; A
j; t
k tAB Rn; i; s
Rm; j; t
dt m¼1 j¼1 t¼1 m¼1 j¼1 t¼1¼
Rn;A i; s ðk tAA l0A þ k tAB l0B Þ ð101ÞFrom this the moment distribution expressions for the number of A units and for the number of A sequences easily follow as:
dFnA
þ¼
FnA ðk tAA l 0A þ k tAB l 0B Þ ð102Þ
dt
dGnA
þ¼
GnA ðk tAA l 0A þ k tAB l 0B Þ ð103Þ
dt
Once the pseudo-distributions have been solved, the average copolymer composi-tion nnA =n and sequence lengths lnA can be calculated with the expressions given in Table 9.19. Note that the average number of B units nnB can simply be derived from nnA , since nnB ¼ n
nnA . For large numbers of sequences per chain we may further assume that these are equal for both monomers: snB ¼ snA . Thus, the average sequence length for B; lnB, can also be inferred easily. Some illustrative calculations have been carried out for a batch copolymerization and the results (together with kinetic and reactor data) are shown in Figure 9.12.

9.6 Pseudo-distribution Approach 479
Copolymer composition
0.8 B
0.6
0.4 A
0.2
0 101 102 103 104 105 106 Chain length Sequence length
10-1
101 102 103 104 105 106 Chain length Fig. 9.12. Average copolymer composition and (kmole s)
1 ; cA0 ¼ 2 kmol m
3 ; cB0 ¼ 2 sequence length as functions of chain length kmol m
3 ; cI0 ¼ 10
2 kmol m
3 ; batch time:for batch copolymerization. Kinetic data: 400 s. Shorter chains (produced early in the kd ¼ 6:25  10
3 s
1 ; k i ¼ 3:75  10
4 m 3 batch) contain equal amounts of A and B with(kmol s)
1 ; k pAA ¼ 1000 m 3 (kmol s
1 ); sequence lengths of 2.5 on the average. Long k pAB ¼ 100 m 3 (kmol s)
1 ; k pAA ¼ 1000 m 3 chains (late in the batch) possess more B, with(kmol s)
1 ; k pBA ¼ 10 6 m 3 (kmol s)
1 ; sequences of up to 2000.k pBB ¼ 10 5 m 3 (kmol s)
1 ; kc ¼ 3  10 5 m 3

9.7
Probability Generating Functions
9.7.1
Introduction Probability generating functions (pgf ) are defined as polynomials in the transfor-mation variable z [where the coefficients p i represent a probability distribution(that is, a length distribution)]:
X
y
GðzÞ ¼ pi z i ð104Þ
i¼0
Distribution moments and coefficients can be obtained from GðzÞ according to:  !
qG q2G
m 0 ¼ Gð1Þ; m1 ¼ ; m2 ¼ ð105Þqz z¼1 qz 2
z¼1
1 q iG
pi ¼ ð106Þ
i! qz i
z¼0
We will discuss pgfs as used in a transformation procedure to solve population bal-ance equations [3], and as employed in the cascade theory of polymer networks[37–44].
9.7.2
Probability Generating Functions in a Transformation Method According to the transformation procedure, the population balance equations in terms of discrete variables (chain length, number of branch points) are trans-formed into a set of equations in z. The transformed set is solved and subsequently inverted to the original discrete variable domain. This process makes use of some interesting properties of certain mathematical expressions as transformed into the z-domain. Transformations and inversions are tabulated in textbooks [3]. As an ex-ample we take linear AB step polymerization in a batch reactor with equal initial end group concentration P0 :
k
Pn þ Pm ! Pnþm ð107Þ
dPn X
n
1 Xy
¼k Pm Pn
m
2kPn Pn ð108Þ
dt m¼1 n¼1
y
The total chain concentration is m 0 ¼ Pn , so by taking the zeroth moment of
n¼1
Eq. (106) we obtain Eq. (109), yielding:

9.7 Probability Generating Functions 481
dm 0
¼
kðm 0 Þ 2 ð109Þ
dt
m 0 ¼ P0 =ð1 þ ktP0 Þ ð110ÞSince by definition [Eq. (104)] Gð1Þ ¼ m 0 , while the convolution property prescribes:
X
n
1
G Pm Pn
m ¼ fGðPn Þg 2 ð111Þ
m¼1
Eq. (108) becomes:
dGðzÞ
¼ kfG 2 ðzÞ
2GðzÞGð1Þg ð112Þ
dt
By putting z ¼ 1, Eq. (110) for the total concentration m 0 is reproduced. To solve Eq. (112) we first realize that under initial conditions the pgf is given as: Gðz; 0Þ ¼P0 z; integration then leads to:GðzÞ ¼ P0 fGð1Þ=P0 g 2 z=½1
zf1
Gð1Þ=P0 g ð113ÞThis expression in the z-domain has a standard inverse form in the chain length domain, which represents a Flory distribution as the well-known solution to this
problem:
Pn ¼ P0 fGð1Þ=P0 g 2 f1
Gð1Þ=P0 g n
1 ð114ÞThis pgf transformation procedure is an elegant method that has found many ap-plications [3]. However, its ability to solve complete distribution problems is re-stricted to cases where explicit inversion is possible. In other cases only the main moments can be calculated, which then often can also be realized directly by the method of moments.
9.7.3
Probability Generating Functions and Cascade Theory The cascade theory has been developed to deal with problems concerning polymer network formation [37–44]. Consider the polymer network made by polycondensa-tion of f -functional monomers, represented as a rooted tree in Figure 9.13.Let end group conversion be given as a; then for a three-functional monomer the pgf of the connectivity between the zeroth and first generation is given by:F0 ðzÞ ¼ ð1
aÞ 3 þ 3að1
aÞ 2 z þ 3a 2 ð1
aÞz 2 þ a 3 z 3 ¼ ð1
a þ azÞ 3 ð115Þ

(1-α) 2z0 2α(1-α)z 1 α2z2 F (z) = (1-α+ αz) 2
Generation
(1-α) 3z0 3α(1-α) 2z1 3α 2(1-α)z 2 α3z3 F0(z) = (1-α+ αz)3 Fig. 9.13. Principle of probability generating functions and cascade theory for a three-functional polymer network.Likewise, the pgf for connectivity between first and second (and all subsequent)generations is given by:F1 ðzÞ ¼ ð1
a þ azÞ 2 ð116ÞIn general, for f -functional monomers we thus have:F0 ðzÞ ¼ ð1
a þ azÞ f
ð117Þ
F1 ðzÞ ¼ ð1
a þ azÞ f
1 From the connectivity pgfs at subsequent generation levels the branch point prob-ability distributions at a certain level can be inferred. For instance, in the example with f ¼ 3 at level 2 the branch point pgf reads as:F0 ðzÞ ¼ ð1
a þ aF1 Þ 3 ð118Þwhere F1 is given by Eq. (116); coefficients of z i in Eq. (118) represent the proba-bilities of finding i branch points at level 2. Thus, at an arbitrary level we have:G0 ðzÞ ¼ F0 ðF1 ðF1 ð. . . F1 ðzÞÞÞÞ ð119ÞFrom Eqs. (117), number- and weight-average chain lengths can be derived using Eqs. (105) [37], but a more elegant method is to construct self-consistent equations[45, 46] to solve the pgf. In the three-functional example above, the probabilities of a given node on an arbitrary generation level being connected to zero, one, or two

9.7 Probability Generating Functions 483
further nodes are expressed by the coefficients for z 0 ; z 1, and z 2 , respectively, in pgf GðzÞ. In addition, being connected to a further node implies the possibility of being connected to even further nodes. This possibility is described by the identical pgf GðzÞ. The connectivity to further nodes if connected to two nodes is expressed by G 2 ðzÞ, since statistics of these connectivities are identical, but independent.Note that the contributions of further connectivities are thus correctly described by the coefficients of the z terms. The self-consistent equation becomes:G ¼ ð1
aÞ 2 þ að1
aÞzG þ a 2 z 2 G 2 ð120ÞWe will illustrate this procedure on a simple linear step-polymerization ( f ¼ 2) to obtain the CLD and on single metallocene ethylene polymerization to find the 2D distribution of chain lengths and numbers of branch points.Linear step-polymerization is treated as above with f ¼ 2, yielding the self-consistent pgf equation:G ¼ ð1
aÞ þ azG ð121Þ
and hence:
G ¼ ð1
aÞ=ð1
azÞ ð122ÞApplying Eqs. (105) to Eq. (121) we find the number- and weight-average for the number of links between monomer units, which after adding 1 yields the familiar expressions for n n and nw already found by Flory [1]:
 
qG a 1
nn ¼ ¼ þ1¼
qz z¼1 1
a 1
a 2    ð123Þq G qG a 2a 2 1þa
nw ¼ ¼ 2
þ1¼
qz 2 z¼1 qz z¼1 1
a ð1
aÞ 1
a Series expansion in z of Eq. (122) yields for G:
X
y
G¼ ð1
aÞa i z i ð124Þ
i¼0
of which the coefficients represent a Flory distribution ð1
aÞa i
1 , realizing again that chain length is one more than the number of links. The derivation of the CLD/DBD 2D distribution for the metallocene system is based on the fact that all segments between branch points possess the same statistics. The pgf method is then applied to the connectivity of segments rather than monomer units as in the polycondensation problem. A pgf is constructed for the probability of finding con-nectivity points (branch points) on a segment going in a direction opposite to the growth direction. The self-consistent pgf equation follows from an argument in

which the probability of a segment having a branch point plays a central role. As-suming steady state and taking the (0; 0)-moment of the population balance equa-tions (35)–(40), we find this branching probability to equal:B kb k p; TDB tl 0 b¼ ¼ ¼ ð125ÞB þ m 0 þ l 0 ð2kb t þ 1Þk p; TDB l 0 þ kb þ 1=t A given segment is connected to an initiation point (hence not to a branch point)with probability 1
b, and it is connected to a branch point with probability b. In the case of a branch point the segment is connected to two further segments obey-ing the same statistics and hence pgf. Thus we have:G ¼ ð1
bÞ þ bzG 2 ð126Þwhich has a quadratic form this time, having Eq. (127) as the solution [45]:qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
1
4bð1
bÞz
G¼ ð127Þ
2b
This can be expanded to give:
X
y
ð2iÞ!G¼ b i ð1
bÞ iþ1 z i ð128Þ
i¼0
i!ði þ 1Þ!where the coefficients for z i represent the probability distribution of the numbers of branch points per molecule. Here, the factor ð2iÞ!=ði þ 1Þ! is called the Catalan number, which has already been derived by Flory [1] and represents the number of different ways a molecule with i branch points can be constructed. Since the num-ber of segments between branch points plus terminal segments per molecule is 2i þ 1, obtaining the length distribution further requires the length distribution of the segments to be taken into account. This should obey a Flory distribution, being dictated by the competition of propagation on the one hand, and termination by b-hydride elimination and TDB incorporation on the other. Hence the number-average segment length and probability distribution of segments with length ni
follow as:
 
kpM 1 ni
nns ¼ p n ¼ exp
ð129Þkb þ k p; TDB m¼0 þ 1=t nns nns From Eq. (129) the probability of finding polymers with i branch points (2i þ 1 segments) is easily inferred. The conditional probability of polymers with 2i þ 1 segments, all with a Flory distribution with average nns , of having total length n is known to be:
 
n 2i n
pðnjiÞ ¼ exp
ð130Þðnns Þ 2iþ1 ð2iÞ! nns

are to be described, typically involving concentrations more than six decades lower than the highest chain concentrations in the system, the numbers of molecules to be generated become excessive, even with primary polymer sampling. Tobita has achieved a great improvement in MC sampling in this respect by applying sam-pling based on weight-fraction instead of number-fraction distributions [11–15].We will now discuss this method as applied on mostly branched radical polymer-ization systems involving transfer to polymer, terminal double bond incorporation,recombination termination, and random scission for both continuous and batch reactors.
9.8.2
Weight-fraction Sampling of Primary Polymers: Batch Reactor, Transfer to Polymer A branched molecule is thought to be composed of primary polymers (pps) in which growth of linear chains has started from monoradicals or from secondary radical sites at other pps, created by transfer to polymer, and stopped by a certain termination mechanism. Primary polymers can possess one or more branch points at which other pps have been growing. Primary polymer sampling is generally based on the assumption of instantaneous growth of pps. In a batch reactor the length distribution of such pps is then determined by the kinetic conditions (con-version) at the ‘‘birth’’ time, y. The MC process starts with sampling such a birth time y, and under number-fraction sampling its length is sampled from the (usu-ally Flory) number-fraction distribution at y. Since molecules may start growing at any moment, we may (but must not necessarily) consider this first sampled pp as the first one of the whole molecule. The first pp may, by transfer to polymer reac-tions, receive one or several branch points in the remaining time between y and the batch end time c, and thus become attached to pps created at later birth times.These first, second, third and further pps may in their turn receive branch points and become attached to more pps at later stages. The MC process stops when the last pps no longer obtain branch points.A slightly modified approach does not regard the first sampled pp as the first one in the molecule. Instead it accounts for the possibility that the first sampled pp has started growing from a branch point created on a previously formed pp (be-tween 0 and y), rather than being initiated from a monoradical. Since in all cases the connectivity is based on the same values for the branching probability at the various birth times, the results are identical.Since most pps are present in short chains, this sampling procedure generates mostly short molecules; hence a very great many molecules have to be created to predict long CLD tails accurately. If, instead, one sampled monomer units on pps and was able to predict their connectivity to other parts of the molecule, then most molecules generated would have sizes around the maximum of the weight-fraction distribution. In other words, such a method is significantly more effective at find-ing long CLD tails. This is the rationale of weight-fraction sampling, since the probability of selecting a monomer unit at random from a pp population is propor-

9.8 Monte Carlo Simulations 487
tional to the length n of the pps. Since in that method a monomer unit is sampled at random in a molecule, it cannot always be on the first pp of the molecule. The MC process consequently proceeds in the second manner described above. Thefirst monomer unit sampled at y forms part of a pp in the zeroth generation, as do the pps to which it (eventually) is connected by branch points from it, on the one hand, and (eventually) to the pp on which it started growing on, on the other.The MC algorithm determines the birth times and lengths of pps in subsequent generations.We will demonstrate the MC procedure on a very simple radical polymerization process with transfer to polymer, while disproportionation is the only termination mechanism (see Figure 9.14). The algorithm employs conversion x, rather than time, as the independent variable. Consequently, it starts with sampling the birth conversion x ¼ y of a zeroth-generation pp by sampling a random number be-tween 0 and end conversion c. Its length is determined by sampling from the weight-fraction distribution (usually according to Flory):
 
n n
wn ðyÞ ¼ exp
ð132ÞnðyÞ 2 nðyÞThe average chain length nðyÞ is given by:nðyÞ ¼ k p MðyÞl 0 ðyÞ=k td l 0 ðyÞ 2 ¼ k p MðyÞ=k td l 0 ðyÞ ð133Þ
pp:
brith conversion z(1),0 < z(1) < z(0) pp with u(1)z(0) < u(1) < ψ branch points sampled with Eq. (138)with ρ (ψ,θ) from Eq. (137)
pp:
brith conversion z(0), initial unit 0 < z(0) < θpp: brith conversion θ,length from wn(θ), Eq. (132)pp with u(0)pp with u(1) θ < u(0) < ψz(0) < u(1) < ψ pp with u(1)u(0) < u(1) < ψpp with u(0)θ < u(0) < ψFig. 9.14. Monte Carlo sampling of primary polymers and their connectivity for radical polymerization with transfer to polymer in a batch reactor. Conversion c, first pp sampled at birth conversion x ¼ y.

Sampling can be realized by using the cumulative distribution of wn ðyÞ, a function of y; cwn ðyÞ, with values between 0 and 1, selecting a random number between 0 and 1 (randð1Þ), and finding y by requiring that cwn ðyÞ ¼ randð1Þ. A faster method utilizes the property that sampling twice from the number-fraction distribution and adding results exactly reproduces a weight-fraction distribution [49]. Sampling from nn can be performed rapidly using the simple formula:n ¼ ceil½nðyÞ lnf1=randð1Þg ð134Þwhere ceil denotes the value obtained when rounding a real number to the nearest higher integer, a standard operation available in most mathematical packages such as MATLAB. Doing this twice and adding the values generates a weight-fraction sample value. The probability of receiving branch points for this pp depends on its average branching density between y and c and is proportional to its now known length, n. The former can be derived from the monomer and branch points balance (Table 9.1):
dM dx
¼
k p l 0 ðtÞM ! ¼ k p l 0 ðtÞð1
xÞ ð135Þ
dt dt
dr
¼ k tp l 0 ðtÞ ð136Þ
dt
Combination and integration between y and c yields the average branching den-sity rðy; cÞ:
 
dr 1 1
y
¼ C tp ! rðy; cÞ ¼ C tp ln ð137Þ
dt 1
r 1
c
The number of branch points m on this pp is sampled from a binomial distribu-tion containing average branching density rðy; cÞ and pp length n:
 
pðmÞ ¼ r m rðn
mÞ ð138ÞSampling can be performed by employing standard random number generators for a binomial distribution (for example, in MATLAB: m ¼ binorndðn; rÞ). Next, to the pps attached at each of these branch points a birth conversion u, y < u < c, and a length must be assigned. The former should follow from the formation intensity distribution of branching density over the conversion interval y–c as given by Eq. (137). This implies that we should sample u from the conditional (given that this branch point exists at a pp formed at y) probability distribution, as expressed by:

9.8 Monte Carlo Simulations 489
   
1
y 1
y
CPa ðujyÞ ¼ ln ln ð139Þ
1
u 1
c
The sampling can be performed by selecting a random number between 0 and 1(randð1Þ) and inferring u from it by requiring that CPa (a function of u between 0 and 1) equals randð1Þ. The shape of Eq. (139) is such that on average u is chosen closer to c than to y. This agrees with the fact that branch formation intensity is higher at high conversion. The length of the pp grown at x ¼ u is sampled from the number-fraction distribution:
 
1 n
nn ðyÞ ¼ exp
; ð140Þ
nðyÞ nðyÞ
since a single chain end (instead of an arbitrary unit on the chain) is chosen at ran-dom as its starting point. Sampling is easily realized by employing Eq. (133). This eventually leads to a number of pps in generation 0 with specified birth conver-sions and lengths. Included in the procedure for this generation is the accounting for the eventuality of the first pp [pp(y)] being attached to a pp created earlier, at z,0 < z < y. This follows from the relative probability of the first pp of being initi-ated by a transfer-to-polymer reaction [Eq. (141)].Pb ¼ k tp l 0 m1 =ðk tp l 0 m1 þ 2kd I2 Þ ¼ k tp m1 =ðk tp m1 þ k td l 0 Þ ð141ÞThe latter equality follows from the quasi-steady-state-assumption. Note that Pb in a batch reactor is a function of conversion. If other transfer mechanisms are pres-ent, the denominator is extended with the corresponding contributions to the initi-ation process. Whether or not the pp is attached to another pp indeed follows by selecting a random number between 0 and 1 and determining whether the in-equality randð1Þ < Pb is false or true. If connected (true) then the birth conversion of the earlier pp simply follows from the conditional distribution (given that thefirst sampled pp is created at x ¼ y and grows from an earlier one):CPi ðzjyÞ ¼ z=y ð142ÞCPi is linear with z since from the perspective of the pp created at x ¼ z its proba-bility of undergoing transfer to polymer is simply linearly proportional to conver-sion. This implies that all values for the birth conversion between 0 and y are equally probable. Sampling can be performed in the same way as described for CPa [Eq. (139)]. Finally, the length of the pp grown at x ¼ u is sampled from the weight-fraction distribution wn ðzÞ, Eq. (132), since any of the monomer units in this pp can undergo branching. This then concludes the MC process at generation
0. If this generation has generated new pps attached in either way to the first one,
the generation number is increased by one. Note that pps attached alongside thefirst pp can only become attached to pps created at higher conversion. In contrast,