Handbook of Polymer Reaction Engineering

Handbook of Polymer Reaction Engineering Edited by Th. Meyer, J. Keurentjes Handbook of Polymer Reaction Engineering...

Author: Thierry Meyer | Jos Keurentjes

654 downloads 5474 Views 19MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Handbook of Polymer 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

Further Titles of Interest E. S. Wilks (Ed.)

Industrial Polymers Handbook Products, Processes, Applications 2000

ISBN 3-527-30260-3

H.-G. Elias (Ed.)

Macromolecules Vols. 1–4 2005

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 2005

ISBN 3-527-31092-4

M. Xanthos (Ed.)

Functional Fillers for Plastics 2005

ISBN 3-527-31054-1

R. C. Advincula, W. J. Brittain, K. C. Caster, J. Ru u˙˙he (Eds.)

Polymer Brushes 2004

ISBN 3-527-31033-9

H.-G. Elias (Ed.)

An Introduction to Plastics Second, Completely Revised Edition 2003

ISBN 3-527-29602-6

Handbook of Polymer Reaction Engineering Edited by Thierry Meyer, Jos Keurentjes

Editors Dr. Thierry Meyer Swiss Federal Institute of Technology Institute of Process Science EPFL, ISP-GPM 1015 Lausanne Switzerland Prof. Jos T. F. Keurentjes Process Development Group Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands

9 All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication 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

Foreword 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 practical, focused on using and exploiting scientiﬁc understanding and scientiﬁc principles 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 examples of polymeric products commercialized without adequate understanding of the chemistry and physics of the underlying polymerization. Polymer reaction engineers, faced with detriments in process safety, product quality or product cost, become the driving force behind many polymer science developments. As such, polymer reaction engineering is more a collaboration of polymer science and reaction engineering. A collaboration where polymer reaction engineers develop a ﬁrm understanding 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 polymer reaction engineering tools and methods. The very important topics of safety and process control are detailed and help frame the conditions through which successful 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-

VI

Foreword

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 engineering 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

Contents Volume 1 Foreword Preface

V XXIX

List of Contributors 1

1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.6 1.7

2

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1

XXXI

Polymer Reaction Engineering, an Integrated Approach Th. Meyer and J. T. F. Keurentjes Polymer Materials 1

1

A Short History of Polymer Reaction Engineering 4 The Position of Polymer Reaction Engineering 5 Toward Integrated Polymer Reaction Engineering 7 The Disciplines in Polymer Reaction Engineering 9 Polymerization Mechanisms 11 Fundamental and Engineering Sciences 12 The Future: Product-inspired Polymer Reaction Engineering Concluding Remarks 15 References 15 Polymer Thermodynamics Theodoor W. de Loos Introduction 17

14

17

Thermodynamics and Phase Behavior of Polymer Solutions 18 Thermodynamic Principles of Phase Equilibria 18 Fugacity and Activity 18 Equilibrium Conditions 20 Low-pressure Vapor–Liquid Equilibria 21 Flory–Huggins Theory and Liquid–Liquid Equilibria 21 High-pressure Liquid–Liquid and Vapor–Liquid Equilibria 25 Activity Coeﬃcient Models 29 Flory–Huggins Theory 30

VIII

Contents

2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.2.1 2.4.2.2 2.4.2.3 2.4.2.4 2.4.2.5 2.5

Hansen Solubility Parameters 32 Correlation of Solvent Activities 34 Group Contribution Models 35 Equation of State Models 39 The Sanchez–Lacombe Lattice Fluid Theory Statistical Associating-ﬂuid Theory 44 SAFT and PC-SAFT Hard Chain Term 44 SAFT Dispersion Term 45 The PC-SAFT Dispersion Term 46 SAFT and PC-SAFT Applications 47 Extension to Copolymers 48 Conclusions 50 Notation 52 References 54

3

Polycondensation 57 Ma´rio Rui P. F. N. Costa and Rolf Bachmann Basic Concepts 57 A Brief History 57

3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.1.4 3.3.1.5 3.3.1.6 3.3.1.7 3.3.2 3.3.2.1 3.3.2.2 3.3.2.3 3.3.3

40

Molecular Weight Growth and Carothers’ Equation 59 The Principle of Equal Reactivity and the Prediction of the Evolution of Functional Group Concentrations 62 Eﬀect of Reaction Media on Equilibrium and Rate Parameters 62 Polycondensation Reactions with Substitution Eﬀects 64 Exchange Reactions 66 Ring-forming Reactions 67 Modeling of Polymerization Schemes 68 Mass Transfer Issues in Polycondensations 69 Removal of Volatile By-products 69 Solid-state Polycondensation 80 Interfacial Polycondensation 82 Polycondensation Processes in Detail 85 Polyesters 85 Structure and Production Processes 85 Acid-catalyzed Esteriﬁcation and Alcoholysis 86 Catalysis by Metallic Compounds 87 Side Reactions in Aromatic Polyester Production 89 Side Reactions in the Formation of Unsaturated Polyesters 90 Modeling of Processes of Aromatic Polyester Production 91 Modeling of Processes for Unsaturated Polyester Production 92 Polycarbonates 93 General Introduction 93 Interfacial Polycondensation 94 Melt Transesteriﬁcation 96 Polyamides 98

Contents

3.3.3.1 3.3.3.2 3.3.3.3 3.3.3.4 3.3.4 3.3.4.1 3.3.4.2 3.3.4.3 3.3.5 3.3.6 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5

4

4.1 4.2 4.3 4.3.1 4.3.1.1 4.3.1.2 4.3.1.3 4.3.2 4.3.2.1 4.3.2.2 4.3.2.3 4.3.3 4.4 4.4.1 4.4.1.1 4.4.1.2 4.4.1.3 4.4.2 4.4.2.1 4.4.2.2 4.4.3 4.4.3.1

Introduction 98 Kinetic Modeling 98 Nonoxidative Thermal Degradation Reactions 100 Process Modeling 101 Polymerizations with Formaldehyde: Amino Resins (Urea and Melamine) and Phenolics 102 Formaldehyde Solutions in Water 102 Amino Resins 102 Phenolic Resins 107 Epoxy Resins 108 Polyurethanes and Polyureas 109 Modeling of Complex Polycondensation Reactions 113 Overview 113 Description of Reactions in Polycondensations of Several Monomers with Substitution Eﬀects 113 Equilibrium Polycondensations with Several Monomers 117 Kinetic Modeling of Irreversible Polycondensations 129 Kinetic Modeling of Linear Reversible Polycondensations 133 Notation 136 References 144 Free-radical Polymerization: Homogeneous Robin A. Hutchinson FRP Properties and Applications 153 Chain Initiation 154

153

Polymerization Mechanisms and Kinetics 156 Homopolymerization 157 Basic Mechanisms 157 Kinetic Coeﬃcients 161 Additional Mechanisms 169 Copolymerization 179 Basic Mechanisms 179 Kinetic Coeﬃcients 183 Additional Mechanisms 187 Diﬀusion-controlled Reactions 190 Polymer Reaction Engineering 193 Types of Industrial Reactors 195 Batch Processes 195 Semi-batch Processes 196 Continuous Processes 196 Mathematical Modeling of FRP Kinetics 197 Method of Moments 198 Modeling of Distributions 201 Reactor Modeling 203 Batch Polymerization 204

IX

X

Contents

4.4.3.2 4.4.3.3 4.4.3.4 4.4.3.5 4.5

Continuous Polymerization 204 Complex Flowsheets 205 Computational Fluid Dynamics (CFD) Model-based Control 206 Summary 206 Notation 206 References 209

5

Free-radical Polymerization: Suspension B. W. Brooks

5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.5 5.5.1 5.5.2 5.6 5.6.1 5.6.2 5.6.3 5.6.4

Key Features of Suspension Polymerization 213 Basic Ideas 213 Essential Process Components 214 Polymerization Kinetics 214 Fluid–Fluid Dispersions and Reactor Type 215 Uses of Products from Suspension Polymerization 216 Stability and Size Control of Drops 216 Stabilizer Types 217 Drop Breakage Mechanisms 218 Drop Coalescence 222 Drop Size Distributions 223 Drop Mixing 224 Events at High Monomer Conversion 228 Breakage of Highly Viscous Drops 229 Polymerization Kinetics in Viscous Drops 229 Consequences of Polymer Precipitation Inside Drops 230 Reaction Engineering for Suspension Polymerization 234 Dispersion Maintenance and Reactor Choice 234 Agitation and Heat Transfer in Suspensions 235 Scaleup Limitations with Suspension Polymerization 237 Reactor Safety with Suspension Processes 238 Component Addition during Polymerization 238 ‘‘Inverse’’ Suspension Polymerization 239 Initiator Types 239 Drop Mixing with Redox Initiators 240 Future Developments 240 Developing Startup Procedures for Batch and Semi-batch Reactors 240 Maintaining Turbulence Uniformity in Batch Reactors 242 Developing Viable Continuous-ﬂow Processes 242 Quantitative Allowance for the Eﬀects of Changes in the Properties of the Continuous Phase 242 Further Study of the Role of Secondary Suspending Agents 243 Further Characterization of Stabilizers from Inorganic Powders 243 Notation 243 References 244

5.6.5 5.6.6

205

213

Contents

6

6.1 6.2 6.2.1 6.2.2 6.2.3 6.3 6.4 6.4.1 6.4.2 6.4.3 6.4.3.1 6.4.3.2 6.4.3.3 6.4.3.4 6.5 6.5.1 6.5.1.1 6.5.1.2 6.5.2

Emulsion Polymerization 249 Jose´ C. de la Cal, Jose´ R. Leiza, Jose M. Asua, Alessandro Butte`, Guiseppe Storti, and Massimo Morbidelli Introduction 249 Features of Emulsion Polymerization 250 Description of the Process 250 Radical Compartmentalization 254 Advantages of Emulsion Polymerization 254 Alternative Polymerization Techniques 256 Kinetics of Emulsion Polymerization 258 Monomer Partitioning 259 Average Number of Radicals per Particle 260 Number of Polymer Particles 264 Heterogeneous Nucleation 264 Homogeneous Nucleation 266 Simultaneous Heterogeneous and Homogeneous Nucleation 267 Coagulative Nucleation 267 Molecular Weights 267 Linear Polymers 268 Zero–One System (Smith–Ewart Cases 1 and 2) 268 Pseudo Bulk System (Smith–Ewart Case 3) 270

Nonlinear Polymers: Branching, Crosslinking, and Gel Formation 272

6.6 6.7 6.7.1 6.7.1.1 6.7.1.2 6.7.1.3 6.7.1.4

Particle Morphology 273 Living Polymerization in Emulsion 275 Chemistry of LRP 275 Nitroxide-mediated Polymerization (NMP) 277 Atom-transfer Radical Polymerization (ATRP) 277 Degenerative Transfer (DT) 278 Reversible Addition–Fragmentation Transfer (RAFT) Polymerization 279

6.7.2 6.7.3 6.7.4 6.7.4.1 6.7.4.2 6.8 6.8.1 6.8.1.1 6.8.1.2 6.8.2 6.8.2.1 6.8.2.2 6.9 6.9.1

Polymerization of LRP in Homogeneous Systems 280 Kinetics of LRP in Heterogeneous Systems 282 Application of LRP in Heterogeneous Systems 284 Ab-initio Emulsion Polymerization 284 Miniemulsion Polymerization 285 Emulsion Polymerization Reactors 286 Reactor Types and Processes 286 Stirred-tank Reactors 286 Tubular Reactors 287 Reactor Equipment 288 Mixing 289 Heat Transfer 290 Reaction Engineering 290 Mass Balances 291

XI

XII

Contents

6.9.2 6.9.3

Heat Balance 292 Polymer Particle Population Balance (Particle Size Distribution) 294

6.9.4 6.10 6.10.1 6.10.1.1 6.10.1.2 6.10.1.3 6.10.1.4 6.10.1.5 6.10.1.6 6.11

Scaleup 295 On-line Monitoring in Emulsion Polymerization Reactors On-line Sensor Selection 297 Latex Gas Chromatography 298 Head-space Gas Chromatography 298 Densimetry 298 Ultrasound 299 Spectroscopic Techniques 299 Reaction Calorimetry 302 Control of Emulsion Polymerization Reactors 305 Notation 312 References 317

7

Ionic Polymerization 323 Klaus-Dieter Hungenberg Introduction 323

7.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.1.3 7.2.1.4 7.2.1.5 7.2.1.6 7.2.2

296

Anionic Polymerization 325 Anionic Polymerization of Hydrocarbon Monomers – Living Polymerization 326 Association Behavior/Kinetics 326 Molecular Weight Distribution of Living Polymers 331 Side Reactions 336 Copolymerization 338 Tailor-made Polymers by Living Polymerization – Optimization 341 Industrial Aspects – Production of Living Polymers 343 Anionic Polymerization of Vinyl Monomers Containing Heteroatoms 344

7.2.3

Anionic Polymerization of Monomers Containing Hetero Double Bonds 346

7.2.4 7.3 7.3.1 7.3.2 7.4

Anionic Polymerization via Ring Opening 346 Cationic Polymerization 351 Cationic Polymerization of Vinyl Monomers 351 Cationic Ring-opening Polymerization 353 Conclusion 356 Notation 357 References 359

8

Coordination Polymerization 365 Joa˜o B. P. Soares and Leonardo C. Simon

8.1 8.1.1 8.1.2

Polyoleﬁn Properties and Applications 365 Introduction 365 Types of Polyoleﬁns and Their Properties 366

Contents

8.1.3 8.2 8.2.1 8.2.2 8.2.3 8.3

The Importance of Proper Microstructural Determination and Control in Polyoleﬁns 369 Catalysts for Oleﬁn Polymerization 372 Ziegler–Natta, Phillips, and Vanadium Catalysts 378 Metallocene Catalysts 379 Late Transition Metal Catalysts 381 Polymerization Kinetics and Mechanism with Coordination Catalysts 383

8.3.1

Comparison between Coordination and Free-radical Polymerization 383

8.3.2 8.3.2.1 8.3.2.2 8.3.3 8.3.4 8.4 8.5 8.5.1 8.5.2 8.6 8.6.1 8.6.2 8.6.3 8.6.4

9

9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.5 9.5.1 9.5.2 9.5.3 9.6 9.6.1 9.6.2

Polymerization Kinetics with Single-site Catalysts 383 Homopolymerization 383 Copolymerization 388 Polymerization Kinetics with Multiple-site Catalysts 392 Long-chain Branch Formation 395 Single Particle Models – Mass- and Heat-transfer Resistances 399 Macroscopic Reactor Modeling – Population Balances and the Method of Moments 408 Homopolymerization 408 Copolymerization 413 Types of Industrial Reactors 416 Gas-phase Reactors 420 Slurry Reactors 422 Solution Reactors 423 Multizone Reactors 425 Notation 425 References 428 Mathematical Methods 431 P. D. Iedema and N. H. Kolhapure Introduction 431

Discrete Galerkin h–p Finite Element Method 432 Method of Moments 435 Introduction 435 Linear Polymerization 435 Nonlinear Polymerization 438 Comparison of Galerkin-FEM and Method of Moments 440 Classes Approach 444 Introduction 444 Computing the CLD of Poly(vinyl acetate) for a Maximum of One TDB per Chain 444 Multiradicals in Radical Polymerization 446 Pseudo-distribution Approach 449 Introduction 449 CLD/DBD for Mixed-metallocene Polymerization of Ethylene 451

XIII

XIV

Contents

9.6.2.1 9.6.2.2 9.6.3 9.6.3.1 9.6.3.2 9.6.3.3

Formulation of Pseudo-distribution Problem 451 Construction of the Full 2D Distribution 456 CLD/Number of Terminal Double Bonds (TDB) Distribution for Poly(vinyl acetate) – More than one TDB per Chain 458 General Case 458 TDB Pseudo-distribution Approach for a Maximum of one TDB per Chain 466 TDB Pseudo-distribution Approach for More than one TDB per Chain 467

9.6.4

Radical Polymerization of Ethylene to Low-density Polyethylene (LDPE) 469

9.6.4.1 9.6.5 9.6.5.1 9.6.5.2 9.7 9.7.1 9.7.2 9.7.3 9.8 9.8.1 9.8.2 9.8.3 9.8.4 9.8.5 9.8.6 9.8.7 9.8.8 9.8.9 9.8.10 9.8.11 9.9 9.9.1 9.9.2 9.9.3 9.9.3.1 9.9.3.2 9.9.3.3 9.9.3.4 9.9.4

Introduction 469 Radical Copolymerization 473 Introduction 473 Balance Equations 474 Probability Generating Functions 480 Introduction 480 Probability Generating Functions in a Transformation Method 480 Probability Generating Functions and Cascade Theory 481 Monte Carlo Simulations 485 Introduction 485 Weight-fraction Sampling of Primary Polymers: Batch Reactor, Transfer to Polymer 486 Example 490 CSTR with Transfer to Polymer 491 Comparison of Galerkin-FEM Classes Model and CSTR with Transfer to Polymer 492 Batch Reactor, Terminal Double Bond Incorporation 493 CSTR, Terminal Double Bond Incorporation 497 Incorporation of Recombination Termination 498 Incorporation of Random Scission, Linear Chains, Batch Reactor 498 Combined Scission/Branching 501 Scission in a CSTR 501 Prediction of Branched Architectures by Conditional Monte Carlo Sampling 502 Introduction 502 Branched Architectures from Radical Polymerization in a CSTR 503 Branched Architectures from Polymerization of Oleﬁns with Single and Mixed Branch-forming Metallocene Catalysts in a CSTR 505 Introduction 505 Single-catalyst System 505 Synthesis of Topology 505 Mixed-catalyst System 508 Mathematical Methods for Characterization of Branched Architectures 510

9.9.4.1

Graph Theoretical Connectivity Matrices

510

Contents

9.9.4.2 9.9.4.3 9.10 9.10.1 9.10.1.1 9.10.2

Characterization of Architectures by Radius of Gyration 511 Characterization of Architectures by Seniorities and Priorities 512 Computational Fluid Dynamics for Polymerization Reactors 517 Introduction 517 Modeling Challenges 517 Development and Optimization of Modern Polymerization Reactors 518

9.10.2.1 9.10.2.2 9.10.3 9.10.3.1 9.10.3.2 9.10.3.3 9.10.4 9.10.4.1 9.10.5

Beneﬁts of CFD 519 Limitations of CFD 519 Integration of CFD with Polymerization Kinetics 520 Classiﬁcation and Complexity of CFD Models 521 Treatment of Polymerization Kinetics 522 Illustration of Homogeneous Reactor Model Formulation Target Applications 523 Illustrative Case Studies 523 Concluding Remarks 528 Acknowledgments 530 References 530

10

Scaleup of Polymerization Processes E. Bruce Nauman

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15

Historic and Economic Perspective 533 The Limits of Scale 533 Scaleup Goals 534 General Approaches 535 Scaleup Factors 537 Stirred-tank Reactors 537 Design Considerations for Stirred Tanks 541 Multiphase Stirred Tanks 542 Stirred Tanks in Series 542 Tubular Reactors 543 Static Mixers 545 Design Considerations for Tubular Reactors 546 Extruder and Extruder-like Reactors 549 Casting Systems 549 Concluding Remarks 550 Notation 550 References 551

522

533

Volume 2 11

11.1 11.2 11.2.1

Safety of Polymerization Processes Francis Stoessel Introduction 553

553

Principles of Chemical Reactor Safety Applied to Polymerization Cooling Failure Scenario 554

554

XV

XVI

Contents

11.2.2 11.2.2.1 11.2.3 11.2.3.1 11.2.3.2 11.2.3.3 11.2.3.4 11.2.3.5 11.2.3.6 11.2.3.7 11.2.4 11.2.5 11.3 11.3.1 11.3.2 11.3.3 11.4 11.4.1 11.4.2 11.4.3 11.5

Criticality Classes Applied to Polymerization Reactors 557 Description of the Criticality Classes 558 Heat Balance of Reactors 559 Heat Production 559 Heat Exchange 560 Heat Accumulation 561 Convective Heat Transport due to Feed 561 Stirrer 561 Heat Losses 562 Simpliﬁed Expression of the Heat Balance 562 Dynamic Control of Reactors 562 Thermal Stability of Polymerization Reaction Masses 563 Speciﬁc Safety Aspects of Polymerization Reactions 564 Kinetic Aspects 564 Thermochemical Aspects 565 Factors Leading to Changing Heat Release Rates 568 Cooling of Polymerization Reactors 570 Indirect Cooling: Heat Exchange Across the Reactor Wall 570 Hot Cooling: Cooling by Evaporation 574 Importance of the Viscosity 578 Chemical Engineering for the Safety of Polymerization Processes 579

11.5.1 11.5.2 11.5.2.1 11.5.2.2 11.5.2.3 11.5.2.4 11.5.3 11.5.3.1 11.5.3.2 11.5.4 11.5.4.1 11.5.4.2 11.5.4.3 11.5.4.4 11.5.4.5 11.5.4.6 11.5.4.7 11.5.4.8 11.5.4.9 11.5.4.10 11.6

Batch Processes 579 Semi-batch Processes 580 Initiation 581 Feed 582 Final Stage 583 Practical Aspects 583 Continuous Processes 584 Concentration Stability 584 Particle Number Stability 584 Design Measures for Safety 585 Process Design 586 Reactor Design 586 Control of Feed 587 Emergency Cooling 587 Inhibition 588 Quenching 588 Dumping 588 Controlled Depressurization 588 Pressure Relief 588 Time Factor 589 Conclusion 589 References 590 Notation 591

Contents

12

12.1 12.1.1 12.1.2 12.2 12.2.1 12.2.1.1 12.2.1.2 12.2.1.3 12.2.1.4 12.2.2 12.2.3 12.2.4 12.2.5 12.2.6 12.2.7 12.2.8 12.2.9 12.2.10 12.2.11 12.3 12.3.1 12.3.2 12.3.3 12.3.4 12.4 12.4.1 12.4.2 12.4.2.1 12.4.2.2 12.4.2.3 12.4.2.4 12.4.2.5 12.4.2.6 12.4.3 12.4.3.1 12.4.3.2 12.4.3.3 12.4.3.4 12.4.3.5 12.4.3.6 12.4.3.7 12.4.3.8 12.4.4

Measurement and Control of Polymerization Reactors John R. Richards and John P. Congalidis Introduction 595 Deﬁnitions 595 Measurement Error 597 Measurement Techniques 598 Temperature 599 Resistance Thermometers 599 Thermocouples 600 Expansion Thermometers 601 Radiation Pyrometers 601 Pressure Measurement 602 Weight 604 Liquid Level 605 Flow 608 Densitometry, Dilatometery, and Gravimetry 617 Viscosity 619 Composition 620 Surface Tension 622 Molecular Weight Distribution (MWD) 622 Particle Size Distribution (PSD) 623 Sensor Signal Processing 625 Sensors and Transmitters 625 Converters 626 Indicators 626 Filtering Techniques 627 Regulatory Control Engineering 627 General 627 Process Dynamics 630 First-order System 631 Second-order System 632 High-order and Dead Time Systems 636 First-order Plus Dead Time System 636 Integrating System 638 Integrator plus Dead Time System 639 Controllers 639 Proportional Control 640 Integral Control 641 Derivative Control 641 PI, PD, and PID Control 642 Digital Controllers 642 Controller Tuning 644 On–Oﬀ Controllers 646 Self-operated Regulators 647 Valve Position Controllers 650

595

XVII

XVIII

Contents

12.4.5 12.4.6 12.4.7 12.5 12.5.1 12.5.1.1 12.5.1.2 12.5.2 12.5.3 12.5.4 12.5.5 12.5.6 12.5.6.1 12.5.6.2 12.5.7

Single-loop Controllers 650 Digital Control Systems 650 Actuators 652 Advanced Control Engineering 656 Feedforward Control 659 Steady-state Model Feedforward Control Ratio Control 660 Cascade Control 661 Feedforward–Feedback Control 663 State Estimation Techniques 666 Model Predictive Control 668 Batch and Semi-batch Control 669 Operation and Variability 669 Statistical Process Control 671 Future Trends 671 Notation 672 References 675

13

Polymer Properties through Structure Uday Shankar Agarwal

13.1 13.1.1 13.1.2 13.1.3 13.1.4 13.1.5 13.1.6 13.1.7 13.1.8

Thermal Properties of Polymers 679 Crystalline and Amorphous Polymers 680 Inﬂuence of Polymer Structure on Crystallizability of Polymers 682 The Glass Transition Temperature 683 Inﬂuence of Polymer Structure on Tg of Polymers 684 The Crystallization Temperature and the Melting Point 686 Tuning Polymer Crystallization for Properties 686 Morphology of Crystalline Polymers 688 Tailoring Polymer Properties through Modiﬁcation, Additives, and Reinforcement 690 New Morphologies through Block Copolymers 691 Polymeric Nanocomposites 692 Polymer Conformation and Related Properties 692 The Chain Conformation 692 Solubility of Polymers 694 Dilute Solution Zero-shear Viscosity 695 Polymers as Dumbbells 696 Polymers as Chains of Beads and Springs 697 Viscosity of Concentrated Solutions and Melts 698 Nonlinear Polymers 699 Rigid Rod-like Polymers 701 Polymer Rheology 702 The Viscous Response: Shear Thinning 702 Normal Stresses during Shear Flow 703 Extensional Thickening 705

13.1.8.1 13.1.8.2 13.2 13.2.1 13.2.2 13.2.3 13.2.3.1 13.2.3.2 13.2.4 13.2.5 13.2.6 13.3 13.3.1 13.3.2 13.3.3

660

679

Contents

13.3.4 13.3.4.1 13.3.4.2 13.3.5 13.3.5.1 13.3.6 13.3.7 13.3.8 13.3.9 13.4

The Elastic Response 706 Ideal Elastic Response 706 Rubberlike Elasticity 706 The Viscoelastic Response 707 Linear Viscoelasticity in Dynamic Oscillatory Flow 709 Inﬂuence of Polymer Branching Architecture in Bulk Polymers 711 Polymers as Rheology Modiﬁers 712 Rheological Control with Block Copolymers 714 Polymer-like Structures through Noncovalent Associations 715 Summary 715 Notation 716 References 718

14

Polymer Mechanical Properties Christopher J. G. Plummer Introduction 721 Long-chain Molecules 721

14.1 14.1.1 14.1.2 14.2 14.2.1 14.2.2 14.2.3 14.3 14.3.1 14.3.2 14.3.3 14.3.4 14.4 14.4.1 14.4.2 14.4.3 14.4.4 14.5

15

15.1 15.2 15.2.1 15.2.1.1 15.2.1.2 15.2.1.3 15.2.1.4 15.2.2

721

Simple Statistical Descriptions of Long-chain Molecules 722 Elasticity 724 Deformation of an Elastic Solid 724 Thermodynamics of Rubber Elasticity 725 Statistical Mechanical Approach to Rubber Elasticity 727 Viscoelasticity 729 Linear Viscoelasticity 729 Time–Temperature Superposition 734 Molecular Models for Polymer Dynamics 736 Nonlinear Viscoelasticity 740 Yield and Fracture 741 Yield in Polymers 741 Models for Yield 744 Semicrystalline Polymers 746 Crazing and Fracture 748 Conclusion 752 References 755 Polymer Degradation and Stabilization Tuan Quoc Nguyen Introduction 757

757

General Features of Polymer Degradation 759 Degradative Reactions 759 Initiation 760 Propagation 760 Chain Branching 761 Termination 762 Some Nonradical Degradation Mechanisms 763

XIX

XX

Contents

15.2.3 15.2.3.1 15.2.3.2 15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.3.4.1 15.3.4.2 15.3.5 15.3.6 15.4 15.4.1 15.4.2 15.4.3 15.4.4 15.4.4.1 15.4.4.2 15.4.4.3 15.4.4.4 15.4.4.5 15.4.4.6 15.4.5 15.4.6 15.4.6.1 15.4.6.2 15.4.6.3 15.4.6.4 15.4.6.5 15.5 15.5.1 15.5.2 15.5.3 15.5.4 15.5.5 15.5.5.1 15.5.5.2 15.5.5.3 15.6 15.6.1 15.6.2 15.6.3 15.6.4 15.6.4.1

Physical Factors 763 Glass Transition Temperature 764 Polymer Morphology 766 Degradation Detection Methods 767 Mechanical Tests 768 Gel Permeation Chromatography 771 Fourier Transform Infrared Spectroscopy 773 Magnetic Resonance Spectroscopy 775 Nuclear Magnetic Resonance (NMR) 775 Electron Spin Resonance (ESR) 776 Oxygen Uptake 776 Chemiluminescence 778 Thermal Degradation 778 Thermal Stability 779 Polymer Structure and Thermal Stability 779 Computer Simulation 780 Thermal Oxidative Degradation of Polypropylene 782 Initiation 782 Propagation 784 Chain Branching 785 Termination 786 Secondary Reactions 786 Formation of Volatile Compounds 788 Homogeneous versus Heterogeneous Kinetics 789 Applications of Thermal Degradation 790 Analytical Pyrolysis 790 Introduction of New Chemical Functionalities 791 Chemical Modiﬁcation of Polymer Structure 791 Metal Injection Molding (MIM) 792 Recycling 792 Photodegradation 793 Absorption of UV Radiation by Polymers 793 The Solar Spectrum 796 Photo-oxidation Proﬁle 796 Inﬂuence of Wavelength: the Activation and Action Spectrum Photodegradation Mechanisms 802 Photoinitiation 802 The Norrish Photoprocesses 803 Photo-Fries Rearrangement 803 Radiolytic Degradation 805 Interaction of High-energy Radiation with Matter 805 Radiation Chemistry 807 Radiolysis Stabilization 810 Applications 811 Radiation Sterilization 812

799

Contents

15.6.4.2 15.7 15.7.1 15.7.1.1 15.7.1.2 15.7.1.3 15.7.2 15.7.3 15.8 15.8.1 15.8.1.1 15.8.1.2 15.8.1.3 15.8.2 15.8.3 15.8.4 15.8.4.1 15.8.4.2 15.9 15.10

Controlled Degradation and Crosslinking 812 Mechanochemical Degradation 813 Initiation by Mechanical Stresses 813 Eﬀect of Tensile Stress on Chemical Reactivity 813 Breaking Strength of a Covalent Bond 814 Rate of Stress-activated Chain Scission 815 Extrusion Degradation 816 Applications 817 Control and Prevention of Aging of Plastic Materials Antioxidants 818 Radical Antioxidants 818 Hindered Amine Stabilizers (HAS) 819 Peroxide Decomposers 821 Photostabilizers 822 PVC Heat Stabilizers 823 Other Classes of Stabilizers 824 Metal Deactivators 824 Antiozonants 824 Lifetime Prediction 824 Conclusions 826 Notation 827 References 830

16

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 Introduction 833 Thermoset Materials 833 Networks 834 Advantages 835 Curing Resins 835 Functionality 835 Formulation 836 Production 837 General Areas of Application 837 Phenolic Resins 838 Introduction 838 Chemistry 838 Resols 840 Novolacs 840 Epoxy-novolacs 841 Discoloration 841 Production 842 Properties and Applications 842 Amino Resins 843

16.1 16.1.1 16.1.2 16.1.3 16.1.4 16.1.5 16.1.6 16.1.7 16.1.8 16.2 16.2.1 16.2.2 16.2.2.1 16.2.2.2 16.2.2.3 16.2.2.4 16.2.3 16.2.4 16.3

818

XXI

XXII

Contents

16.3.1 16.3.2 16.3.2.1 16.3.3 16.3.4 16.4 16.4.1 16.4.2 16.4.2.1 16.4.3 16.4.3.1 16.4.3.2 16.4.3.3 16.4.4 16.5 16.5.1 16.5.2 16.5.2.1 16.5.2.2 16.5.3 16.5.4 16.5.4.1 16.5.4.2 16.5.4.3 16.5.5 16.5.5.1 16.6 16.6.1 16.6.2 16.6.3 16.6.3.1 16.6.4 16.6.5 16.6.5.1 16.6.5.2 16.6.5.3 16.7 16.7.1 16.7.2 16.7.2.1 16.7.2.2 16.7.2.3 16.7.3 16.7.4 16.7.5

Introduction 843 Chemistry 843 Polymerization Chemistry 845 Production 848 Properties and Applications 849 Epoxy Resins 849 Introduction 849 Chemistry 850 Cure 851 Production 853 Standard Liquid 853 Taﬀy Process 854 Advancement Process 854 Properties and Applications 855 Alkyd Resins 855 Introduction 855 Chemistry 856 The Alkyd Constant 858 Autoxidative Drying 858 Production 859 Properties and Applications 861 Short Oil Alkyds 861 Long Oil Alkyds 861 Medium Oil Alkyds 861 Alkyd Emulsions 861 The Inversion Process 862 Saturated Polyester Resins 862 Introduction 862 Chemistry 863 Production 865 Monitoring the Reaction 865 Properties and Applications 866 Powder Coatings 866 Application 867 Crosslinking 868 Advantages 869 Unsaturated Polyester Resins and Composites Introduction 869 Chemistry 869 Crosslinking 871 Styrene Emission 871 Vinyl Ester Resins 873 Production 874 Reinforcement 875 Fillers 878

869

Contents

16.7.6 16.7.6.1 16.7.6.2 16.7.6.3 16.7.6.4 16.7.6.5 16.7.6.6 16.7.6.7 16.7.6.8 16.7.6.9 16.7.6.10

Processing 879 Hand Lay-up and Spray-up 882 Continuous Lamination 882 Filament Winding 882 Centrifugal Casting 882 Pultrusion 883 Cold-press Molding 883 Resin Infusion 883 Resin-transfer Molding 883 Hot-press Molding 883 Casting, Encapsulation, and Coating (Non-reinforced Applications) 886

16.7.7 16.8 16.8.1 16.8.2 16.8.3 16.8.3.1 16.8.3.2 16.8.3.3 16.8.3.4 16.8.3.5 16.8.3.6 16.8.3.7 16.8.3.8 16.8.3.9 16.8.4 16.8.5 16.8.5.1 16.8.5.2 16.8.5.3 16.8.5.4 16.8.5.5 16.8.5.6 16.8.5.7 16.9 16.9.1 16.9.1.1 16.9.2 16.9.3 16.9.3.1 16.9.3.2 16.9.3.3 16.9.4 16.9.4.1

Design Considerations: Mechanical Properties of Composites 886 Acrylate Resins and UV Curing 889 Introduction 889 Chemistry 890 Production 891 Epoxy Acrylates 891 Polyester Acrylates 891 Urethane Acrylates 892 Inside-out 893 Outside-in 894 Comparing Inside-out with Outside-in 894 Stabilization 894 Dilution 895 Safety 895 Properties 895 Introduction to UV Curing 896 General Introduction to UV-initiated Radical Polymerization 896 Photoinitiators for Radical Polymerization 897 Resin 897 Reactive Diluent 898 Curing Process 899 Cationic Curing 900 Base-mediated Curing 901 Rubber 901 Introduction 901 Types of Rubber 902 Polymerization 903 Crosslinking 904 Sulfur Vulcanization 904 Peroxide Curing 905 Processing 906 Properties and Applications 907 Advantages and Disadvantages 907

XXIII

XXIV

Contents

16.9.4.2

Thermoplastic Vulcanizates Notation 908 References 909

17

Fibers 911 J. A. Juijn

17.1 17.1.1 17.1.2 17.2 17.2.1 17.2.2 17.2.3 17.2.4 17.2.5 17.2.6 17.2.7 17.3 17.3.1 17.3.2 17.3.3 17.4 17.4.1 17.4.2 17.4.3 17.4.4 17.4.4.1 17.4.4.2 17.4.5 17.4.6 17.4.7 17.4.8 17.4.9 17.4.10 17.4.11 17.4.12 17.4.12.1 17.4.12.2 17.4.12.3 17.4.13 17.4.13.1 17.4.13.2 17.4.13.3 17.4.13.4

Introduction 911 A Fiber World 911 Scope of this Chapter 912 Fiber Terminology 912 Deﬁnitions: Fibers, Filaments, Spinning 912 Synthetic Yarns 914 Titer: Tex and Denier 914 Tenacity and Modulus: g denierC1, N texC1 , or GPa 915 Yarn Appearance 916 Textile, Carpet, and Industrial Yarns 917 Physical Structure 918 Fiber Polymers: Choice of Spinning Process 920 Polymer Requirements 920 Selection of Spinning Process 920 Spinnability 922 Melt Spinning 923 Extrusion 923 Polymer Lines and Spin-box 924 Spinning Pumps 925 Spinning Assembly 926 Filtration 926 Spinning Plate 926 Quenching 928 Finish 929 Spinning Speed 931 Winding 931 Drawing 931 Relaxation and Stabilization 934 Process Integration 934 Rheology 934 Shear Viscosity 934 Elasticity 936 Elongational Viscosity 936 Process Calculations 936 Mass Flow 937 Volume Flow 937 Extrusion Speed and Elongation in the Spin-line 937 Pressure Drop over the Spinning Holes 938

907

Contents

17.4.14 17.4.14.1 17.4.14.2 17.4.14.3 17.4.14.4 17.4.14.5 17.4.15 17.4.15.1 17.4.15.2 17.4.15.3 17.4.15.4 17.4.15.5 17.4.16 17.4.16.1 17.4.16.2 17.4.16.3 17.4.16.4 17.4.16.5 17.5 17.5.1 17.5.2 17.5.2.1 17.5.2.2 17.5.2.3 17.5.3 17.5.3.1 17.5.3.2 17.5.3.3 17.6 17.7 17.7.1 17.7.1.1 17.7.1.2 17.7.2 17.7.2.1 17.7.2.2 17.7.2.3 17.7.3 17.7.3.1 17.7.3.2 17.7.3.3 17.7.4

Polyester (Poly(ethylene terephthalate), PET) 938 PET Polymer 938 Spinning of PET 939 PET Staple Fiber 939 PET Textile Filament Yarns 940 PET Industrial Yarns 940 Polyamide (PA6 and PA66) 941 PA Polymer 941 PA Spinning 941 PA Staple Fiber 942 PA Textile Filament Yarns 942 PA Industrial Yarns 942 Polypropylene (PP) 943 PP Polymer 943 PP Spinning 943 PP Staple Fiber 943 PP Split Fiber 943 PP Filament Yarns 944 Solution Spinning 944 Preparation of Spinning Dope 944 Dry Spinning 944 Cellulose Acetate 945 Acrylics 946 Poly(vinyl alcohol) 946 Wet Spinning 946 Viscose Rayon 948 Acrylics 951 Poly(vinyl alcohol) 952 Comparison of Melt and Solution Spinning 953 High-modulus, High-strength Fibers 956 Air-gap Spinning 956 Aramids 956 Other Liquid-crystalline Polymers 960 Gel Spinning 961 Theory 961 Gel Spinning of Polyethylene 962 Other Gel-spun or Superdrawn Fibers 964 Carbon Fiber 965 Carbon Fiber from PAN 965 Carbon Fiber from Pitch 966 Applications of Carbon Fibers 966 Other Advanced Fibers 966 Notation 967 Acknowledgments 969 References 969

XXV

XXVI

Contents

18

18.1 18.2 18.2.1 18.2.1.1 18.2.1.2 18.2.1.3 18.3 18.4 18.4.1 18.4.1.1 18.4.2 18.4.2.1 18.4.2.2 18.4.2.3 18.4.2.4 18.4.3 18.4.4 18.5

19

19.1 19.1.1 19.1.2 19.2 19.3 19.4 19.5 19.6 19.7

20

20.1 20.2 20.2.1 20.2.2 20.2.3

Removal of Monomers and VOCs from Polymers Marı´a J. Barandiaran and Jose´ M. Asua Introduction 971 Polymer Melts and Solutions 972 Devolatilization 973 Fundamentals 973 Implementation of Devolatilization 975 Equipment 975 Polyoleﬁns 979 Waterborne Dispersions 979 Post-polymerization 980 Modeling Post-polymerization 981 Devolatilization 981 Modeling 982 Rate-limiting Steps 985

Devolatilization under Equilibrium Conditions Equipment 986 Combined Processes 988 Alternative Processes 989 Summary 989 Notation 990 References 991

971

986

Nano- and Microstructuring of Polymers 995 Christiane de Witz, Carlos Sa´nchez, Cees Bastiaansen, and Dirk J. Broer Introduction 995 Patterning Techniques 996 Photoembossing 998 Materials and their Photoresponsive Behavior 999 Single-exposure Photoembossing 1001 Dual-exposure Photoembossing 1007

Complex Surface Structures from Interfering UV Laser Beams Surface Structure Development under Fluids 1010 Conclusion 1012 Acknowledgments 1012 Notation 1013 References 1013 Chemical Analysis for Polymer Engineers 1015 Peter Schoenmakers and Petra Aarnoutse Introduction 1015 Process Analysis 1017 Near-infrared Spectroscopy 1017 In-situ Raman Spectroscopy 1018 At-line Conversion Measurements 1020

1007

Contents

20.3 20.3.1 20.3.1.1 20.3.2 20.3.2.1 20.3.2.2 20.3.2.3 20.3.3 20.3.3.1 20.3.3.2 20.3.3.3 20.3.3.4 20.3.3.5

Polymer Analysis 1022 Basic Laboratory Measurements 1022 Conversion 1022 Detailed Molecular Analysis 1023 FTIR Spectroscopy 1023 NMR Spectroscopy 1024 Mass Spectrometry 1025 Polymer Distributions 1030 Molecular Weight Distributions 1030 Functionality-type Distributions 1034 Chemical Composition Distributions (CCDs) 1037 Degree of Branching Distributions 1040 Complex Polymers (Multiple Distributions) 1041 Notation 1044 References 1045

21

Recent Developments in Polymer Processes Maartje Kemmere Introduction 1047

21.1 21.2 21.2.1 21.2.1.1 21.2.1.2 21.2.1.3 21.2.2 21.2.3 21.2.3.1 21.2.3.2 21.3 21.3.1 21.3.2 21.3.3 21.3.3.1 21.3.3.2 21.3.3.3 21.3.4 21.3.5 21.4

1047

Polymer Processes in Supercritical Carbon Dioxide 1048 Interactions of Carbon Dioxide with Polymers and Monomers 1050 Solubility in Carbon Dioxide 1051 Sorption and Swelling of Polymers 1052 Phase Behavior of Monomer–Polymer–Carbon Dioxide Systems 1054 Polymerization Processes in Supercritical Carbon Dioxide 1055 Polymer Processing in Supercritical Carbon Dioxide 1058 Extraction 1060 Impregnation and Dyeing 1061 Ultrasound-induced Radical Polymerization 1062 Ultrasound and Cavitation in Liquids 1063 Radical Formation by Cavitation 1065 Cavitation-induced Polymerization 1067 Bulk Polymerization 1067 Precipitation Polymerization 1069 Emulsion Polymerization 1070 Cavitation-induced Polymer Scission 1072 Synthesis of Block Copolymers 1073 Concluding Remarks and Outlook for the Future 1074 Acknowledgments 1076 Notation 1076 References 1077 Index

1083

XXVII

XXIX

Preface Freshly started as chairman and secretary of the Working Party on Polymer Reaction 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 engineering 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 institutions. 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 individual 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 activities 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 starting point for further reading. Reading books, you often read the preface as well. We have seen numerous examples from which the frustration is quite obvious. Of course things may not always 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

XXX

Preface

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

XXXI

List of Contributors P. Aarnoutse Polymer-Analysis Group Department of Chemical Engineering (ITS) Faculty of Science, University of Amsterdam Nieuwe Achtergracht 166 1018 WV Amsterdam The Netherlands Dr. U. S. Agarwal Polymer Technology Group Department of Chemical Engineering and Chemistry Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands Prof. J. M. Asua The University of the Basque Country Institute for Polymer Materials (POLYMAT) Paseo Manuel Lardizabal 3 20018 Donostia-San Sebastia´n Spain Dr. R. Bachmann Bayer AG ZT-TE-SVT 51368 Leverkusen Germany

Prof. M. J. Barandiaran The University of the Basque Country Institute for Polymer Materials (POLYMAT) Manuel Lardizabal, 3 20018 Donostia-San Sebastia´n Spain Dr. C. W. M. Bastiaansen Eindhoven University of Technology Den Dolech 2 5600 MB Eindhoven The Netherlands Prof. D. J. Broer Philips Research Laboratories Prof. Holstlaan 4 5656 AA Eindhoven The Netherlands and Eindhoven University of Technology Den Dolech 2 5600 MB Eindhoven The Netherlands and Dutch Polymer Institute (DPI) P.O. Box 902 5600 AX Eindhoven The Netherlands

XXXII

List of Contributors

Polymer Technology Department of Chemical Engineering and Chemistry Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands

Prof. J. C. de la Cal The University of the Basque Country Institute for Polymer Materials (POLYMAT) Paseo Manuel Lardizabal, 3 20018 Donostia-San Sebastia´n Spain

Prof. B. W. Brooks Loughborough University Department of Chemical Engineering Loughborough Leicestershire, LE11 3TU United Kingdom

Dr. A. J. de Koning DSM Research Oude Postbaan 1 6167 RG Geleen The Netherlands

Dr. A. Butte` Swiss Federal Institute of Technology Zurich, ETHZ Institut fu¨r Chemie- und Bioingenieurwissenschaﬀten Gruppe Morbidelli ETH Hoenggerberg/HCI F135 8093 Zurich Switzerland Dr. J. P. Congalidis E.I. du Pont de Nemours and Company DuPont Central Research and Development Experimental Station Wilmington, DE 19880 USA Dr. M. R. P. F. N. Costa Faculty of Engineering University of Porto Rua Roberto Frias, s/n 4200-465 Porto Portugal

Dr. T. W. de Loos Delft University of Technology Faculty of Applied Sciences Department Chemical Technology Julianalaan 136 2628 BL Delft The Netherlands C. de Witz Philips Research Laboratories Prof. Holstlaan 4 5656 AA Eindhoven The Netherlands Dr. L. J. Evers DSM Melamine Oude Postbaan 1 6167 RG Geleen The Netherlands Dr. A. Hoﬂand DSM Coating Resins Ceintuurbaan 5 8022 AW Zwolle The Netherlands Dr. K.-D. Hungenberg BASF AG Polymer Technology, B1 67056 Ludwigshafen Germany

List of Contributors

Prof. R. A. Hutchinson Department of Chemical Engineering Queen’s University Dupuis Hall, 19 Division St. Kingston, Ontario K7M 2G9 Canada

N. H. Kolhapure DuPont Engineering Research and Technology 1007 N. Market St. Wilmington, DE 19898-0001 USA

Dr. P. D. Iedema Department of Chemical Engineering University of Amsterdam Nieuwe Achtergracht 166 1018 WV Amsterdam The Netherlands

Prof. J. R. Leiza The University of the Basque Country Institute for Polymer Materials (POLYMAT) Paseo Manuel Lardizabal 3 20018 Donostia-San Sebastia´n Spain

Dr. J. F. G. A. Jansen DSM Research Oude Postbaan 1 6167 RG Geleen The Netherlands Dr. J. A. Juijn Research Institute Department QRI P.O. Box 9600 6800 TC Arnheim The Netherlands Dr. M. F. Kemmere Process Development Group Department of Chemical Engineering and Chemistry Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands Prof. J. T. F. Keurentjes Process Development Group Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands

J. Mattheij DSM Melamine Oude Postbaan 1 6167 RG Geleen The Netherlands Dr. T. Meyer Swiss Federal Institute of Technology Institute of Process Science EPFL, ISP-GPM 1015 Lausanne Switzerland L. J. Molhoek DSM Coating Resins Ceintuurbaan 5 8022 AW Zwolle The Netherlands Prof. M. Morbidelli Swiss Federal Institute of Technology Zurich, ETHZ Institut fu¨r Chemie- und Bioingenieurwissenschaften Gruppe Morbidelli ETH Hoenggerberg/HCI F135 8093 Zurich Switzerland

XXXIII

XXXIV

List of Contributors

Prof. E. B. Nauman The Isermann Department of Chemical and Biological Engineering Rensselaer Polytechnic Institute Troy, NY 12180 USA Dr. Q. T. Nguyen Laboratory of Polymers (LP) Ecole Polytechnique Fe´de´rale de Lausanne 1015 Lausanne Switzerland J. C. Plummer Laboratory of Composite and Polymer Technology (LTC) Ecole Polytechnique Fe´de´rale de Lausanne 1015 Lausanne Switzerland Dr. J. R. Richards E. I. du pont de Nemours and Company DuPont Engineering and Research Technology Experimental Station Wilmington, DE 19880 USA Dr. C. Sa´nchez Eindhoven University of Technology Den Dolech 2 5600 MB Eindhoven The Netherlands and Dutch Polymer Institute (DPI) P.O. Box 902 5600 AX Eindhoven The Netherlands

Prof. P. J. Schoenmakers Polymer-Analysis Group Department of Chemical Engineering (ITS) Faculty of Science, University of Amsterdam Nieuwe Achtergracht 166 1018 WV Amsterdam The Netherlands Prof. L. C. Simon Department of Chemical Engineering University of Waterloo 200 University Avenue West Waterloo, Ontario N2L 3G1 Canada Prof. J. B. P. Soares Department of Chemical Engineering University of Waterloo 200 University Avenue West Waterloo, Ontario N2L 3G1 Canada Prof. F. Stoessel Swiss Institute for the Promotion of Safety and Security Chemical Process Safety Consulting Klybeckstrasse 141 WKL-32.322 4002 Basel Switzerland Prof. G. Storti Swiss Federal Institute of Technology Zurich, ETHZ Institut fu¨r Chemie- und Bioingenieurwissenschaften Gruppe Morbidelli ETH Hoenggerberg/HCI F125 8093 Zurich Switzerland

List of Contributors

Prof. R. A. T .M. van Benthem Coating Technology Department of Chemical Engineering and Chemistry Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands

Dr. M. van Duin DSM Research Oude Postbaan 1 6167 RG Geleen The Netherlands

XXXV

1

1

Polymer Reaction Engineering, an Integrated Approach 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 consumption per capita has also increased over the years to a worldwide average of approximately 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 insulators for electricity and heat, cover a wide range of properties from soft packaging materials to ﬁbers stronger than steel, and allow for relatively easy processing.

25 120

Annual Production 106 to/an

20

100 Consumption, kg/hab

15

80 60

10 40 5

20

World Population , 109 people

0 1940 Fig. 1.1.

1950

1960

1970 1980 Year

1990

2000

0 2010

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 [10 3 tonnes]

Applications

LDPE

7935

PP

7803

PVC

5792

HDPE

5269

PET PS/EPS

3234 3279

PA

1399

pallet and agricultural ﬁlm, bags, toys, coatings, containers, pipes ﬁlm, battery cases, microwave-proof containers, crates, automotive parts, electrical components window frames, pipes, ﬂooring, wallpaper, bottles, cling ﬁlm, toys, guttering, cable insulation, credit cards, medical products containers, toys, housewares, industrial wrappings and ﬁlms, pipes bottles, textile ﬁbers, ﬁlm food packaging electrical appliances, thermal insulation, tape cassettes, cups and plates, toys ﬁlm for food packaging (oil, cheese, ‘‘boil-in-bag’’), hightemperature engineering applications, textile ﬁbers general appliance moldings transparent all-weather sheet, electrical insulators, bathroom units, automotive parts

ABS/SAN PMMA

788 317

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 major thermoplastics include high-density polyethylene (HDPE), low-density polyethylene (LDPE), polyethylene terephthalate (PET), polypropylene (PP), polystyrene (PS and EPS), poly(vinyl chloride) (PVC), polyamide (PA), poly(methyl methacrylate) (PMMA) and styrene copolymers (ABS, SAN). The most important applications 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 applications 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 Phenolics

3089 912

coatings, ﬁnishes, cushions, mattresses, vehicle seats general appliance moldings, adhesives, appliances, automotive parts, electrical components adhesives, automotive components, E&E components, sports equipment, boats

Epoxy resins

420

1.1 Polymer Materials Domestic 22.3% Automotive 7.0% Large industry 5.2% Electrical and electronics 7.3%

Building and construction 17.6%

Fig. 1.2.

Agriculture 2.5%

Packaging 38.1%

Plastic consumption in 2002 by industry sectors in Western Europe [1].

however, that about one-third of the market for thermosets is for relatively smallscale 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 reliability, safety, comfort, competitive pricing, fuel eﬃciency, and, increasingly, reassurance about the impact on the environment. Lightweight polymeric materials are increasingly used in this sector (Daimler Benz’s Smart is a nice example), also contributing to a 10% reduction in passenger fuel consumption across Europe. Building and construction Polymeric materials are used in the building and construction 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 holographic 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 polymers, 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 synthetic polymers consumed in Europe, they play only a marginal role. Irrigation and

3

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 socalled ‘‘super absorbers’’ for increased irrigation eﬃciency in arid areas can be considered an important emerging market.

1.2

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 cellulose, Hevea brasiliensis latex (natural rubber), and starch, were processed to manufacture 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 macromolecules was developed by Staudinger [2]. At the same time, new methods were developed 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 methodology that was further developed in the 1940s leading to more complex and comprehensive models, some of which are still being used today.

Tab. 1.3.

The history of polymers in brief.

19th century 1920 1930–1940

1940–1950 1950–1960

1960–1970 1970–1980 1980–1990 1990–2000 2000–

natural polymer and derivatives (vulcanized rubber, celluloid) concept of macromolecules postulated by Staudinger ﬁrst systematic synthesis of polymers synthesis of polyamides (nylon) by Carothers at DuPont discovery of polyethylene at ICI (Fawcett and Gibson) synthetic rubbers and synthetic ﬁbers stereospeciﬁc polymerizations by Ziegler and Natta, the birth of polypropylene discovery of polymer single crystals (Keller, Fischer, Till) development of polycarbonate discovery of PPO at GE by Hay and commercialization of PPO/PS blends (Noryl2 ) liquid-crystalline polymers superstrong ﬁbers (Aramid2 , polyethylene) functional polymers (conductive, light-emitting) metallocene-based catalysts; novel polyoleﬁns hybrid systems (polymer/ceramic, polymer/metals) Nature-inspired catalysts synthesis of polymers by bacteria and plants

1.3 The Position of Polymer Reaction Engineering

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 concepts 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 engineering (PRE). 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 development 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 followed by condensation outside the reactor. In the same period, polycarbonate and (somewhat later) poly(propylene oxide) (PPO) were developed. The main characteristic of the polymers developed so far was that they were bulk materials, to be produced in extremely large quantities. In the 1970s, a paradigm shift occurred when polymers with more speciﬁc properties 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 polymer (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 synthesis of quite complex polymers like natural silk. Also, bacteria and plants are being 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 scientiﬁc principles (from disciplines such as chemistry, physics, biology, and mathematics) and engineering knowledge (transfer of heat, mass, and momentum) to

5

6

1 Polymer Reaction Engineering, an Integrated Approach

Energy Raw material Safety

Natural gas

Coal, Oil

Capacity

Pollution control

Water

Process Scientific methodology

Quality control Air

Flexible production

Recycling Clean processes

Optimization

Green solvents

Intensification

From empiricism to strategy

1980

Renewable feedstock Integrated and inherent safety

Plant operation

Market

Fig. 1.3.

Less energy demanding processes

Integrated heat recovery

1990

Multidisciplinarity

2000

2010

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 requires a shift from heat loss minimization toward novel intensiﬁed process concepts 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 be ﬂexible 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 developments, 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, physics, and process technology. PRE problems are often of a multi-scale and multifunctional 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 information across the boundaries of diﬀerent scale regions and to provide a comprehensive 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 integration 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 Time scale System integration Environmental, Global modeling

Years Engineering design Process models

Days

Continuum Models Heterogeneous

Minutes

Phenomenological Models Microstructure

Milliseconds

Molecular level Elementary reactions Molecular modeling Chemical equilibrium

Nanoseconds

Picoseconds

Atomic level 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 directions, both toward a more fundamental understanding and at the same time toward 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 the ﬁeld, rather, of process or plant engineers. Today, it becomes obvious that only using a multidisciplinary, parallel, and synergetic approach can lead to successful developments. 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 disciplines 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 multidisciplinary approach. In general terms, PRE can be deﬁned as the science that

7

8

1 Polymer Reaction Engineering, an Integrated Approach

Materials sciences Thermodynamics

Novel processes

Inherent safety

Materials Application

Environment Recycling, Disposal

Modeling and simulation

Polymer Reaction Engineering (PRE)

Process integration optimization

New products

Polymer chemistry Reaction kinetics

Post-reaction processes

Novel processes

Nano-, MicroBio-

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 involved 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 minimizing 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].

Raw materials

Fig. 1.6.

Polymer Reaction Engineering

PRE as a black box process.

End use product

1.5 The Disciplines in Polymer Reaction Engineering

9

Sustainability

Materials sciences Thermodynamics

Raw materials

Novel processes

Inherent safety

Materials Application

Energies Needs

Environment Recycling, Disposal

Profit Modeling and simulation

Laws Economy

Satisfaction

Integrated PRE Knowledge

Process integration optimization

New products

Polymer chemistry Reaction kinetics

Post-reaction processes

Nano-, MicroBio-

Novel processes Measurement and control

Renewable

Fig. 1.7.

Products

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 different process and raw material alternatives, as well as the ﬁnal product requirements. LCA is an instrument driven by environmental considerations against a background of technical and economic speciﬁcations, and involves the so-called 3P 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 aspects 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

waste management

Specification - technical - economic - ecologic - safety

recycling

Balances

raw material

product use

- energy - material - emission - waste - sewage

synthesis

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 disciplines are interconnected by a synergetic and multidisciplinary approach, and com-

Academia Fundamental kinetics

Novel technologies

Thermodynamics

Molecular modeling

Measurement and control Reactor design

Environment

Polymer physics Polymer chemistry

Safety Quality assurance

Applied modeling

Process modeling

Market economics

Industry Fig. 1.9.

Overlap of industrial and academic disciplines.

1.5 The Disciplines in Polymer Reaction Engineering

11

Polycondensation

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 Figure 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 polymerization), macromolecules grow to full size in a much shorter time (seconds being the order of magnitude) than required for high monomer conversion. High molecular 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 monomers, with or without the formation of a condensation by-product (Chapter 3). Examples 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 condensation 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 nonlinear, depending on the number of functional groups per monomer. The polymerization process can be performed in bulk (liquid or solid state) or as an interfacial polymerization. Free-radical polymerization (FRP) can be performed homogeneously (in bulk, solution, or suspension; Chapters 4 and 5) or heterogeneously (emulsion, precipitation; 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 homolytic bond rupture of initiators (molecules sensitive to homolytic cleavage, such as peroxides, photosensitive molecules, or bisazo compounds) or by complex mechanisms 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 heterolytic process leads to charged parts of molecules that can induce the polymerization 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 reactions 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). Polyoleﬁns 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 research 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 provides information on the interaction between the various components present in the reaction mixture, from which phenomena such as phase behavior and partitioning can be derived. This information is usually accessible by using the appropriate equation of state for a given system studied. A close collaboration between chemical physicists, chemists, and chemical engineers is required to take full advantage 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

electron microscopy, chromatography, electrophoresis, viscometry, calorimetry, and laser diﬀraction are widely used to determine polymer properties, often in combination. 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 the ﬁnal 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 speciﬁcally for two application areas, namely the use of thermosets for coating applications (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 diﬀraction or electrical methods. New control schemes appear where the reaction is performed 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 consideration also. Modern process development intrinsically includes safety and environmental 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, semiempirical or empirical, descriptive or more fundamental. To describe the development 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 begins with a scaledown approach in order to have reliable and representative equipment already at the laboratory scale. Scaleup is always dependent on the system studied and requires a proper understanding of the performance of process equipment 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 simulation 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 product properties and legislation (FDA approval in the USA, for example). All the processes aiming to lower the residual VOC content in the product are denoted as ‘‘removable’’ (Chapter 18). These processes can diﬀer from each other, depending on

13

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 mechanic stresses are the main contributors to product degradation. Currently, there is a strong emphasis on the synthesis of novel functional polymers 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 implemented 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 industry, when product concepts are still being created, the rate of product innovation 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 polymer reaction engineering in the past. In recent discussions, however, it has been emphasized that a need exists to refocus chemical engineering toward productdriven 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 properties 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 followed by a morphological analysis [26]. Finally, a conceptual process design exercise is performed to generate possible process routes to achieve the desired product properties. This sequence of events is the core of product-inspired polymer reaction 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.

References

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 multidisciplinary 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 combination with tools such as life cycle analysis and product-inspired PRE, this will allow the development of sustainable new polymer processes.

References 1 Association of Plastics Manufacturers 2 3 4 5 6 7 8 9 10 11

12 13

14

in Europe, Annual report, 2002. H. Staudinger, Chem. Ber., 1920, 53, 1073. W. Kuhn, Berichte de Deutschen Chem. Gesellsch., 1930, 63, 1503. W. H. Chalmers, J. Am. Chem. Soc., 1934, 56, 912. H. Dostal, H. Mark, Trans. Faraday Soc., 1936, 32, 54. G. V. Schulz, Z. Physik. Chem., 1935, B30, 379. P. J. Flory, J. Am. Chem. Soc., 1936, 58, 1877. W. H. Carothers, US Patent 2 130 948, 1937. E. W. Fawcett, R. O. Gibson, J. Chem. Soc., 1934, 386. K. G. Denbigh, Trans. Faraday Soc., 1947, 43, 648. K. Ziegler, E. Holzkamp, H. Breil, H. Martin, Angew. Chem., 1955, 67, 541. P. W. Morgan, S. L. Kwolek, Macromolecules, 1975, 8, 104. H. Sasabe, T. Wada, Polymers for electronic applications, in: Comprehensive Polymer Science, vol. 7, S. L. Aggarwal (Ed.), Pergamon Press, Oxford, 1989. J. K. J. van Duren, J. Loos, F. Morrissey, C. M. Leewis, K. P. H.

15

16

17 18 19 20 21 22 23 24

25 26

Kivits, L. J. IJzendoorn, M. T. Rispens, J. C. Hummelen, R. A. J. Janssen, Adv. Funct. Mater., 2002, 12, 665. W. Kaminsky, H. Sinn (Eds.) Transition metals and organometallics as catalysts for oleﬁn polymerisation, Springer Verlag, Berlin, 1987. J. M. Benedikt, B. L. Goodall, Metallocene-catalyzed polymers, B.F. Goodrich, Brecksville, 1998. J. Sinke, Appl. Comp. Mater., 2003, 10, 293. E. R. Howells, Chem. Ind., 1982, 508. A. Penlidis, Can. J. Chem. Eng., 1994, 72, 385. A. Sapre, J. R. Katzer, Ind. Eng. Chem. Res., 1995, 34, 2202. A. Azapagic, Chem. Eng. J., 1999, 73, 1. P. Eyerer, J. Polym. Eng., 1996, 15, 197. W. J. Abernathy, J. M. Utterback, Technol. Rev., 1978, 80, 40. E. L. Cussler, G. D. Moggeridge, Chemical product design, Cambridge University Press, Cambridge, 2001. E. L. Cussler, J. Wei, AIChE J., 2003, 49, 1072. C. J. King, AIChE Monograph Ser., 1974, 70, 1.

15

17

2

Polymer Thermodynamics1 Theodoor W. de Loos 2.1

Introduction

The phase behavior of polymer solutions plays an important role in polymer production and processing. Many polymers are produced by solution polymerization. Solvent choice, solvent recovery and the removal of traces of solvent from the polymer product are important factors in these processes. An example is the production 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 conditions 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 polyethylene (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 pressures are higher than 200 MPa. Since the conversion of ethylene to LDPE is incomplete, 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 involve a high-pressure vapor–liquid ﬂash. From a thermodynamic point of view polymer solutions are complicated solutions. A polymer is not a single component but a multicomponent mixture characterized 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 example 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

18

2 Polymer Thermodynamics

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 thermodynamics 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 conditions given by Eq. (2) can easily be derived for a system of p phases and N components [7]. X

nai ¼ constant

ð1Þ

a

mai ¼ mib ¼ ¼ mpi

for i ¼ 1; 2; . . . ; N

ð2Þ

The 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 i @ A 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 component i in a mixture, fi , is deﬁned by Eq. (4a) together with (4b). C dmi 1 RT ln fi at constant T C fi lim ¼ 1 P!0 Pi

ð4aÞ ð4bÞ

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

C According to this deﬁnition, fi is equal to the partial pressure Pi in the case of an C ideal gas. The fugacity coeﬃcient fi is deﬁned by Eq. (5) and is a measure of the deviation from ideal gas behavior. C C fi fi ¼ Pi

ð5Þ

The fugacity coeﬃcient can be calculated from an equation of state by Eq. (6) or (7) [8]. C ð P RT dP ð6Þ Vi RT ln fi ¼ P 0 0X 1 " # n i RT ð V C @ A qP RT þ RT ln i RT ln fi ¼ ð7Þ qn V PV i V; T; nj0i y 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

ð8Þ

C The 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)]. C fi ðP; T; xÞ ai 1 0 fi ðP; T; x 0 Þ

ð9Þ

An ideal solution is deﬁned by Eq. (10). a iid 1 x i

ð10Þ

The 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

ai a iid

ð11Þ

C fi ¼ x i gi fi 0

ð12Þ

The activity coeﬃcient gi can be calculated from a model for the molar excess Gibbs energy g E, Eq. (13). 0 X 1 E q n g i B C @ A i ð13Þ RT ln gi ¼ qn i P; T; n j0i

19

20

2 Polymer Thermodynamics

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

ð14Þ

From Eq. (4a) we obtain Eq. (15), where viL is the molar volume of pure liquid i. fi L ðP; TÞ ¼ fi L ðPisat ; TÞ exp

ðP

dmiL Pisat RT

! ¼ fi L ðPisat ; TÞ exp

ðP

viL dP Pisat RT

! ð15Þ

Combining Eqs. (14) and (15), we get Eq. (16). L

fi ðP; TÞ ¼

fVi ðPisat ; TÞPisat

exp

ðP

viL dP sat RT Pi

! ð16Þ

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) C and ji with Eq. (6) or (7); x i and yi are the liquid-phase and vapor-phase mole fraction of component i, respectively. C C fi L ¼ fi V

ð18Þ

C x i gi fi ¼ yi ji P

ð19Þ

0

The truncated virial equation or the ideal gas equation is often used as the equation 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

ð20Þ

In 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

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. (21) reduces to Eq. (22). ðx i gi Þ 0 ¼ ðx i gi Þ 00

ð22Þ

For the calculation of high-pressure vapor–liquid equilibria or liquid–liquid equilibria an equation of state is always used for both phases and the equilibrium condition used is given by Eq. (23). C C fi a ¼ fi b

or

C C ðx i ji PÞa ¼ ðx i ji PÞb

or

C C 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

ð24Þ

P ¼ ð1 wp ÞWs Pssat

ð25Þ

The relation between a mole fraction based activity coeﬃcient gi and a weight fraction 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). N X wj gi wi ¼ ¼ Mi Wi x i M j j¼1

ð26Þ

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 RT

ð27Þ

21

22

2 Polymer Thermodynamics

The ﬁrst two terms on the right-hand side of Eq. (27) represent the so-called combinatorial entropy of mixing, and the third term is an approximation for the enthalpy 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).

js ¼

Ns Ns þ rNp

and jp ¼

rNp Ns þ rNp

ð28Þ

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.

wz

ð2esp ess epp Þ T

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 using 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 þ 1 jp þ jp2 w ln gs ¼ ln ð1 jp Þ þ r r

ð30Þ

ln gp ¼ ln½ð1 jp Þr þ jp ðr 1Þð1 jp Þ þ rð1 jp Þ 2 w

ð31Þ

As can be seen from Eq. (30), the activity coeﬃcient of the solvent is strongly dependent on r for low values of r, but at high values of r gs becomes practically independent 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 polymer. 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 0.40

0.50

r=1000 0.60

χ

r=100 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 temperature, 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 qjp

! ¼ P; T

! q2G ¼0 qjp2

ð32Þ

This leads to Eqs. (33) and (34) for the Flory–Huggins expression [Eq. (27)]. w

crit

1 1 2 1 þ pﬃﬃ ¼ 2 r

jpcrit ¼

1 pﬃﬃ 1þ r

ð33Þ ð34Þ

23

24

2 Polymer Thermodynamics

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 using 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 behavior is found for polystyrene–methylcyclohexane [14]; see Figure 2.5. Another type of LCST is connected to speciﬁc interactions like hydrogen bonding. 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.

T

L1+L2

L

L1+L2 Solvent

φp

Polymer

Liquid–liquid equilibria showing a low-temperature two-phase region with a UCST and a high-temperature twophase region with a LCST.

Fig. 2.2.

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

T

L1+L2

L

L

L1+L2 Solvent Fig. 2.3.

φp

Polymer

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 twophase 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 polymer 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 region. 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 decreases. Stronger polymer–polymer or solvent–solvent interactions lead to a lower mutual solubility, while stronger polymer–solvent interactions lead to a higher mutual solubility [15]. As discussed in Section 2.2.5, increasing molecular weight of

25

26

2 Polymer Thermodynamics

Liquid–liquid equilibria in the polystyrene–acetone system at the indicated polystyrene molecular weights (in g mol1 ). Reproduced with permission from Ref. 13.

Fig. 2.4.

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 pressure 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 pressures for the system ethylene þ LDPE [18, 19] are higher than for the system

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

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 Ref. 14.

ethane þ LDPE. For this type of systems it is not possible to make a clear distinction between vapor–liquid equilibria and liquid–liquid equilibria. The polymer-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 LCSTtype cloud-point curve to higher temperatures. An increased degree of branching of the polymer leads to a better mutual solubility [21].

27

28

2 Polymer Thermodynamics

T

L1+L2

φp

Solvent Fig. 2.6.

L

Polymer

Closed-loop liquid–liquid equilibria.

c

P b

a

a

T P; T phase diagram for a constant composition polymer–solvent system: (a) system with separate low- and hightemperature regions of demixing; (b) system in which the low- and high temperature regions of demixing have merged, showing a minimum

Fig. 2.7.

cloud pressure; (c) system in which the lowand high-temperature regions of demixing have merged, without a minimum cloud pressure. The full curves are cloud-point curves; the broken curves are liquid–liquid vapor curves.

2.3 Activity Coeﬃcient Models

Fig. 2.8. Liquid–liquid equilibria in the polystyrene–acetone system at the indicated pressures. M ¼ 20400 g mol1 . Reproduced with permission from Ref. 16.

The addition of an anti-solvent such as a supercritical gas also produces changes in the phase diagram to one like Figure 2.7 [22–25]. However, in this case the liquid–liquid–vapor equilibrium is represented by a region instead of by a curve.

2.3

Activity Coeﬃcient Models

In practice the original Flory–Huggins theory is not accurate enough for a quantitative representation of polymer–solvent phase equilibria. To improve this situation a concentration-dependent w-parameter can be introduced and also the temperature dependence of w can be made more complicated than in Eq. (29). The terms representing the combinatorial entropy of mixing, the ﬁrst two terms in Eq. (27), can be replaced by results from more accurate theories [4, 26] and the diﬀerence

29

30

2 Polymer Thermodynamics

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.

Fig. 2.9.

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 polydispersity of the polymer. The system-dependent parameters in these models can be adjusted to experimental 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 solvent, the Flory–Huggins expression for the Gibbs energy of mixing per mole of lat-

2.3 Activity Coeﬃcient Models

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, m P ji is the segment fraction of polymer component i, and jp ¼ ji ¼ ð1 js Þ is the i¼1 overall polymer segment fraction. m X Dmix G ¼ js ln js þ ji ri1 ln ji þ js jp g sp RT i¼1

ð35Þ

g sp ¼ f ðT; jp Þ

ð36Þ

To 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

b þ cT þ djp þ ejp2 T

c T g sp ¼ a þ 1 djp

ð37aÞ

bþ

ð37bÞ

31

32

2 Polymer Thermodynamics

aþ g sp ¼

b þ c ln T T 1 djp

ð37cÞ

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 polymer. ! qg sp 2 1 ð38Þ j lnðxs gs Þ ¼ ln js þ 1 j þ g sp js rn p qjp p ri jp qg sp 2 lnðx i gi Þ ¼ ln ji þ 1 j ri js þ ri g sp jp ð39Þ rn qjs i m X

rn ¼

n i ri

i¼1 m X

ð40Þ ni

i¼1

Using Eqs. (37) and (38), vapor–liquid and liquid–liquid equilibria can be calculated as discussed above. Reference 5 gives details of liquid–liquid equilibrium calculations in polydisperse polymer–solvent systems. In Figure 2.11 the experimentally 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 approximated 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 involved, the results of this model can be considered as only a rough estimate. Following the slogan ‘‘like dissolves like’’, a good solvent for a polymer is a solvent for which ds and dp have similar values. w¼

vs ðds dp Þ 2 RT

ð41Þ

2.3 Activity Coeﬃcient Models

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

vs ½ðds; d dp; d Þ 2 þ ðds; p dp; p Þ 2 þ ðds; hb dp; hb Þ 2 RT

ð42Þ

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

vs ½ðds; d dp; d Þ 2 þ 0:25ðds; p dp; p Þ 2 þ 0:25ðds; hb dp; hb Þ 2 RT

ð43Þ

33

34

2 Polymer Thermodynamics

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 fractions 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 equations is not very successful for describing the phase behavior of polymer–solvent systems with strong interactive species and of systems which only diﬀer in molecular weight. Since the mid-1990s activity coeﬃcient models have been proposed based on a combination of the Flory–Huggins type of expression for the combinatorial 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 nonrandom 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 solutions. In 2004, Pedrosa et al. [44] suggested use of the UNIQUAC theory [45] instead of the NRTL theory and tested various combinations of combinatorial contributions 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 produce 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

ð44Þ

The combinatorial/free-volume term is given by Eq. (45). ln gscombfv ¼ ln

fsfv f fv þ1 s xs xs

The free-volume fraction fsfv is calculated from Eq. (46).

ð45Þ

2.3 Activity Coeﬃcient Models fv

xiv fv fi ¼ X i fv xj vi

ð46Þ

j

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]. fv

vi ¼ ðvi vi; vdW Þ p

ð47Þ

vs vp

ð48Þ

p¼1

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. " gsres

2 tps Gps

tsp Gsp2

þ ðXs þ X p Gps Þ 2 ðX p þ Xs Gsp Þ 2 a ij tij ¼ exp and Gij ¼ expðatij Þ RT ln

¼

qs X p2

# ð49Þ ð50Þ

X i is the eﬀective mole fraction of segments of species i and qi is the eﬀective segment 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 xj q j

ð51Þ

j

1 qi ¼ ri 1 2a 1 ri

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

35

36

2 Polymer Thermodynamics

of experimental data. This makes the group contribution methods purely predictive. However, since details of molecular structure are not considered, group contribution 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

ð53Þ

The 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 conditions, this term is usually negligible. The residual contribution gsres accounts for energy interactions. In the approach of Oishi and Prausnitz, the combinatorial contribution 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 entropic 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 interaction 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 expression, Eq. (54). ln gicomb ¼ ln

fi f þ1 i xi xi

ð54Þ

The free-volume and residual terms are calculated from a modiﬁcation of the original Flory equation of state, Eq. (55), where ~v is the reduced volume, deﬁned by Eq. (56). P¼

RT v~1/3 þ C E attr v v~1/3 1 v

ð55Þ

2.3 Activity Coeﬃcient Models

v~ ¼

v v

ð56Þ

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)]. v ¼

X

x i vi

ð57Þ

x i Ci

ð58Þ

i

C¼

X i

The 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

Ci ¼

X

uðiÞ n

CT0 ; n þ CT; n

n

1 1 T T0

þ

X n

R X n Cn0 Rm

ð60Þ

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 E attr ¼

X1 i

2

X

yj expðDeji /RTÞDeji

6 6 j X zqi x i 6 4eii þ

3

7 7 7 yk expðDeki /RTÞ 5

k

1/3 v~ 1 X X ðiÞ xi un CT; n 3R ln v~ n i qi ¼

X

vðiÞ n Qn

ð61Þ ð62Þ

n

Q n is the normalized van der Waals surface of group n, as in UNIFAC. The interaction energy parameters eji and Deji are given by Eqs. (63), in which eij0 is calculated from a group contribution expression [Eq. (64), together with Eq. (65)]. eij ¼

eij0 v~

and

Deij ¼ eij eii

ð63Þ

37

38

2 Polymer Thermodynamics

eji0 ¼

X

yðiÞ m

X

m

yðn jÞ enm

ð64Þ

n

enm ¼ ðenn emm Þ 1/2 þ Denm

ð65Þ

In these expressions the volume fraction fi of molecule i, the segment fraction yi of molecule i, and the segment fraction yðiÞ n of n in molecule i are deﬁned by Eqs. (66)–(68). xiv fi ¼ X i xj vj

ð66Þ

j

x i qi yi ¼ X xj q j

ð67Þ

j ðiÞ

u Q Xn n yðiÞ n ¼ uðiÞ m Qm

ð68Þ

m

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 contribution in the activity coeﬃcient are Eqs. (69) and (70). ln

fv gi

ln gires

! 1/3 v~i 1 v~i Ci ln ¼ 3ð1 þ Ci Þ ln 1/3 v~ 1 v~ 2 X 1 4 1 ½eii ð~ ¼ zqi vÞ eii ð~ vi Þ þ 1 ln yj expðDeji /RTÞ 2 RT j 3 X yj expðDeji /RTÞ 5 X y expðDe /RTÞ k ki j

ð69Þ

ð70Þ

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

2.4 Equation of State Models

Fig. 2.12. Activity of benzene in the poly(propylene oxide)– benzene system at 347.85 K. Mn ¼ 500 kg mol1 . Symbols: experimental data; curves: UNIQUAC correlation and GC–Flory prediction. Reproduced with permission from Ref. 54.

coeﬃcients at inﬁnite dilution and predictions by the GC–Flory model [54] for homopolymer–solvent systems, demonstrating that the results of predictions are good. The predictions for copolymer solutions are slightly worse [55].

2.4

Equation of State Models

High-pressure phase equilibria in systems of polymers, solvents, and supercritical gases are in almost all cases modeled using equations of state. A review of equations of state for polymer systems, including a discussion of their theoretical background, has been given by Lambert et al. [6]. One of the ﬁrst equations of state that was used to model the high-pressure phase behavior of polymer–solvent systems was the Flory equation of state [11, 12]. Patterson and Delmas [10] showed that this equation of state can be used to describe both LCST and UCST phase behavior. The perturbed hard-chain theory (PCHT) was developed by Prausnitz and coworkers [57–59]. It can be considered as an improvement of the approach of Flory

39

2 Polymer Thermodynamics

log Ω∞cal

40

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 temperature [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 occupy 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

The resulting equation of state for a pure component is given by Eq. (71). 1 1 p~v~ 1 ¼ 1 þ v~ ln 1 ~ r v~ T v~T~

ð71Þ

The reduced volume v~, the reduced pressure p~, and the reduced temperature T~ are deﬁned by Eqs. (72a)–(72c). v~ ¼

v n0 þ rn ¼ v rn

ð72aÞ

p~ ¼

P Pv ¼ P e

ð72bÞ

kT T~ ¼ T

ð72cÞ

The parameters with an asterisk ( ) are the so-called characteristic parameters. In practice r, the number of segments per molecule, e , the interaction energy parameter, and v , the molar volume of a lattice site, are used as the independent purecomponent 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 parameters r; e , and v are composition-dependent. Neau [65] gives an overview of diﬀerent 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. r¼

X

v ¼ e ¼

ð73Þ

x i ri

X

ji vi

XX

ji jj

ð74Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃ eii ejj ð1 k ij Þ

ð75Þ

The segment fraction ji of component i is given by Eq. (76). x i ri ji ¼ X x i ri

ð76Þ

41

42

2 Polymer Thermodynamics

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

1 1 þ ln j^i ¼ ln z þ ri 2 ln 1 v~ v~T~ " # 1 nr qe v~T~ e qn i nj 0n i

z1 X xj r j

!"

nr qv v qn i

#

nj 0n i

ð77Þ

The compressibility factor z is given by Eq. (78). z¼

p~v~ Pv ¼ r RT T~

ð78Þ

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

10 9 8

P / MPa

7 6 5 4 3 2 1 0 0.00

450 K 470 K 490 K Critical Point Cloud Point Spinodal Critical Point 0.05

0.10

0.15

0.20

0.25

Weight fraction LLDPE Isothermal cloud-point curves of LLDPE þ n-hexane. Symbols: experiments; curves: modiﬁed Sanchez–Lacombe ﬁt. [66]. Fig. 2.15.

0.30

0.35

43

44

2 Polymer Thermodynamics

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 it ﬁrst 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 energy a res, which is the diﬀerence between the molar Helmholtz energy of the system 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

ð79Þ

From this expression the pressure and chemical potential can be derived [Eqs. (80) and (81), respectively] using standard thermodynamic relationships P (A ¼ n i a).

qA p¼ qV T; n qA mi ¼ qn i V; T; nj0i

ð80Þ ð81Þ

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. SAFT and PC-SAFT Hard Chain Term In the molecular picture behind SAFT a chain consists of mi hard-sphere segments. These hard-sphere segments are bonded by covalent bonds. The hardsphere 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 2.4.2.1

2.4 Equation of State Models

given by Eq. (82), where m is the average chain length of the molecules in the mixture [Eq. (83)] [64]. a hc a hs a chain ¼m þ RT RT RT m¼

X

x i mi

ð82Þ ð83Þ

The hard-sphere contribution a hs is represented by the Boublik–Mansoori hardsphere equation of state for mixtures of hard spheres, Eq. (84) [74, 75]. " # 3 a hs 6 z23 þ 3z1 z2 z3 3z1 z2 z32 z2 ¼ þ 2 z0 lnð1 z3 Þ RT pr z3 z3 ð1 z3 Þ 2

ð84Þ

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 parameters. p X x i mi ðdii Þ k zk ¼ r 6 3ei dii ðTÞ ¼ si 1 0:12 exp kT

ð85Þ ð86Þ

The chain term a chain is given by Eq. (87), where gðdii Þ hs is the so-called hardsphere radial distribution function at close contact [Eq. (88)] [74, 75]. a chain X ¼ x i ð1 mi Þ ln½gðdii Þ hs RT gijhs

didj d i d j 2 3z22 1 3z2 þ þ ¼ ð1 z3 Þ d i þ d j ð1 z3 Þ d i þ d j ð1 z3 Þ 3

ð87Þ ð88Þ

SAFT Dispersion Term In the Huang and Radosz version of SAFT [71, 72] the Chen–Kreglewski dispersion 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 constants Dij are given by Chen and Kreglewski [76]. 2.4.2.2

i j 4 X 9 a disp X u z3 Dij ¼ kT RT t i¼1 j¼1

ð89Þ

45

46

2 Polymer Thermodynamics

For mixtures, the van der Waals one-ﬂuid mixing rules or the volume fraction mixing 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 u ¼ kT

i

j

XX

uij 3 d kT ij ð90Þ

x i xj dij3

j

i

dij ¼

x i xj

di þ dj 2

ð91Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uij ¼ ð1 k ij Þ uii ujj

ð92Þ

The volume fraction mixing rules are given by Eq. (93), with volume fractions deﬁned by Eq. (94). XX uij u ¼ fi fj kT kT j i

ð93Þ

x i mi d 3i fi ¼ X xj m j d 3j

ð94Þ

j

The combing rule for uij is again given by Eq. (92). The PC-SAFT Dispersion Term In SAFT the dispersion term represents the interactions between individual segments, 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). 2.4.2.3

a disp a1 a2 ¼ þ RT RT RT

ð95Þ

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

2

u kT

2

s3

6 X i¼0

bi ðmÞh i

ð96Þ

ð97Þ

2.4 Equation of State Models

The reduced ﬂuid density h is deﬁned by Eq. (98), in which NAv is Avogadro’s number. h¼

pNAv rmd 3 6

ð98Þ

The parameters a i and bi are dependent on m, as Eqs. (99) and (100) state. a i ðmÞ ¼ a 0i þ

m1 ðm 1Þðm 2Þ a 1i þ a 2i m m2

ð99Þ

bi ðmÞ ¼ b0i þ

m1 ðm 1Þðm 2Þ b1i þ b2i m m2

ð100Þ

The constants a 0i –b2i are ﬁtted to the thermophysical properties of n-alkanes and are given by Gross and Sadowski [73]. SAFT and PC-SAFT Applications Both SAFT and PC-SAFT contain pure component parameters: the energy parameter e or u, the hard-sphere diameter s, or the hard-sphere volume v 00 , and the number 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 polydispersity 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 PCSAFT. 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 polymer 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 2.4.2.4

47

48

2 Polymer Thermodynamics

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 concentration. Extension to Copolymers For the modeling of the phase behavior of copolymer–solvent systems, the copolymer 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 disadvantage of this approach is that the pure component polymer parameters depend on the type and composition of the copolymer. Pure component polymer parameters 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 molecular weight and structure only. For an AB-type copolymer there are three binary interaction parameters, the interaction parameters between A segments and segments of the solvent molecule, the interaction parameter between B segments 2.4.2.5

2.4 Equation of State Models

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 be ﬁtted to some copolymer–solvent data. Once these parameters are known, predictions 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 PCSAFT. 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 modeling of the ethylene þ poly(E-co-EA) and ethane þ poly(E-co-BA) systems, respectively [85], at a polymer concentration of 5 wt.%. The model correctly predicts the change in the location of the cloud point with changing comonomer concentration 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.

49

50

2 Polymer Thermodynamics

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 adjustable parameters in these models are ﬁtted to experimental data, reliable predictions 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

2.5 Conclusions

Fig. 2.19. Constant composition cloud-point curves for poly(E-co-EA)–ethylene systems with diﬀerent repeat-unit compositions at 5 wt.% polymer. The copolymer molecular weight is in

the range 113–157 kg mol1 . Symbols: experimental data; curves: PC-SAFT calculations. Reproduced with permission from 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 promising models for high-pressure phase equilibria of polymer solutions are the SAFT and the PC-SAFT ones.

51

52

2 Polymer Thermodynamics

Fig. 2.20. Constant composition cloud-point curves for poly(E-co-BA)–ethylene systems with diﬀerent repeat-unit compositions at 5 wt.% polymer. The copolymer molecular weight is in

the range 155–283 kg mol1 . Symbols: experimental data; curves: PC-SAFT calculations. Reproduced with permission from Ref. 85.

Notation

Symbols a A C d DC f g G Gij G k k ij m M n

activity, molar Helmholtz energy Helmholtz energy number of external degrees of freedom temperature-dependent hard-sphere diameter constant fugacity molar Gibbs energy; interaction parameter; radial distribution function Gibbs energy NRTL parameter Gibbs energy per mole of lattice sites Boltzmann’s constant binary interaction parameter number of polymer components; number of segments molecular weight number of moles

Notation

N p P q Q r rn R Rm ; Rn T u v V w x; y X z

number of components correction factor pressure eﬀective segment number; surface area normalized van der Waals surface number of segments in a molecule number-average chain length gas constant normalized van der Waals volume absolute temperature energy parameter molar volume volume weight fraction mole fraction segment fraction lattice coordination number; compressibility factor

Greek a a; b; p g d e y jC f m W s t z h u w

NRTL nonrandomness parameter phase activity coeﬃcient (mole fraction basis) solubility parameter; density energy parameter theta temperature; segment fraction segment fraction; volume fraction fugacity coeﬃcient chemical potential activity coeﬃcient (weight fraction basis) temperature-independent hard-sphere diameter NRTL interaction parameter; constant in SAFT density-related variable [Eq. (85)] reduced density number of groups interaction parameter

Subscripts d hb i; j mix p s vdW

dispersion hydrogen bonding component mixing polymer; polar solvent van der Waals

53

54

2 Polymer Thermodynamics

Superscripts 0

standard state hard core, indicates characteristic parameter reduced association attraction chain formation contribution combinatorial critical dispersion excess free volume hard chain hard sphere ideal mixture ideal gas liquid residual saturated gas; vapor

@ assoc attr chain comb crit disp E fv hc hs id ig L res sat V Acronyms LCST LDPE LLDPE LLE NRTL PCHT poly(E-co-BA) poly(E-co-EA) SAFT S–L UCST VLE

lower critical solution temperature low-density polyethylene linear low-density polyethylene liquid–liquid equilibria nonrandom two-liquid perturbed hard-chain theory poly(ethylene-co-BA) poly(ethylene-co-EA) statistical associating-ﬂuid theory Sanchez–Lacombe lattice ﬂuid model upper critical solution temperature vapor–liquid equilibria

References 1 P. J. Flory, Principles of Polymer

Chemistry, Cornell University Press, USA, 1953. 2 H. Tompa, Polymer Solutions, Butterworth, UK, 1956.

3 H. Fujita, Polymer Solutions, Elsevier,

The Netherlands, 1990. 4 R. P. Danner, M. S. High, Handbook

of Polymer Solution Thermodynamics, DIPPR/AIChE, USA, 1993.

References 5 R. Koningsveld, W. H. Stockmayer,

6

7

8

9

10 11 12 13 14

15 16 17 18 19

20

21

22 23

E. Nies, Polymer Phase Diagrams, Oxford University Press, Oxford, 2001. S. M. Lambert, Y. Song, J. M. Prausnitz in Equations of State for Fluids and Fluid Mixtures, J. V. Sengers, R. F. Kayser, C. J. Peters, H. J. White (Eds.), Elsevier, The Netherlands, 2000, Chapter 14. J. M. Smith, H. C. Van Ness, M. M. Abbott; Introduction to Chemical Engineering Thermodynamics; 6th ed., McGraw-Hill, USA, 2001. J. M. Prausnitz, R. N. Lichtenthaler, Edmundo Gomes de Azevedo; Molecular Thermodynamics of Fluid Phase Equilibria, 3rd ed. Prentice-Hall Inc., Englewood Cliﬀs, USA, 1999. J. S. Rowlinson, F. L. Swinton, Liquids and Liquid Mixtures, 3rd ed., Butterworth, UK, 1982. D. Patterson, G. Delmas, Trans. Faraday Soc. 1969, 65, 708. P. J. Flory, R. A. Orwoll, A. Vrij, J. Am. Chem. Soc. 1964, 86, 3507. P. J. Flory, J. Am. Chem. Soc. 1965, 87, 1833. K. S. Siow, G. Delmas, D. Patterson, Macromolecules 1972, 5, 29. S. Saeki, N. Kuwahara, S. Konno, M. Kaneko, Macromolecules 1973, 6, 246. C. F. Kirby, M. A. McHugh, Chem. Rev. 1999, 99, 565. L. Zeman, D. Patterson, J. Phys. Chem. 1972, 76, 1214. P. Ehrlich, J. J. Kurpen, J. Polym. Sci. Pt. A. 1963, 1, 3217. P. Ehrlich, J. Polym. Sci. Pt. A. 1965, 3, 131. T. Swelheim, J. de Swaan Arons, G. A. M. Diepen, Recl. Trav. Chim. Pays-Bas 1965, 84, 261. Th. W. de Loos, J. de Graaf, J. de Swaan Arons, Fluid Phase Equilib. 1996, 117, 40. Th. W. de Loos, W. Poot, R. N. Lichtenthaler, J. Supercrit. Fluids 1995, 8, 282. M. A. McHugh, T. L. Guckes, Macromolcules 1988, 34, 9. E. Kiran, W. Zhuang, Y. L. Sen, J. Appl. Polym. Sci. 1993, 47, 895.

24 N. Koak, Th. W. de Loos, R. A.

25

26

27

28 29

30

31

32 33 34

35

36

37 38 39

40 41

42 43

Heidemann, Fluid Phase Equilib. 1998, 145, 311. M. H. ter Horst, S. Behme, G. Sadowski, Th. W. de Loos, J. Supercrit. Fluids 2002, 23, 181. N. Pedrosa, J. Gao, I. M. Marrucho, J. A. P. Coutinho, Fluid Phase Equilib. 2004, 219, 127. S. Vanhee, R. Koningsveld, H. Berghmans, K. Solc, J. Plolym. Sci. B., Polym. Phys. Ed. 1994, 32, 2307. R. Koningsveld, L. A. Kleintjens, Macromolecules 1971, 4, 637. Y. C. Bae, J. J. Shim, D. S. Soane, J. M. Prausnitz, J. Appl. Polym. Sci. 1993, 47, 1193. Th. W. de Loos, R. N. Lichtenthaler, G. A. M. Diepen, Macromolecules 1983, 16, 117. C. Qian, S. J. Mumby, B. E. Eichinger, Macromolecules 1991, 24, 1655. S. Enders, Th. W. de Loos, Fluid Phase Equilib. 1997, 139, 335. C. M. Hansen, J. Paint Technol. 1967, 39, 104. T. Lindvig, M. L. Michelsen, G. M. Kontogeorgis, AIChE J. 2001, 47, 2573. T. Lindvig, M. L. Michelsen, G. M. Kontogeorgis, Fluid Phase Equilib. 2002, 203, 247. A. F. M. Barton, Handbook of Solubility Parameters and Other Cohesion Parameters, CRC Press, USA, 1990. C.-C. Chen, Fluid Phase Equilib. 1993, 83, 301. H. Renon, J. M. Prausnitz, AIChE J. 1968, 14, 135. Y.-T. Wu, Z.-Q. Zhu, D.-Q. Lin, L.-H. Mei, Fluid Phase Equilib. 1996, 121, 125. M. G. Bawendi, K. F. Freed, J. Chem. Phys. 1988, 88, 2741. J. Dudowicz, K. F. Freed, W. G. Madden, Macromolecules 1990, 23, 4803. A. Haghtalab, J. H. Vera, AIChE J. 1988, 34, 803. M. T. Zafarani-Moattar, R. Sadeghi, Fluid Phase Equilib. 2002, 34, 803.

55

56

2 Polymer Thermodynamics 44 N. Pedrosa, J. Gao, I. M. Marrucho,

45

46

47 48

49 50

51 52

53

54 55 56

57 58 59 60 61 62 63

64

J. A. P. Coutinho, Fluid Phase Equilib. 2004, 219, 127. G. M. Kontogeorgis, P. Coutsikos, D. Tassios, A. Fredenslund, Fluid Phase Equilib. 1994, 92, 35. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968. T. Oishi, J. M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 1978, 17, 333. A. Fredenslund, J. Gmehling, M. L. Michelsen, P. Rasmussen, J. M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 1977, 16, 450. P. J. Flory, Discuss. Faraday Soc. 1970, 49, 7. F. Chen, A. Fredenslund, P. Rasmussen, Ind. Eng. Chem. Res. 1990, 29, 875. M. S. High, R. P. Danner, AIChE J. 1990, 36, 1625. H. S. Elbro, A. Fredenslund, P. Rasmussen, Macromolecules 1990, 93, 2927. G. M. Kontogeorgis, A. Fredenslund, D. Tassios, Ind. Eng. Chem. Res. 1993, 32, 362. G. Bogdanic, A. Fredenslund, Ind. Eng. Chem. Res. 1994, 33, 1331. G. Bogdanic, A. Fredenslund, Ind. Eng. Chem. Res. 1995, 34, 324. H. S. Elbro, A. Fredenslund, P. Rasmussen, Ind. Eng. Chem. Res. 1991, 30, 2576. S. Beret, J. M. Prausnitz, Macromolecules 1975, 6, 878. S. Beret, J. M. Prausnitz, AIChE J. 1975, 21, 1123. M. D. Donohue, J. M. Praunitz, AIChE J. 1978, 24, 849. D. D. Liu, J. M. Pruasnitz, Ind. Eng. Chem. Process Des. Dev. 1980, 19, 205. I. C. Sanchez, R. H. Lacombe, J. Phys. Chem. 1976, 80, 2352. I. C. Sanchez, R. H. Lacombe, Macromolecules 1978, 11, 1145. W. G. Chapman, K. E. Gubbins, G. Jackson, M. Radosz, Fluid Phase Equilib. 1989, 52, 31. W. G. Chapman, K. E. Gubbins, G. Jackson, M. Radosz, Ind. Eng, Chem. Res. 1990, 29, 1709.

65 E. Neau, Fluid Phase Equilib. 2002,

203, 133. 66 Th. W. de Loos, I. Nagy, R. A. Krenz,

67 68 69 70 71 72 73 74 75

76 77

78

79 80

81

82 83 84

85

R. A. Heidemann, Proceedings 7th International Symposium on Supercritical Fluids, Trieste, Italy, June 13–16, 2004, ISASF, 2004. K. Gauter, R. A. Heidemann, Ind. Eng. Chem. Res. 2000, 39, 1115. N. Koak, R. A. Heidemann, AIChE J. 2001, 47, 1219. M. S. Wertheim, J. Chem. Phys. 1987, 87, 7323. E. A. Muller, K. E. Gubbins, Ind. Eng. Chem. Res. 2001, 40, 2193. S. H. Huang, M. Radosz, Ind. Eng. Chem. Res. 1990, 29, 2284. S. H. Huang, M. Radosz, Ind. Eng. Chem. Res. 1991, 30, 1994. J. Gross, G. Sadowski, Fluid Phase Equilib. 2000, 168, 183. T. Boublik, J. Chem. Phys. 1970, 53, 471. G. A. Mansoori, T. W. Leland, N. F. Carnahan, K. E. Starling, J. Chem. Phys, 1971, 54, 1523. S. S. Chen, A. Kreglewski, Ber. Bunsenges Phys. Chem. 1977, 81, 1048. N. Koak, R. M. Visser, Th. W. de Loos, Fluid Phase Equilib. 1999, 158– 160, 835. F. Tumakaka, J. Gross, G. Sadowski, Fluid Phase Equilib. 2002, 194–197, 541. B. M. Hasch, M. A. McHugh, J. Polym. Sci. B 1995, 33, 715. S. H. Lee, B. M. Hasch, M. A. McHugh, Fluid Phase Equilib. 1996, 117, 61. M. Banaszak, C. K. Chen, M. Radosz, Macromolecules, 1996, 29, 6481. C. Pan, M. Radosz, Ind. Eng. Chem. Res., 1998, 37, 3169. C. Pan, B. Folie, M. Radosz, Ind. Eng. Chem. Res., 1999, 38, 2842. J. Gross, O. Spuhl, F. Tumakaka, G. Sadowski, Ind. Eng. Chem. Res., 2003, 42, 1266. F. Becker, M. Buback, H. Latz, G. Sadowski, F. Tumakaka, Fluid Phase Equilib. 2004, 215, 263.

57

3

Polycondensation1 Ma´rio 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 between 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) consisting of an entirely synthetic polymer [2]; its formation was also an example of polycondensation; but this early development was purely empirical and contributed 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 hydrogen atoms with epoxides and isocyanates do not lead to the splitting of a byproduct; he proposed a classiﬁcation into ‘‘step growth’’ and ‘‘chain growth polymerization’’. 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

58

3 Polycondensation Tab. 3.1.

A survey of technically important polycondensation reactions.

Functional groups or monomers

Connecting group

By-product

aCOOH Carboxylic acid

aOH hydroxyl

aCOOa ester

H2 O

aCOOR Ester

aOH hydroxyl

aCOOa ester

RaOH

aCOOH Carboxylic acid

aNH2 primary amine

aCONHa amide

H2 O

ClCOCl/aCOCl Phosgene/chloroformate

aOH hydroxyl

O

O C

HCl O

carbonate Na2 S/aSNa Di-sodium sulﬁde monomer/sodium sulﬁde end group

aCl chloro end group (as in ClC6 H4 Cl)

aSa sulﬁde

NaCl

NaOa Sodium alkoxide/phenoxide, e.g.,

aCl chloro end group, e.g.,

aOa ether

NaCl

NaO

CH3 C CH3

aNbCbO Isocyanate

Cl ONa

O S O

aOH hydroxyl

Cl

O

O C

none NH

urethane aNbCbO Isocyanate

aNH2 primary amine

O C NH

none NH

urea CH CH2 O

1,2-epoxide aCH2 OH Methylol

aNHa secondary amine

none

OH N CH2 CH

b-hydroxy-t-amine (NH2 )2 CO/NH2 CONHa urea/N-alkylurea

CH2NH

O C

H2 O NH

N,N 0 -dialkylurea AraCH2 OH Aromatic methylol

Ar 0 aCH2 OH aromatic methylol

AraCH2 aAr 0

HCHO

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 Basic Concepts

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 hydroxyl 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

ð1Þ

If 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 (comprising 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 byproduct formation, p MW , yielding Eq. (2), allowing the number-average molecular weight (mass of polymer/moles of polymer) to be predicted.

Mn ¼

MAXB pMW 1 p

ð2Þ

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).

Mn ¼

MAXA þ rMBYB 2pMW 1 þ r 2p

ð3Þ

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 polymerization 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.

Mn ¼

MAXA ð1 pc Þ þ ðr pc ÞMBYB 2ð p pc ÞMW 1 þ r 2ð p pc Þ

ð4Þ

59

60

3 Polycondensation

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.

Fig. 3.1.

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 negligible, except when ﬁve- or six-membered rings (or higher in the case of rings containing 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 irreversible 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 polycondensations 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 monomer units present in each ﬁnite polymer molecules (Eq. (5), where Px is the set of isomers with x repeating units X). x½Px /½X ¼

f ð fx xÞ! xp x1 ð1 pÞ xð f 2Þþ2 x!ð fx 2x þ 2Þ!

ð5Þ

3.1 Basic Concepts

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 x1

ð6Þ

This latter distribution has a relative maximum for x G x n, in contrast to the general 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 stoichiometry, and possible change of functionality due to side reactions, is essential to predict 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

61

62

3 Polycondensation

with monomers [8] or even with recycled polymer [9], by dilution with inert solvents and a suitable catalyst (see also Section 3.1.7). The discussion of these otherwise 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 reaction through use of high vacuum. Further research carried out by his former collaborator 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 reactivity. 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

ð7Þ

Equation (7) states that a single apparent second-order rate constant k should describe all intermolecular reactions of the involved groups A and B for a certain concentration 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 reaction 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

ð8Þ

In 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 formation of rings. If the end groups have polarities and/or hydrogen acceptor or do-

3.1 Basic Concepts

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 conversion or stoichiometric ratio. Known examples are the apparent increase in reactivity with conversion in esteriﬁcations 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, knowledge 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). K ¼ K0

gA gB ½Z½W ¼ g Z gW ½A½B

ð10Þ

This 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 contribution methods [12]. Multiple kinetic experiments with diﬀerent media compositions 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 transition temperature Tg is attained (the glass eﬀect). This situation occurs with many thermosetting systems of practical interest, epoxies being the most studied, as discussed in the reviews by Mita and Horie [15] and Dusˇek [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 covalent bonds by polymerization decreases the number of available statistical conﬁgurations of the chemical system, thus decreasing conﬁgurational entropy, suggesting Eq. (11) relating structural relaxation time tr to number-average degree of polymerization x n . tr ¼ t0 exp½b g ðTÞx n bg ¼

b g0 T T0

ð11Þ

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.

63

64

3 Polycondensation

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

ð12Þ

In Dusˇek’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 conversion, 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 esteriﬁcation 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 functional 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 terephthalic 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 k1

AXA þ BYB T AXZY B þ W k1Z k3

AXZ þ BYB T BYZX Z þ W k3Z

k2

AXA þ BYZ T AXZY Z þ W k2Z

k4

ð13Þ

AXZ þ BYZ T ZXZY Z þ W 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 ¼

ki kiZ ½W

ð14Þ

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 possible 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 þ ½ZA 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 sequences 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 distribution; 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 ¼ 2½AXZA 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 Þ RBYB ¼ 4k1 ½AXA½BYB 2k3 ½BYBð½ZD þ ½ZA Þ þ k Z ð½ZD þ ½ZB Þ RZD ¼ 4k1 ½AXA½BYB ½ZD ½2k2 ½AXA þ 2k3 ½BYB þ k4 ð2½ZD þ ½ZA þ ½ZB Þ k Z ð½ZD ½ZA ½ZB Þ

ð15Þ

RZA 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

65

66

3 Polycondensation

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 distributions 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

XA þ YC T XZ1 Y þ W k1Z

E k12

k2

XB þ YC T XZ2 Y þ W k2Z

ð16Þ

XA þ X 0 Z2 Y T XZ1 Y þ X 0 B E 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 functional 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). E k12 K1 ¼ E k21 K2

ð17Þ

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 reasoning 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

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 counterparts. 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) polycondensations 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 appreciable 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 character 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 factor for molecular weight increase becomes ring formation, not closeness to stoichiometry 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 reaction 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 cZ

ð18Þ

k

k aE

L m þ Cn ! L mþn cE

ð19Þ

k 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 exchange 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 concentration) k c and k cE should be related to the corresponding parameters for intermolecular relations through Eqs. (20)–(21). kjc ðnÞ ¼ kj k c ðnÞ

ð20Þ

67

68

3 Polycondensation

kijcE ðnÞ ¼ kijE k c ðnÞ k c ðnÞ ¼

1 ð3/2pnb 2 Þ 3/2 NAL

ð21Þ ð22Þ

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– Lo¨schmidt number. If there is only one kind of bond, an equilibrium constant of cyclization K c ðnÞ can be deﬁned from the equilibrium in Eq. (23). K c ðnÞ ¼

½Cn ½L m ¼ k c ðnÞ/n ½L nþm

ð23Þ

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 Sections 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 reactions starting from monomers can sometimes be described by the same solutions, 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 intramolecular 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

Reversible polycondensations can be tackled using the concept of molecular fragments 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 polycondensations 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 alternating polyesteriﬁcations and polyamidations) is nearly always fairly close to equilibrium, the main discrepancies occurring as far as the concentrations of the ﬁrst linear oligomers are concerned. So, a method which predicts those concentrations correctly is what is really needed in practice. The main factor controlling the CLD should be the presence of multifunctional impurities, which should be carefully 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 simulate 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 (devolatilization, 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 polycondensations, 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.

69

70

3 Polycondensation

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 product entrainment. Equilibrium constants are of the order of unity for polyesters and polycarbonates, 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 104 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 equilibrium solubility decreases, but even so partial pressures of some millibars for polyester 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 polyS 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. S PW ¼ HW wW ¼ PW WW wW

ð24Þ

The second part of Eq. (24) is useful below the critical temperature of the byproduct, 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). WW ¼

rP expð1 þ wÞ rW

ð25Þ

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. 0 1 W Yi ¼

N N Y þ1 f Y þ1 Y þ1 N Y þ1 X X X B C f NX rP Yj expB VYi þ fYi wij Yi yYi fYk wjk C @ A rYi V y Y Y j i j¼1 k¼ jþ1 j¼1 j¼1

ð26Þ

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 109 to 1011 m 2 s1 ), 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

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 columns. In either case, these reacting mixtures at the beginning of the processes resemble 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 mechanisms 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 being a common choice. They are also used in the ﬁnal stages of reversible polycondensations [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 remixes 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 s1 ) in the gaps between their tips and the wall. This high shear is favorable to conveying pseudoplastic, high-viscosity polymers, 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.

71

72

3 Polycondensation

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 considers polymer molecules immobile. The ﬂux of small molecules has a negligible convective 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 diﬀusive ﬂux 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 definitions [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 a ¼ kfYi ð½Yi ½Yi Þ ¼ kfY ðaYw aYi Þ N_ Yi ¼ DYi i qy jy¼0

ð28Þ

Combining this expression with the microscopic mass balances in the polymer ﬁlm, 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

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 0

½Yi jy¼0 ¼ ½Yi ;

q½Yi ¼0 qy jy¼L

DYi

ð tf

kfYi t f ¼

0

ð29Þ

q½Yi ð y; tÞ dt qy jy¼0

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 pt f

ð30Þ

kfYi ¼ 2

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 accepted). 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 y JT

Low M polymer inlet

Polymer outlet

z=0

Fig. 3.5.

Simple model for a WFR.

z

z=L y

73

74

3 Polycondensation

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 superﬁcial 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. uz

d½Yi ¼ RYi kfYi a v ð½Yi ½Yi Þ dz

uz

d½A i ¼ RA i dz

ð31Þ

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. tf ¼

Lx pd T n_

ð32Þ

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 concentrations of the species in the ﬁlm. There is no need to consider cylindrical geometry for the ﬁlm, since its thickness is low. q½Yi f qt

þ ux

q½Yi f qx

¼ DYi

q 2 ½Yi f qy 2 ðL

q½Yi q½Yi ux þ uz ¼ R Yi þ qt qz L

q½A i q½A i ux þ uz ¼ RA i þ qt qz L ½Yi jz¼0 ¼ ½Yi 0 ðtÞ

0

þ RYi

q½A i f qt

þ ux

q½A i f qx

¼ RA i

½Yi f ðt; yÞ dy u z ½Yi

ðL 0

½A i f ðt; yÞ dy u z ½A i

ð33Þ

½A i jz¼0 ¼ ½A i 0 ðtÞ

½Yi 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 qy

¼0 jy¼L

½Yi f ðt; 0Þ ¼ ½Yi

With some modiﬁcations, a model of single-screw vented extruders can also be developed. We will present here a slightly extended version of the treatment by Roberts [49] and Biesenberger and Sebastian [50]. Fundamental studies on ﬂuid mechanics and mass transfer without reaction are reported in Refs. 51 and 52.

3.2 Mass Transfer Issues in Polycondensations

Unwrapped views: 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 illustrating 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). ð34Þ

vT ¼ pd T n_ sin y

The 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

ð35Þ

t f ¼ ð1 fL Þ/n_

ð36Þ

The 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

75

76

3 Polycondensation

Assuming plug ﬂow in the channel (a trivial change would be to add an axial dispersion 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).

uz

kfpYi d½Yi vT h ¼ RYi þ ð½Yi ½Yi Þ ð½Yi f ½Yi Þ dz fL LW H fL LW

uz

d½A i vT h ¼ RA i þ ð½A i f ½A i Þ dz fL LW H

½Yi jz¼0 ¼ ½Yi 0

ð38Þ

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).

t fP ¼

H pd T n_ sin y

ð39Þ

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 penetration theory and experiments have been done with a transparent device (for observing whether bubbles were present or not in the devolatilization zone). Observed 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 predictions, 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 liquid hold-ups Vm j can be approximated as CSTRs. Taking into account the possibility of back-ﬂow, the overall volume ﬂow rate leaving the jth CSTR, Q j , will be divided 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 compartment ( j ¼ 1, J) may be written as in Eqs. (40).

3.2 Mass Transfer Issues in Polycondensations

Fig. 3.7.

Staged model of an RDC.

Q j1 ð1 b j1 Þ½A i j1 þ Q jþ1 b jþ1 ½A i jþ1 ¼ Q j ½A i j RA i Vm j Q j1 ð1 b j1 Þ½Yi j1 þ Q jþ1 b jþ1 ½Yi jþ1

ð40Þ

¼ Q j ½Yi j RYi Vm j þ kfYij avj Vm j ð½Yi j ½Yi Þ Mass transfer coeﬃcients can be computed using penetration theory as described above. 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 predicted from the mixing time. This dead space should obviously be kept to a minimum, 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 properties (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)]. av ¼

J 1:72 1:87H/d T LT 0:085 þ 0:955H/d T

ð41Þ

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 correlation 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_ ¼ 1:59 R Sh ¼ D D

ð42Þ

Falling-ﬁlm or falling-strand devolatilizators receive increasing attention as they require 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

77

78

3 Polycondensation

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 further 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 discrepancy has been linked to the presence of bubbles. The Laplace–Kelvin equation predicts that an isolated gas bubble should redissolve if its size is below a critical threshold, and conversely it should grow if its radius r b exceeds the critical value given by Eq. (43) [50], where s is the surface tension, 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. rbc ¼

2s Pb Pme

ð43Þ

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 heterogeneous 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 nor ﬁlm 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 efﬁciency. 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

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) polycarbonate. 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 15mQ g 3/4 4p 2rg 3 Q g d b 1/2 2gd b ub ¼ 1þ Cd u b Vb Vb ¼

Cd ¼

ð44Þ

16 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 exposure time was computed [Eq. (45)] as the rising time of a bubble in the melt (at height hR above the gas injection point). tf ¼

hR ub

ð45Þ

The interfacial area per unit volume was obtained from the number of bubbles Nb and the melt volume Vm [Eqs. (46)]. a v ¼ pdb2

Nb 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.

79

80

3 Polycondensation

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 temperature 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 particle 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 sufﬁciently 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 precrystallization 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

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 diﬀusioncontrolled regime. 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). DYie ¼ DYi

1 fcr tD

ð47Þ

Notice that volume and mass fractions wcr of the crystalline phase are diﬀerent because 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 crystalline phase, which aﬀects mass transfer and chemical reactions by increasing functional group concentrations in the amorphous phase. The rate of crystallization is often described by the Avrami equation [Eq. (48)]. wcr ¼ 1 expðkcr t ncr Þ

ð48Þ

The 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-

81

82

3 Polycondensation

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 components given in Eqs. (27) has to be modiﬁed in order to take into account the variable 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 particle ½A 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 fcr ¼ ‘ 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 successful 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 residence 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 solution (in alkaline media in both cases), and the other monomer, with low water solubility (acid chloride, phosgene), is usually dissolved in an organic solvent. Either 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

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 production. Also, the organic phase is continuous in this latter case, which is a rather exceptional 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 microcapsules [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 ‘‘interfacial’’ 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 consumption of water-soluble monomer and consequent rate of ﬁlm growth [83–87]. Although a possible framework for describing the simpler case of polycarbonate formation has been presented by Mills [88], the ﬁrst attempt at predicting molecular weight distributions for more typical interfacial polycondensation systems 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 m3 , the characteristic reaction time for consumption of monomer AXA is of the order of 104 to 102 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 1010 to 1011 m 2 s1 , and this yields a characteristic width of reaction zone LR ¼ 108 to 107 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, Karode 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 assumptions are given in Eqs. (50).

83

84

3 Polycondensation

[AXA]blk

[BYB]blk

[BYB]blk

He [AXA]blk [AXA]

[BYB]

Hi [AXA]

[BYB]

LP Swollen

aqueous

polymer

phase

film

Reaction zone

Fig. 3.9.

Bulk

LR

x Bulk organic phase

Model for interfacial polycondensation.

LR

d½AXA He ½AXAblk Hi ½AXA ¼ LR RAXA þ DAXA dt Lp

LR

d½BYB ¼ LR RBYB þ kf BYB ð½BYBblk ½BYBÞ dt

ð50Þ

d½AXAblk He ½AXAblk Hi ½AXA Vaq ¼ av; aq DAXA dt Lp Vorg

d½BYBblk 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 concentration 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 resistance is assumed to exist outside the polymer ﬁlm (although it can be easily included), 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

ing the rates of reaction of polymer species (see Section 3.4.4), their mass balances can be written as Eq. (51). IJ

d½Pn ¼ RPnIJ k nuc; n ð½PnIJ ½PnIJ sat Þ I; J ¼ A; B; C dt

ð51Þ

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 independent of molecular weight for n > 1) [93]. The ‘‘saturation’’ concentrations of polymer 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 earlier 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 nucleation k nuc , considered as an adjustable parameter, it also tries to ﬁt the diﬀusion coeﬃcient 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 complexity of the phenomena involved), this approach has given an important new insight on these processes. Film permeation properties should depend on the mode of phase separation, nucleation giving better crystallinity. Molecular weight depends 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 monomer 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 Structure and Production Processes Two kinds of polyesters will be discussed in this section: 3.3.1.1

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);

85

86

3 Polycondensation

Unsaturated low molecular weight (Mn between 1000 and 10 000), often branched, used as macromonomers for synthesis of thermosets (polyester resins), or thermosetting materials by themselves (alkyd resins). They are prepared from several monomers, namely phthalic and maleic anhydrides, adipic acid, isophthalic acid, natural fatty acids or triglycerides, and a great variety of multifunctional 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.

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 þ alcohol 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 established 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)]. 3.3.1.2

R COOH ¼ kscat ½OH½COOH 2

ðno foreign acidÞ

R COOH ¼ kfcat ½OH½COOH½Cat-H ðstrong foreign acid Cat-HÞ

ð52Þ

However, 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 reaction 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 Mare´chal [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 extend the validity of Eq. (52) to the whole range of concentrations of functional

3.3 Polycondensation Processes in Detail

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

ð53Þ

In 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 constant of the carboxylic acid on the dielectric constant of the medium and to the dependence 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 esteriﬁcations 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 Lehtonen et al. [103]. Zetterlund et al. [101] have also presented interesting experimental 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 reaction 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. Catalysis by Metallic Compounds Self-catalyzed esteriﬁcation is often too slow to be of practical use, especially because 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 methanesulfonic 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]: 3.3.1.3

Metals of groups II–III (such as Zn, Mn, Ce, Pb), usually introduced as carboxylates, 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 Ma´lek [105, 106] and Chung [107], who found volcano-shaped relationships with diﬀerent optima of activity of the several metals in terms of metal electronegativity according to Tanaka [108] for the glycolysis of dimethyl terephthalate

87

88

3 Polycondensation

(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 properties, because it is little inhibited by carboxyls and hydroxyls (which poison Sb) and also eﬃciently catalyzes esteriﬁcation. It is also active at surprisingly low concentrations [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 acidcatalyzed reactions. Alcoholysis of DMT, BHET, and other aromatic and aliphatic esters is now becoming better understood. A key step was to recognize that in systems such as DMT þ ethylene glycol (EG) there are two reversible alcoholysis reactions, 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 produces oligomers detected by HPLC [Eq. (54)]. kEG

HOCH2 CH2 OH þ XCOOCH3 ! XCOOCH2 CH2 OH þ CH3 OH kEG =KEG

kHE

ð54Þ

! 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 synergistic eﬀects [117], which cannot be explained unless polynuclear complexes participate in the reaction. It is noteworthy that, in the similar system dimethyl 2,6-naphthalenedicarboxylate þ 1,3-propanediol, the reactivity of hydroxyl groups in monomer and in hydroxypropyl chain ends is the same [118]. There are also studies for DMT þ 1,4-butanediol with Ti and divalent metal catalysts, for which an order of one with respect to Ti and the hydroxyl and ester

3.3 Polycondensation Processes in Detail

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 formation (see below). Models for Ti-catalyzed esteriﬁcation are still more complex. Titanates are hydrolyzed by water and form oligomeric a(RO, R 0 O)aTiOa structures (unless the hydroxyl excess is large). These structures are more active than the monomeric titanate [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, hydroxyl, 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 observed 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 esteriﬁcation and alcoholysis similar to Ti and Zr [123]. The SnaC bond is fairly stable, 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 possibility 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. 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 chromophoric products. Carboxyl end groups promote hydrolytic and thermal instability. 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 conditions 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): 3.3.1.4

O C O

O H C CH O CH2

Scheme 3.1.

OH C O

CH O

+

CH2

Vinyl end group formation from PET.

O C

89

90

3 Polycondensation

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 presence 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 absence of acid catalysts [126]. Reimschuessel [127] has suggested the attack of ethylene 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

O

O C OH

Scheme 3.2.

C

CH2 O O CH2

+

H O

+

CH2 CH2 OX

O C CH2 O CH2 CH2 O CH2 OX

Formation of DEG moieties in PET.

O C OCH2CH2 O OH

O +

C OH

CH2 CH2 O O CH2 CH2

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 production of PBT by direct esteriﬁcation of terephthalic acid and 1,4-butanediol disadvantageous. 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. 3.3.1.5

3.3 Polycondensation Processes in Detail

O H CH2 CH2 C O CH CH2 C O CH2 O

OH + C O

H2C CH CH CH2 Scheme 3.4.

CH2 O H2C CH CH2

+

HO

O C

O C

Vinyl end group formation from PBT.

O C O Scheme 3.5.

CH2 CH2 CH2 CH2 O

THF splitting-oﬀ from PBT.

HC CH R OH + COOR'' R'OOC Scheme 3.6.

OH + C O

CH2 CH2 CH2 CH2 HO

RO HC CH2 R'OOC COOR''

Ordelt reaction.

Modeling of Processes of Aromatic Polyester Production Continuous processes are currently used in the manufacture of PET. Several models have been developed, with the aim of contributing to a better design and operation. A brief discussion of their main assumptions and predictive capacities follows. 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 inﬂuence of variables, such as the initial stoichiometric ratio, reactor temperatures, and average residence times for CSTRs in series connected with distillation columns, 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 (Figure 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 3.3.1.6

91

92

3 Polycondensation

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 conversion and minimize side reactions by a choice of average reaction times, temperatures, and pressures. There are, however, no actual plant data to validate these conclusions. The kinetic model uses a ‘‘fragment’’ approach similar to what was recommended in previous sections, although without taking into account the inﬂuence of alcoholysis and acidolysis on monomer concentrations. Also, no thermodynamic model has been used for predicting the solubility of TPA, only an interpolation 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 esteriﬁcation and pre-polycondensation reactors, with very little change afterwards. Minimizing vinyl ester formation, thus improving molecular weight and product quality, 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. 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 order 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 predicting 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 consideration of the losses of volatile monomers (lack of reliable vapor–liquid equilibria data is a problem), and the aforementioned problems of taking into account the low solubility of isophthalic and terephthalic acids are also pending. 3.3.1.7

3.3 Polycondensation Processes in Detail

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 approximations 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 overcome 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 g1 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 instance, to fatty acid content. New compositions are often sought in order to improve 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 satisfy 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 accurate, 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 General Introduction Polycarbonates are polyesters of carbonic acid formed by reaction of diols (aromatic, 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 Hedenstro¨m in 1902 [156] obtained the same aromatic polycarbonates via transesteriﬁcation with diphenyl carbonate (DPC). Thus the main routes to polycarbonates were established early, but the properties of the products seemed uninteresting. 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 3.3.2.1

93

94

3 Polycondensation

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(4hydroxyphenyl)propane] by Schnell at Bayer [159, 160] and shortly afterwards by Fox at General Electric [161]. BPA polycarbonate (BPA-PC) proved to be an outstanding engineering thermoplastic that diﬀers from the other polyesters in that it is noncrystalline with a high glass transition temperature (about 150 C) and retains 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, inherent 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 commercially most important polycarbonate. The original processes – phosgenation in pyridine solution and melt transesteriﬁcation – were soon replaced by interfacial polycondensation with phosgene, which proceeds at low temperatures and allows the easy production of high molecular weight polymer. It still remains the predominant production process although interest in the simpler, non–phosgene-based and environmentally more attractive transesteriﬁcation process revived in the 1990s; problems with the earlier melt carbonates, in particular the poor resin color, could be overcome. Polycarbonate demand has enjoyed steady growth and total production capacity in 2003 was about 2.5 million tons per year. Approximately 12% is produced by transesteriﬁcation 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. Interfacial Polycondensation In the interfacial process, BPA and phosgene react at the boundary of two immiscible liquids, an aqueous alkaline BPA solution and an organic phase containing phosgene. The overall reaction is shown in Scheme 3.7. 3.3.2.2

CH3 C CH3

n NaO

Scheme 3.7.

ONa + n COCl2

-2n NaCl O

CH3 C CH3

O C O

n

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

*

...

*

...

*

...

m

m

m

O C Cl

+

O C Cl

+ NaO

O C Cl

+

Scheme 3.8.

...

*

n

Na

4 NaOH

*

...

*

...

*

...

m

m

m

O C

...

n

*

O

+ NaCl

+ NaCl

Na + NaCl + Na2CO3 + H2O

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 monochlorobenzene) 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% phenol, 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 react with the chloroformate end groups to form intermediate quaternary acylium salts which then react with phenolate to form carbonate and OH ions, hydrolyzing 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 reactions 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 precipitation, steam precipitation, spray drying, falling-strand devolatilization, and vacuum 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:

95

96

3 Polycondensation

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 diﬀerent phases and the partitioning of BPA between them. Monofunctional phenols with better solubility in the organic phase show a better eﬃciency as chain terminators. Mass transfer to and across the boundaries. Intrinsic (kinetic) reaction rates.

Due to the low reaction temperature and the use of chain terminator, the molecular 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 polycarbonate. Varying the ratio of mass transfer/kinetic rates, they show how mass transfer limitations can lead to a higher polydispersity or a higher oligomer content. Melt Transesteriﬁcation The melt process is based on the transesteriﬁcation of diphenylcarbonate (DPC) and BPA (see Scheme 3.9). 3.3.2.3

O n HO

OH + n

OCO

O O

OC

+ 2n

OH

n 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 process, however, requires high temperatures (190–320 C) so that the polymer remains molten and the viscosity can still be handled. With improved quality of the starting materials and improved high-viscosity reactors these temperatures no longer aﬀect polycarbonate quality and most grades can now be produced via the melt process. The transesteriﬁcation is a reversible reaction with an equilibrium constant close to unity, so that the by-product phenol has to be removed for the reaction to proceed. Pressures below 0.1 hPa may have to be applied for the ﬁnal stage.

3.3 Polycondensation Processes in Detail

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. Simple 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 equilibrium, 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 solubility of DPC and phenol in polycarbonate by Webb [153]. The liquid–gas equilibrium 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 assumption 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

ð55Þ

Mass 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 kinetically 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 beginning 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 reduced with increasing viscosity. Reactors are needed that provide a high rate of surface renewal. These may be wiped-ﬁlm evaporators, single- or twin-screw extruders, 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 reactors with disks that contain holes. Apart from surface renewal, mass transfer is de-

97

98

3 Polycondensation

termined by diﬀusion. Measurement of diﬀusion coeﬃcients is diﬃcult because of the parallel reaction. An alternative is the determination of diﬀusion constants using molecular dynamics simulations [148]. 3.3.3

Polyamides 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 ﬁbers and composites, which are produced by solution or interfacial processes from acid chlorides and amines. 3.3.3.1

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) complicate 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 3.3.3.2

3.3 Polycondensation Processes in Detail

expressed by Eq. (56), where bA and mA are assumed to be constant parameters; activity coeﬃcients are based on mole fractions. gCOOH gNH2 ¼ bA þ mA ½H2 O gCONH

ð56Þ

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 relationship into the mass action law provides a relationship between apparent equilibrium constant K a and true thermodynamic constant K 0 [Eq. (57)]. K a ¼ K 0 gH2 O ðbA þ mA ½H2 OÞ

ð57Þ

It 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). 3613 ðNylon 6,12Þ gH2 O ¼ exp 9:624 T gH2 O

2258 ðNylon 6,6Þ ¼ exp 6:390 T

ð58Þ

Choosing a reference temperature T0 ¼ 549 K, Eq. (57) is rewritten as Eq. (59).

K a ¼ K a0

K a0

1 þ g A ½H2 O DH 1 1 exp gH2 O /gH2 O ðT0 Þ R T T0

bA DH DS exp ¼ þ RT0 R gH2 O ðT0 Þ

ð59Þ

Parameters describing equilibrium (DH; K a0 , and g A ) have been ﬁtted simultaneously with a mass transfer time constant for water k m, appropriate to their experimental set-up, and activation energy Ec and pre-exponential factors kc0 or ku0 describing forward reaction through a third- or second-order rate law:

R CONH R CONH

Ec 1 1 ½H2 O½CONH ¼ kc0 exp ½COOH ½NH2 ½COOH Ka R T T0 Ec 1 1 ½H2 O½CONH ½NH2 ½COOH ð60Þ ¼ ku0 exp Ka R T T0

99

100

3 Polycondensation Tab. 3.2.

Equilibrium and rate parameters for nylon formation.

Parameter

Units

Estimate

95% conﬁdence interval

DH K a0 gA Ec kc0 km

kcal mol1 – g mmol1 kcal mol1 g 2 mmol2 h1 h1

1.82 63.1 2:03 102 22.6 3:19 104 24.3

G1.42 G6.2 G0:45 102 G7.7 G0:71 104 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 mol1 h1 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 writing. 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 mol1 , whereas aminolysis has the much smaller activation energy of 13 kcal mol1 . 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 because 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 caprolactam 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]. 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 3.3.3.3

3.3 Polycondensation Processes in Detail

Scheme 3.10) [193, 194]. A slow deamination reaction has also been shown to occur. X

X

HO C N OH O C

CH2

CH2

N

X CH2 k2

H2C

CH2 CH2 Scheme 3.10.

N

O C

- H2 O

CH2

H2C

C

O

k1

CH2 - CO 2 CH2

C H2C H2C

CH2 CH2 CH2

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 k1 O

NH Y O

O C X NH

k2

C

O

CH2

CH2

C

k3

- CO2

N

Scheme 3.11.

X N

k4 + 2 NH2

Z

Y

+ NH2

XNH

CH2 CH2 X

- H2O

Z

Z

+ 2 NH3

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. Process Modeling Recent progress in kinetic and reactor modeling make it possible to assist the reactor 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 problem 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 3.3.3.4

101

102

3 Polycondensation

the autocatalytic nature of the caprolactam hydrolysis, back-mixing increases conversion. A simple and widely used reactor is the VK column (simpliﬁed continuous column), 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, consisting in one or two CSTRs in series, followed by a plug-ﬂow reactor. Other designs [200] have improved performance by preventing water evaporation 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 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. 3.3.4.1

HCHO + H2O (CH2O)xOH + HOCH2OH HCHO + CH3OH (CH2O)xOH+ HOCH2OCH3 Scheme 3.12.

HOCH2OH (CH2O)x+1OH + H2O HOCH2OCH3 (CH2O)x+1OCH3 + CH3OH

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. Amino Resins Reaction between a water solution of formaldehyde with urea, melamine, and similar molecules (such as acrylamide) leads to hydroxymethylation of the nitrogens 3.3.4.2

3.3 Polycondensation Processes in Detail

[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 melamine (1). XaNH2 þ HCHO ! XaNHaCH2 OH

ðbÞ

XaNHaCH2 OH þ YaNH2 ! XaNHaCH2 Y þ H2 O

ðcÞ

NH2 N H2N

N N NH2

1 Scheme 3.13.

Melamine.

Since melamine is made from urea and ammonia, it is more expensive. Melamine 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. Laminated sheets (tables and counter tops) are a major application for melamine resins, which stay in the outer decorative surface. Molding compounds, their ﬁrst big application, 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 employed, 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, formaldehyde, 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 proposed an FSSE model for the early stage of this polycondensation. A modiﬁed version of that model introduces the groups presented in Scheme 3.14, where their ﬁve 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 negligible. 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 bimolecular, which seems to be reasonable. Available experimental data could not decide 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-

103

104

3 Polycondensation

U0 =

O C

H2N

O C

U3= H2N F0 = H

O C

NH2

U1 =

U4 = N

H2N

O C

U2 =

NH-

-HN

O C NH-

O C -HN

N

F2 = -CH2-

F1 = -CH2OH 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.

U 0 + F0

k1 →

←  k

U 1 + F1 + W

k6 →

U 0 + F1

h2

k

U 1 + F0

2 → U + F1 + W ←  2 k

k

U 1 + F1

h1

7 → U + F2 + W ←   2 k h2

k

U 1 + F0

5 →

←  k

k

U 3 + F1 + W

U 1 + F1

h1

U 2 + F0

k4 →

←  k → h1

Scheme 3.15.

←   k

U 3 + F2 + W

k

U 4 + F1 + W

U 2 + F1

h1

←  k

8 →

h2

9 → U + F2 + W ←   4 k h2

k5

U 3 + F0

U 1 + F2 + W

←   k

h1

k10

U 4 + F1 + W

U 3 + F1

→ ←   k

U 4 + F2 + W

h2

Kinetic scheme of FSSE model of urea/formaldehyde polycondensation.

A possible simpliﬁcation consists in distinguishing only rate constants of forward 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 formaldehyde 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 constants 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

However, that ratio is 6.9 according to de Jong and de Jonge; the activation energy of the hydrolysis is the same (20 kcal mol1 ), though, according to both research studies. 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 forward rate constants of the successive substitutions. Interestingly, the rate constants for the reaction of methylenediurea with formaldehyde or monomethylolurea 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

ð61Þ

Rate 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, reaction 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).

2

HOCH2N

N HOCH2

O C O H

Scheme 3.16.

O C

N

N CH2

N

N CH2

O C NH-CH2OCH2N O C O

N CH2

O C

N

+ H2O

+ H2O

2 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.

105

106

3 Polycondensation

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 thermostated 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 daunting complexity of the chemistry. Notice that it should be integrated with the modeling of the cure stage (a complex combination of heat, mass, and mechanical modeling, in the case of wood panel manufacture). Cure is performed with ammonium chloride as catalyst, which also acts as a formaldehyde scavenger. Its chemistry 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 reduced importance of hydrolysis reactions. They are prepared using two stages, alkaline 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 hydroxymethylation, but also of the formation of methylene and methylene ether bridges using polarography. The functionality of melamine is six, with a negative substitution effect 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 formation based on Tomita’s kinetic scheme [225] has been presented [227] and afterwards 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 overcome the astronomical number of reaction possibilities in the rates of formation of individual oligomers.

3.3 Polycondensation Processes in Detail

Phenolic Resins Two subclasses have to be distinguished [229]: 3.3.4.3

resols, which are highly branched, low molecular weight (150–1500) polymers with a formaldehyde/phenol stoichiometric ratio between 1.2 to 3, formed at alkaline pH; novolacs, made at acid pH, with a formaldehyde/phenol stoichiometric ratio between 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 crosslinked with hexamethylenetetramine. Phenolic resins are mainly used as wood adhesives, laminates, molded parts, insulating varnishes, abrasives, and rigid foams. Novolacs can be made using either strong acid catalysts (sulfuric acid is preferred) 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 omethylolphenol in a ﬁrst step. Subsequent addition steps may also be orthodirected (Scheme 3.17), or more randomly distributed (in the case of Zn). H

O

M2+ OH CH2 OH

OH

OH - M2+

CH2OH

- H2 O

- M2+

OH CH2

...

- H2 O +

OH

Formation of ortho-directed methylene bridges.

Scheme 3.17.

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 molecular conformation (Scheme 3.18). Nonterminal units are often assumed to stay

CH2 O O

H O

H H

CH2

Scheme 3.18.

O

CH2 H O H H O

CH2

CH2

Hypothetical intramolecular hydrogen bonds in novolac resins.

107

108

3 Polycondensation

preferentially inside the molecular coils and decrease their reactivity because of this. Mathematical models describing formation of novolacs both in batch and continuous 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 the ﬁrst time be successfully described, thanks to Zavitsas and collaborators [233]. Resol formation in further reactions is a complex process, owing to the several diﬀerent aromatic reaction sites and substitution eﬀects [234]; a total of 19 can be distinguished [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 extensive use of 13 C NMR and synthesis of model compounds. The research information [237, 238] allows modeling of these reacting systems to be elaborated with better chemical support. Fairly good agreement of model and experimentally measured functional group concentrations in resol formation at various stoichiometric ratios has been claimed [239] and it is expected that a trustworthy quantitative description of these systems may eventually be achieved. Validation of these predictions 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 mol1 ). 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 operation in accident situations. In novolac production, formaldehyde can be fed continuously 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, phenols, 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

ClCH2CH CH2 O 3

+

CH3 C CH3

CH2 CHCH2O O Scheme 3.19.

CH3 C CH3

HO

-NaCl

OH

OCH2CHCH2O n

CH3 C CH3

OCH2CH CH2 O

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 solution of bisphenol A with ECH in stoichiometric excess [241]. The main reaction as described above is accompanied by side reactions, such as hydrolysis and alcoholysis 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 catalyst, 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 mol1 ) than that of chain extension, and both constants have been determined both for the catalyzed and noncatalyzed reaction. 3.3.6

Polyurethanes and Polyureas

Urethane polymers were discovered by Baeyer in 1937 [246, 247], using the addition of alcohols to isocyanates leading to carbamates (or urethanes), 4 in Scheme 3.20.

R NCO

+ HO

R'

R

N

O C O R' H

Scheme 3.20.

Addition of alcohols to isocyanates.

4

109

110

3 Polycondensation

The analogous, much faster, reaction with primary amines produces Nsubstituted ureas 5 (Scheme 3.21).

R NCO

+ H2N

R

R'

N

O C NH R'

5

H 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).

R NCO

R

+ H2O

N

O C O H

H R

N

Scheme 3.22.

+ CO2

H

H 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).

R NCO + R

N

O C O R'

R

N

H

C N O

Scheme 3.23.

R

6

H

Reaction of urethanes with isocyanates leading to allophanates.

R NCO + R

N

O C NH R'

R

N

H

O C NH R' C N

O Scheme 3.24.

O C O R'

R

7

H

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 reaction, 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 precipitation of insoluble ureas, which stabilize the foam even before the system

3.3 Polycondensation Processes in Detail

R

N

O C O C N

O

R' R'

+ R NCO

R

N

H

O C

O O

C N O

R

R

R

H N C O

R O

N

N N R

R + R' OH O

8 Scheme 3.25.

Formation of isocyanurates.

O C 2 R NCO

R N

N R C O

Scheme 3.26.

111

9

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 starshaped 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.

112

3 Polycondensation

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 excellent book by Macosko [249] extensively covers this technology, which is particularly 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 production cycle can take less than one minute from injection to demolding. As with the other thermosets in this chapter, these processes will not be discussed in this section. We will nevertheless give a few hints of the reaction engineering of the production of polymers and macromonomers based on this chemistry; 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, selfcatalyzed) reactions may sometimes be fast enough in practice: isocyanate reactions 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 ineffective 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 catalysts, 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 described 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 provided 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 (including 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

reactors for very low molecular weight or solvent-based processes, to tubular reactors 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 too important. 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, including more complex molecular fragments (monads, dyads and so on) from mass balance equations are established in Section 3.4.2 for irreversible reactions or systems with at most FSSEs if reversible reactions exist. Stoichiometric coeﬃcients are introduced 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 Section 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 considerably in the presence of SSSEs. There seems to be no alternative to Monte Carlo methods for dealing with reversible nonlinear polycondensations or even linear 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-

113

114

3 Polycondensation

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 A gi ; A giþ We i Z iR Zei ; Zeþi

monomer units, h i A 1 . . . NX functional groups which react forming bond ZiR ; indices gi ; giþ A 1 . . . NA by-products, e i A 1 . . . NW bonds, i A 1 . . . NR þ ‘‘half-bonds’’, e i ; ei A 1 . . . NZ

The NA functional (or end) groups A i will be distinguished (even if chemically similar) 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 isocyanate 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 reaction 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 possible 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 silanols 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 considered 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 Zei and Zeþi , using yet another 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 zei . 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

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 Eq. (63). kiZ ¼

½W½i ki Ki

ð63Þ

This may look rather artiﬁcial, but it helps in situations such as the thermal decomposition 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: NW NA NZ X X X ðninA þ ninAþ ÞAn þ nW ðninZ þ ninZþ ÞZn ¼ 0 in Wn þ n¼1

n¼1

ð65Þ

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 Z No FSSE means that, for every reaction i, nigA ¼ n þ ¼ 1, nie ¼ n þ ¼ 1, and, igi iei i i moreover, all other stoichiometric coeﬃcients are nil. Likewise, it is useful to introduce 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

þ þþ X A½i þ X þ Z½ j Z½ jþ X þþ ! X Z½i Z½iþ X þ A½ j X E kji

X A½iþ þ X

þ

Z½ j Z½ jþ X

þþ

kijEþ

ð66Þ

! X Z½i Z½iþ X Eþ

þ

þ A½ jþ X

þþ

kji

In the presence of FSSEs, they would be written with an algebraic notation as in Eqs. (67).

115

116

3 Polycondensation NA X

AE AEþ AEþþ ðnijn þ nijn þ njin ÞAn þ

n¼1

NZ X ZE ZEþ ZEþþ ðnijn þ nijn þ nijn ÞZn ¼ 0 n¼1

ð67Þ NA X

NZ X AEþ AEþþ AEþþþ ZEþ ZEþþ ZEþþþ ðnijn 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 AEþþ ZEþþ AEþþþ ZEþþþ ; nijn ; nijn ; nijn are the changes in A½i or A½iþ were attached, and nijn 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 coefAEþ ZEþ AEþþ ZEþþ ; nijn ; nijn ; nijn are the changes in the numbers of groups ﬁcients nijn attached to root unit X þ which gets connected to the unit where stood the attacking 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 functional groups or bonds, as shown by the cancellation of the stoichiometric coeﬃcients 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 using Eqs. (68)–(70).

R An ¼

NR X ðninA þ ninAþ Þðki ½A½i ½A½iþ kiZ ½Z½iþ Þ i¼1

þ

NR X NR X

e

EA EAþ EAþþ de ij ½kijE ½A½i ½Z½ jþ ðnijn þ nijn þ nijn Þ

i¼1 j¼iþ1 EA EAþ EAþþ þ njin þ njin Þ þ kjiE ½A½ j ½Z½iþ ðnjin

R Zn ¼

ð68Þ

NR X ðninZ þ ninZþ Þðki ½A½i ½A½iþ kiZ ½Z½iþ Þ i¼1

þ

NR X NR X

e

ZA ZAþ ZAþþ de ij ½kijE ½A½i ½Z½ jþ ðnijn þ nijn þ nijn Þ

i¼1 j¼iþ1 ZA ZAþ ZAþþ þ njin þ njin Þ þ kjiE ½A½ j ½Z½iþ ðnjin

RWn ¼

NR X

Z k i nW in ð½A½i ½A½iþ ki ½Z½iþ Þ

ð69Þ ð70Þ

i¼1

The reaction volume changes mostly because of by-product removal, and little because of density changes. The relative rate of change of reaction volume RV can be

3.4 Modeling of Complex Polycondensation Reactions

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 written 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 equilibrium 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 programs. The goal is to obtain a set of formulae for predicting average molecular weights, molecular weight distributions, and other polymer properties, valid for generic 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 distributions of the numbers of bonds connecting the fragments, for each kind of fragment. In fact, one may need only some of the moments of the aforementioned distributions 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 fragments. This has been one of our motivations for presenting the formalism in Section 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 meaningless: it would be impossible to know the counts of the monomer units according 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 independent 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 .

117

118

3 Polycondensation

Vector xX containing the counts of the monomer units can be obtained from xZ through Eq. (72). xX ¼ ðZ ZX Þ t xZ

ð72Þ

ZZX is a matrix containing the incidence vectors z deﬁned by Eq. (73). ZijZX ¼

1

if j ¼ zi

0

if j 0 zi

ð73Þ

The chemical system is further described through knowledge of the molar fractions of monomers or monomer units, yX i , (summing to 1) and of the molecular 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 Þ can be computed through Eq. (74).

M½Vi ðxZ ; xA Þ ¼

NZ NA X X ðMX½ jZ þ MZ j ÞxZ j þ MA j xA j j¼1

ð74Þ

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 concentration 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

ð75Þ

Let 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)]. y X

F J ðsZ ; sA Þ ¼

xZ1 ¼0 y X

y X

y X

xZN ¼0 xA1 ¼0 Z

y X xAN ¼0 A

xZ

xAN

PfJ is connected to xZ ; xA groupsgsZ1 1 sAN A A

xAN ¼0 A

J ¼ Xi; Ai; Zi

ð76Þ

These pgf values are mutually related through Eqs. (77) and (78), in which 1N means a vector with N components equal to 1. Aj

F ðsÞ ¼

s1 Aj

qF X½ jA q log sA j

qF X½ jA q log sA j js¼1

X

X

½ jA ½ jA ¼ s1 A j LA j ðsÞ/lA j NZA

ð77Þ

3.4 Modeling of Complex Polycondensation Reactions

F Z j ðsÞ ¼ s1 Zj

qF X½ jZ q log sZ j

qF X½ jZ q log szj js¼1

X

X

½ jZ ½ jZ ¼ s1 Z j LZ j ðsÞ/lZ j

ð78Þ

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

ð79Þ

The 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). lZX ij ¼ lAX ij ¼

qF X i q log sZ j js¼1

NZA

qF X i q log sA j js¼1

NZA

ð80Þ

ð81Þ

The mass of polymer per mole of monomer units, MP , can therefore be computed using Eq. (82).

MP ¼

NX X i¼1

" y X i MX i þ

NZ X

lZX ij MZ j

þ

j¼1

NA X

# lAX ij MA j

ð82Þ

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 lZX ¼ f p

lAX ¼ f ð1 pÞ

ð83Þ

MP 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.

119

120

3 Polycondensation

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. " # " # ½ð1 pA ÞsA þ pAB sZAB þ pAC sZAC f F XX X F ¼ ½ð1 pB ÞsB þ pB sZBA g ½ð1 pC ÞsC þ pC sZCA F XY

2

3 F ZAB 6 ZAC 7 6F 7 FZ ¼6 7 4 F ZBA 5 F ZCA 2 AA 3 F 6 A 7 A 6 F ¼4 F B 7 5

2

Gelation condition

gpBg pACg þ ðg 1Þ pBg pABg þ pCg pABg ¼

F AC

½ð1 6 ½ð1 6 6 4 ½ð1 ½ð1 2 ½ð1 6 4 ½ð1 ½ð1

3 pB sZBA g1 ½ð1 pC ÞsC þ pC sZCA 7 pB sZBA g 7 7 f 1 5 pAB sZAB þ pAC sZAC f 1 pAB sZAB þ pAC sZAC 3 pA ÞsA þ pAB sZAB þ pAC sZAC f 1 7 pB ÞsB þ pB sZBA g1 ½ð1 pC ÞsC þ pC sZCA 5 g pB ÞsB þ pB sZBA pB ÞsB þ pB ÞsB þ pA ÞsA þ pA ÞsA þ

1 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

ð84Þ

The 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, implying that its Jacobian becomes nil [Eq. (85)]. " # qF Z i I ¼ j½LzZj i ð1NZA Þ Ij ¼ 0 qnj

ð85Þ

For the polycondensation of a single monomer, this leads to the well-known result of Eq. (86).

3.4 Modeling of Complex Polycondensation Reactions

pg ¼

1 f 1

ð86Þ

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 processes through Eqs. (87) and (88). GYi ðsZ ; sA Þ ¼ sYi F Yi ½VðsZ ; sA Þ; sA Y ¼ Z; A

ð87Þ

G X i ðsx ; sA Þ ¼ sX i F X i ½VðZZX sX ; sA Þ; sA

ð88Þ

The 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 directed 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 relative 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

ð89Þ

Prediction 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 intersection 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 deﬁned. 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).

121

122

3 Polycondensation

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.

( NX NZ X X me ¼ ½X i 1 F X i ðv; 1NA Þ nj ð1 nj ÞLZXji ðv; 1NA Þ j¼1

i¼1

) NZ X NZ 1X Xi nj nk ð1 nj Þð1 nk ÞLZ j Zk ðv; 1NA Þ 2 j¼1 k¼1 ne ¼

ð90Þ

( NX NZ X 1X ½X i ½lZX ij nj ð1 nj ÞLZXji ðv; 1NA Þ 2 i¼1 j¼1

NZ X NZ X

) 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

ð92Þ

In 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

F^Yi ðsZ ; sA Þ ¼ F Yi ðn1 s Z1 ; . . . ; nNZ sZNZ ; sA Þ/ySYi ¼ F Yi ðv5sZ ; sA Þ/ySYi Y ¼ X; Z; A

ð93Þ

^ ðsZ ; sA Þ; sA i ¼ 1; NZ V^i ðsZ ; sA Þ ¼ sZ i F^Z i ½V

ð94Þ

So, the various pgf values with respect to the diﬀerent kinds of groups in the ^ X i ðsZ ; sA Þ; G ^Z i ðsZ ; sA Þ; G ^A i ðsZ ; sA Þ, can be computed using molecules of the sol, G Eqs. (95). ^Yi ðsZ ; sA Þ ¼ F^Yi bV ^ ðsZ ; sA Þ; sA c G

Y ¼ X; Z; A

ð95Þ

After 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

ð96Þ

Pgf 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): M

ZN MZ þM V^Mi ðsM Þ ¼ V^i ðsM 1 X½1Z ; . . . ; sM Z

þMX ½NZ Z

MA

MAN

; sM 1 ; . . . ; sM

A

M þMX z

Þ ¼ V^i ðsM z

MA ; 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. GM ðsM Þ ¼

y X

X

Mi sM wi ¼

NY X

MY MA ^ M ðsM Þ; sM wYi sM i F^Yi ½V

ð98Þ

Y ¼X; Z; A i¼1

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). wS ¼

NX X

y X i ySX i MX i þ

j¼1

i¼1

wX i ¼

y X i ySX i MX i MP wS MZ i

wZ i ¼

NZ X

NX X

l^ZX ij MZ j þ

NA X

!, l^AX ij MA j

MP

ð99Þ

j¼1

ð100Þ

X y X j ySX j l^Z ji

j¼1

MP wS

ð101Þ

123

124

3 Polycondensation

MA i

NX X

X 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 molecular 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 molecule is consumed, the number of moles of polymer molecules per mole of monomer units before gelation is given by Eq. (103).

yP ¼ 1

NX NZ X 1X yXi lZX ij 2 i¼1 j¼1

ð103Þ

After gelation, the more general expression [Eq. (104)] is needed.

yP ¼

NX X

y X i ySX i

i¼1

NZ 1X 1 l^X i 2 j¼1 Z j

! ð104Þ

The number-average molecular weight of the sol can thereafter be computed through Eq. (105). NX X

Mn ¼

y X i ySX i MX i þ

NZ X

MZ i l^ZX ij

j¼1

i¼1 NX X

y X i ySX i

i¼1

þ

NA X j¼1

NZ 1X 1 l^X i 2 j¼1 Z j

!

! MA i l^AX ij ð105Þ

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)]. NX X

xn ¼

y X i ySX i

i¼1 NX X i¼1

y X i ySX i

NZ 1X l^X i 1 2 j¼1 Z j

!

ð106Þ

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 j

½miZ ¼ ½di l^ZZ ij 1 ½MZ i þ MX½iZ ½miA

¼

j ½di

ð107Þ

l^ZZ ij 1 ½l^ZAji ½MA i

ð108Þ

" ½miZZ

¼

j ½di

l^ZZ ij 1

2ðMZ i þ MX½iZ Þ

NZ X

mZl l^ZZli

þ

¼

j ½di

" ¼

j ½di

ð109Þ

" # NA NZ X NA X X Z i 1 Zi Zi Z ^ ^ ^ lZ j ðMZ i þ MXX½iZ Þ MAm lAk þ m l MAm lZl Am

l^ZZ ij 1

NZ X NA X

ð110Þ

l¼1 m ¼1

m¼1

½miAA

# Z ^Z i mZl mm lZl Zm

l ¼1 m¼1

l ¼1

½miZA

NZ X NZ X

mlA MAm l^ZZliAm

þ

l¼1 m¼1

NA X NA X

# MAl MAm l^AZliAm

ð111Þ

l¼1 m¼1

j

ð112Þ

½miMA ¼ ½di l^ZZ ij 1 ½MA2 i

ð113Þ

½miMZX ¼ ½di l^ZZ ij 1 ½ðMX½iZ þ MZ i Þ 2 j

Weight-average and z-average molecular weights are now obtained explicitly through Eqs. (114) and (115). qGM ¼ Mw q log sM jsM ¼1 " # NY NZ NA X X X X Y Z A ^Yi ¼ w Yi M Yi þ ðmj þ mj ÞlZ j þ MA j l^A ji

lM ¼

Y ¼X; Z; A i¼1

j¼1

ð114Þ

j¼1

lMM ¼ Mw MZ ¼

X Y¼X; Z; A

(

NY X

" w Yi

MY2i

j¼1

i¼1

þ

þ 2MYi

NZ NA X X ðmjZ þ mjA Þl^YZij þ MA j l^YZij

!

j¼1

NZ X NZ X ðmjZ þ mjA ÞðmkZ þ mkA Þl^YZij Zk j¼1 k¼1

þ

NZ X ðmjZZ þ 2mjZA þ mjAA þ mjMXZ þ mjMA Þl^YZij j¼1

þ2

NZ X NA X ðmjZ þ mjA ÞMAk l^YZij Ak j¼1 k¼1

þ

NA X NA X j¼1 k¼1

MA j MAk l^AYjiAk þ

NA X

#) MA2 j l^AYji

ð115Þ

j¼1

Except for extremely simple polycondensations, formulae for predicting average molecular weights are very cumbersome and numerical evaluation is a must.

125

126

3 Polycondensation

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 polycondensation of a single monomer XA f , this leads to Eqs. (116) and (117). xw ¼

1 þ pð2v 1Þ 1 p½1 þ vð f 2Þ

xz ¼ 1 þ

þ

ð116Þ

2f pv þ f p 2 vð f 1Þð1 p þ pvÞ f 2 ð1 p þ 2pvÞ½1 pð f 1Þð1 p þ pvÞ f 2 ð f 1Þð f 2Þ p 2 v 2

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 stoichiometric reasoning as previously discussed [Eq. (118)]. xn ¼

1 p þ pv 1 p þ pvð1 f /2Þ

ð118Þ

Values 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 setting 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 derivatives with respect to dummy Laplace variables on s ¼ 0 is feasible with the simplest 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 y P s x ½Px usually a better way to predict them. Unless the cost of evaluating GðsÞ ¼ x ¼0

is too high and more sophisticated methods based upon Laplace transform inversion are needed, an accurate evaluation can be obtained using the method independently developed by Mills [262] and ourselves [263] (see also Ref. 264 for a thorough 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 centered on the origin with radius jsj slightly below 1, using the trapezium rule and Fast Fourier Transform for evaluating the sums [Eq. (119)].

½Px ¼

N 1 y X jsjx X 2pimx Gðsm Þ exp ½PxþNn jsj Nn N N m¼0 n¼1

ð119Þ

3.4 Modeling of Complex Polycondensation Reactions

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 transform 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 properties, 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 functions [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 experimental results concerning the POCl3 /P2 O5 system have been successfully described. More general calculations are better carried out using the method described 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, summing 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 function 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). V ¼ F Zl ðVÞ ¼ b 0 sC0 ½sA ð1 ar Þ þ ar V f 1 þ

y X

b n sCn ½sA ð1 ar Þ þ ar V nð f 2Þ1

n¼1

G ¼ ð1 yc ÞsC0 ½sA ð1 ar Þ þ ar V f þ

y X

y Xc ðnÞsCn ½sA ð1 ar Þ þ ar V nð f 2Þ

n¼1

b0 ¼

f ð1 yc Þ ; f 2yc

bn ¼

ð f 2Þ y Xc ðnÞ f 2yc

ð120Þ

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

127

128

3 Polycondensation

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 order 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 Stockmayer in their classic paper [21] for f ¼ 2, is Eq. (121). ½Px ¼ ½X ð f 2yc Þ y Xc ðnÞ ¼ y X n¼1

½xð f 1Þ!arx1 ð1 ar Þ xð f 2Þþ2 f ð1 yc Þ x f 2yc x!½xð f 2Þ þ 2!

nK c ðnÞ n q ; ½X

nK c ðnÞq n ¼ ½X yc ;

q¼

f ð f 1Þar ð1 yc Þ f 2yc

yc ¼

f ð p ar Þ 2ð1 ar Þ

ð121Þ

A solution is also known for the analogous system of the alternating polycondensation XA f þ YBg [267]. Dilution with an inert solvent will make [X] decrease without aﬀecting the cyclization 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 affected (that eﬀect using TBP has been modeled by Dusˇek 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 gelation theory described here considers a uniform distribution in space of all the chemical properties. Stauﬀer, one of the main contributors to the progress of percolation 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 argument [271], claiming his theory to be universal and capable of describing gel formation 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 generating 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

tems, and we leave it here only for reference, as even a basic but comprehensible description would be too extensive. 3.4.4

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)]. ( x1 d½Px 1X ¼k ½ yð f 2Þ þ 2½ðx yÞð f 2Þ dt 2 y¼1 þ 2½Py ½Pxy ½A½xð f 2Þ þ 2½Px d½A ¼ k½A 2 ; dt

½Px jt¼0 ¼ ½X ;

) ð122Þ

½Ajt¼0 ¼ f ½X

The two solutions are identical. Hence, for a long time no importance was attributed to the use of a kinetic approach for describing batch polycondensations starting from monomers, and the statistical approach was preferred. Of course, chemical 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 presenting FSSEs, kinetic and statistical approaches give distinct results for average molecular weights and gel points. Dusˇek [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 considering 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 equilibrium 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.

129

130

3 Polycondensation

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 considering unimolecular reactions and new fake functional groups, which are pairs of groups [Eq. (123)]. ki

A½i !

N XZA X

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 unimolecular reactions deﬁned above, the new general rate equation becomes Eq. (124).

R An

NR NR X X þ ¼ ðn þ n Þk ½A ½A þ nin ki ½A½i i ½i ½iþ in in i¼1

ð124Þ

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)] NR X þ GRP ðsÞ ¼ ki G ni Gni i¼1

qG qG qG qG ½A½iþ ½A½i q log s½i q log s½iþ q log s½i q log s½iþ

þ

NR X i¼1

ki

qG ðG 1Þ q log s½i ni

ð125Þ

where

J

Gni ¼

N XZA Y

n

J

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

NR qG X qG qG qG qG þ ¼ ki G G ½A ½A ½iþ ½i ni ni qt q log s½i q log s½iþ q log s½i q log s½iþ i¼1

þ

NR X

ki

i¼1

qG GF ðsÞ GðsÞ ðGni 1Þ RV G þ q log s½i t

GðsÞjt¼0 ¼ G0 ðsÞ

ð127Þ

Solution 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 reproduced here, for the sake of brevity. If GðsÞ is to be evaluated, in order to take advantage of the numerical inversion formula Eq. (119), or if average degrees of polymerization 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 P , in the prediction of the moments of ½Px with respect to the counts of groups, lmn... 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 dt

¼

NR X

þ þ k m ½ðn mk þ nmk Þðnmj þ nmj Þ½A½m ½A½mþ

m¼1 þ P P þ ðn mj þ nmj Þðl½mk ½A½mþ þ l½mþk ½A½m Þ þ P P P P P P þ ðn mk þ nmk Þðl½m j ½A½mþ þ L½mþ j ½A½m Þ þ l½m j l½mþk þ l½mþ j l½mk

þ

NR X

P P km ðl½m j nmk þ l½m k nmj þ ½A½m nmj nmk Þ þ

m¼1

ljkPF ljkP t

RV ljkP

ð128Þ

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)]. NR X ½Pf ½P q½P ¼ ki ½A½i ½A½iþ RV ½P þ t qt i¼1

½Pjt¼0 ¼ ½P0

ð129Þ

Number-average and weight-average molecular weights are found by the wellknown expressions (130).

131

132

3 Polycondensation N ZXA X

Mn ¼

N ZXA N ZXA X X

MAn ½An

n¼1

Mw ¼

½P

m¼1 n¼1 N ZXA X

P MAm MAn lmn

ð130Þ MAn ½An

n¼1

This approach can only deal with ring-forming reactions either for a limited number of the smallest rings, or alternatively, for linear polycondensations. The important 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: n1 X

RPnAA ¼ k 4

RPnBB ¼ k 4

AA ½PmAA ½Pnm

þ2

m¼1

m¼1

n1 X

n1 X

BB ½PmBB ½Pnm

þ

m¼1

" RPnAB ¼ k

n1 X

n X

BB 4 ½PmAA ½Pnmþ1 m¼1

! AB ½PmAA ½Pnm

2½PnBB ½A

½PnAB ð½A

!

AB 2 ½PmBB ½Pnm m¼1

þ

2½PnAA ½B

#

n1 X

AB ½PmAB ½Pnm m¼1

þ ½BÞ kc ðnÞ½PnAB

RCn ¼ kc ðnÞ½PnAB RPnAC ¼ k 2

n X

BC ½PmAA ½Pnmþ1

m¼1

RPnBC ¼ k 2

n1 X

n1 X

! AC ½PmAB ½Pnm

½PnAC ½B

m¼1

AC ½PmBB ½Pnm

m¼1

RPnCC ¼ k

þ

n1 X

þ

n1 X

! BC ½PmAB ½Pnm

½PnBC ½A

m¼1

BC ½PmAC ½Pnm

m¼1

RA ¼ RB ¼ k½A½B

y X

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

equation of Eqs. (131) may be done accurately. A more elegant method uses generating 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 provide no information about polymer or network properties. More recently, a description 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 discussing 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

ð132Þ

þ 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:

AB RPnAA ¼ 2k2 ½P1AA ½Pn1 þ 2k4

n2 X

AA ½PmAB ½Pnm 2½PnAA ½2k3 ½P1BB þ k4 ð½P1AB þ ½ZB Þ

m¼1

"

½PnAA ½2k2Z

þ

2k4Z ðn

2Þ þ

k3Z ½PnAB

þ

k4Z

½ZA

n X

# ð½PmAB

þ

2½PmAA Þ

m¼2

AB þ 2k4 RPnBB ¼ 2k3 ½P1BB ½Pn1

n2 X

BB ½PmAB ½Pnm 2½PnBB ½2k2 ½P1AA þ k4 ð½P1AB þ ½ZA Þ

m ¼1

"

½PnBB ½2k3Z

þ

2k4Z ðn

2Þ þ

k2Z ½PnAB

þ

k4Z

½ZB

n X m¼2

# ð½PmAB

þ

2½PmBB Þ

133

134

3 Polycondensation

RPnAB ¼ 4k2 ½P1AA ½PnBB þ 4k3 ½P1BB ½PnAA n1 X

þ k4

AB ½PmAB ½Pnm

þ4

n1 X

! BB ½PmAA ½Pnmþ1

m¼2

m¼1

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

" þ k4Z ½ZA þ ½ZB 2

n X

½PnAB 2

m¼2

nþ1 X

# ð½PmAA þ ½PmBB Þ

ð133Þ

m¼2

It may be observed from these expressions that, in general, it is not possible to obtain 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 abovementioned 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).

GAA ¼

y X

s n2 ½PnAA GBB ¼

n¼2

y X

s n2 ½PnBB GAB ¼

n¼2

y X

s n2 ½PnAB

½ZA ¼ 2GAA ð1Þ þ GAB ð1Þ ½ZB ¼ 2GBB ð1Þ þ GAB ð1Þ ½P2AA ¼ GAA ð0Þ ½P2BB ¼ GBB ð0Þ

ð134Þ

n¼2

½P2AB ¼ GAB ð0Þ

ð135Þ

The 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 þ k4 ð½P1AB þ ½ZB Þ 2k2Z GAA 2k4Z s þ k4Z

qGAA þ k3Z GAB qs

½ZA 2GAA GAB 1s

RGBB ¼ 2k3 ½P1BB ðsGAB þ ½P1AB Þ þ k4 s 2 GBB GAB 2GBB ½2k2 ½P1AA þ k4 ð½P1AB þ ½ZA Þ 2k3Z GBB 2k4Z s þ k4Z

½ZB 2GBB GAB 1s

qGBB þ k2Z GAB qs

3.4 Modeling of Complex Polycondensation Reactions

RGAB ¼ 4k2 ½P1AA GBB þ 4k3 ½P1BB GAA þ k4 ½2½P1AB ðsGAB þ ½P1AB Þ 2 þ 4sGAA GBB ½2k2 ½P1AA þ 2k3 ½P1BB þ k4 ð2½P1AB þ ½ZA þ ½ZB Þ þ s 2 GAB

þ k2Z þ k3Z þ k4Z GAB 2k4Z s þ 2ðk2Z k4Z Þ

qGAB ½ZA þ ½ZB 2ðGAB þ GAA þ GAB Þ þ k4Z qs 1s

GAA ½P2AA GBB ½P2BB þ 2ðk3Z k4Z Þ s s

ð136Þ

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 5 þ k4Z ½X þ ½Y þ ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2 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 5 þ k4Z ½X þ ½Y þ ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2

ð137Þ

A 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 k2Z ½ZA k3Z ½ZB k4Z

y X

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 3 k4Z ½X þ ½Y ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2

ð138Þ

Insertion 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)]. ds ¼ 2sk4Z dt

ð139Þ

The relationship of Eq. (138) allows integration along characteristics, relating initial 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,

135

136

3 Polycondensation

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 numerical estimation of oligomer concentrations through Eq. (140). GI ð0Þ ¼

GI ðsþ Þ þ GI ðs Þ þ Oðes2 Þ I ¼ AA; BB; AB 2

ð140Þ

Parameter es should be a small value (say 106 to 108 ). Owing to Eq. (139), it is easy to relate the auxiliary characteristics to the ‘‘main’’ characteristic sðtÞ starting 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 described above is no longer useful, as there is no simple way of describing formation 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 av b

vector with N components equal to one ﬁlm area per unit volume [m1 ] length of a Kuhn segment connecting two half-bonds in the same repeating unit [m]

Notation

bA Cd Cg D db dR dT Ec ei

empirical parameter in Eq. (56) (dimensionless) drag coeﬃcient (dimensionless) parameter in Eq. (12) [s1 ] binary diﬀusivity [m 2 s1 ] bubble diameter [m] diameter of ring (in a RDC devolatilizer) [m] inner diameter of tube, or of the barrel of an extruder [m] activation energy of amidation reaction [J mol1 ] index of by-product formed by reaction leading to bond Z iR ; no byproduct if it is nil þ indices of ‘‘half-bonds’’ of bond Z iR e i ; ei 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 unreacted group A i , or a half-bond Z i f monomer functionality (number of functional groups it contains) equilibrium zero strain shear modulus of polymer network [Pa] Ge GðsÞ moment generating function of molar concentrations of polymer molecules with respect to their counts of monomer units and unreacted functional groups J generating functions of stoichiometric coeﬃcients of reaction Gni i ð J ¼ þ; ; Þ, as deﬁned by Eq. (126) Xi G ðsX ; sA Þ; G A i ðsX ; sA Þ; G Z i ðsX ; sA Þ pgf of trees starting, respectively, with a prescribed 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 s2 ] ratio mA /bA [m 3 mol1 ] gA þ indices of functional groups A gi ; A giþ reacting to create bond Z i gi ; gi H depth of liquid in the channel of a vented extruder [m] partition coeﬃcients of hydrophilic monomer in interfacial polymerHe ; Hi ization relating, respectively, the bulk concentration in the water phase and the concentration in the outer interface of the polymer ﬁlm, and the concentrations in organic phase in the two interfaces of polymer ﬁlm (dimensionless) Henry constant of component W in terms of pressure versus weight HW fraction [Pa] h clearance between the screw and barrel in a vented extruder [m] index of a repeating unit carrying functional group A i hi 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 mass transfer rate per unit volume of component j [mol m3 s1 ] Jj K apparent equilibrium ratio of bond formation from end groups (dimensionless) thermodynamic equilibrium constant of bond formation from end K0 groups (dimensionless)

137

138

3 Polycondensation

Ka K a0 K c ðnÞ k k0 kA k aE k c ðnÞ k cE ðnÞ kcr k cZ kc0 kfYi a kfY i

ki0 kijE kijE ; kijEþ kiH kiOH km k nuc; n kscat ; kfcat ku0 kZ kiZ ki L Lp LR LW Lx

apparent equilibrium ratio of amidation (dimensionless) reference value of the apparent equilibrium ratio of amidation (dimensionless) apparent equilibrium ratio of cyclization [mol m3 ] apparent second-order rate constant [m 3 mol1 s1 ] parameter in Eq. (12) [m 3 mol1 s1 ] empirical parameter in Chen–Wu relation, Eq. (53) [mol m3 s1 ] rate constant of breakage of a bond in a ring molecule through reaction with a functional group [mol m3 s1 ] ﬁrst-order rate constant of L n cyclization [s1 ] rate constant of formation of Cn by the back-biting cyclization reaction [s1 ] rate coeﬃcient of crystallization in Avrami equation [sncr ] (see below) apparent ﬁrst-order rate constant of breakage of a bond in a ring molecule [s1 ] third-order pre-exponential factors of amidation rate constant [kg 2 mol2 s1 ] mass transfer coeﬃcient of species Yi in terms of molar concentrations [m s1 ] mass transfer coeﬃcient of species Yi in terms of activities [mol m2 s1 ] noncatalytic contribution to the rate constant in Eq. (61) [mol m3 s1 ] rate constant of exchange reaction forming bond Z iR and destroying bond Z jR rate constants of exchange reaction forming bond Z iR and replacing bond Z jR , when the attacking group is, respectively, A½i or A½iþ acid-catalyzed reaction constant in Eq. (61) [m 6 mol2 s1 ] catalytic term of hydroxyls in Eq. (61) [m 6 mol2 s1 ] water mass transfer rate constant [s1 ] rate coeﬃcients of nucleation of polymer species with chain length n [s1 ] rate constants of self-catalyzed and foreign acid-catalyzed esteriﬁcation [m 6 mol2 s1 ] second-order pre-exponential factor of amidation rate constant [m 3 mol1 s1 ] apparent ﬁrst-order rate constant of bond group breakage [s1 ] apparent ﬁrst-order rate constant of breakage of bond group ZiR [s1 ] ﬁrst-order rate constant of intramolecular reaction i destroying group A½i [s1 ] ﬁlm thickness [m] thickness of polymer ﬁlm in interfacial polymerization [m] thickness of reaction zone [m] width of the channel in an extruder [m] ﬁlm perimeter [m]

Notation

M Mi

relative molecular weight relative molecular weight of generic polymer species i, where i is an arbitrary y index y P P Mn ¼ Mi ½Pi ½Pi number-average molecular weight of polymer i¼1

i¼1

mass of polymer per mole of monomer units [kg mol1 ] relative ymolecular weight of by-product W y P P Mw ¼ Mi2 ½Pi Mi ½Pi weight-average molecular weight of polymer i¼1 i¼1 y y P P Mz ¼ Mi3 ½Pi Mi2 ½Pi z-average molecular weight of polymer MP MW

i¼1

i¼1

empirical parameter in Eq. (56) [mol1 m 3 ] mA number of functional (or end) groups NA Avogadro–Lo¨schmidt number [mol1 ] NAL Nb number of bubbles ﬂux of component j [mol m2 s1 ] N_ j number of distinctive chemical bonds NR number of symmetrical chemical bonds NRs number of kinds of monomer units NX number of by-products NW number of volatile (or precipitating) components NY number of half-bonds NZ 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) [s1 ] Avrami exponent ncr pressure inside a bubble [Pa] Pb local pressure over a bubble [Pa] Pme vapor pressure of W [Pa] PW S PW vapor pressure of pure W [Pa] p conversion of reference functional groups A conversion of functional groups A having given rise to rings pc Q volumetric ﬂow rate of polymer [m 3 s1 ] volumetric ﬂow rate of gas [m 3 s1 ] Qg R ideal gas constant [J mol1 K1 ] Re Reynolds number for the rising movement of a bubble rate of formation by chemical reaction of species j [mol m3 s1 ] Rj RV combined relative rate of change of reaction volume by chemical reaction and transfer of by-product [s1 ] r stoichiometric ratio in alternating polycondensations radius of a bubble [m] rb s vector of dummy Laplace variables associating sA and sZ (deﬁned below)

139

140

3 Polycondensation

sA

vector of dummy Laplace variables associated with the counts of unreacted functional groups dummy Laplace variables associated with the counts of rings (includsCn 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 vector of dummy Laplace variables associated with the counts of halfsZ bonds dummy Laplace variable associated with xZy s Zy sZl dummy Laplace variable associated with the counts of bonds not belonging to rings for the equilibrium polycondensation of a single monomer XA f T absolute temperature [K] reference temperature [K] T0 t time [s] exposure time [s] tf exposure time of liquid in the pool of a vented extruder [s] t fP bubble rising speed [m s1 ] ub average velocity in directions x; z [m s1 ] ux ; uz V volume of reacting mixture [m 3 ] volume of a bubble [m 3 ] Vb 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 mol1 ] ^ pgf of ﬁnite pendent chains Vi with respect to molecular weight VMi ðsM Þ overall melt volume [m 3 ] Vm volume of liquid in compartment j in a staged model of an RDC devoVmj latilizer [m 3 ] v vector of the NZ probabilities of extinction (the fractions of ﬁnite pendent chains) average transverse velocity of the screw dragging the ﬁlm [m s1 ] vT wcr weight fraction of crystalline material weight fraction of j wj weight fraction of the sol, relative to the overall mass of the polymer wS 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] vector storing the counts of functional groups xA xi degree of polymerization of generic polymer species i, where i is an arbitrary y index y P P xn ¼ x i ½Pi ½Pi number-average degree of polymerization of polymer i¼1 i¼1 y y P P xw ¼ xi2 ½Pi x i ½Pi weight-average degree of polymerization of polymer i¼1

i¼1

Notation

xX xZ xZ y y yc yP ySX i ; ySZ i y Xc ðnÞ

ZZX z zi

vector containing the counts of monomer units vector storing the counts of half-bonds number of inﬁnite pendent chains connected to a junction Cartesian coordinate normal to the interface [m] overall molar fraction of repeating units X in rings for a single monomer polycondensation number of moles of polymer molecules per mole of monomer units before gelation and ySA i fractions of units and groups of the various kinds in ﬁnite molecules (in a sol), after gelation 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 polycondensation matrix deﬁned in Eq. (73) from incidence vectors z deﬁned below Cartesian coordinate parallel to the interface [m] index of the monomer unit to which half-bond Z i points

Greek empirical parameter of Chen–Wu relation [Eq. (53)] (dimensionless) probability of reaction of unreacted functional groups after ring formation in the ﬁrst stage (absence of open-chain molecules) of a hypothetical two-stage equilibration of XA f b g ; b g0 parameters in Eq. (11) describing the glass eﬀect [K] fraction of volume ﬂow rate in compartment j going to the preceding bj compartment j 1 activity coeﬃcient of species j gj Flory–Huggins parameter (multicomponent mixture) wij DH enthalpy of reaction [J mol1 ] DS entropy of reaction [J mol1 K1 ] d upstream distance of re-entry of ﬁlm wiped by the screw in a vented extruder [m] small starting value of s in characteristic line es y screw angle ratio of rate constants of similar intramolecular and intermolecular k c ðnÞ reactions [mol m3 ] q . . . qF I LIJ...K ðsÞ ¼ derivatives of pgf F I ðsÞ (or other moment generq log sJ . . . q log sK ating functions) with respect to logarithms of dummy Laplace parameters y y P P q . . . qF I I lJ...K ¼ ð1N Þ ¼ ... xJ . . . xJ IðxÞ moments of distribuq log sJ . . . q log sK x1 ¼0 xN ¼0 I tions, where F ðsÞ is the moment generating function of vector distribution IðxÞ a ar

141

142

3 Polycondensation

ﬁrst and second moments of molecular weight of distribution of polymer m dynamic viscosity [N m1 s1 ] concentrations of elastically active network junctions (EANJ) me [mol m3 ] n stoichiometric coeﬃcient (the change in number of moles caused by chemical reaction) stoichiometric coeﬃcients of functional groups An and oriented bonds ninA ; ninZ 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 , respectively, 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 AEþþ ZEþþ AEþþþ ZEþþþ nijn ; 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 AEþ ZEþ 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 expense 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 concentration of elastically active network chains [mol m3 ] ne density of j [kg m3 ] rj s surface tension [N m1 ] t space time based upon feed ﬂow rate, Vm /Q F for a CSTR [s] tr structural relaxation time [s] parameter in Eq. (11) [s] t0 tortuosity for by-product diﬀusion in a solid polymer particle tD (dimensionless) volume fraction of crystalline material fcr volume fraction of j fj fraction of channel extruder ﬁlled with liquid fL w Flory–Huggins parameter (binary mixture) (dimensionless) weight fraction activity coeﬃcient of W WW lM ; lMM

Chemical symbols A; B functional groups A gi 1 A½i ; A giþ 1 A½iþ functional groups reacting in bond Z iR formation

Notation

Aj

active component (molecular species or fragment) j; nonvolatile, nonprecipitating ring molecule with n bonds Cn linear polymer molecule with n 1 bonds and a pair of active end Ln groups length of tube LT generic polymer species, where i is an arbitrary index Pi IJ Pn linear polymer molecule with end groups I and J and n repeating units of the most frequent kind ½U 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 volatile (or precipitating) component j Yj Z bond group; Z iR is the ith distinctive bond in the chemical system bond group in fragment AXZA YZ (polycondensation AXA þ BYB) ZA bond group in fragment BYZB XZ (polycondensation AXA þ BYB) ZB bond group in dimer AXZD YB (polycondensation AXA þ BYB) ZD Zei 1 Z½i ; Zeþi 1 Z½iþ ‘‘half-bonds’’ of bond Z iR ZP bond group in fragment ZXZP Z (polycondensation AXA þ BYB) Subscripts aq b blk c cr F g org sat

aqueous of a bubble bulk critical crystalline in feed at gel point organic at saturation

Superscripts * x

used for describing values of concentration at the interface average value of x

Acronyms BHET BPA

bis(2-hydroxyethyl) terephthalate bisphenol A

143

144

3 Polycondensation

CLD CSTR DEG DMT DPC EANC EANJ ECH EG FSSE ODE PBT PDE PET pgf RDC RIM SSP SSSE TBP TPA UF WFR

chain length distribution continuous stirred tank reactor diethylene glycol dimethyl terephthalate diphenyl carbonate elastically active network chains, the chains connecting EANJs (see below) elastically active network junctions, the intersection of at least three chains leading to a gel epichlorohydrin ethylene glycol ﬁrst-shell substitution eﬀect ordinary diﬀerential equation poly(butylene terephthalate) partial diﬀerential equation poly(ethylene terephthalate) probability generating function rotating disk contactor reaction injection molding solid-state polycondensation second-shell substitution eﬀect theory of branching processes terephthalic acid urea–formaldehyde polymer wiped-ﬁlm reactor

References 1 A. V. Lourenc ¸ o, Comptes Rendus

2

3 4 5

6

7

He´ bd. Se´ances Acad. Sciences 1859, 49, 619; A. V. Lourenc¸o, Comptes Rendus He´ bd. Se´ances Acad. Sciences 1860, 50, 91. L. H. Baekeland, J. Ind. Eng. Chem. 1909, 1, 149; reprinted in Chemtech, 1976, 6, 40. H. Staudinger, Ber. 1920, 53, 1073. W. H. Carothers, J. Am. Chem. Soc. 1929, 51, 2548. P. J. Flory, Principles of Polymer Chemistry, 1st ed., Cornell University Press, Ithaca, New York, 1953. J. L. Stanford, R. F. T. Stepto, J. Chem. Soc., Faraday Trans. 1975, 71, 1292. W. H. Stockmayer, J. Chem. Phys. 1943, 11, 45.

8 D. J. Brunelle, J. E. Bradt, US

Pat. 5 039 783 to General Electric, 1991. 9 R. B. Burch, S. R. Lustig, M. Spinu,

Macromolecules 2000, 33, 5053. 10 G. V. Schulz, Z. Phys. Chem. 1938,

A182, 127. 11 M. R. P. F. N. Costa, J. Villermaux,

Ind. Eng. Chem. Res. 1989, 28, 711. 12 M. H. Abraham, Pure Appl. Chem.

1985, 57, 1055. 13 E. A. Macedo, R. C. Gonc ¸ alves, A. N.

˜ es, C. M. Ventura, L. C. Simo Albuquerque, Ind. Eng. Chem. Res. 1995, 34, 1910. 14 K. Dusˇek, Polym. Gels Networks 1996, 4, 383. 15 I. Mita, K. Horie, J. Macromol. Sci. Rev. Macromol. Chem. Phys. 1987, C27, 91.

References 16 S. Corezzi, D. Fioretto, D. Puglia,

17 18 19 20 21 22 23 24

25

26

27 28 29 30 31 32 33 34 35 36 37

38

39

J. M. Kenny, Macromolecules 2003, 36, 5271. G. Adam, J. Gibbs, J. Chem. Phys. 1965, 43, 139. S. Lunˇa´k, J. Vladyka, K. Dusˇek, Polymer 1978, 19, 931. A. M. Kotliar, J. Polym. Sci.: Macromol. Revs. 1981, 16, 367. M. Gordon, G. R. Scantlebury, Trans. Far. Soc. 1964, 60, 604. H. Jacobson, W. Stockmayer, J. Chem. Phys. 1950, 18, 1600. P. J. Flory, J. A. Semlyen, J. Am. Chem. Soc. 1966, 88, 14. P. J. Flory, U. W. Suter, M. Mutter, J. Am. Chem. Soc. 1976, 98, 5733. H. R. Kricheldorf, M. Rabenstein, M. Maskos, M. Schmidt, Macromolecules 2001, 34, 713. H. R. Kricheldorf, S. Bo¨hme, G. Schwarz, Macromolecules 2001, 34, 8879. C. Puglisi, F. Samperi, R. Alicata, G. Montaudo, Macromolecules 2002, 35, 3000. T. E. Harris, The Theory of Branching Processes, Springer, Berlin, 1963. M. Gordon, Proc. Roy. Soc. 1962, A268, 240. G. R. Dobson, M. Gordon, J. Chem. Phys. 1965, 43, 705. M. Gordon, G. N. Malcolm, Proc. Roy. Soc. 1966, A295, 29. K. Kajiwara, J. Chem. Phys. 1971, 54, 296. K. Kajiwara, M. Gordon, J. Chem. Phys. 1973, 59, 3623. W. Burchard, Adv. Polym. Sci. 1982, 48, 1. C. W. Macosko, D. R. Miller, Macromolecules 1976, 9, 199. C. Sarmoria, D. R. Miller, Macromolecules 1991, 24, 1833. K. Platkowski, K. H. Reichert, Polymer 1999, 40, 1057. B. E. Eichinger, O. Akgyray, chap. 8 in Computer Simulation of Polymers, E. A. Colbourn (Ed.), Longman, Harlow, UK, 1994. H. Tobita, N. Hamashima, J. Polym. Sci. B: Polym. Phys. 2000, 38, 2009. J. E. Crawford, R. W. Edwards, E. D.

40 41

42

43 44 45 46 47 48 49 50

51 52 53 54 55

56

57 58 59 60 61

Henze, W. C. L. Wu, U.S. Pat. 3 522 214, 1970. K. Ravindranath, R. A. Mashelkar, Chem. Eng. Sci. 1986, 41, 2969. R. Albalak, An Introduction to Devolatilization, in Polymer Devolatilization, R. J. Albalak (Ed.), Marcel Dekker, New York, 1996. P. S. Mehta, An Overview of Devolatilizers, in Polymer Devolatilization, R. J. Albalak (Ed.), Marcel Dekker, New York, 1996. F. Widmer, Adv. Chem. Ser. 1973, 128, 51. R. M. Secor, AIChE J. 1969, 15, 861. T. M. Pell Jr, T. G. Davies, J. Polym. Sci. B: Polym. Phys. 1973, 11, 1671. J. W. Ault, D. A. Mellichamp, Chem. Eng. Sci. 1972, 27, 2233. M. Amon, C. D. Denson, Ind. Eng. Chem. Fundam. 1980, 19, 415. B. R. Choi, H. H. Lee, Ind. Eng. Chem. Fundam. 1996, 35, 1550. G. W. Roberts, AIChE J. 1970, 16, 878. J. A. Biesenberger, D. H. Sebastian, Principles of Polymerization Engineering, chap. 6, Wiley, New York, 1983. G. A. Latinen, Adv. Chem. Ser. 1962, 34, 235. R. W. Coughlin, G. P. Canevari, AIChE J. 1969, 15, 560. W. A. Mack, R. Herter, Chem. Eng. Prog. 1976, 72(1), 64. G. P. Collins, C. D. Denson, G. Astarita, AIChE J. 1985, 31, 1288. K. Ravindranath, R. A. Mashelkar, in Frontiers in Chemical Engineering, Vol. 1, p. 652, L. K. Doraiswamy, R. A. Mashelkar (Eds.), Wiley Eastern, Delhi, 1984. Y. Murakami, K. Fujimoto, K. Kakimoto, M. Sekino, J. Chem. Eng. Jpn. 1972, 5, 257. K. Vijayaraghvan, J. P. Gupta, Ind. Eng. Chem. Fundam. 1982, 21, 333. M. Dietze, H. Kuhne, Chemiefasern 1969, 3, 19. H. Brod, Chem. Eng. Technol. 1988, 21, 937–942. D. D. Ravetkar, D. D. Cale, Chem. Eng. Sci. 1981, 36, 399. R. L. Denecour, A. N. Gent, J. Polym. Sci. A-2 1968, 6, 1853.

145

146

3 Polycondensation 62 S. T. Lee, J. A. Biesenberger, Polym. 63 64 65 66

67 68 69 70 71

72

73 74 75 76

77

78

79 80 81

82 83 84

Eng. Sci. 1989, 29, 782. M. Favelukis, Z. Tadmor, R. Semiat, AIChE J. 1999, 45, 691. A. Yarin, D. Lastochkin, Y. Talmon, Z. Tadmor, AIChE J. 1999, 45, 2590. I. Gestring, D. Mewes, Chem. Eng. Sci. 2002, 57, 3415. C.-T. Yang, T. G. Smith, D. I. Bigio, C. Anolick, AIChE J. 1997, 43, 1861 and 1874. K. H. Song, S. H. Lee, K. H. Park, Korean Pat. Appl. 1999-73 161, 1999. B.-G. Woo, K. Y. Choi, K. H. Song, Ind. Eng. Chem. Res. 2001, 40, 3459. ¨ler, J. F. Davidson, B. O. G. Schu Trans. Inst. Chem. Eng. 1960, 38, S105. P. Snabre, F. Magnifotcham, Eur. Phys. J. 1998, B4, 369. F. Pilati, Solid-State Polymerization, in Comprehensive Polymer Science, p. 201, G. Allen, J. C. Bevington (Eds.), Pergamon Press, Oxford, 1989. C. Gostoli, F. Pilati, G. C. Sarti, B. Di Giacomo, J. Appl. Polym. Sci. 1984, 29, 2873. F. Pilati, C. Gostoli, G. C. Sarti, Polym. Proc. Eng. 1986, 4, 303. F. K. Mallon, W. H. Ray, J. Appl. Polym. Sci. 1998, 69, 1233. F. K. Mallon, W. H. Ray, J. Appl. Polym. Sci. 1998, 69, 1775. K. Z. Yao, K. B. McAuley, D. Berg, E. K. Marchildon, Chem. Eng. Sci. 2001, 56, 4801. P. W. Morgan, Condensation Polymers: by Interfacial and Solution Methods, Wiley, New York, 1965. J. H. Bradbury, P. J. Crawford, A. N. Hambly, Trans. Faraday Soc. 1968, 65, 1338. F. MacRitchie, Trans. Faraday Soc. 1969, 65, 2503. V. Enkelmann, G. Wegner, Makromol. Chem. 1976, 177, 3177. J. Cadotte, R. King, R. Majerle, R. Petersen, J. Macromol. Sci. Chem. 1981, A15, 727. R. J. Petersen, J. Membrane Sci. 1993, 83, 81. R. G. Pearson, E. L. Williams, J. Polym. Sci. 1985, 23, 9. M. Sirdesai, K. C. Khilar, Can. J. Chem. Eng. 1988, 42, 2616.

85 S. K. Yadav, A. K. Suresh, K. C.

Khilar, AIChE J. 1990, 36, 431. 86 S. K. Yadav, K. C. Khilar, A. K.

Suresh, AIChE J. 1996, 42, 2616. 87 M. Kubo, Y. Harada, T. Kawatsu, Y.

88 89

90

91

92 93 94

95 96

97 98 99 100

101

102

103

104

105

Yonemoto, J. Chem. Eng. Jpn. 2001, 34, 1506. P. L. Mills, Chem. Eng. Sci. 1986, 41, 1045. S. K. Karode, S. S. Kulkarni, A. K. Suresh, R. A. Mashelkar, Chem. Eng. Sci. 1997, 52, 3243. S. K. Karode, S. S. Kulkarni, A. K. Suresh, R. A. Mashelkar, Chem. Eng. Sci. 1998, 53, 2649. E. D. Johnson, A study of the kinetics and mechanism of interfacial polymerization, Ph.D. dissertation, CarnegieMellon University, Pittsburgh, 1985. K. Kamide, H. Ijima, S. Matsuda, Polym. J. 1993, 25, 1113. L. J. Gonzalez-Ortiz, J. M. Asu´a, Macromolecules 1996, 29, 4520. J. Huang, J.-P. Leblanc, H. K. J. Hall, J. Polym. Sci. A: Polym. Chem. 1992, 30, 345. M. Berthelot, L. Pe´au de SaintGilles, Ann. Chim. Phys. 1863, 68, 225. S. Hamann, D. H. Solomon, J. Swift, J. Macromol. Sci. Chem. 1968, A-2, 153. A. Fradet, E. Mare´chal, Adv. Polym. Sci. 1982, 43, 51. S.-A. Chen, K. C. Wu, J. Polym. Sci. A: Polym. Chem. 1982, 20, 1819. C. T. Kuo, S.-A. Chen, J. Polym. Sci. A: Polym. Chem. 1989, 27, 2793. D. Beigzadeh, S. Sajjadi, F. A. Taromi, J. Polym. Sci. A: Polym. Chem. 1995, 33, 1505. P. B. Zetterlund, W. Weaver, A. F. Johnson, Polym. React. Eng. 2002, 10, 41. E. Paatero, K. Narhi, T. Salmi, P. Nyholm, K. Immonen, Chem. Eng. Sci. 1994, 49, 3601. J. Lehtonen, T. Salmi, K. Immonen, E. Paatero, P. Nyholm, Ind. Eng. Chem. Res. 1996, 35, 3951. J. Lehtonen, K. Immonen, T. Salmi, E. Paatero, P. Nyholm, Chem. Eng. Sci. 1996, 51, 2799. O. M. O. Habib, J. Ma´lek, Coll. Czech. Chem. Commun. 1976, 41, 2724.

References 106 O. M. O. Habib, J. Ma´lek, Coll. Czech. 107 108 109 110

111

112 113 114

115

116

117

118 119

120

121

122 123

Chem. Commun. 1976, 41, 3077. J. S. Chung, Kor. J. Chem. Eng. 1989, 6, 354. K. Tanaka, A. Ozaki, J. Catalysis 1967, 8, 1. S. C. Hovenkamp, J. Polym. Sci. A: Polym. Chem. 1971, 9, 3617. J. Otton, S. Ratton, V. Vasnev, G. D. Markova, K. M. Nametov, V. I. Bakhtumov, S. V. Vinogradova, V. V. Korshak, J. Polym. Sci. A: Polym. Chem. 1989, 27, 3535. L. I. Komarova, N. N. Lapina, B. V. Lokshin, V. I. Bakhmutov, G. D. Markova, V. A. Vasnev, Bull. Acad. Sci. USSR Div. Chem. Sci. 1991, 39, 1767. J.-M. Besnoin, G. D. Lei, K. Y. Choi, AIChE J. 1989, 35, 1445. G. D. Lei, K. Y. Choi, Ind. Eng. Chem. Res. Sci. 1992, 31, 769. M. J. Barandiara´n, J. M. Asu´a, J. Polym. Sci. A: Polym. Chem. 1989, 27, 4241. E. Santacesaria, F. Trulli, L. Minervini, M. Di Serio, R. Tesser, S. Contessa, J. Appl. Polym. Sci. 1994, 54, 1371. M. Di Serio, R. Tesser, F. Trulli, E. Santacesaria, J. Appl. Polym. Sci. 1996, 62, 409. B. Apicella, M. Di Serio, L. Fiocca, R. Po, E. Santacesaria, J. Appl. Polym. Sci. 1998, 69, 2423. U. Stier, F. Ga¨hr, W. Opermann, J. Appl. Polym. Sci. 2001, 80, 2039. F. Pilati, P. Manaresi, B. Fortunato, A. Munari, V. Passalacqua, Polym. 1983, 24, 1479. S. I. Chervina, E. G. Maksimenko, R. S. Barshtein, N. V. Shabanova, A. Kh. Bulai, Yu. I. Kotov, I. Ya. Slonim, Kin. Cat. 1987, 28, 947. J. Otton, S. Ratton, V. Vasnev, G. D. Markova, K. M. Nametov, V. I. Bakhtumov, L. I. Komarova, S. V. Vinogradova, V. V. Korshak, J. Polym. Sci. A: Polym. Chem. 1988, 26, 2199. F. Leverd, A. Fradet, E. Mare´chal, Eur. Polym. J. 1987, 23, 699 and 705. A. Kh. Bulai, I. Ya. Slonim, B. M. Arshava, R. S. Barshtein, V. G.

124 125 126 127 128

129

130 131 132 133 134 135 136 137

138 139

140 141

142

143 144

Gorbunova, Kin. Cat. 1990, 32, 541. J. Otera, N. Dan-oh, H. Nozaki, J. Org. Chem. 1991, 56, 5307. L. H. Buxbaum, Angew. Chem. Int. Ed. Engl. 1968, 7, 182. A. V. Popoola, J. Appl. Polym. Sci. 1991, 43, 1875. H. K. Reimschuessel, Ind. Eng. Chem. Prod. Res. Dev. 1980, 19, 117. F. Pilati, P. Manaresi, B. Fortunato, A. Munari, V. Passalacqua, Polym. 1981, 22, 1566. M. Malik, V. Choudhary, I. K. Varma, J. Macromol. Sci. Rev. Macromol. Chem. Phys. 2000, C40, 139. J. Grobelny, Polymer 1995, 36, 4215. Z. Ordelt, Makromol. Chem. 1963, 63, 153. Z. Ordelt, Makromol. Chem. 1963, 68, 166. A. Fradet, E. Mare´chal, Makromol. Chem. 1982, 183, 319. K. Ravindranath, R. A. Mashelkar, J. Appl. Polym. Sci. 1982, 26, 471. K. Ravindranath, R. A. Mashelkar, J. Appl. Polym. Sci. 1984, 29, 437. T. Yamada, Y. Imamura, O. Makimura, Polym. Eng. Sci. 1985, 25, 788. C.-K. Kang, B. C. Lee, D. W. Ihm, D. A. Tremblay, J. Appl. Polym. Sci. 1997, 63, 163. Z. Yao, W. H. Ray, AIChE J. 2001, 47, 401. H. Nava, Polyesters, Unsaturated, in Kirk-Othmer Encyclopedia of Chemical Technology (on-line version), John Wiley, New York, 2004; http://www. mrw.interscience.wiley.com./kirk/ articles/polysell.a01/sect4.html Y. S. Yang, J. P. Pascault, J. Appl. Polym. Sci. 1997, 64, 133 and 147. F. N. Jones, Alkyd Resins, in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim, 2001. N. Tuck, The Chemistry of Alkyds, in Waterborne and Solvent Based Alkyds and their End User Applications, chap. I, John Wiley, New York, 2000. R. H. Kienle, Ind. Eng. Chem. 1949, 41, 726. M. Jonason, J. Appl. Polym. Sci. 1960, 11, 129.

147

148

3 Polycondensation ¨ hling, K. Schebesta, U. 145 S. Ku

146

147 148

149

150

151 152

153 154

155 156 157 158 159

160 161 162 163

164

Hucks, H. Schuchardt, R. Bachmann, T. Fischer, K. ¨ber, F. F. Rhiel, G. Zaby, Kohlgru EP 0 798 329 to Bayer-AG, 2001. P. J. McCloskey, A. Nisoli, W. W. Reilly, US Pat. 6 399 739 to General Electric Company, 2002. K. Komiya, Y. Kawakami, H. Okamoto, US Pat. 5 596 067 to Asahi, 1997. ¨llerO. Hahn, D. Mooney, F. Mu Plathe, K. Kremer, J. Chem. Phys. 1999, 111, 6061. K. Tominari, A. Kanezawa, T. Sakashita, K. Miura, T. Shimoda, US Pat. 5 525 701 to General Electric Company, 1996. ¨ hling, U. Hucks, G. Zaby, C. S. Ku Wulff, US Pat. 5 652 313 to BayerAG, 1997. K. Komiya, S. Fukuoka, EP 0 892 001 to Asahi, 2002. B.-G. Woo, K. Y. Choi, W. H. Song, S. H. Lee, J. Appl. Polym. Sci. 2001, 80, 1253. S. W. Webb, AIChE J. 1993, 39, 701– 706. U. Hucks, R. Bachmann, F. F. Rhiel, T. Fischer, S. Kratschmer, WO 02/077 066 A1, 2002. A. Einhorn, Liebigs Ann. Chem. 1898, 300, 135. ¨ m, C. A. Bischoff, A. van Hedenstro Ber. 1902, 35, 3431. W. H. Carothers, F. J. van Natta, J. Am. Chem. Soc. 1930, 52, 314. J. R. Whinﬁeld, J. D. Dickson, Br. Pat. 578 079, 1946. H. Schnell, L. Bottenbruch, H. Krimm, Belg. Pat. 532 543 to Bayer AG, 1954. H. Schnell, Angew. Chem. 1956, 68, 633. D. W. Fox, Australian Pat. 221 192 to General Electric Company, 1959. D. W. Fox, US Pat. 3 153 008 to General Electric Company, 1964. D. Freytag, G. Fengler, L. Morbitzer, Angew. Chem. Int. Ed. Engl. 1991, 30, 1598. H. Schnell, The Chemistry and Physics of Polycarbonates (Polymer Reviews, Vol. 9), Interscience Publishers, New York, 1964.

165 H. Vernaleken, Polycarbonates,

166

167

168

169

170

171

172

173 174 175 176

177 178

Interfacial Synthesis, in Polymer Applications and Technology, Vol. 2, p. 65, F. Millich, C. E. Carrahers (Eds.), Marcel Dekker, New York, 1977. D. W. Fox, Polycarbonates, in KirkOthmer Encyclopedia of Chemical Technology, 3rd ed., Vol. 18, p. 479, John Wiley, New York, 1982. ¨ ller, D. Freytag, U. Grigo, P. R. Mu W. Nouvertne´, Polycarbonates, in Encyclopedia of Polymer Science and Engineering, 2nd ed., Vol. 11, pp. 651– 692, H. Mark, G. Menges, N. Bikales, C. G. Overberger, J. I. Kroschwitz (Eds.), Wiley, New York, 1988. D. C. Clagett, S. J. Shafer, Polycarbonates, in Comprehensive Polymer Science; The Synthesis, Characterization, Reactions, and Applications of Polymers, Vol. 5, p. 345, G. Allen, J. C. Bevington (Eds.), Pergamon Press, Oxford, 1989. V. Serini, Polycarbonates, in Ullmann’s Encyclopedia of Industrial Chemistry, 5th ed., p. 207, B. Elvers, S. Hawkins, G. Schulz (Eds.), VCH, Weinheim, 1992. D. G. LeGrand, J. T. Bendler (Eds.), Handbook of Polycarbonate Science and Technology, Marcel Dekker, New York, 2000. D. J. Brunelle, in Encyclopedia of Polymer Science and Engineering, 3rd ed., Vol. 7, pp. 396–426, H. Mark, N. Bikales, J. I. Kroschwitz (Eds.), Wiley, New York, 2003. P. G. Kosky, J. M. Silva, E. A. Guggenheim, Ind. Eng. Chem. Res. 1991, 30, 462. E. Aquino, W. J. Brittain, D. J. Brunelle, Polymer Int. 1994, 33, 161. P. G. Kosky, E. P. Boden, J. Polym. Sci. A: Polym. Chem. 1990, 28, 1507. J.-T. Gu, C. S. Wang, J. Appl. Polym. Sci. 1992, 44, 849. C. Wulff, H. Kauth, G. Weymans, Z. Kricsfalussy, W. Alewelt, J. Heuser, EP 0 517 044 to Bayer AG, 1995. Ya. I. Korenman, A. A. Gorokhov, J. Appl. Chem. USSR, 1974, 46, 2571. J. M. Silva, P. K. Kosky, Ind. Eng. Chem. Res. 1991, 30, 468–474.

References 179 P. L. Mills, Chem. Eng. Sci. 1985, 41, 180 181 182

183 184 185 186 187

188

189

190

191 192 193 194

195

196

197

198

2939. S. Munjal, Polym. Eng. Sci. 1994, 34, 93–101. W. H. Carothers, G. T. Berchet, J. Am. Chem. Soc. 1930, 52, 5289. J. N. Weber, Aromatic, Polyamides in Kirk-Othmer Encyclopedia of Chemical Technology (on-line version), John Wiley, New York, 2002; http://www. mrw.interscience.wiley.com./kirk/ articles/genewebe.a01/sect10-fs.html N. Ogata, Makromol. Chem. 1960, 42, 52. N. Ogata, Makromol. Chem. 1961, 43, 117. ¨ nchen) F. Wiloth, Z. Phys. Chem. (Mu 1955, 4, 66. C. Giori, B. T. Hayes, J. Polym. Sci. A: Polym. Chem. Ed. 1970, 8, 335 and 351. K. Tai, H. Teranishi, Y. Arai, T. Tagawa, J. Appl. Polym. Sci. 1979, 24, 211. D. D. Steppan, M. F. Doherty, M. F. Malone, J. Appl. Polym. Sci. 1987, 33, 2333. M. A. Schaffer, K. B. McAuley, M. F. Cunningham, E. K. Marchildon, Ind. Eng. Chem. Res. 2003, 42, 2946. M. A. Schaffer, E. K. Marchildon, K. B. McAuley, M. F. Cunningham, Polym. Eng. Sci. 2003, 43, 639. F. K. Mallon, W. H. Ray, J. Appl. Polym. Sci. 1998, 69, 1213. I. K. Miller, J. Polym. Sci. A: Polym. Chem. Ed. 1976, 14, 1403. H. K. Reimschuessel, J. Polym. Sci., Macromol. Rev. 1977, 12, 65. H. K. Reimschuessel, G. J. Dege, J. Polym. Sci.: Polym. Chem. Ed. 1970, 8, 3265. D. D. Steppan, M. F. Doherty, M. F. Malone, J. Appl. Polym. Sci. 1991, 42, 1009. M. A. Schaffer, E. K. Marchildon, K. B. McAuley, M. F. Cunningham, J. Macromol. Sci. Rev., Macromol. Chem. Phys. 2000, C40(4), 233. H. K. Reimschuessel, K. Nagasubramanian, Chem. Eng. Sci. 1972, 1119. A. Kumar, S. K. Gupta, J. Macromol. Sci. Rev., Macromol. Chem. Phys. 1986, C26(2), 183.

199 H. Jacobs, C. Schweigman, in

200 201

202

203

204 205 206 207

208 209 210 211 212

213 214 215 216 217 218 219

Proceedings of the 5th European/2nd International Symposium on Chemical Reaction Engineering, Amsterdam, May 2–4, p. B7.1, 1972. German Pat. 1 167 021 to Vereinigte Glanzstoﬀe Fabriken, 1962. C. S. Wang, N. X. Huang, Z. L. Tang, K.-D. Hungenberger, Macromol. Theory Simul. 2002, 11, 687. M. Albert, I. Hahnenstein, H. Hasse, G. Maurer, AIChE J. 1996, 42, 1741. L. L. Williams, Amino Resins and Plastics, in Kirk-Othmer Encyclopedia of Chemical Technology (on-line version), John Wiley, New York, 2002; http:// www.mrw.interscience.wiley.com./ kirk/articles/aminwill.a01/sect1.html J. I. de Jong, J. de Jonge, Rec. Trav. Chem. 1952, 71, 643. J. I. de Jong, J. de Jonge, Rec. Trav. Chem. 1952, 71, 661. J. I. de Jong, J. de Jonge, Rec. Trav. Chem. 1953, 72, 139. A. F. Price, A. R. Cooper, A. S. Meskin, J. Appl. Polym. Sci. 1980, 25, 2597. A. Kumar, A. Sood, J. Appl. Polym. Sci. 1990, 40, 1473. N. Landqvist, Acta Chem. Scand. 1956, 10, 244. J. I. de Jong, J. de Jonge, Rec. Trav. Chem. 1953, 72, 213. J. I. de Jong, J. de Jonge, Rec. Trav. Chem. 1953, 72, 207. R. M. Rammon, W. E. Johns, J. Magnuson, A. K. Dunker, J. Adhes. 1986, 19, 115. M. G. Kim, L. W. Amos, Ind. Eng. Chem. Res. 1990, 29, 208. M. G. Kim, J. Polym. Sci. A: Polym. Chem. Ed. 1999, 37, 995. M. G. Kim, J. Appl. Polym. Sci. 2000, 75, 1243. M. G. Kim, J. Appl. Polym. Sci. 2001, 80, 2800. S. Katusˇca´k, M. Toma´sˇ, O. Schiessl, J. Appl. Polym. Sci. 1981, 26, 381. C. Soulard, C. Kamoun, A. Pizzi, J. Appl. Polym. Sci. 1999, 72, 277. T. J. Pratt, W. E. John, R. M. Rammon, W. L. Plagemann, J. Adhes. 1985, 17, 275.

149

150

3 Polycondensation 220 A. K. Dunker, W. E. John, R. M.

221 222 223

224 225 226

227

228 229

230

231 232

233

234 235 236 237

Rammon, B. Farmer, S. J. Johns, J. Adhes. 1986, 19, 153. K. Siimer, T. Pehk, P. Christjanson, Macromol. Symp. 1999, 148, 149. G. E. Myers, Holzforschung 1990, 44, 117. R. Nastke, K. Dietrich, G. Reinisch, G. Raﬂer, J. Macromol. Sci.–Chem. 1986, A23, 579. M. Gordon, A. Alliwell, T. Wilson, J. Appl. Polym. Sci. 1966, 10, 1153. B. Tomita, J. Polym. Sci. A: Polym. Chem. Ed. 1977, 15, 2347. M. L. Scheepers, P. J. Adriansens, J. M. Gelan, R. A. Carleer, D. J. Vanderzande, N. K. de Vries, P. M. Brandts, J. Polym. Sci. A: Polym. Chem. Ed. 1985, 33, 915. S. K. Gupta, A. K. Gupta, A. K. Ghosh, A. Kumar, Modeling of Melamine Formaldehyde Polymerization, in Frontiers in Chemical Engineering, p. 824, L. K. Doraiswamy, R. A. Mashelkar (Eds.), Wiley Eastern, Delhi, 1984. A. Kumar, J. Appl. Polym. Sci. 1987, 34, 1367 and 1383. P. W. Koft, Phenolic Resins, in KirkOthmer Encyclopedia of Chemical Technology (on-line version), John Wiley, New York, 2003; http://www. mrw.interscience.wiley.com./kirk/ articles/phenkoft.a01/abstract-fs.html P. M. Frontini, T. R. Cuadrado, R. J. J. Williams, Polymer 1982, 23, 267. A. Kumar, S. K. Gupta, N. Somu, B. Kumar, Polymer 1982, 23, 215. S. Ishida, Y. Tsutsumi, K. Kaneko, J. Polym. Sci. A: Polym. Chem. Ed. 1981, 19, 1609. A. A. Zavitsas, R. D. Beaulieu, J. R. LeBlanc, J. Polym. Sci. A: Polym. Chem. Ed. 1968, 6, 2541. S. K. Gupta, A. Kumar, P. K. Pal, Polymer 1981, 22, 1691. P. K. Pal, A. Kumar, S. K. Gupta, Brit. Polym. J. 1980, 12, 21. M. G. Kim, L. W. Amos, E. E. Barnes, Ind. Eng. Chem. Res. 1990, 29, 2032. G. Astarloa-Aierbe, J. M. Echeverria, J. L. Egiburu, M. Ormaetxea, I. Mondragon, Polymer 1998, 39, 3147.

238 C. C. Riccardi, G. Astarloa-Aierbe,

239

240 241

242

243 244 245 246 247

248 249

250 251 252

253

254 255 256 257

J. M. Echeverria, I. Mondragon, Polym. 2002, 43, 1631. L. B. Manfredi, C. C. Riccardi, O. de la Osa, A. Va´zquez, Polym. Intl. 2001, 50, 796. E. Kumpinsky, Ind. Eng. Chem. Res. 1994, 33, 285. H. Lee, K. Neville, Handbook of Epoxy Resins, 1st ed., McGraw-Hill, New York, 1967. N. S. Enikolopyan, M. A. Markevitch, L. S. Sakhonenko, S. Z. Rogovina, V. G. Oshmyan, J. Polym. Sci. A: Polym. Chem. Ed. 1982, 20, 1231. J. G. Gao, J. Appl. Polym. Sci. 1993, 48, 237. P. A. Oyanguren, R. J. J. Williams, Polym. 1992, 33, 2736. M. E. Smith, H. Ishida, Macromolecules 1994, 27, 2701. O. Baeyer, Angew. Chem. 1947, A59, 257. H. Ulrich, Urethane Polymers, in Kirk-Othmer Encyclopedia of Chemical Technology (on-line version), John Wiley, New York, 2000; http://www. mrw.interscience.wiley.com./kirk/ articles/uretulri.a01/sect2.html G. Oertel, Polyurethane Handbook, 2nd ed., Hanser, Mu¨nich, 1993. C. W. Macosko, RIM, fundamentals of Reaction Injection Molding, 1st ed., Hanser, Mu¨nich, 1989. A.-C. Draye, J. J. Tondeur, J. Mol. Cat. A. Chem. 1999, 138, 135. K. Schwetlick, R. Noack, F. Stebner, J. Chem. Soc. Perkin Trans. 2, 1994, 599. M. Sˇpı´rkova´, M. Kubı´n, P. Sˇpacek, I. Krakovsky´, K. Dusˇek, J. Appl. Polym. Sci. 1994, 52, 895. M. Sˇpı´rkova´, M. Kubı´n, P. Sˇpacek, I. Krakovsky´, K. Dusˇek, J. Appl. Polym. Sci. 1994, 53, 1435. K. Schwetlick, R. Noack, J. Chem. Soc. Perkin Trans. 2 1995, 395. M. C. Pannone, C. W. Macosko, J. Appl. Polym. Sci. 1987, 34, 2409. M. C. Pannone, C. W. Macosko, Polym. Eng. Sci. 1988, 28, 660. T. A. Speckhard, J. A. Miller, S. L. Cooper, Macromolecules 1986, 19, 1558.

References 258 J. A. Miller, T. A. Speckhard, S. L. 259

260 261 262 263 264 265

266 267

268 269

270 271 272 273

274 275 276

Cooper, Macromolecules 1986, 19, 1568. J. A. Miller, T. A. Speckhard, J. G. Homan, S. L. Cooper, Polymer 1987, 28, 758. L. C. Case, J. Polym. Sci. 1960, 45, 397. J. Scanlan, J. Polym. Sci. 1960, 43, 501. P. L. Mills, Comp. Chem. Eng. 1986, 10, 399. M. R. P. F. N. Costa, J. Villermaux, Ind. Eng. Chem. Res. 1988, 27, 421. J. Abate, W. Whitt, Oper. Res. Lett. 1992, 12, 245. A. Brandolin, M. Asteasuain, C. Sarmoria, A. Lo´pez-Rodriguez, K. S. Whiteley, B. del Amo Ferna´ndez, Polym. Eng. Sci. 2001, 41, 1156. S. I. Kuchanov, S. V. Korolev, M. G. Slin’ko, Polym. J. 1983, 15, 785. S. I. Kuchanov, S. V. Korolev, M. G. Slin’ko, Vysokomol. Soyed. 1982, A42, 2160; Polym. Sci. USSR 1982, 24, 2477. M. Gordon, G. R. Scantlebury, Proc. Roy. Soc. 1966, A292, 380. K. Dusˇek, M. Gordon, S. B. RossMurphy, Macromolecules 1978, 11, 236. D. Stauffer, Pure Appl. Chem. 1981, 53, 1479. M. Gordon, J. A. Torkington, Pure Appl. Chem. 1981, 53, 1461. S. I. Kuchanov, Dokl. Akad. Nauk. SSSR 1987, 294, 633. S. I. Kuchanov, S. V. Korolev, S. V. Panyukov, Adv. Chem. Phys. 1988, 72, 115. W. H. Abraham, Ind. Eng. Chem. Fundam. 1963, 2, 221. H. Kilkson, Ind. Eng. Chem. Fundam. 1964, 3, 281. H. Kilkson, Ind. Eng. Chem. Fundam. 1968, 7, 354.

277 S. I. Kuchanov, Methods of Kinetic

278

279 280 281

282 283

284 285 286 287 288

289 290

291

Calculations in Polymer Chemistry, Khimiya, Moscow, 1978 (in Russian). S. I. Kuchanov, Ye. S. Povolotskaya, Vysokomol. Soyed. 1982, A24, 2190; Polym. Sci. USSR 1982, 24, 2512. K. Dusˇek, J. Sˇomva´rsky, Polym. Bull. 1983, 13, 313. J. Mikesˇ, K. Dusˇek, Macromolecules 1982, 15, 93. S. I. Kuchanov, A. G. Kholostyakov, J. Polym. Sci. A: Polym. Chem. 1999, 37, 2145. M. R. P. F. N. Costa, R. C. S. Dias, Chem. Eng. Sci. 1994, 49, 491. R. Courant, D. Hilbert, Partial Diﬀerential Equations (Methods of Mathematical Physics, Vol. II), pp. 97– 105, Interscience Publishers, New York, 1962. L. M. Pis’men, S. I. Kuchanov, Polym. Sci. USSR 1971, 13, 890. S. I. Kuchanov, L. M. Pis’men, Polym. Sci. USSR 1972, 14, 147. S. I. Kuchanov, L. M. Pis’men, Polym. Sci. USSR, 1972, 14, 985. H. Galina, A. Szustalewicz, Macromolecules 1989, 22, 3124. F. D. Magalha˜es, M. R. P. F. N. Costa, in 3rd Berlin International Workshop on Polymer Reaction Engineering, p. 311, K. H. Reichert, W. Geiseler (Eds.), DECHEMA Monographs, VCH, Weinheim, 1989. S. I. Kuchanov, T. V. Zharnikov, J. Stat. Phys. 2003, 111, 1273. M. R. P. F. N. Costa, R. C. S. Dias, in 6th International Workshop on Polymer Reaction Engineering, DECHEMA Monographs, Vol. 134, pp. 197–205, K. H. Reichert, H.-U. Moritz (Eds.), Wiley-VCH, Weinheim, 1998. M. R. P. F. N. Costa, J. Villermaux, Ind. Eng. Chem. Res. 1989, 28, 702.

151

153

4

Free-radical Polymerization: Homogeneous1 Robin A. Hutchinson

Free-radical polymerization (FRP) is one of the most important commercial processes 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 monomers can be easily copolymerized via FRP, leading to the preparation of an endless range of copolymers with properties dependent on the proportion of the incorporated comonomers. This chapter will provide an overview of the kinetics and mechanisms, and the techniques used to construct mathematical representations of bulk and solution FRP. The description will also serve as a good base for the suspension 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 elsewhere (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. Homopolymer 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 rubber. Acrylic- and methacrylic-based polymers. Poly(methyl methacrylate) (PMMA), due to its transparency and weatherability, is used extensively in signs, lighting ﬁxtures, 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 conﬁgurations ranging from tubular to well-mixed tanks (and everything in between) in processes that may be continuous, batch, or semi-batch. FRP kinetics together with reactor 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 properties such as branch points and branch lengths. The processing and end-use properties 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 fundamental kinetic mechanisms of free-radical polymerization to these distributed characteristics.

4.2

Chain Initiation

Free-radical polymerization, like other chain growth mechanisms, involves the sequential 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 reactions), and is considered ‘‘dead’’. These dead chains have a residence time of minutes 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 thermal or photochemical homolytic cleavage of covalent bonds. Commercial initiators include azo and peroxy compounds. The driving force for the dissociation of azo

4.2 Chain Initiation

155

initiators is the formation of the stable nitrogen molecule and resonance stabilized tertiary radicals, as shown for 2,2 0 -azobisisobutyronitrile (AIBN) in Scheme 4.1.

N N

H3C

N2

CH3

+

2 C

CH3

CH3

CH3

CH3 Scheme 4.1.

CN

CN

CN

Decomposition of AIBN.

Many types of peroxides (R-O-O-R 0 ) are also utilized, including diacyl peroxides, peroxydicarbonates, peroxyesters, dialkyl peroxides, and inorganic peroxides such as persulfate, the latter being used mainly in water-based systems. The rate of peroxide decomposition as well as the subsequent reaction pathway is greatly aﬀected by the nature of the peroxide chemical structure, as illustrated for tert-butyl peroxyesters in Scheme 4.2. Pathway (a), the formation of an acyloxy and an alkoxy radical via single bond scission, is favored for structures in which the carbon atom in the a-position to the carbonyl group is primary (for example, tert-butyl peroxyacetate, R ¼ CH3 ). Pathway (b), concerted two-bond scission, occurs for secondary and tertiary peroxyesters (for example, tert-butyl peroxypivalate, R ¼ C(CH3 )3 ) [1, 2]. The tert-butoxy radical formed in both pathways may decompose to acetone and a methyl radical, or abstract a hydrogen atom to form tert-butanol. The exact nature of the decomposition pathway is important, as it plays a role in the eﬃciency of the primary radicals in initiating new polymer chains. When an initiator decomposes, the primary radicals are nearest neighbors surrounded by a ‘‘cage’’ of other molecules through which they must diﬀuse to escape from each other before they recombine. Once one radical leaves the cage it is extremely unlikely that the pair will encounter each other again. In-cage reactions for pathway (a) in Scheme 4.2 will lead to the re-formation of the original initiator molecule, while the in-cage reactions for pathway (b) lead to the formation of an ether which is not capable of forming free radicals by another dissociation step.

R

O

CH3

O O

+O

(a) CH3

CH3

CH3

O R

O

(b) CH3

. R

CH3

+ CO2 +

CH3

Decomposition of tert-butyl peroxyesters. The preferred pathway depends on the nature of the R substituent. Scheme 4.2.

In general, peroxide decomposition can yield both carbon- and oxygen-centered primary radicals that add to the carbon–carbon double monomer bond to form a new propagating chain, abstract hydrogen atoms from other molecules (including

O

CH3 CH3

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 endgroup structures of the polymeric products, aﬀecting ﬁnal properties such as thermal 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 primary 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.

f ¼

Initiation Rate of Propagating Chains nðRate of Initiator DisappearanceÞ

ð1Þ

The 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 additional mechanism capable of forming primary radicals at higher temperatures, as discussed for styrene (see Section 4.3.1.3). Other initiation systems are also available, and bear a brief mention at this point. Initiators with multiple peroxide linkages ðn > 2Þ have been the subject of recent academic study [4]. Photoinitiators that produce radicals by ultraviolet irradiation are commonly used to initiate crosslinking 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 redox (reduction–oxidation) process is often used to initiate chains in emulsion polymerization (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

4.3.1

Homopolymerization

Many commercial polymers, including polystyrene, PMMA, and LDPE, are synthesized via homogeneous free-radical polymerization of a single monomer. Homopolymer properties are controlled by average chain length and chain-length distribution as well as, in some cases, structural characteristics such as branching level. 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. 4.3.1.1

k

Initiator Decomposition

d I  →2 f I ∗

Chain Initiation

i I ∗ + M  → P1

Chain Propagation

Pn + M → Pn +1

k

kp

Chain Termination k

By Combination

tc Pn + Pm  → Dn+m

By Disproportionation

td Pn + Pm  → Dn + Dm

k

Chain Transfer k mon

To Monomer

tr Pn + M  → Dn + M *

k mon

i M * + M  → P1

k sol

To Solvent or Agent Scheme 4.3.

tr Pn + S → Dn + S *

k sol

i S * + M → P1

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) unimolecularly decomposes (with rate coeﬃcient kd ) to form two primary radicals (I ) with eﬃciency f. Chain initiation occurs when the primary radical adds to monomer M, and chain growth continues via successive addition of monomer units to the radical center (chain propagation, with rate coeﬃcient k p ). Bimolecular coupling 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 (termination by disproportionation, k td ) dead polymer chains. Chain stoppage may also occur via a transfer mechanism, where the growing radical abstracts a weakly bonded

157

158

4 Free-radical Polymerization: Homogeneous

atom (usually hydrogen) from monomer or other molecules (solvent or chaintransfer 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 termination, 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 reaction system. The nature of this diﬀusion control makes termination the most complex reaction in the polymerization process since the apparent rate coeﬃcient can vary greatly with system conditions such as monomer conversion and solution viscosity (see Section 4.3.3). Although termination may also exhibit some chainlength dependence, most engineering treatments of FRP neglect this complex dependence. 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

ð2Þ

Chain Initiation

R init ¼ 2f kd ½I

ð3Þ

Chain Propagation

R p ¼ k p ½M½Ptot

ð4Þ 2

Chain Termination

R term ¼ ðk tc þ k td Þ½Ptot ¼ k t ½Ptot

Chain Transfer

R tr ¼ ðktrmon ½M þ ktrsol ½SÞ½Ptot

2

ð5Þ ð6Þ

Here Ptot represents the concentration of all polymer radicals in the system [Eq. (7)].

½Ptot ¼

y X 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 t ¼ k tc þ k td ;

d¼

k td k tc þ k td

ð8Þ

The following assumptions are widely accepted and usually valid in FRP systems:

4.3 Polymerization Mechanisms and Kinetics

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 conditions [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 expression for total radical concentration [Eq. (9)].

½Ptot ¼

R init kt

1=2

¼

2f kd ½I kt

1=2 ð9Þ

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 polymerization (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.

1=2

R pol

2f kd ½I ¼ k p ½M½Ptot ¼ k p ½M kt

u¼

Rp k p ½M ¼ R term þ R tr k t ½Ptot þ ktrmon ½M þ ktrsol ½S

DPninst ¼

k p ½M ðk td þ 0:5k tc Þ½Ptot þ ktrmon ½M þ ktrsol ½S

ð10Þ

ð11Þ

ð12Þ

The diﬀerence between Eqs. (11) and (12) arises because termination by combination 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 system at the same instant. Table 4.1 lists the range of concentration and rate coeﬃcient values typically encountered in homogeneous free-radical polymerization systems at low conversion. These can be combined with Eqs. (9)–(12) to illustrate the tradeoﬀs involved be-

159

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 ] f k p [L mol1 s1 ] k t [L mol1 s1 ][a] ktrmon =k p ktrsol =k p [I] [mol L1 ] [M] [mol L1 ] [S] [mol L1 ]

106 –104 0.4–0.9 10 2 –10 4 10 6 –10 8 106 –104 106 –103 104 –102 1–10 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 second, 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 calculated 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 approached in homogeneous FRP by reducing rates of polymerization to extremely low levels. For most systems both termination and transfer events play an important role in controlling polymer MW. To produce high MW polymer (DPn of 10 2 –10 4 ) it is necessary to keep total radical concentration low, so that [Ptot ] is of the order 108 –106 mol L1 . This dictates 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 polymer 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

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 increasing 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 system, 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 system. 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 complicated 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 necessitate 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 controlling rate and MW, and are very useful for a qualitative examination of FRP systems. Kinetic Coeﬃcients The coupling of polymer MW and polymerization rate is further illustrated via rearrangement of Eq. (12) to give Eq. (13). 4.3.1.2

1 0:5k tc ½Ptot k td ½Ptot ktrmon ktrsol ½S ¼ þ þ þ inst k p ½M k p ½M k p ½M kp DPn ¼

0:5k tc R pol kp2 ½M 2

þ

k td R pol kp2 ½M 2

þ

ktrmon ktrsol ½S þ k p ½M kp

ð13Þ

R 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 resolve the quantities into estimates for individual rate coeﬃcients. Even the estimation of the ratio k 2p =k t from R pol requires independent knowledge of initiator characteristics 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 considerable scatter in rate coeﬃcients reported in the literature (for example, Ref. 7), especially for individual rate coeﬃcients [8, 9]. Yet individual values, and knowledge of how they vary with temperature, are required for model development and an accurate 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

161

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 generating 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 at t ¼ 0.

½I ¼ ½I0 expðkd tÞ

ð15Þ

The 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

ð16Þ

The 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 during 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 energies 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 dependence 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 measuring 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. Decomposition kinetics are summarized for a wide range of initiators in the Polymer Handbook [7] and in trade literature available from commercial suppliers. Of special 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 pathway and initiator eﬃciency [1, 2]. Propagation In general, chain growth or propagation proceeds in a highly selective 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 competing transfer and termination events.

4.3 Polymerization Mechanisms and Kinetics

R1 CH2

R1

+

C R2

Scheme 4.4.

R1 CH2 C

H2C R2

R2

R1 CH2

C R2

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 exclusion chromatographic (SEC) analysis of the resulting polymer [10], has greatly improved the reliability of k p data. In this technique, a mixture of monomer and photoinitiator 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

ð17Þ

Since radicals have a certain probability of surviving the laser ﬂash and of terminating 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, illustrate the good agreement between facilities (typically 10–20%), and make recommendations 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 arising from additional secondary mechanisms that occur (see Section 4.3.1.3). What

163

4 Free-radical Polymerization: Homogeneous 1.2

2DP0

1 wt log(MW)

164

0.8

DP0

0.6 0.4 0.2 0 3.5

4

4.5

5

5.5

6

6.5

Log MW

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. Fig. 4.1.

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 methacrylates at 50 C increase with increasing size of the alkyl ester group (methyl to dodecyl) 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 to 12 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 [kJ molC1 ]

DVp [cm 3 molC1 ]

Ap [L molC1 sC1 ]

Ethylene Styrene Methyl methacrylate Butyl methacrylate Dodecyl methacrylate Glycidyl methacrylate Cyclohexyl methacrylate 2-Hydroxypropyl methacrylate Vinyl acetate Methyl acrylate Butyl acrylate Dodecyl acrylate

34.3 32.5 22.4 22.9 21.0 22.9 23.0 20.8

27.0 12.1 16.7 16.5 16.0 15.0 16.2 n.d.

1:88 10 7 4:27 10 7 2:67 10 6 3:78 10 6 2:50 10 6 6:19 10 6 6:29 10 6 3:51 10 6

54 238 648 757 995 1230 1204 1504

20.7 17.7 17.4 17.0

10.7 11.7 n.d. 11.7

1:47 10 7 1:66 10 7 1:81 10 7 1:79 10 7

6625 22 900 27 900 32 000

[a] All

values taken from Ref. 15; n.d., not determined.

k p at 50 C/1 atm [L molC1 sC1 ]

4.3 Polymerization Mechanisms and Kinetics

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 kinetics 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 coeﬃcient with the structural characteristics of the propagating radicals and monomers is receiving attention. While some progress has been made in relating activation 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-laserinduced 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. Nonetheless, 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 [kJ molC1 ]

DVt [cm 3 molC1 ]

At [L molC1 sC1 ]

k t at 50 C/1 atm [L molC1 sC1 ]

Ethylene Styrene Methyl methacrylate Butyl methacrylate Dodecyl methacrylate Vinyl acetate Methyl acrylate Butyl acrylate Dodecyl acrylate

4.6 6.5 4.1 4.1[b] 4.1[b] – 6.7 4.0 1.7

15.6 14 15 15[b] 15[b] – 20 16 21

1:6 10 9 2:2 10 9 4:3 10 8 8:5 10 7 2:8 10 7 – 6:0 10 9 5:1 10 8 2:1 10 7

2:9 10 8 2:0 10 8 9:4 10 7 1:9 10 7 6:2 10 6 (5–50) 10 7 [c] 5:1 10 8 1:2 10 8 1:1 10 7

[a] All

values taken from Ref. 15, unless otherwise indicated. equal to MMA value. [c] From Ref. 7. [b] Assumed

165

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 termination rate. Note the large diﬀerence in magnitude between values for styrene, 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 architecture of the polymer formed, and thus some of its properties. Polymer polydispersity (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 unsaturated end group that may undergo further reaction.

R1

R1 CH2

+

C R2

C R2

CH2

R1

R1

CH2 C

C

R2

R2 Scheme 4.5.

R2 R1

R1 CH2 CH

CH2

+

C

CH

R2

Free-radical chain termination by combination and disproportionation.

The relative importance of termination by disproportionation versus combination 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 monomers 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 reduction 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

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 examining 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 concentration 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 ½S ¼ þ tr DPn ðDPn Þ0 k p ½M

ð18Þ

A 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 absence of transfer agent. The form of this equation leads to the deﬁnition of a chaintransfer constant C for each species, including monomer, in Eq. (19). Ctrsol ¼

ktrsol ; kp

Ctrmon ¼

ktrmon kp

ð19Þ

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 transfer. In other systems, a small amount of chain-transfer agent (CTA) is added speciﬁcally 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 ratio of ½S=½M will cause a corresponding drift in polymer MW. In such cases careful control of CTA addition to the system is required. If it is added at higher concentrations, 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

167

168

4 Free-radical Polymerization: Homogeneous Tab. 4.4.

Transfer constants (k trsol =k p ) to solvents and transfer agents at 50 C.[a]

Benzene Toluene Ethyl acetate Triethylamine CCl 4 1-Butanethiol

Styrene

Methyl methacrylate

Methyl acrylate

Vinyl acetate

Ethylene[b]

2 106 1 105 5 104 5 104 1 102 20

4 106 2 105 1 105 8 104 2 104 0.6

2 105 1 104 6 105 4 102 2 104 1.5

1 104 2 103 2 104 4 102 0.8 50

9 104 [b] 1:3 102 [b] 5 103 [b] 1:8 102 [b] 1.0[b] 6.0[b]

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]. [a] Representative

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 eﬀective transfer agents than those that produce secondary radicals. Halomethanes and thiols, used as MW-controlling agents due to their high transfer rates, react 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 absence 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

CH2

O

O

C CH3

C

+

H Scheme 4.6.

O H2C

C

O O

C CH3 CH2

O

C CH3

CH2

+

H Free-radical chain transfer to monomer (vinyl acetate).

O H2C

C H

C CH2

4.3 Polymerization Mechanisms and Kinetics Tab. 4.5.

Transfer constants (k trmon =k p ) to monomer.[a]

Styrene Methyl methacrylate Butyl acrylate Vinyl acetate Ethylene[b]

C trmon

E tr C E p [kJ molC1 ]

Reference

5 105 2 105 6 105 2 104 4 104 [b]

– 23.7 15.2 – 43.9

7 23 24 7 25

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. [a] Representative

the range of values found in the literature. Ctrmon increases with temperature, with (Etrmon Ep ) in the range of 10–40 kJ mol1 . 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 operating conditions. 4.3.1.3

Thermal initiation Primary radicals in most FRP systems are generated by scission 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 initiation is suﬃcient to make an industrially viable process [26]. Acrylates and methacrylates have also been reported to undergo thermal polymerization, but at a significantly slower rate than styrene. The mechanism behind the thermal polymerization of styrene is still under debate, 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 styrene molecule to form two radicals that can initiate polymerization (Scheme 4.7). This complex mechanism can be approximated by a third-order dependency on styrene concentration [Eq. (20)] [27]. k therm

3M ! 2I R therm ¼ k therm ½M 3 ;

k therm

L2 mol 2 s

¼ 2:2 10 5 expð114:8=RTÞ

ð20Þ

169

170

4 Free-radical Polymerization: Homogeneous

+

CH CH

+

+

Scheme 4.7.

Thermal initiation of styrene.

This expression from the mid-1970s has stood the test of time, although the preexponential factor has been increased by a factor of 2–3 to ﬁt 260–340 C data obtained in a more recent study [26]. Retardation and inhibition Some substances retard or suppress free-radical polymerization by reacting with primary or polymer radicals to yield nonradical products or radicals that are too stable to add further monomer. By decreasing the concentration of reactive radicals in the system, polymerization rate is slowed (retardation) or stopped completely (inhibition). Phenolic inhibitors (for example, hydroquinone, monomethylhydroquinone) are commonly added to monomers at parts-per-million levels in order to prevent polymerization 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

ð21Þ

Ideally 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 expressions (Section 4.3.1.1) cannot be applied since termination is no longer the

4.3 Polymerization Mechanisms and Kinetics

Conversion

Normal Polymerization

Inhibition

a

b

Retardation

Time Conversion–time plots for normal, retarded, and inhibited free-radical polymerizations. Retardation cases a and b are described in the text. Fig. 4.2.

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 transfer (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 balances 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 polymerization (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

tr Pn + S → Dn + S *

k sol

i S *+ M → P1

k sol

t 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 radicals that are generally unreactive. However, subsequent monomer addition can occur in some systems; this copolymerization forms polymeric peroxides that aﬀect the thermal stability of the ﬁnal polymer product. Good practice requires the removal of air by sparging or freeze–thaw cycles before a reaction is started, and exclusion of air during polymerization by operating under an inert atmosphere (for example, nitrogen) or reﬂuxing solvent.

171

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)]. kp

Pn þ M Ð Pnþ1 ; kdep

R dep ¼ kdep ½Ptot

ð22Þ

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 represents the entropy change upon propagation; polymerization can only occur spontaneously when DGp is negative. With most common monomers, depropagation is negligible at typical reaction temperatures. For some systems, however, the enthalpy and entropy terms for propagation are delicately balanced so that depropagation 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 kpeﬀ ¼ k p

ð23Þ

kdep ½M

Examining the reaction from a thermodynamic viewpoint leads to the deﬁnition of ½Meq , the monomer concentration for a particular temperature at which the eﬀeceﬀ tive propagation rate coeﬃcient (k p ) and polymerization rate become zero [Eq. (24)]. K eq ¼

kp DGp DHp DSp 1 ¼ exp ¼ exp ¼ ½Meq kdep RT RT R

ð24Þ

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: Tc ¼

DHp DSp þ R ln½M

ð25Þ

It also provides a link between the thermodynamic and kinetic coeﬃcients according to Eqs. (26). DHp ¼ Ep Edep ;

DSp ¼ R ln

Ap A dep

þ R lnð½MÞ

ð26Þ

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 typically 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 Depropagation behavior of common monomers.

Tab. 4.6.

a-Methylstyrene Styrene Methyl methacrylate Methyl acrylate Vinyl acetate Ethylene [a] From

˚

CDH p [kJ molC1 ] [a]

Tc [ C][b]

35 73 56 80 88 102

19 335 194 394 460 577

Ref. 7. for ½M ¼ 1 mol L1 , assuming DSp ¼ 120 J mol1 K1 .

[b] Calculated

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 is 1 1 1 1 4000 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 propagation (Table 4.2), the temperature dependence of kdep may be estimated directly from

4 3.8

log[kpeff (L/mol-s)]

3.6 3.4 3.2 3 2.8 2.6 2.4 2.0

2.2

2.4

2.6

2.8

3.0

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.

3.2

3.4

3.6

173

4 Free-radical Polymerization: Homogeneous

174

these data. The corresponding values for DHp (54 kJ mol1 ) and DSp (123 J mol1 K1 ) agree well with thermodynamic estimates. The depropagation behavior of many methacrylate monomers is similar [28], and depropagation must be considered at temperatures above 120 C, especially for systems with low monomer concentration [29]. 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 component to manipulation and control of polymer structure. Branches can be formed either by intramolecular reactions, discussed in the next subsection, or intermolecular 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 structures, 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 number of long-chain branches formed [Eq. (27)]. pol

mk tr

Pn; b þ Dm; c ! Dn; b þ Pm; cþ1 ;

pol

pol

R tr ¼ k tr ½Ptot m1

ð27Þ

This reaction does not change the number of monomer units that have been polymerized 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 CH2

+

C

CH2 C

CH2

CH2

CH2

+

H

H

R1

R1 CH2 C

R1

R1

CH2

+

H2C

CH R1

CH2 C

CH2

H2C R1 CH

Scheme 4.9.

Long-chain branch formation by chain transfer to polymer.

CH2 C

CH2

4.3 Polymerization Mechanisms and Kinetics

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)].

m1 ¼

y X y X

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 branching to monomer consumption. pol

pol pol k tr x p R tr k ½Ptot m1 ¼ tr ¼ R pol k p ½Ptot ½M k p ð1 x p Þ

ð29Þ

Thus polymerizations operating at high monomer conversion will have signiﬁcantly higher branching than low-conversion systems. Second, longer polymer chains – those with more repeat units – are more likely to participate in a branching 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 compared 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 radicals such as ethylene [30–32], vinyl acetate [33–35], and acrylate [36, 37] polymerizations, but seldom occurs in styrene and methacrylate systems. Transfer to polymer 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 increases with temperature. Reaction with unsaturated polymer chains Termination by disproportionation, transfer 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 Pn; b þ Dm; c ! Pnþm; bþcþ1 ;

pol ¼ R pol p ¼ k p ½Ptot ½Dtot

ð30Þ

175

4 Free-radical Polymerization: Homogeneous

176

R1 CH2

+

C

R1

R1 H2C

CH2 C

C

R1 CH2

C

R2

R2 Long-chain branch formation by addition to a terminally unsaturated polymer chain.

Scheme 4.10.

This reaction is fundamentally diﬀerent than transfer to polymer in several respects. 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 =½M according to Eq. (31). to propagation is controlled by the ratio ½Dtot pol

pol

¼ Rp k p ½Ptot ½Dtot ¼ k p ½Ptot ½M R pol

ð31Þ

Unlike 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 highertemperature methacrylate [3] polymerizations. Reaction with multifunctional monomers Another way to introduce branching is through addition of a multifunctional monomer to the polymer system. An example 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-

O C

OCH3

C

+

C

CH2

O

O CH2CH2 O

C O

C CH2

H2C C

C

CH3

H3C

H3C

CH3 C CH2

O

CH2

O CH2CH2 O

C

O

C H3C CH3

CH3 C

O C CH2

C H3C

OCH3

+

O C H2C C

C CH2

C

C O O CH2 CH2 O

O O CH3 C C

CH2

H3C

CH3 Addition of ethylene glycol dimethacrylate to a methyl methacrylate radical. The pendent double bond is attacked by another polymer radical to form a crosslink.

Scheme 4.11.

CH2

O O H2C CH2 O O C C CH3

CH2

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 interconnected 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 properties; 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 common butyl branch in LDPE, ethyl and 2-ethylhexyl branches have been identiﬁed using 13 C NMR [30, 41, 42].

COOBu COOBu COOBu R CH

CH

CH

C

COOBu COOBu COOBu

COOBu R

CH H HC COOBu

CH

COOBu CH

C H CH

COOBu

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.

Scheme 4.12.

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 hydrogen 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 intramolecular (back-biting) mechanism is also important in acrylate polymerizations. 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– 340 C) [26]. Chain scission The radical structure formed by intra- or intermolecular transfer to polymer is less reactive than a chain-end radical. Under higher-temperature conditions, 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 CH2 C CH CH

COOBu COOBu COOBu CH

CH

COOBu

+

HC

COOBu CH2

COOBu CH

C H

COOBu COOBu

CH COOBu

CH

CH

+

COOBu COOBu H2C

C

HC

COOBu CH2

b-Scission of butyl acrylate mid-chain radical.

Scheme 4.13.

Scission events can occur in any system where mid-chain radicals are formed. However, scission is more temperature-activated than H-abstraction and thus becomes important only at elevated temperatures. The reaction is not believed to occur 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 midchain 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. Equations (32)–(34) are a network of reactions developed to treat intramolecular transfer, short-chain branch formation, and b-scission for butyl acrylate polymerization [47]. kbb

Pn ! Q n ;

R bb ¼ kbb ½Ptot

kptert

Q n þ M ! Pnþ1 þ SCB;

ð32Þ RSCB ¼

kptert ½M½Q tot

ð33Þ

0:5kb

¼ Q n ! Dn2 þ P2 0:5kb

Q n ! Pn3 þ D3¼ ;

R b ¼ kb ½Q tot

ð34Þ

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

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 reaction engineering challenge is to consider the set of mechanisms required to represent the basic rate behavior and polymer architecture of the system without introducing 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 improved 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, reactivity of a monomer in copolymerization cannot be predicted from its behavior in homopolymerization. Vinyl acetate polymerizes about 30 times more quickly than styrene (see Table 4.2), yet the product is almost pure polystyrene if the two monomers are copolymerized together in a 50:50 mixture. a-Methylstyrene cannot be homopolymerized 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 copolymerization mechanisms and kinetics. Basic Mechanisms In copolymerization, the presence of more than one type of monomer adds an extra 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 terminal 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 polymeriza4.3.2.1

179

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

Initiator Decomposition

d I  →2 f I ∗

Chain Initiation

j I ∗ + M j  → P1 j

Chain Propagation

ij Pni + M j → Pn +j 1

ki

kp

Chain Termination ktc

By Combination

ij Pni + Pmj → Dn + m

By Disproportionation

ij Pni + Pmj → Dn + Dm

ktd

Chain Transfer To Monomer

ktrmon

ij Pni + M j  → Dn + P1 j

ktrsol

To Solvent or Agent

i Pni + S → Dn + S *

k isol

j S * + M j → P1 j

Basic free-radical copolymerization mechanism, assuming terminal radical kinetics.

Scheme 4.14.

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 dependent 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 hypothesis, the consumption rates of the two monomers are written as in Eqs. (35). 1 2 ½M1 þ k p21 ½Ptot ½M1 R pol1 ¼ k p11 ½Ptot 1 2 R pol2 ¼ k p12 ½Ptot ½M2 þ k p22 ½Ptot ½M2

ð35Þ

4.3 Polymerization Mechanisms and Kinetics i where Ptot represents the concentration of all polymer radicals of type-i in the system [Eq. (36)].

i ½Ptot ¼

y X ½Pni

ð36Þ

n¼1

The ratio of the two consumption rates dictates the instantaneous composition ) of the polymer being formed [Eq. (37)]. (Fpinst i Fpinst 1 Fpinst 2

¼

1 2 R pol1 k p11 ½Ptot ½M1 þ k p21 ½Ptot ½M1 ¼ 1 2 ½M R pol2 k p12 ½Ptot ½M2 þ k p22 ½Ptot 2

ð37Þ

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 polymer 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 deﬁned as k p11 =k p12 and k p22 =k p21 . 1 ½Ptot k p21 ½M1 2 ¼ k ½M ½Ptot p12 2

¼ Fpinst 1

ð38Þ

r1 f12 þ f1 f2 þ 2 f1 f2 þ r2 f22

ð39Þ

r1 f12

Equation (39) deﬁnes the composition of the copolymer formed at any instant during polymerization, and is dependent only on the ratios of the propagation rate coeﬃcients 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 copolymer 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)].

P11 ¼

1 k p11 ½Ptot ½M1 1 1 ½M k p11 ½Ptot ½M1 þ k p12 ½Ptot 2

¼

r1 f1 r1 f1 þ f2

ð40Þ

The probability P12 that an M2 unit follows an M1 is given by Eq. (41). P12 ¼ 1 P11 ¼

f2 r1 f1 þ f2

ð41Þ

Similar expressions, Eqs. (42) and (43), can be derived for addition to radical-2:

181

182

4 Free-radical Polymerization: Homogeneous

P22 ¼

2 k p22 ½Ptot ½M2 2 2 ½M k p22 ½Ptot ½M2 þ k p21 ½Ptot 1

P21 ¼ 1 P22 ¼

¼

r2 f2 r2 f2 þ f1

f1 r2 f2 þ f1

ð42Þ

ð43Þ

These probabilities can be used to calculate NðM1 ; n i Þ, the fraction of all M1 sequences that are exactly n i units long. This is simply the probability of having ðn i 1ÞM1 M1 linkages followed by an M1 M2 linkage, according to Eq. (44). n i 1 NðM1 ; n i Þ ¼ P11 P12 ;

n 1

NðM2 ; nj Þ ¼ P22j P21

ð44Þ

Thus 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 2 P12 , and so forth. The number-average length of M1 sequences (N 1 ) is given by P11 Eq. (45) and the fraction of all M1 units contained in a sequence of length n i is n i 1 2 2 P12 ; that is, the fraction of M1 contained in isolated diads is 2P11 P12 . n i P11 N1 ¼

1 1 ¼ ; 1 P11 P12

N2 ¼

1 1 ¼ 1 P22 P21

ð45Þ

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 linkages. Thus the ratio of M1 to M2 units in the chain must equal the ratio of the respective average sequence lengths [Eq. (46)]. Fpinst 1 Fpinst 2

¼

N1 N2

ð46Þ

Substitution and rearrangement of this equation yields the polymer composition equation, Eq. (39). Thus it is possible to estimate reactivity ratios for binary copolymers 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 ). 1 2 1 2 R pol ¼ k p11 ½Ptot ½M1 þ k p21 ½Ptot ½M1 þ k p12 ½Ptot ½M2 þ k p22 ½Ptot ½M2 ! 2 2 XX 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

fri ¼

i Ptot Pi ¼ tot 2 þ Ptot Ptot

1 Ptot

ð48Þ

[Ptot ], the total radical concentration, is calculated from an overall radical balance similar to Eq. (9) and given by Eq. (49).

½Ptot ¼

R init 1=2 2f kd ½I 1=2 ¼ cop cop kt kt

ð49Þ

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). k pcop ¼

Nmon X Nmon X

k pij fri fj

ð50Þ

i¼1 j¼1

For a two-monomer system, application of the QSSA [Eq. (38)] and simpliﬁcation lead to Eq. (51). k pcop ¼

r1 f12 þ 2f1 f2 þ r2 f22 ðr1 f1 =k p11 Þ þ ðr2 f2 =k p22 Þ

ð51Þ

Using 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. 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 effects. There is a large body of published r1 –r2 data for monomer pairings, summarized 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 suﬃciently 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 system does not remain homogeneous (often observed with acid monomers). 4.3.2.2

183

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 Alkyl methacrylate Alkyl acrylate Vinyl acetate [a] Representative

Styrene

Methacrylate

Acrylate

Vinyl acetate

– 0.4 0.2 0.02

0.6 – 0.4 0.03

0.8 2.2 – 0.03

40 20 6 –

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 resulting polymers exhibit an alternating structure (for example, styrene–maleic anhydride), or the observed reactivity ratios in the system vary with monomer concentration 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 represent 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 reactivity 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 systems 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 various 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 Methacrylic acid Alkyl acrylates Acrylic acid Vinyl acetate [a]

0.058 0.058 0.058 0.058 1.0

7 11 4 8 1.0

[a] From

Ref. 7.

4.3 Polymerization Mechanisms and Kinetics 1 0.9 0.8 0.7

Fp 1

0.6 0.5 0.4 0.3 MMA-Sty MMA-MA MMA-VAc Eth-VAc

0.2 0.1 0 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 1 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 polymerization. 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, methacrylates, 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 demonstrated 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].

185

4 Free-radical Polymerization: Homogeneous

1000.0

cop

(L/mol-s)

10000.0

kp

186

100.0

10.0 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 propagation 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], however, demonstrated that the rate deviations are due to the inadequacy of the terminal model to describe propagation, with further evidence obtained by applicacop 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 terminal 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 propagation kinetics [51], provides a good representation of this behavior [Eqs. (52)]. k pcop ¼ k p11

r1 f12 þ 2f1 f2 þ r2 f22 ðr1 f1 =k p11 Þ þ ðr2 f2 =k p22 Þ

k p ½r1 f1 þ f2 ¼ 111 r1 f1 þ ½ f2 =s1

k p22

k p ½r2 f2 þ f1 ¼ 222 r2 f2 þ ½ f1 =s2

ð52Þ

4.3 Polymerization Mechanisms and Kinetics

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)]. k p211 k p111

s1 ¼

s2 ¼

k p122 k p222

ð53Þ

A value of greater than unity for si indicates that a comonomer unit-j in the penultimate 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 diﬃcult 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

kt

¼

Nmon X Nmon X i¼1 j¼1

k tij fri frj ;

F¼

k t 12 k t 11 k t22

ð54Þ

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 homotermination 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 experimental 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

f þ kt0:5 f þ kt0:5 f þ kt0:5 f ðkt Þ 0:5 ¼ kt0:5 11; 11 r11 21; 21 r21 22; 22 r22 12; 12 r12

ð55Þ

While the terminal model and reactivity ratios provide a good description of copolymer composition, the kinetic studies summarized in this section indicate that adcop 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. 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 transfer 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. 4.3.2.3

187

4 Free-radical Polymerization: Homogeneous kp

11 → ~~~ M 1M 1 . ~~~ M 1 ⋅ + M 1 

 →

188

kdep 11 kp12

~~~ M 1 ⋅ + M 2 → ~~~ M 1M 2 . kp

22 ~~~ M 2 ⋅ + M 2  → ~~~ M 2 M 2 .

kp

21 ~~~ M 2 ⋅ + M 1   → ~~~ M 2 M 1 .

Depropagation in copolymerization for the case where M2 does not depropagate and M1 depropagates only when an M1 -unit is in the penultimate position. Scheme 4.15.

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 illustrated 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. Furthermore, depropagation of monomer-1 will occur only if an M1 unit is also in the penultimate 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 probabilities, 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 depropagation).

Fpinst 1 Fpinst 2

R pol1 ¼ ¼ R pol2

fr11 ½P 1 fr11 þ fr21 tot 1 ½M þ k ½P 2 ½M k p12 ½Ptot 2 p22 tot 2

1 2 k p11 ½Ptot ½M1 þ k p21 ½Ptot ½M1 kdep11

ð56Þ

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 increases. Lowry [68] ﬁrst derived the composition expressions for the situation where only one monomer depropagates, and general expressions have been developed 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 highertemperature 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 j transfer (long-chain branching), this is represented by Eq. (57), where m1 represents the total concentration of polymerized monomer-j units in the system and

4.3 Polymerization Mechanisms and Kinetics

nj represents the number of monomer-j units on a particular chain of length n [Eqs. (58)]. pol

mj k tr

j

ij

pol

Pn;i b þ Dm; c ! Dn; b þ Pm; cþ1 ;

j

m1 ¼

y X y X

nj ½Dn; b ;

n¼

n¼1 b¼0

Nmon X

R tr ¼

Nmon X Nmon X

j

pol

i k trij ½Ptot m1

ð57Þ

i¼1 j¼1

ð58Þ

nj

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 lessreactive 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 occur 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 C

C

R

H

COOBu

COOBu COOBu

CH

CH

C

C R

HC

H CH

COOBu

COOBu COOBu C

C

R

H

H3C

COOBu C

COOBu

COOBu COOBu C R

HC COOBu Back-biting reaction in high temperature copolymerization of butyl acrylate and butyl methacrylate (R ¼ H or CH3 ).

Scheme 4.16.

COOBu

C

H3C H

COOBu C

CH COOBu

189

190

4 Free-radical Polymerization: Homogeneous 1 R bb ¼ kbb ½Ptot ðP11 P11 þ P12 P21 Þ

ð59Þ

Back-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. Harada et al. [72] studied the copolymerization of cyclohexyl acrylate and methyl methacrylate 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 (methacrylate chain end) radical species, a result in agreement with the high rate of fragmentation observed in methacrylate macromonomer systems [73].

CH3

CH3

C CH2

HC

C CH2

fast

CH2

C

+

CH2

H2C

CO2CH3

CO2CH3

CO2CH3 CO2CH3 CO2CH3

CH2 C

HC CO2CH3

CH3

slow

C CH2

C CH2

CO2CH3

CH2 CO2CH3

+

HC CO2CH3

Fragmentation of a mid-chain radical with adjacent MA and MMA units (after Ref. 72).

Scheme 4.17.

Fragmentation after intramolecular transfer results in the formation of a longchain 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 homopolymerization 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 independent of the sizes of the radicals involved in the termination reaction. This is not

4.3 Polymerization Mechanisms and Kinetics 1.0E+09

1.0E+08

kt (L/mol-s)

1.0E+07

1.0E+06

1.0E+05

MMA nBA DA

1.0E+04

1.0E+03

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 difﬁculties 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 neglecting 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 together quickly. The rate of reaction is controlled by segmental diﬀusion, the internal reorganization of the chain that is required to bring the reactive ends together. 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].

191

192

4 Free-radical Polymerization: Homogeneous

Lower values of 10 6 L mol1 s1 for dodecyl (meth)acrylate is attributed to shielding 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 monomers 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 represented 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 magnitude 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 diﬀusion, 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)]. 1 1 1 ¼ þ k t k t; SD k t; TD þ k t; RD

ð60Þ

The subscripts SD; TD, and RD refer to segmental diﬀusion, translational diﬀusion, and reaction diﬀusion. k t; SD can be set to the low conversion values summarized 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

ð61Þ

Many 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 experimental data. k t; TD z C1 expðC2 vf Þ

ð62Þ

This 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 1 0.9 0.8

Conversion

0.7 0.6 0.5 0.4 0.3 0.2

MMA nBA

0.1 0 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 homogeneous batch FRP systems. The large decrease in k t at intermediate conversion 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 comparing the nBA and MMA rate proﬁles in the ﬁgure. The decrease also leads to an increase 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 understanding 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-

193

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 features 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. Knowing 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 recycled or altered by downstream unit-operations. The diﬃculty of characterizing polymer structure on-line makes design of a robust easy-to-operate process especially 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 versus 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 disturbance leading to a thermal runaway condition, the heat generation rate can exceed the heat removal rate to such an extent that the reaction behaves close to adiabatically. 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 therefore it takes longer for desired changes (for example decreasing the reactor temperature) to occur. This may in itself not be a serious problem, provided the reactor

4.4 Polymer Reaction Engineering

cooling system is able to maintain control, albeit at a higher than desired temperature. However, large variations in the temperature proﬁle can translate into diﬀerences in the ﬁnal product properties, particularly molecular weight distribution. The monograph by Biesenberger and Sebastian [80] provides an excellent discussion 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 gradients 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 become increasingly coupled. Homogeneous FRP reactions oﬀer a particularly challenging problem because of the large increase in viscosity accompanying the conversion from bulk monomer (@1 cp for liquids) to polymer (> 10 5 cp). The increase 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 designed to not require mixing. For example, PMMA can be polymerized in large sheets. By having large surface areas available for heat transfer, adequate temperature 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 example, concentrations of individual species, including monomers, initiators, chaintransfer agents, and so on), and is therefore a potentially powerful design tool for controlling properties such as molecular weight distribution, copolymer composition distribution, and degree of branching. Because of the ability to manipulate local monomer concentrations, the rate of polymerization can also be controlled, thereby providing safer and more robust operation. 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 polymerization is complete, the contents are discharged and the reactor prepared for 4.4.1.1

195

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 monomer, 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 polymerizations is that reactant concentrations change throughout the polymerization. Molecular 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 discharging steps to minimize this interval. Automation has also improved reproducibility of batch reactions. For operations where changes to the formulation or the polymerization conditions are common, batch processes have the advantage of being ﬂexible and readily adaptable to new products. Semi-batch Processes These processes (also called semi-continuous) are similar to batch processes, except that reactants can be added and/or products removed during the polymerization. Usually only a portion of the total reactant charge is initially fed into the reactor. The polymerization is then started, and reactants are added during reaction in order to control a desired property (for example molecular weight distribution, copolymer 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 homogeneous FRP are control of copolymer composition distribution and control of reaction rate. In a batch reaction, copolymer composition drifts according to the inherent 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 terpolymers (for example coatings) is also readily done using this type of approach. Initiator and monomers are continuously fed in the desired ratio to provide composition and MW control while maintaining starved conditions (low monomer concentration) 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 potential 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. 4.4.1.2

Continuous Processes Reactants are fed, and products and unconsumed reactants are removed, continuously. The process may take place in a single reactor or in a train (series) of reactors in which the monomer conversion is gradually increased. Most continuous processes are operated at ‘‘steady-state’’ conditions, meaning all reactant concentrations 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 4.4.1.3

4.4 Polymer Reaction Engineering

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 polymerization 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 process development from lab to pilot-plant to full-scale production, to help discriminate 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 inappropriate for design of a new process or to choose operating conditions for a new polymer 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 optimization. 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 application. Although a few historical references are included, the main focus is on advances made since the mid-1990s. The objective of kinetic modeling is to build a description of how polymer architecture and polymerization rate depend on reaction conditions (temperature, pressure) and species concentrations from a deﬁned set of kinetic mechanisms; a dynamic 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

197

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 termination. To track polymer molecular weight, all mechanisms that include radical transfer must also be included. Additional balances are needed to follow other molecular properties, such as the density of short- or long-chain branches, end-group functionality, and the creation and reaction of terminal double bonds. 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 substituting 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 reduce 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 systems it is possible to track the number-average (B n ) and weight-average (B w ) number of branches per chain. Only the basics of the mathematical treatment for a homopolymerization 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 radical (l j ) and dead (zj ) polymer distributions are deﬁned in Eqs. (63) and (64). 4.4.2.1

lj ¼

y X

n j ½Pn

ð63Þ

n j ½Dn

ð64Þ

n¼1

vj ¼

y X n¼1

It is also helpful to deﬁne moments for the bulk polymer (mj ), the total polymer in the system including live radicals [Eq. (65)].

mj ¼

y X

n j ð½Dn þ ½Pn Þ

ð65Þ

n¼1

With ½Dn g ½Pn there is little diﬀerence in magnitude between mj and zj . Its introduction, 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

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 concentration 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 [Eqs. (66)]. Mn ¼ m w

m1 ; m0

Mw ¼ m w

m2 ; m1

Mz ¼ m w

m3 m2

ð66Þ

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)]. ( RPn ¼

2f kd ½I þ

ktrmon ½M

) y y X X sol ½Pj þ ktr ½S ½Pj dðn 1Þ j¼1

j¼1

þ k p ½Mð½Pn1 ½Pn Þ ( ktrmon ½M

þ

ktrsol ½S

) y X þ ðk td þ k tc Þ ½Pj ½Pn j¼1

y y X X pol pol þ k tr n½Dn ½Pj k tr ½Pn j½Dj j¼1

( R Dn ¼

ktrmon ½M pol

þ k tr ½Pn ( RPn þDn ¼

þ

ktrsol ½S

y X

) y n1 X 1 X þ k td ½Pj ½Pn þ k tc ½Pj ½Pnj 2 j¼1 j¼1 pol

j½Dj k tr n½Dn

j¼1

2f kd ½I þ

ð67Þ

j¼1

y X

½Pj

ð68Þ

j¼1

ktrmon ½M

) y y X X sol ½Pj þ ktr ½S ½Pj dðn 1Þ j¼1

j¼1

þ k p ½Mð½Pn1 ½Pn Þ ! y n1 X 1 X ½Pj ½Pn þ k tc ½Pj ½Pnj k tc 2 j¼1 j¼1

ð69Þ

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 polymeric radicals (P1 ), and the terms for transfer to polymer account for the probabil-

199

200

4 Free-radical Polymerization: Homogeneous

ity that transfer to a certain chain Dn is proportional to chain length n. The expression 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 summations, and leads to Eqs. (70)–(78) for the moments. Live moments: Rl 0 ¼ 2 f kd ½I ðk td þ k tc Þl02 Rl1 ¼ 2f kd ½I þ

ktrmon ½Ml 0

þ

ð70Þ 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 Rl2 ¼ 2f kd ½I þ

ktrmon ½Ml 0

þ

ktrsol ½Sl 0

ð71Þ

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

ð72Þ

Dead moments: 1 R v0 ¼ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k td l02 þ k tc l02 2

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 R v2 ¼

fktrmon ½M

þ

ktrsol ½S

þ ðk td þ k tc Þl 0 gl2 þ

k tc l12

pol k tr ðl 0 v3

ð74Þ l2 v1 Þ

ð75Þ

Bulk moments: 1 Rm 0 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 k tc l02 2

ð76Þ

Rm1 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Ml 0

ð77Þ

Rm 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

ð78Þ

The 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 approximate 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)]. v3 ¼

v2 ð2v0 v2 v12 Þ v0 v 1

ð79Þ

4.4 Polymer Reaction Engineering

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, pol

RLCB ¼ k tr l 0 m1

ð80Þ

Thus, a set of six equations [Eqs. (70), (71), (76)–(78), and (80)] can be used to collapse 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 additional 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 copolymerization 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 the ﬁnal step of model development, substitution of the kinetic expressions for the moments into reactor balances, see Section 4.4.3. 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 termination kinetics into the mechanistic scheme. In such cases, the kinetic and modeling assumptions can be tested more rigorously through a detailed comparison 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 systems, 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 distribution of polymer formed at any instant in time [92]. 4.4.2.2

wninst ¼ ðt þ 0:5ðn 1Þbðt þ bÞÞðt þ bÞn

t¼

k td ½l 0 þ ktrmon ½M þ ktrsol ½S ; k p ½M

b¼

1 1þbþt

k tc ½l 0 k p ½M

n ð81Þ

ð82Þ

201

202

4 Free-radical Polymerization: Homogeneous

Assuming the formation of long-chain polymer (t þ b f 1), the values for instantaneous DPn and DPw are given by Eqs. (83). DPninst ¼

1 ; t þ 0:5b

DPwinst ¼

2t þ 3b ðt þ bÞ 2

ð83Þ

The instantaneous PDI is 2 for a system where there is no termination by combination (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 polymerization, crosslinking). The methods for modeling MWDs with branching can be divided 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 calculated, 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 succession 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 systems. 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 second by joining (through crosslinking, terminal double bond polymerization, or termination 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, Butte´ 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

The commercial software Predici2 package uses yet another numerical technique, 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 perform detailed kinetic modeling. These new modeling capabilities, in combination with improved characterization techniques, will hasten progress to a better understanding 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 wellmixed reactor system, Eqs. (84)–(89) comprise the general system of equations for homopolymerization. dðV½MÞ ¼ ðqin ½Min Þ ðqout ½MÞ VR pol dt

ð84Þ

dðV½IÞ ¼ ðqin ½Iin Þ ðqout ½IÞ VR d dt

ð85Þ

dðV½SÞ ¼ ðqin ½Sin Þ ðqout ½SÞ VRtrsol dt

ð86Þ

dðV½l i Þ ¼ ðqin ½li; in Þ ðqout ½l i Þ þ VRl i dt

ð87Þ

dðV½m i Þ ¼ ðqin ½m i; in Þ ðqout ½m i Þ þ VRmi dt

ð88Þ

dðVrc p ðT Tref ÞÞ ¼ ðqin ðrc p Þin ðTin Tref ÞÞ ðqout ðrc p Þout ðT Tref ÞÞ dt þ ðDHp ÞVR pol Q

ð89Þ

In 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.

203

204

4 Free-radical Polymerization: Homogeneous

Batch Polymerization Inﬂow and outﬂow terms are set to zero, but the change in volume with conversion must be considered for the constant-mass system [Eq. (90)]. 4.4.3.1

dV V dr ¼ dt r dt

ð90Þ

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 solving 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. dx p ¼ dt

kp 1=2

kt

!

2f kd ½I0 1 ex p

!1=2

kd t ð1 x p Þ exp 2

ð91Þ

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 radicals, Eq. (92) results for l 0. 4.4.3.2

ðqout ½l 0 Þ ¼ Vð2f kd ½I k t ½l 0 2 Þ

ð92Þ

For most cases, the radical lifetime is much shorter than the average residence time. Thus outﬂow of radicals can be neglected, resulting in the familiar expression, Eq. (93), for l 0.

½l 0 ¼

2f kd ½I 1=2 kt

ð93Þ

[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 fractional density change between inlet and outlet streams.

½I ¼

qin ½Iin ½Iin ¼ qout þ Vkd ð1 þ ykd Þð1 eÞ

ð94Þ

Substitution of these expressions into the monomer balance leads to a nonlinear relationship between conversion and y [Eq. (95)].

4.4 Polymer Reaction Engineering

x p ¼ yk p l 0 ¼ y

kp 1=2 kt

!

2f kd ½Iin ð1 eÞð1 þ ykd Þ

1=2 ð95Þ

Assuming 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 signiﬁcantly [81].

Complex Flowsheets These are often constructed to represent systems with nonideal mixing and fast reaction. A classic example is the high-pressure high-temperature free radical production of ethylene copolymers, generally conducted in a homogeneous phase consisting largely of supercritical ethylene monomer. These conditions make for very fast reactions (for example, initiator half-life of <1 s), promote numerous side reactions (long-chain branching, short-chain branching, and chain scission), and introduce the potential of thermal runaway. Recent models of these systems combine a detailed description of polymerization kinetics with a complex ﬂowsheet of continuous 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. 4.4.3.3

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 prediction of temperature and conversion proﬁles in the reactors; to simplify the calculational 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 branching 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 article 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. 4.4.3.4

205

206

4 Free-radical Polymerization: Homogeneous

Model-based Control Model-based control of polymerization systems has also garnered its share of attention. 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 unmeasured 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 combined with limited on-line measurements (for example, in Refs. 109–112) to improve control performance. Fundamental models can also be used to test empirical models developed for control purposes [113]. 4.4.3.5

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 processes are often limited by transport eﬀects. In many cases, the ﬁnal reaction rate, molecular weight, and copolymer composition are determined by the coupled effects 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 utility in the control of polymerization reactors, and can be an important contributor to product quality and process robustness. Continuing development of new measurement 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 industrial 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 combined with fundamental reaction engineering principles. Notation

Ai CtrXXX

pre-exponential factor for mechanism-i ratio of chain-transfer rate coeﬃcient (k trXXX , XXX ¼ mon, pol, sol) to propagation [Eq. (19)]

Notation

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

207

208

4 Free-radical Polymerization: Homogeneous

Pn Pn; b [Ptot ] i ] [Ptot Qn qin qout Rd R dep R inhib R init RLCB Rp pol Rp R pol R term R therm R trXXX Rl i ri [S] si T Tc t 1=2 V wn xp [Z]

polymer radical of length n polymer radical of length n with b branch points total concentration of radicals [mol L1 ] total concentration of polymer radicals of type-i [mol L1 ] mid-chain polymer radical of length-n (formed by intramolecular or intermolecular H-abstraction) reactor inlet volumetric ﬂow [L s1 ] reactor outlet volumetric ﬂow [L s1 ] rate of initiator disappearance [mol L1 s1 ] rate of depropagation [mol L1 s1 ] rate of inhibition [mol L1 s1 ] rate of radical generation from initiator [mol L1 s1 ] rate of long-chain branch formation [mol L1 s1 ] rate of propagation [mol L1 s1 ] addition rate of terminally unsaturated polymer chains mol L1 s1 ] total rate of monomer consumption [mol L1 s1 ] rate of radical–radical termination [mol L1 s1 ] rate of thermal initiation of styrene [mol L1 s1 ] rate of transfer to species XXX (mon, pol, sol) [mol L1 s1 ] rate of change of moment l i [mol L1 s1 ] monomer reactivity ratio for radical-i (for example, r1 ¼ k p11 =k p12 ) solvent (transfer agent) concentration [mol L1 ] radical reactivity ratio for monomer-i in the penultimate copolymerization model (for example, s1 ¼ k p211 =k p111 ) temperature [K] ceiling temperature of monomer [ C] initiator half-life [s1 ] reactor volume [L] weight fraction of chains of length-n fractional monomer conversion to polymer inhibitor concentration [mol L1 ]

Greek b DGp DHp DSp DVi d e zk y lk mk

ratio of chain termination (combination) to propagation [Eq. (82)] free energy change of polymerization [kJ mol1 ] heat of polymerization [kJ mol1 ] entropy of polymerization [J mol1 K1 ] activation volume for mechanism-i [cm 3 mol1 ] fraction of termination by disproportionation volume contraction factor kth moment of the dead polymer chains [mol L1 ] average residence time [s] kth moment of the polymer radicals [mol L1 ] kth moment of the total (dead þ radical) polymer chains [mol L1 ]

References

kinetic chain length of polymer radicals density [kg L1 ] ratio of chain termination (disproportionation) and transfer to propagation [Eq. (82)] average number of monomer units on a living chain copolymerization cross-termination factor

n r t u F Acronyms AIBN BMA CFD CTA DMA EGDMA FRP LCB LCH LDPE MA MMA MW MWD nBA PDI PLP PMMA QSSA SCB SEC Sty

2,2 0 -azobisisobutyronitrile butyl methacrylate computational ﬂuid dynamics chain-transfer agent dodecyl methacrylate ethylene glycol dimethacrylate free radical polymerization long-chain branching long-chain hypothesis low-density polyethylene methyl acrylate methyl methacrylate molecular weight molecular weight distribution Butyl acrylate polydispersity index (Mw/Mn ) pulsed-laser polymerization poly(methyl methacrylate) quasi-steady-state assumption short-chain branching size exclusion chromatography styrene

References 1 M. Buback, S. Klingbeil, J.

Sandmann, M.-B. Sderra, H.-P. Vo¨gele, H. Wackerbarth, L. Wittkowski, Z. Phys. Chem. 1999, 210, 199–221. 2 M. Buback, J. Sandmann, Z. Phys. Chem. 2000, 214, 583–607. 3 G. Moad, D. H. Solomon, The Chemistry of Free Radical Polymerization, Elsevier Science, New York, 1995.

4 M. Asteasuain, A. Brandolin, C.

Sarmoria, Polymer 2004, 45, 321–335. 5 A. A. Gridnev, S. D. Ittel,

Macromolecules 1996, 29, 5864–5874. 6 H. Tobita, A. E. Hamielec, Polymer

1991, 32, 2641–2647. 7 J. Brandrup, E. H. Immergut, E. A.

Grulke (Eds.), Polymer Handbook, 4th Edition, John Wiley & Sons, New York, 1999. 8 M. Buback, L. H. Garcia-Rubio, R. G.

209

210

4 Free-radical Polymerization: Homogeneous

9

10

11

12

13

14

15 16 17 18

19

20

Gilbert, D. H. Napper, J. Guillot, A. E. Hamielec, D. Hill, K. F. O’Driscoll, O. F. Olaj, J. Shen, D. Solomon, G. Moad, M. Stickler, M. Tirrell, M. A. Winnik, J. Polym. Sci.: Part C: Polym. Lett. 1988, 26, 293–297. M. Buback, R. G. Gilbert, G. T. Russell, D. J. T. Hill, G. Moad, K. F. O’Driscoll, J. Shen, M. A. Winnik, J. Polym. Sci., Polym. Chem. Ed. 1992, 30, 851–864. O. F. Olaj, I. Bitai, F. Hinkelmann, Makromol. Chem. 1987, 188, 1689– 1702. M. Buback, R. G. Gilbert, R. A. Hutchinson, B. Klumperman, F.-D. Kuchta, B. G. Manders, K. F. O’Driscoll, G. T. Russell, J. Schweer, Macromol. Chem. Phys. 1995, 196, 3267–3280. S. Beuermann, M. Buback, T. P. Davis, R. G. Gilbert, R. A. Hutchinson, O. F. Olaj, G. T. Russell, J. Schweer, A. M. van Herk, Macromol. Chem. Phys. 1997, 198, 1545–1560. S. Beuermann, M. Buback, T. P. Davis, R. G. Gilbert, R. A. Hutchinson, A. Kajiwara, B. Klumperman, G. T. Russell, Macromol. Chem. Phys. 2000, 201, 1355–1364. S. Beuermann, M. Buback, T. P. Davis, R. G. Gilbert, R. A. Hutchinson, A. Kajiwara, M. Kamachi, I. Lacı´k, G. T. Russell, Macromol. Chem. Phys. 2003, 204, 1338–1350. S. Beuermann, M. Buback, Prog. Polym. Sci. 2002, 27, 191–254. I. Lacı´k, S. Beuermann, M. Buback, Macromolecules 2003, 36, 9355–9363. H. Fischer, L. Radom, Angew. Chem. Int. Ed. 2001, 40, 1340–1371. M. Buback, M. Egorov, R. G. Gilbert, V. Kaminsky, O. F. Olaj, G. T. Russell, P. Vana, G. Zifferer, Macromol. Chem. Phys. 2002, 203, 2570–2582. M. D. Zammit, T. P. Davis, D. M. Haddleton, K. G. Suddaby, Macromolecules 1997, 30, 1915–1920. G. A. Mortimer, J. Polym. Sci.: Part A-1 1966, 4, 881–900.

21 P. V. T. Raghuram, U. S. Nandi, J.

22 23

24 25

26

27 28

29

30 31 32

33

34 35 36

37

38 39 40 41

Polym. Sci.: Part A-1 1969, 7, 2379– 2385. G. A. Mortimer, J. Polym. Sci.: Part A-1 1970, 8, 1535–1542. M. Stickler, G. Meyerhoff, Makromol. Chem. 1978, 179, 2729– 2745. S. Maeder, R. G. Gilbert, Macromolecules 1998, 31, 4410–4418. M. Buback, C.-R. Choe, E.-U. Franck, Makromol. Chem. 1984, 185, 1685–1697. J. D. Campbell, F. Teymour, M. Morbidelli, Macromolecules 2003, 36, 5491–5501. A. W. Hui, A. E. Hamielec, J. Appl. Polym. Sci. 1972, 16, 749–769. R. A. Hutchinson, D. A. Paquet, Jr., S. Beuermann, J. H. McMinn, Ind. Eng. Chem. Res. 1998, 37, 3567– 3574. M. C. Grady, W. J. Simonsick, Jr., R. A. Hutchinson, Macromol. Symp. 2002, 182, 149–168. P. Ehrlich, G. A. Mortimer, Adv. Polym. Sci. 1970, 7, 386–448. P. Feucht, B. Tilger, G. Luft, Chem. Eng. Sci. 1985, 40, 1935–1942. L. I. Kulin, N. L. Meijerink, P. Starck, Pure Appl. Chem. 1988, 60, 1403–1415. W. W. Graessley, R. D. Hartung, W. C. Uy, J. Polym. Sci.: Part A-2 1969, 7, 1919–1935. D. Britton, F. Heatley, P. A. Lovell, Macromolecules 1998, 31, 2828–2837. S. Grcev, P. Schoenmakers, P. Iedema, Polymer 2004, 45, 39–48. N. M. Ahmad, F. Heatley, P. A. Lovell, Macromolecules 1998, 31, 2822–2827. C. Plessis, G. Arzamendi, J. R. Leiza, H. A. S. Schoonbrood, D. Charmot, J. M. Asua, Ind. Eng. Chem. Res. 2001, 40, 3883–3894. P. Tackx, J. C. J. F. Tacx, Polymer 1998, 39, 3109–3113. W.-H. Li, A. E. Hamielec, C. M. Crowe, Polymer 1989, 30, 1518–1523. R. A. Hutchinson, Polym. React. Eng. 1993, 1, 521–577. T. Usami, S. Takayama, Macromolecules 1984, 17, 1756–1761.

References 42 J. C. Randall, JMS – Rev. Macromol. 43

44 45

46 47 48 49 50 51 52

53

54

55 56

57 58

59

60

61 62

Chem. Phys. 1989, C29, 201–317. E. F. McCord, W. H. Shaw, Jr., R. A. Hutchinson, Macromolecules 1997, 30, 246–256. T. Hirano, B. Yamada, Polymer 2003, 44, 347–354. J. Chiefari, J. Jeffery, R. T. A. Mayadunne, G. Moad, E. Rizzardo, S. N. Thang, Macromolecules 1999, 32, 7700–7702. R. A. Hutchinson, Macromol. Theory Simul. 2001, 10, 144–157. A. F. N. Peck. MSc Thesis, Queen’s University, Kingston, ON, 2003. P. W. Tidwell, G. A. Mortimer, J. Polym. Sci.: Part A 1965, 3, 369–387. A. M. van Herk, T. Dro¨ge, Macromol. Theory Simul. 1997, 6, 1263–1276. V. E. Meyer, G. G. Lowry, J. Polym. Sci.: Part A 1965, 3, 2843–2851. T. Fukuda, K. Kubo, Y.-D. Ma, Prog. Polym. Sci. 1992, 17, 875–916. M. Buback, A. Feldermann, C. Barner-Kowollik, I. Lacı´k, Macromolecules 2001, 34, 5439–5448. M. Buback, L. Wittkowski, Macromol. Chem. Phys. 2000, 201, 419–426. M. Buback, L. Wittkowski, S. A. Lehmann, F.-O. Ma¨hling, Macromol. Chem. Phys. 1999, 200, 1935–1941. L. K. Kostanski, A. E. Hamielec, Polymer 1992, 33, 3706–3710. M. Hakim, V. Verhoeven, N. T. McManus, M. A. Dube´, A. Penlidis, J. Appl. Polym. Sci. 2000, 77, 602–609. M. Buback, H. Dietzsch, Macromol. Chem. Phys. 2001, 202, 1173–1181. A. D. Jenkins, K. F. O’Driscoll, J. Polym. Sci.: Part A: Polym. Chem. 1995, 33, 1189–1190. M. L. Coote, L. P. M. Johnston, T. P. Davis, Macromolecules 1997, 30, 8191– 8204. T. P. Davis, K. F. O’Driscoll, M. C. Piton, M. A. Winnik, Polym. Int. 1991, 24, 65–70. Y.-D. Ma, P.-S. Kim, K. Kubo, T. Fukuda, Polymer 1994, 35, 1375–1381. R. A. Hutchinson, J. H. McMinn, D. A. Paquet, Jr., S. Beuermann, C. Jackson, Ind. Eng. Chem. Res. 1997, 36, 1103–1113.

63 M. Buback, C. Kowollik,

Macromolecules 1999, 32, 1445–1452. 64 M. Buback, A. Feldermann, Aust. J.

Chem. 2002, 55, 475–481. 65 H. A. S. Schoonbrood, S. C. J.

66 67 68 69 70 71

72

73

74

75

76

77

78 79

80

81

82 83

Pierik, B. van den Reijen, J. P. A. Heuts, A. L. German, Macromolecules 1996, 29, 6717–6723. P. Wittmer, ACS Adv Chem Ser. 1971, 99, 140–174. N. T. McManus, A. Penlidis, M. A. Dube´, Polymer 2002, 43, 1607–1614. G. G. Lowry, J. Polym. Sci. 1960, 42, 463–477. C. Badeen, M. A. Dube´, Polym. React. Eng. 2003, 11, 53–77. D. Li, MSc Thesis, Queen’s University, Kingston, ON, 2003. C. Plessis, G. Arzamendi, J. R. Leiza, H. A. S. Schoonbrood, C. Charmot, J. M. Asua, Macromolecules 2001, 34, 5147–5157. T. Harada, P. B. Zetterlund, B. Yamada, Polym. Prepr. 2002, 43(2), 160–161. C. L. Moad, G. Moad, E. Rizzardo, S. H. Thang, Macromolecules 1996, 29, 7717–7726. P. Vana, L. H. Yee, C. BarnerKowollik, J. P. A. Heuts, T. P. Davis, Macromolecules 2002, 35, 1651–1657. G. T. Russell, D. H. Napper, R. G. Gilbert, Macromolecules 1988, 21, 2141–2148. S. K. Soh, D. C. Sundberg, J. Polym. Sci.: Polym. Chem. Ed. 1982, 20, 1345– 1371. B. M. Louie, G. M. Carratt, D. S. Soong, J. Appl. Polym. Sci. 1985, 30, 3985–4012. D. Achilias, C. Kiparissides, Macromolecules 1992, 25, 3739–3750. N. Tefera, G. Weickert, K. R. Westerterp, J. Appl. Polym. Sci. 1997, 63, 1649–1660. J. A. Biesenberger, D. H. Sebastian, Principles of Polymerization Engineering, John Wiley & Sons, New York, 1983. A. Baltsas, D. S. Achilias, C. Kiparissides, Macromol. Theory Simul. 1996, 5, 477–497. W. H. Ray, J. Macromol. Sci. – Rev. Macromol. Chem. 1972, C8, 1–56. D. S. Achilias, C. Kiparissides, J.

211

212

4 Free-radical Polymerization: Homogeneous

84

85 86 87 88 89

90

91

92 93 94 95

96 97 98

Macromol. Sci. – Rev. Macromol. Chem. Phys. 1992, C32, 183–234. M. A. Dube´, J. B. P. Soares, A. Penlidis, A. E. Hamielec, Ind. Eng. Chem. Res. 1997, 36, 966–1015. D. J. Arriola, PhD Thesis, University of Wisconsin-Madison, 1989. T. Xie, A. E. Hamielec, Macromol. Chem. Theory Simul. 1993, 2, 421–454. T. Xie, A. E. Hamielec, Macromol. Chem. 1993, 2, 455–483. H. M. Hulburt, S. Katz, Chem. Eng. Sci. 1964, 19, 555–574. R. C. Zabisky, W. M. Chan, P. E. Gloor, A. E. Hamielec, Polymer 1992, 33, 2243–2262. S. Ochs, P. Rosendorf, I. Hyanek, S. X. Zhang, W. H. Ray, Comp. Chem. Eng. 1996, 20, 657–663. A. Pertsinidis, E. Papadopoulos, C. Kiparissides, Comp. Chem. Eng. 1996, 20, S449–S454. S. T. Balke, A. E. Hamielec, J. Appl. Polym. Sci. 1973, 17, 905–949. H. Tobita, J. Polym. Sci.: Part B: Polym. Phys. 1993, 31, 1363–1371. F. Teymour, J. D. Campbell, Macromolecules 1994, 27, 2460–2469. A. Butte´, A. Ghielmi, G. Storti, M. Morbidelli, Macromol. Theory Simul. 1999, 8, 498–512. H. Tobita, Macromol. Theory Simul. 1996, 5, 129–144. M. Wulkow, Macromol. Theory Simul. 1996, 5, 393–416. P. D. Iedema, M. Wulkow, H. C. J. Hoefsloot, Macromolecules 2000, 33, 498–512.

99 M. Busch, Macromol. Theory Simul.

2001, 10, 408–429. 100 W.-M. Chan, P. E. Gloor, A. E.

101

102 103

104 105 106 107

108 109

110

111

112 113

Hamielec, AIChE J. 1993, 39, 111– 126. H. Nordhus, O. Moen, P. Singstad, J. Macromol. Sci. – Pure Appl. Chem. 1997, A34, 1017–1043. P. Pladis, C. Kiparissides, J. Appl. Polym. Sci. 1999, 73, 2327–2348. A. Brandolin, M. H. Lacunz, P. E. Ugrin, N. J. Capiati, Polym. React. Eng. 1996, 4, 193–241. K. Tsai, R. O. Fox, AIChE J. 1996, 42, 2926–2940. N. K. Read, S. X. Zhang, W. H. Ray, AIChE J. 1997, 43, 104–117. N. H. Kolhapure, R. O. Fox, Chem. Eng. Sci. 1999, 54, 3233–3242. F. Bezzo, S. Macchietto, C. C. Pantelides, Comp. Chem. Eng. 2000, 24, 653–658. J. P. Congalidis, J. R. Richards, Polym. React. Eng. 1998, 6, 71–111. R. K. Mutha, W. R. Cluett, A. Penlidis, Ind. Eng. Chem. Res. 1997, 36, 1036–1047. S.-M. Ahn, M.-J. Park, H.-K. Rhee, Ind. Eng. Chem. Res. 1999, 38, 3942– 3949. Y. Yabuki, T. Nagasawa, J. F. MacGregor, Comp. Chem. Eng. 2000, 24, 585–590. M.-J. Park, H.-K. Rhee, Chem. Eng. Sci. 2004, 58, 603–611. S.-S. Na, H.-K. Rhee, J. Appl. Polym. Sci. 2000, 76, 1889–1902.

213

5

Free-radical Polymerization: Suspension1 B. W. Brooks 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 ionexchange resins and specialist materials. The annual polymer production from suspension 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 product must have the correct molecular weight distribution. The conditions that are necessary to achieve those goals can be predicted from the kinetics of homogeneous 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 dispersed 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 remains 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 copolymer composition. 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 processes because the vessels often contain about 50% of the continuous phase. Suspension 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 suspension processes than in bulk processes because suspensions are more mobile than molten polymers. Therefore, simple rate expressions may not be applicable because 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 dispersed 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. However, 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 composition of the drops may then be unknown. The eﬀective monomer concentrations,

5.1 Key Features of Suspension Polymerization Tab. 5.1. Apparent reactivity ratios in solution and suspension copolymerization (reproduced with permission from Ref. 4).

Monomer 1 Monomer 2

AS S

AN MA

VC VA

MMA MAA

r1 (soln) r2 (soln) r1 (susp) r2 (susp) Temp [ C]

1.18 0.85 1.0 1.0 60

1.02 0.7 0.75 1.54 50

1.68 0.23 2.47 1.99 70

0.35 1.63 0.63 1.07 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 copolymerization of styrene and acrylonitrile [9]. Apparent reactivity ratios obtained directly from suspension polymerization experiments may not be identical to those expected from the equivalent bulk processes 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 material 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 dispersions are applicable to suspension polymerization. But sometimes their use is limited because they do not account for changing physical properties of drops or for the presence of drop stabilizers.

215

216

5 Free-radical Polymerization: Suspension

5.1.5

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 control. In suspension polymerization, however, that problem is avoided. There, coagulation is largely conﬁned to the drop interiors and aggregation of polymerizing drops is restricted. That is why suspension polymerization is used for the largescale production of PVC. In that case, the initial drop diameters, and the ﬁnal particle sizes, range between 10 and 100 mm. If a polymer product is required in particulate form, then suspension polymerization 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 granulated 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 important. 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 condition 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 enhanced 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

5.2.1

Stabilizer Types

Many drop stabilizers in suspension polymerization are water-miscible polymers. These materials are sometimes called protective colloids. They include naturally occurring substances, such as gelatin and pectin, and a wide range of synthetic polymers such as partially hydrolyzed poly(vinyl acetate)s. Modiﬁed natural products such as cellulose ether derivatives are also widely used. Munzer and Trommsdorﬀ provide 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 sometimes 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 continuous 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 behavior 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 reduced [18]. But when the DH is less than 60%, drop sizes become sensitive to reduction 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 morphology 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 suspension [20]. There are many reports of inorganic solids (such as calcium phosphate, aluminium hydroxide, and zinc phosphate) being used in the suspension polymerization 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 electrolytes can also be important [21]. Inorganic stabilizers are advantageous when only low levels of contamination are required because they can sometimes be removed 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 suspension polymerization, other solids are not stabilizers and may even be destabilizers. O’Shima and Tanaka [22] suggested that the contact angle between dispersed liquid and inorganic solid is a crucial factor in determining whether an inorganic

217

218

5 Free-radical Polymerization: Suspension

solid is a stabilizer or a destabilizer in suspension polymerization. Solids that provide a relatively large contact angle (such as aluminium hydroxide) would be stabilizers 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 suggested 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 Wolters et al. [23] used hydroxyapatite and calcium carbonate in the suspension polymerization 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 dispersion ceased. If some particles remain in the continuous phase, it is not feasible for the coverage 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 dispersion 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 exceeds a critical value, a drop will break into smaller drops. The ﬂuids in agitated vessels are often turbulent. If the turbulence in local regions 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

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

ð1Þ

Rushton 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 impeller 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

ð2Þ

Pressure ﬂuctuations can deform the drops and they may break if the inertial forces exceed the interfacial tension forces. Kolmogoroﬀ [27] and Hinze [31] derived 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

ð3Þ

The inertial forces are related to e. Thus, We crit is given by Eq. (4). 5=3 We crit ¼ ðC1 rc e 2=3 d max 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 d max ¼ C4 ðs=rc Þ

ð5Þ 3=5

N

6=5

d max =D ¼ C4 ðWeÞ0:6

4=5

D

ð6Þ ð7Þ

In 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].

219

5 Free-radical Polymerization: Suspension

30 Volume fraction of dispersed phase

4

d 32 /D ( x 10 )

220

0.01 0.05 0.1 0.2 0.3 0.4

20

10

0 0

10 0.027(1 + 3.06

20 )We

- 0.6

30 4

( x 10 )

Correlation of vinyl chloride drop size with volume fraction j and Weber number We. Reproduced by permission of Elsevier Science Ltd. From Ref. 35.

Fig. 5.1.

In the suspension polymerization of vinyl chloride, Zerfa and Brooks found the relationship in Eq. (8). 0:58 d32 ¼ d max

ð8Þ

Equation (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 noncoalescing systems [28]. But, unlike many laboratory studies, real suspension polymerizations are operated with a high volume fraction of dispersed phase, j. Then, drop coalescence and breakage happen simultaneously and damping of the turbulent 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 polymerization of vinyl chloride, d32 was given by Eq. (9). d32 =D ¼ 0:027ðþ3:06jÞWe0:6

ð9Þ

Equation (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 coalescence is reduced by addition of an appropriate drop stabilizer. If the surface coverage 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

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

ð10Þ

Shinnar and Church [26, 32] demonstrated that for locally isotropic ﬂow where ðqu=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

ð11Þ

d max z ðsnc1=2 =mc ÞN 3=2 D1 f ðmd =mc Þ

ð12Þ

When viscosity-increasing agents were present in the continuous phase for the suspension 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 vessels 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 accompanied by the formation of a number of smaller satellite drops. Drop stabilizers 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

221

5 Free-radical Polymerization: Suspension 1.6

A

1.2

Ca crit

222

0.8 0.4

B

0 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 wellcharacterized 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, Kuriyama 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 reduction 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 diameters 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

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 quantify. 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 compatible with experimental results from nonpolymerizing model liquid–liquid dispersions. 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 coalescence is negligible, the equilibrium drop size distribution broadened considerably as dispersed-phase viscosity increased. If no chemical reaction occurs, then the rates of drop breakage and drop coalescence 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 physical properties of the drops change with monomer conversion. In a batch process, early increases in drop viscosity reduce the rates of breakage and coalescence before 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

223

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, because 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 developed from satellite drops which are formed during the breakage process. The existence of small satellite drops is compatible with the ideas of Chatzi and Kiparissides [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 induce some emulsion polymerization. That can by important with monomers that are more water-soluble, such as methyl methacrylate, vinyl acetate, and vinyl chloride. 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 reduction in stirrer speed led to a broadening of the DSD. There, the DSD became similar 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

Drop number

200

A

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

B

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

C

150 100 50 0 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 copolymer 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-

225

5 Free-radical Polymerization: Suspension

Drop number

200

A

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

B

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

200

Drop number

226

150

C

100 50 0 15

35

55

75

95

115

Drop diameter (microns) 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]. Fig. 5.4.

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 material 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

the original drops. The new monomer cannot mix with existing drops (which contain 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 danger 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 stabilized drops. The drops were composed of polystyrene solutions in styrene. The initial 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 enthalpy of polymerization for most of the vinyl monomers that are used in suspension polymerization ranges between 30 and 90 kJ mol1 . Therefore, a high heat removal rate is usually necessary to maintain a constant reactor temperature. This is diﬃcult to achieve by heat transfer through the reactor walls in commercial operations 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 polymerization process is to be maintained, drops of returning monomer must gain access to initiator and drop stabilizer. With the suspension polymerization of vinyl chloride, Zerfa and Brooks [60] found that monomer returning from a reﬂux condenser 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 absence 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 dispersed in the aqueous phase of a stabilized suspension. Then the subsequent mixing 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 nonuniformity in drop behavior aﬀected the ﬁnal polymer properties. Also, addition of initiator via the aqueous phase promoted simultaneous emulsion polymerization and modiﬁed the PSD. Drop mixing rates were quantiﬁed by using dyed monomer

227

228

5 Free-radical Polymerization: Suspension

Eﬀect of the method of initiator addition to vinyl chloride: (A) initiator predissolved in drop phase; (B) initiator predispersed in continuous phase. Panels:

Fig. 5.5.

1, 5 min; 2, 20 min; 3, 60 min. N ¼ 350 rpm (5.8 s1 ); j ¼ 0:1; PVA concentration ¼ 0.06%. Reproduced by permission of John Wiley & Sons, Inc. From Ref. 62.

drops in the experiments. The mixing rate was found to depend on both the concentration of the PVA and the grade that was used [61].

5.3

Events at High Monomer Conversion

Suspension polymerization provides a practical method of achieving high monomer conversions. Therefore, studies of polymerization and drop behavior that deal with events at low monomer conversion may not be applicable to suspension polymerization.

5.3 Events at High Monomer Conversion

5.3.1

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 aspects 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 conversion 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

ð13Þ

If 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

ð14Þ

Chain termination is often diﬀusion-controlled and the value of k t diminishes substantially 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 polymerization rate, a phenomenon often described as a ‘‘gel eﬀect’’. Although most workers agree that the chain termination reaction is diﬀusioncontrolled, 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 steadystate 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).

229

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 becomes 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 conversion 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 conversion 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 describe 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 polymerization and temperature increases can lead to a marked reduction in initiator concentration. As the monomer conversion increases, the glass transition temperature 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 monomers 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 provide a means of investigating copolymerization kinetics at high conversion. The monomer sequence in styrene–methyl methacrylate copolymers at high conversion 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 polymerization. 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

Fig. 5.6.

Schematic representation of polymer formation inside vinyl chloride drops.

polymerization can inﬂuence the properties of the ﬁnal product, especially the porosity, which in the case of PVC aﬀects the ability of the polymer to take up plasticizers. Structural changes that occur inside the drops of polymerizing vinyl chloride 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 precipitating polymer aggregates to form unstable microdomains, which aggregate further to give domains. Subsequent aggregation of domains results in the formation 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 particles 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 primary particles [82]. Those components are sometimes called secondary suspending agents (SSAs). As a direct consequence of the process described above, and of the density diﬀerence 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-

231

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 substructure 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 porosity) of the resulting PVC was related to the surface activity of the suspending agent, regardless of type. Ormondroyd [86] demonstrated the eﬀect of PVA structure on the particle size, cold plasticizer absorption, and bulk density of the PVC produced. Cheng [87] and Cheng and Langsam [88] used hydroxypropyl methylcellulose (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 concentration inﬂuenced the particle properties of PVC. SSAs are regularly used in the production 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 commercially 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 increases quickly as the concentration of nonionic surfactant increases [82]. The degree of agglomeration of primary particles increases with polymerization temperature and with conversion. A nonionic surfactant with a greater hydrophilelyophile 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 increased coalescence of VCM/PVC droplets during the polymerization process, as a result of a lowering of water-drop interfacial tension by the SSA. Nonionic surfactants 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

CPA (g DOP/100g PVC)

5.3 Events at High Monomer Conversion 100 90 80 70 60 50 40 30 20 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 because the porosity of PVC particles usually decreases linearly with monomer conversion, 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 solubility in VCM) is more eﬀective in raising the product porosity. The surface of primary 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 controlled by using blends of PVA suspending agents with diﬀering degrees of hydrolysis [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 propylene [95]. In that case, the SSA was more soluble in the organic phase than the primary stabilizer (a cellulose ether). In vinyl chloride polymerization, particle porosity facilitates the subsequent uptake 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-

233

234

5 Free-radical Polymerization: Suspension

ration inside the drops [96–99]. Appropriate functional groups can be incorporated in the monomers.

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, extrapolating 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 vessels 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 feasible 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 industrial suspension polymerizations are described as ‘‘batch’’ processes, they are not genuine batch operations because some material enters the reactor after polymerization has started. In some cases, substantial reﬂux of monomers occurs and condensed monomer returns to the reactor continuously. In others, a semi-batch process is used to control product composition. In both these situations, unconverted monomer with low viscosity is being mixed with drops of higher viscosity. The complexities 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 distribution. Ni et al. [106] developed an oscillatory baﬄed reactor for batch suspension 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

trolled by choosing appropriate values for oscillation frequency, oscillation amplitude, 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 operations, some continuous-ﬂow processes have been proposed [107]. Use of ﬂow reactors 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 molecular 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 the ﬂuid 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 suspension 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 formation. Vortices are undesirable because the centrifugal eﬀect favors drop congregation 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

235

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 junction. 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 input 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 reality, 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 predicted 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 suspension 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, especially after the startup period in batch operation when a gel eﬀect causes autoacceleration in the polymerization. If the monomers are volatile at the working pressure of the reactor, then some heat can be removed relatively quickly by monomer 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 polymerization 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 reaction time for a given monomer conversion. It should be remembered that the overall 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 between 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

higher than that of their surroundings when drop diameters exceeded 1 mm. Temperature diﬀerences were large enough to cause changes to polymer quality. 5.4.3

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 consideration must be given to maintaining the required drop/particle size distribution. 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 coalescence 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. Sometimes 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 turbulence 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 region (between laminar and turbulent ﬂow). In such cases, the drop breakage mechanism 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 evaporation, 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).

237

238

5 Free-radical Polymerization: Suspension

5.4.4

Reactor Safety with Suspension Processes

The potential for thermal runaway exists with most commercial free-radical polymerization processes that employ batch reactors. This arises because the heat of reaction is high and there is no eﬀective steady state. Although temperature control 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 temperature 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 polymerization 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 accumulate 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 maintained at a positive pressure because a volatile blowing agent (for example, pentane) 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 pressure 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 premature 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

dures when a large reactor is used. Many scaleup criteria do not apply to transient 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 during the drop creation process.

5.5

‘‘Inverse’’ Suspension Polymerization

When a water-miscible polymer is to be made via a suspension process, the continuous phase is a water-immiscible ﬂuid, often a hydrocarbon. In such circumstances 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 initiator. 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 microsuspension’’ or ‘‘inverse dispersion’’ polymerization. Here, as with conventional suspension polymerization, the polymerization reaction occurs inside the monomercontaining drops. The drop stabilizers are initially dispersed in the continuous (nonaqueous phase). If particulate solids are used for drop stabilization, the surfaces of the small particles must be rendered hydrophobic. Inverse dispersion polymerization is used to make water-soluble polymers and copolymers from monomers 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 polymerization of acrylamide by using an oscillatory baﬄed reactor. 5.5.1

Initiator Types

The thermal decomposition of single-component initiators (such as inorganic persulfates) can be used to generate the free radicals in the aqueous drops, but in

239

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 circumstances, 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 dispersed 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 begin. Liu and Brooks [123] used a freeze–fracture technique with electron microscopy to show how ‘‘large’’ reductant drops became mixed with ‘‘small’’ monomercontaining oxidant drops in the early stages of acrylic acid polymerization (Figure 5.8). The actual volumes and concentrations used for the two aqueous solutions affect 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 development of new processes requires further fundamental investigation. Improvements 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

5.6 Future Developments

Fig. 5.8. Inverse dispersion polymerization of acrylic acid. Freeze–fracture SEM image of dispersion 2 min after adding aqueous reductant. Reproduced by permission of Elsevier Science Ltd. From Ref. 123.

reactor startup procedures. The initial sequence in which the initiator and drop stabilizer are put into the reactor can aﬀect copolymer composition and polymerization rates because inter-drop mixing can be restricted during polymerization. That is especially important when the initiator and monomer are not premixed before entry to the reactor. Good control of copolymer composition will continue to be difﬁcult when monomer reactivity ratios vary widely. If late addition of some monomer is used to maintain constant composition at high conversions, then drop segregation must be overcome. Here, a careful choice of drop stabilizer, and possibly the use of solvents, may be helpful. Desirable changes in PSD may be obtained by exercising better control of drop viscosity in the initial drop formation stage. That might be achieved by careful use of radical inhibitors which reduce initial polymerization rates (and associated viscosity increases) and allow the initial DSD to develop in a desired way. The decision to remove (or not remove) radical inhibitors from monomers, before they are put into the reactor, should not be made in an arbitrary way. Radical inhibitors that dissolve preferentially in the continuous phase might also be used to eliminate the initial formation of emulsion particles. That could be important because those emulsion particles, which represent a small mass fraction of the drop phase, have

241

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. 5.6.2

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 undesirable 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 conditions. 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 reactors 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 regions of localized turbulence; restricted overall back-mixing; and good heat transfer. New designs of oscillatory baﬄed reactors [106] may make such processes feasible 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 number 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 important. More work is required to determine the relationships between drop sizes and agitation conditions when the ﬂuid ﬂow is not fully turbulent. Maintenance

Notation

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 between the phases, especially on the copolymer composition, will be important. Prediction 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 inside 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 effect 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 C1 ; C3 ; C4 CI CM d d max d min d32 D

baﬄe width [m] constants (dimensionless) initiator concentration [mol m3 ] monomer concentration [mol m3 ] drop diameter [m] maximum drop diameter [m] minimum drop diameter [m] Sauter mean diameter [m] impeller diameter [m]

243

244

5 Free-radical Polymerization: Suspension

f G H kd kp kt N Re Ri Rp T We We crit

eﬃciency factor (dimensionless) clearance underneath impeller [m] height of the total suspension [m] initiator decomposition rate coeﬃcient [s1 ] propagation rate coeﬃcient [m 3 mol1 s1 ] termination rate coeﬃcient [m 3 mol1 s1 ] impeller speed [s1 ] Reynolds number (dimensionless) initiation rate [mol m3 s1 ] polymerization rate [mol m3 s1 ] tank diameter [m] Weber number (dimensionless) critical Weber number (dimensionless)

Greek parameter to deﬁne ﬂow (dimensionless) local energy dissipation rate per unit mass of ﬂuid [m 2 s3 ] Kolmogorov length [m] viscosity [kg m1 s1 ] kinematic viscosity [m 2 s1 ] density [kg m3 ] interfacial tension [kg m2 ] volume fraction of dispersed phase (dimensionless)

a e h m n r s j Subscripts c d

continuous phase dispersed phase

References 1 M. Munzer, E. Trommsdorff, in

Polymerization Processes, Ed. C. E. Schildknecht, I. Skeist, Chapter 5, p. 106, Wiley. New York, 1977. 2 H. Bieringer, K. Flatan, D. Reese, Angew. Makromol. Chem., 1984, 123, 307–334. 3 H. Warson, Polym. Paint. Col. J., 1983, 24, 541–544. 4 B. W. Brooks, Makromol. Chem. Makromol. Symp., 1990, 35/36, 121– 140.

5 H. G. Yuan, G. Kalfas, W. H. Ray,

6

7 8 9

Rev. Makromol. Chem. Phys., 1991, C31(2/3), 215–299. E. Vivaldo-Lima, P. E. Wood, A. E. Hamielec, Ind. Eng. Chem. Rev. 1997, 36, 939–965. R. Ashady, Colloid Polym. Sci., 1992, 270, 717–732. M. Langsam, J. T. Cheng, J. Appl. Polym. Sci., 1985, 30, 1285–1308. C. G. Hagberg, Polym. Mater. Sci. Eng., 1988, 58, 614–621.

References 10 R. Ashady, G. W. Kenner, A.

11 12

13

14

15

16

17

18 19

20 21 22 23

24 25

26 27 28 29 30

Ledwith, J. Polym. Sci., Polym. Chem. Ed., 1974, 12, 2017–2025. Z. Izumi, H. Kitagawa, J. Polym. Sci. A-1, 1967, 5, 1967–1975. A. Lopez, B. W. Brooks, Instn. Chem. Engrs Research Event, Nottingham, 1997, pp. 1225–1228. S. X. Zhou, Z. X. Weng, Z. M. Huang, Z. R. Pan, J. Appl. Polym. Sci., 2001, 79, 1431–1438. M. J. Rosen, Surfactants and Interfacial Phenomena, 2nd ed., Wiley, New York, 1989. R. P. Borwankar, S. I. Chung, D. T. Wasan, J. Appl. Polym. Sci., 1986, 32, 5749–5762. P. J. Dowding, B. Vincent, Colloids and Surfaces A, Phys. Eng. Aspects, 2000, 161, 259–269. M. Zerfa, B. W. Brooks, Colloids and Surfaces A, Phys. Eng. Aspects, 1998, 132, 267–273. M. Zerfa, B. W. Brooks, Chem. Eng. Sci., 1996, 51, 3591–3611. M. D. Horrill, B. W. Brooks, Inst. Chem. Engrs Research Event, Leeds, 1996, 545–547. S. U. Pickering, J. Chem. Soc., 1907, 2001. Z. Wang, B. W. Brooks, Polymer International, 1992, 28, 239–244. E. O’Shima, M. Tanaka, Kagaku Kogaku Ronbunshu, 1982, 8, 188. D. Wolters, W. Meyer-Zaika, F. Bandermann, Macromolecular Materials and Engineering, 2001, 286, 94–106. Z. Wang, B. W. Brooks, Polymer International, 1993, 30, 317–326. R. Aveyard, B. P. Binks, J. H. Clint, Advances in Colloid and Interface Science, 2003, 100, 503–546. R. Shinnar, J. M. Church, Ind. Engng. Chem., 1960, 52, 253–256. A. N. Kolmogorov, Dokl. Acad. Nauk SSSR, 1949, 66, 825–828. C. A. Coulaloglou, L. L. Tavlarides, A.I.Ch.E.J., 1976, 32, 289–297. A. N. Kolmogorov, Dokl. Acad. Nauk SSSR, 1941, 30, 301–304. J. H. Rushton, E. W. Costich, H. J. Everett, Chem. Engng. Prog., 1950, 46, 395–404 and 467–476.

31 J. O. Hinze, A.I.Ch.E., 1955, 1, 289–

295. 32 R. Shinnar, J. Fluid, Mech., 1961, 10,

259–275. 33 F. B. Sprow, Chem. Engng. Sci., 1967,

22, 435–442. 34 M. Kuriyama, M. Ono, H. Tokanai,

35 36

37

38

39 40 41

42 43 44 45

46 47 48 49 50

51

52

H. Konno, Chem. Engng. Res. Des., 1996, 74, 431–437. M. Zerfa, B. W. Brooks, Chem. Eng. Sci., 1996, 51, 3223–3233. B. W. Brooks, H. N. Richmond, Chem. Engng Sci., 1994, 49, 1053– 1064. F. Jahanzad, S. Sajjadi, B. W. Brooks, Macromolecular Symposia, 2004, 206, 255–262. F. Langner, H. U. Moritz, K. H. Reichert, Chem. Ing. Tech., 1979, 51, 746–747. D. E. Leng, G. J. Quarderer, Chem. Engng, Commun., 1982, 14, 177–201. G. I. Taylor, Proc. R. Soc. Lond., 1934, A 146, 501–523. C. Jegat, A. Bois, M. Camps, J. Polym. Sci.: Polym. Phys. Ed, 2001, 39, 201– 210. H. P. Grace, Chem. Eng. Commun., 1982, 14, 225–277. B. J. Bentley, L. G. Leal, J. Fluid. Mech., 1986, 167, 241–283. E. G. Chatzi, C. Kiparissides, A.I.Ch.E.J., 1992, 41, 1640–1652. K. Arai, M. Konno, Y. Matunaga, S. Saito, J. Chem. Eng. Japan, 1977, 10, 325–330. C. Y. Wang, R. V. Calabrese, A.I.Ch.E.J., 1986, 32, 667–676. M. Konno, K. Arai, S. Saito, J. Chem. Eng. Japan, 1982, 15, 131–135. F. B. Sprow, A.I.Ch.E.J., 1967, 13, 954–998. W. J. Howarth, Chem. Eng. Sci., 1964, 19, 33–38. J. Alvarez, J. J. Alvarez, M. A. Hernandez, Chem. Eng. Sci., 1994, 49, 99–113. D. Maggioris, A. Goulas, A. H. Alexopoulos, E. G. Chatzi, C. Kiparissides, Chem. Eng. Sci., 2000, 55, 4611–4627. B. Yang, K. Takahashi, M. Takeishi, Ind. Eng. Chem. Res., 2000, 39, 2085– 2090.

245

246

5 Free-radical Polymerization: Suspension 53 R. V. Calabrese, C. Y. Wang, N. P.

54

55

56

57 58 59 60 61 62 63

64 65 66

67 68

69 70

71 72

73

Bryner, A.I.Ch.E.J., 1986, 32, 677– 681. P. O. Hong, J. M. Lee, Ind. Eng. Chem. Proc. Des. Dev., 1985, 24, 868– 872. B. W. Brooks, H. N. Richmond, Chem. Engng. Sci., 1994, 49, 1843– 1853. A. N. Sathyagal, D. Ramkrishna, Chem. Engng. Sci., 1996, 51, 1377– 1391. M. F. Cunningham, Polym. React. Eng., 1999, 7, 231–257. F. W. Billmeyer, Textbook of Polymer Science, Wiley, New York, 1971. S. Hashim, B. W. Brooks, Chem. Engng. Sci., 2002, 57, 3703–3714. M. Zerfa, B. W. Brooks, Chem. Engng. Sci., 1997, 52, 2421–2427. M. Zerfa, B. W. Brooks, J. Appl. Polym. Sci., 1996, 60, 2077–2086. M. Zerfa, B. W. Brooks, J. Appl. Polym. Sci., 1997, 65, 127–134. C. H. Bamford, W. G. Barb, A. D. Jenkins, P. F. Onyon, The Kinetics of Vinyl Polymerisation by Radical Mechanisms, Butterworths, London, 1958. J. H. Duerksen, A. E. Hamielec, J. W. Hodgins, A.I.Ch.E.J., 1967, 13, 1081. B. W. Brooks, Proc. R. Soc. Lond., 1977, A 357, 183–192. J. T. S. Bogunjoko, B. W. Brooks, Makromol. Chem., 1983, 184, 1603– 1621. N. Tefera, G. Weickert, Makromol. Chem. Phys., 1994, 195, 3067–3085. O. F. Olaj, I. Bitai, H. Hinklemann, Makromol. Chem., 1987, 188, 1689– 1702. M. Buback, C. Kowollik, Macromolecules, 1998, 31, 3211–3215. N. Tefera, G. Weickert, R. Bloodworth, J. Schweer, Macromol. Chem. Phys., 1994, 195, 3067–3085. D. S. Achilias, C. Kiparissides, Polymer, 1994, 35, 1714–1721. N. Tefera, G. Weickert, K. R. Westerterp, J. Appl. Polym. Sci., 1997, 63, 1649–1661. G. A. ONeil, M. B. Wisnudel, J. M. Torkelson, Macromolecules, 1996, 29, 7477–7490.

74 A. Faldi, M. Tirrell, T. P. Lodge, E.

75 76 77 78

79

80 81

82 83 84 85 86 87 88

89

90 91

92

93

94

Vonmeerwall, Macromolecules, 1994, 27, 4184–4192. J. G. Qin, W. P. Guo, Z. Zhang, Polymer, 2002, 43, 1163–1170. D. S. Achilias, C. Kiparissides, Macromolecules, 1992, 25, 3739–3750. J. San Roman, E. L. Madruga, Angew. Makromol. Chem., 1980, 86, 17–31. M. W. Allsopp, in Manufacture and Processing of PVC, Ed. R. H. Burgess, Chapter 7, Applied Science, London, 1982. R. C. Stephenson, P. V. Smallwood, in Encyclopedia of Polymer Science and Engineering, 2nd ed., Vol. 17 and Supplement, Wiley, New York, 1989. P. V. Smallwood, Polymer, 1986, 27, 1609–1618. A. F. Cebollada, M. J. Schmidt, J. N. Farber, N. J. Capiati, E. M. Valles, J. Appl. Polym. Sci., 1989, 37, 145–166. Y. Z. Bao, B. W. Brooks, J. Appl. Polym. Sci., 2002, 85, 1544–1552. M. H. Lewis, G. R. Johnson, J. Vinyl Technol., 1981, 3, 102. D. H. Lee, J. Chem. Eng. Japan, 1999, 32, 97–103. F. Wolf, I. Schuessler, Plaste und Kautschuk, 1971, 18, 492–494. S. Ormondroyd, Br. Polym. J., 1989, 20, 353–359. J. T. Cheng, J. Macromol. Sci., Phys., 1981, 3B20, 65–380. J. T. Cheng, M. Langsam, J. Macromol. Sci., Chem., 1984, A21, 395–409. M. Clark, in Particulate Nature of PVC, Ed. G. Butters, Applied Science, London, 1982. B. E. Toernell, J. M. Uustalu, J. Vinyl Technol., 1982, 4, 53–56. H. Nilsson, C. Silvergen, B. Tornell, J. Vinyl Technol., 1985, 7, 123–127. C. Kiparissides, I. Moustakis, A. Hamielec, J. Appl. Polym. Sci., 1993, 49, 445–459. J. A. Davidson, D. E. Witenhafer, J. Polym. Sci. Polym. Phys. Ed., 1980, 18, 51–69. Y. Z. Bao, J. G. Liao, Z. M. Huang, Z. X. Weng, J. Appl. Polym. Sci., 2003, 90, 3848–3855.

References 95 M. Langsam, P. A. Mango, J. Appl. 96 97

98

99

100

101

102 103 104

105 106 107

108

109 110

Polym. Sci., 1986, 31, 2361–2376. W. P. Hohnstein, H. Mark, J. Polym. Sci., 1946, 1, 127–145. O. Okay, E. Soner, A. Gungor, T. I. Balkas, J. Appl. Polym. Sci., 1985, 30, 2065–2074. Y. Ohtsuka, H. Kagaguchi, T. Hamasaki, J. Appl. Polym. Sci., 1982, 27, 1771–1782. W. Kangwansupamonkon, S. Damronglerd, S. Kiatkamjornwong, J. Appl. Polym. Sci., 2002, 85, 654–669. J. Y. Oldshue, D. O. Mechler, D. W. Grinnell, Chem. Eng. Prog., 1982, 198, 68–74. E. Vivaldo-Lima, P. E. Wood, A. E. Hamielec, A. Penlidis, Canad. J. Chem. Eng., 1998, 76, 495–505. M. Tanaka, T. Izumi, J. Chem. Eng. Japan, 1985, 18, 354–358. M. Tanaka, E. O’shima, Canad. J. Chem. Eng., 1988, 66, 29–35. Y. Sato, Y. Murakami, T. Hirose, Y. Hashiguchi, S. Ono, M. Ichikawa, J. Chem. Eng. Japan, 1979, 12, 448– 453. M. Tanaka, T. Takahashi, I. Kimura, Chem. Eng. Tech., 1996, 19, 97–102. X. Ni, Y. Zhang, I. Mustafa, Chem. Engng. Sci., 1999, 54, 841–850. K. H. Reichert, H. U. Moritz, C. Gabel, G. Deiringer, in Polymer Reaction Engineering, Ed. K. H. Reichert, W. Geiseler, pp. 153–174, Hanser, Munich, 1983. W. Baade, H. U. Moritz, K. H. Reichert, J. Appl. Polym. Sci., 1982, 27, 2249–2267. T. W. Taylor, K. H. Reichert, J. Appl. Polym. Sci., 1985, 30, 227–250. X. Ni, K. R. Murray, Y. M. Zhang,

111 112

113

114

115

116 117 118

119

120 121 122

123 124

D. Bennett, T. Howes, Powder Technol., 2002, 124, 281–286. J. M. Pinto, R. Giudici, Chem. Engng. Sci., 2001, 56, 1021–1028. T. Mejdell, T. Pettersen, C. Naustdal, H. F. Svendsen, Chem. Engng. Sci., 1999, 54, 2459–2466. G. D. Longeway, D. E. Witenhafer, J. Vinyl Additive Technol., 2000, 6, 100– 103. R. H. Perry, D. W. Green, Perry’s Chemical Engineers’ Handbook, 7th ed., McGraw Hill, New York, Section 1127, 1997. N. Lazrak, A. Ricard, 5th International Workshop on Polymer Reaction Engineering, Dechema Monograph 131, pp. 291–299, VCHVerlagsgesellschaft, Weinheim, 1995. S. Nemeth, F. Thyrion, Chem. Eng. Technol., 1995, 18, 315–323. B. W. Brooks, J. Appl. Polym. Sci., 1983, 28, 619–623. J. W. Vanderhoff, E. B. Bradford, H. L. Tarkowski, J. B. Shaﬂer, R. M. Wiley, ACS Adv. Chem. Ser., 1962, 34, 32. X. Ni, J. C. Johnstone, K. C. Symes, B. D. Grey, D. C. Bennett, A.I.Ch.E.J., 2001, 47, 1746–1757. Z. Liu, B. W. Brooks, J. Appl. Polym. Sci., 1997, 66, 2191–2197. Z. Liu, B. W. Brooks, J. Polym. Sci., 1999, A37, 313–324. M. V. Dimonie, N. N. Marinescu, D. S. Vasilescu, O. Shaffer, H. Caldararu, A. Caragheorgheopol, J. J. Languri, Macromol. Sci., 1992, A29, 277. Z. Liu, B. W. Brooks, Polymer, 1999, 40, 2181–2188. Z. Liu, B. W. Brooks, J. Polym. Sci., 1997, 66, 2191–2197.

247

249

6

Emulsion Polymerization1 Jose´ C. de la Cal, Jose´ R. Leiza, Jose´ M. Asua, Alessandro Butte`, 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, textiles, and construction materials, impact modiﬁers for plastic matrices, and diagnostic 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 substitution of solvent-based system by waterborne products. In a scenario of reduction of margins, increasing competition, and public sensitivity to environmental issues, emulsion polymer producers are forced to achieve eﬃcient production of high-quality materials in a consistent, safe, and environmentally 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 microstructure (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 eco1) 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 polymer microstructure should be achieved in the reactor. This involves a deep understanding of the emulsion polymerization process, the highest level of understanding being the development of predictive mathematical models. In addition, eﬃcient, safe, and consistent production requires accurate on-line monitoring, optimization, 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 process. 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 included in the formulation as they provide some special characteristics, such as improved 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 Tab. 6.1.

Typical formulation for emulsion polymerization.

Ingredient

Content [wt.%]

Monomer(s) styrene methyl methacrylate vinyl chloride vinyl acetate butadiene butyl acrylate 2-ethylhexyl acrylate Veova 10 ethylene (meth)acrylic acid crosslinking monomers Deionized water Initiators Emulsiﬁers Chain-transfer agents

50–55

45 0.5 0.5–3

Typically, emulsion polymerization is carried out in stirred-tank reactors, which commonly operate in a semicontinuous mode, although both batch and continuous 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 formulations, the amount of surfactant exceeds that needed to completely cover the monomer 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 thermal, 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 monomer in the aqueous phase. After adding some monomer units, the oligoradicals become hydrophobic enough to be able to enter the micelles (entry into the monomer 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

251

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 polymer 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 hydrophobic 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

neous and heterogeneous nucleation may be operative in a given system. In general, 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 homogeneous) 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 nucleation, 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 monomer swollen polymer particles coexist in the reactor (Figure 6.2(b)). Polymer particles compete eﬃciently for radicals, and hence monomer is consumed by polymerization inside the polymer particles. It is worth mentioning that polymerization proceeds according to the same mechanisms as are discussed in Chapter 4 for free-radical polymerization in homogeneous systems. The monomer that is consumed by polymerization in the polymer particles is replaced by monomer that diffuses 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 radical and also because they are destroyed to provide surfactant to stabilize the increasing 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 nucleation is not possible and there is no free surfactant available in the system to stabilize the particles formed by homogeneous nucleation. The stage of the batch emulsion 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 during the rest of the process. In Interval II, the system is composed of monomer droplets and polymer particles (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 solubility 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 monomer 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-

253

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 polymer 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 polymerizes in Interval III (Figure 6.2(d)). In this interval, monomer concentration in the polymer particles decreases continuously. The ﬁnal product is a waterborne, concentrated (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 surfactant 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 terminate 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

Fig. 6.3.

Species present in semicontinuous emulsion polymerization.

tion is advantageous because the low viscosity of the latex allows high heat removal 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 combination 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 compartmentalization of the radicals, emulsion polymerization overcomes the limita-

255

256

6 Emulsion Polymerization

tions imposed by the kinetics of bulk and solution polymerization, and high polymerization 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 materials, 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 polymer. The main disadvantage of emulsion polymerization is that the product contains emulsiﬁer and residues of initiator, which give it water sensitivity.

6.3

Alternative Polymerization Techniques

Dispersed polymers are also achieved by inverse emulsion polymerization, miniemulsion polymerization, dispersion polymerization, and microemulsion polymerization. Inverse emulsion polymerization is used in the preparation of solventbased 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 organic solvent [10]. In miniemulsion polymerization [11], the size of the monomer droplets is substantially reduced (dd ¼ 50–1000 nm) by combining a suitable emulsiﬁer and an eﬃcient emulsiﬁcation apparatus, and stabilizing the resulting monomer miniemulsion against diﬀusional degradation by using a costabilizer (hydrophobic, 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, catalytic chain-transfer agents, inorganic materials, agents for controlled radical polymerization) 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

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 stabilized 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 stable 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 continuous to the dispersed phase. Dispersion polymerization allows the production of monodispersed micron-size particles, which are too large for emulsion polymerization and too small for suspension polymerization. Microemulsion polymerization [13] involves the polymerization of oil-in-water and water-in-oil monomer microemulsions. Microemulsions are thermodynamically stable and isotropic dispersions, whose stability is due to the very low interfacial tension achieved using appropriate emulsiﬁers. Particle nucleation occurs upon entry of a radical into a microemulsion droplet. Microemulsion polymerization allows the production of particles smaller than those obtained by emulsion polymerization. This leads to a higher number of polymer particles, which results in a more compartmentalized system. Under these conditions the lifetime of the polymer chains increases, leading to ultra-high molecular weights.

257

258

6 Emulsion Polymerization

6.4

Kinetics of Emulsion Polymerization

Conventional emulsion polymerization occurs in a three-phase system (polymer particles, aqueous phase, and monomer droplets) and polymerization may, in principle, occur in any phase. However, the concentrations of both monomer and radicals 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 monomer droplets is very low because the monomer droplets are not eﬃcient at capturing 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 propagation 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

n NA

ðmol/particle sÞ

ð1Þ

The overall polymerization rate Rp takes into account the existence of many polymer particles in the system according to Eq. (2), where Np is the number of polymer particles in the reactor, and V the volume of the reactor. Rp ¼ k p ½Mp

n Np NA V

ðmol/m 3 sÞ

ð2Þ

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 ðmol/m 3 sÞ kpji Pj ½Mi p ð3Þ Rp i ¼ N A V j P1 ¼

kp21 ½M1 p kp21 ½M1 p þ kp12 ½M2 p

;

P2 ¼ 1 P1

ð4Þ

In order to calculate the polymerization rate, ½Mi p ; n and Np should be available.

6.4 Kinetics of Emulsion Polymerization

6.4.1

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 monomer that diﬀuses from the monomer droplets through the aqueous phase. In Interval III, there are no droplets and the monomer is mostly located in the polymer particles. In semibatch processes, monomers are continuously fed into the reactor, 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 polymerization 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 particles 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 solubility of the monomer, the higher the equilibrium swelling. The swelling equilibrium depends on the particle size, but for particle sizes common in emulsion polymerization 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 thermodynamic 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]. p

Monomer

fM [a]

Styrene n-Butyl methacrylate n-Butyl acrylate Methyl methacrylate Vinyl acetate Methyl acrylate

0.6 0.6 0.65 0.73 0.85 0.85

[a] fp

M

¼ volume fraction of monomer in the polymer particles.

259

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 usually available, and it has been shown [22] that for solids contents typical of commercial latexes (> 50 wt.%) the use of the Morton–Flory–Huggins equation does not provide signiﬁcant advantages over the use of partition coeﬃcients. In the latj 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, j 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

j

Ki ¼

fi fiw

j ¼ polymer particles; droplets

ð5Þ

Material balances: fpp þ

X

p

fi ¼ 1

i

fww þ

X

fiw ¼ 1

i

X

fid ¼ 1

ð6Þ

i p

Vp 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 Nn

ð7Þ

Figure 6.5 illustrates the mechanisms controlling n. In most emulsion polymerization systems, radicals are produced in the aqueous phase from thermal or redox

6.4 Kinetics of Emulsion Polymerization

Fig. 6.5.

Processes controlling the average number of radicals per particle.

initiators. Often, these radicals are too hydrophilic and cannot directly enter the hydrophobic 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 eﬃciency. In conventional emulsion polymerization, the total area of the polymer particles 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 diﬀusional model is more general, whereas the propagational model is a simpler approach applicable to many homopolymerizations. The rate of radical entry can 3 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

ð8Þ

Equation (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-

261

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 terminate between them. The rate of termination in a particle with n radicals is given by Eq. (9). 2kt nðn 1Þ ¼ 2cnðn 1Þ Rate of Termination ¼ vp NA

radicals particle s

ð9Þ

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 formation of small and mobile radicals that can exit the polymer particle. Radical desorption leads to a decrease in the average number of radicals per particle. Equation (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 radicals 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 coeﬃcient 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

lNp gNA kd ¼ l ðs1 Þ 1 hm lNp þ k p ½Mw þ 2k tw ½Rw

ð11Þ

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 0 1 n lN p A gNA @ n kdðnÞ ¼ l 1 lNp þ k p ½Mw þ 2k tw ½Rw hm

ð12Þ

ðs1 Þ

ð13Þ

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

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 decomposition [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 3 1 1 s ]. [maqueous phase mol 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 Þ n ¼ 0; 1; 2; 3 . . .

ð particles/sÞ

Np Np d½Rw ¼ 2f kI ½I þ kd n 2k tw ½Rw2 ka ½Rw dt NA Vw NA Vw

ð14Þ 3 ðmol/maqueous phase sÞ ð15Þ

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]. n¼

C¼

2ka ½Rw kd þ ðkd2 þ 4ka ½Rw CcÞ 0:5 2ð2ka ½Rw þ kd Þ 2ka ½Rw þ kd þ c

ð16Þ

ð17Þ

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 pioneering 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 particle 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).

263

264

6 Emulsion Polymerization

n¼

ka ½Rw 2ka ½Rw þ kd

ð18Þ

Under Smith–Ewart Case 1 conditions, for a given solids content and if termination 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 aqueous phase is signiﬁcant, the polymerization rate increases with the number of polymer particles. For large particles (dp > 200 nm), high initiator concentrations or redox initiators, and slow termination rates (gel eﬀect), n g 0:5 (Smith–Ewart Case 3) and n can be calculated as follows: ka ½Rw 0:5 n¼ 2c

ð19Þ

For a given solids content under Smith–Ewart Case 3 kinetics, n is inversely proportional 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. Heterogeneous Nucleation The rate of formation of polymer particles by heterogeneous nucleation is expressed 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). 6.4.3.1

1 dNp Nm ¼ R nuc ¼ kam ½Rw V dt V Nm ¼

ðSw CMCVw ÞNA nm

ð particles/m 3 sÞ

ðmicellesÞ

ð20Þ ð21Þ

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 micelle); and Sw [mol] is the amount of surfactant that is in the aqueous phase forming 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

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). Sw ¼ ST Ap ¼

Ap as

4:83Np0:33

ð22Þ Vpol fpp

!0:66 ð23Þ

265

266

6 Emulsion Polymerization

Assuming no termination of radicals in the aqueous phase and that during nucleation 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 concentrations, where rv [m 3 s1 ] is the volumetric growth rate of one polymer particle.

2f kI ½INA fpp 0:4 as ST 0:6 Np o Vw rv Vw

ð24Þ

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 nucleation. 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.

2f kI ½INA fpp 1z as ST z Np o Vw rv Vw

0:6 a z a 1

ð25Þ

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 formation 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 concentration 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. 6.4.3.2

R nuc ¼ k p ½Mw

ðR jcrit þ Micrit Þ V

ð particles/m 3 sÞ

ð26Þ

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 a2

R jcrit ¼ a1 Micrit ¼

kd nNp icrit a k p ½Mw 1

2f kI INA k p ½Mw

ðradicalsÞ

ðradicalsÞ

ð27Þ

ð28Þ

6.5 Molecular Weights

In these equations, d is the critical length for entry of radicals generated from the initiator, a1 is the probability of propagation of radicals able to enter the polymer particles (generated from desorbed radicals, and from the initiator, with lengths equal or greater than d) and a2 the probability of propagation of radicals generated from the initiator, with length smaller than d). These probabilities are given by Eqs. (29) and (30). k p ½Mw

a1 ¼

k p ½Mw þ 2k tw ½Rw þ ka a2 ¼

Np Nm þ kam NA Vw NA Vw

k p ½Mw k p ½Mw þ 2k tw ½Rw

ð29Þ

ð30Þ

Simultaneous Heterogeneous and Homogeneous Nucleation In systems including rather water-soluble monomers and surfactant concentrations high enough for micelles to be present, particle nucleation may be formed by both heterogeneous and homogeneous mechanisms. In this case, the overall nucleation rate is given by Eq. (31). 6.4.3.3

R nuc ¼ ½kam ½Rw Nm þ k p ½Mw ðR jcrit þ Micrit Þ

1 ð particles/m 3 sÞ V

ð31Þ

Coagulative Nucleation In this mechanism, the species formed by entry of radicals into micelles and by self-precipitation of radicals are considered nonstable precursor particles. The actual particles are formed by coagulation and polymerization growth. The detailed modeling of coagulative nucleation is rather complex [31, 32]. A simpliﬁed model can be derived by considering that the precursor particles disappear by mutual coagulation, that they are captured by the existing polymer particles, and that the rate of nucleation of mature particles is proportional to a k4 th power of the precursor concentration [Eqs. (32) and (33), where k1 ; k2 ; k3 , and k4 are adjustable parameters that depend on the type and amount of surfactant] [33]. 6.4.3.4

Npr2 dNpr Np ¼ kam ½Rw Nm þ k p ½Mw ðR jcrit þ Micrit Þ k1 Npr k2 dt V V k4 Npr R nuc ¼ k3 V

ð32Þ ð33Þ

6.5

Molecular Weights

The molecular weight distribution of the polymer has a profound eﬀect on its ﬁnal properties [34, 35]. Emulsion polymerization is a compartmentalized system in

267

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 polymerization 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. Zero–One System (Smith–Ewart Cases 1 and 2) In a zero–one system, the inactive polymer chains are formed in particles containing 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 . 6.5.1.1

dMm ¼ k tr; M ½Mp R m þ ka ½Rw R m dt

m ¼ 1; 2; 3; . . .

ðmacromolecules/sÞ

ð34Þ

Integration of Eq. (34) allows calculation of the whole molecular weight distribution. However, this is computationally demanding and often the MWD is represented in terms of the moments of the distribution, the kth-order moment of the distribution being deﬁned by Eq. (35). nk ¼

y X m¼1

m k Mm

ð35Þ

6.5 Molecular Weights

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 kthorder moment of the distribution of active chains.

Ruk ¼ ðk tr; M ½Mp þ ka ½Rw Þ

mk V

ð36Þ

The balance for active chains of length m in particles with one radical (N1 ) is obtained from Eq. (37). X dR m 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

ð37Þ

The 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)].

m0 ¼

m1 ¼

X

R m ¼ N1 ¼ nNp

ka ½Rw N0 þ ðk tr; M ½Mp þ k p ½Mp kd ÞN1

m 2 ¼ m1 1 þ

ka ½Rw þ k tr; M ½Mp ! 2k p ½Mp

ð38Þ

ka 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.

Mn ¼

n1 PM ; n0

Mw ¼

n2 PM n1

ð39Þ

On some occasions, it is interesting to know the average molecular weights produced at a given moment in the process. These instantaneous average molecular weights can be calculated by means of Eqs. (40).

Mni ¼

Rv1 PM ; Rv0

Mwi ¼

Rv2 PM Rv1

Combination of Eqs. (36), (38), and (40) yields Eqs. (41).

ð40Þ

269

270

6 Emulsion Polymerization

M ni ¼

A

M wi ¼

ka ½Rw N0 þ ðk tr; M ½Mp þ k p ½Mp kd Þ m1 PM ¼ PM m0 ka ½Rw þ k tr; M ½Mp k p ½Mp ka ½Rw þ k tr; M ½Mp

PM

ð41Þ

2k p ½Mp m2 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. M ni A

k p ½Mp ka ½Rw

PM

ð42Þ

If radical termination in the aqueous phase is negligible, from Eq. (15) we obtain Eq. (43) and hence Eq. (44).

ka ½Rw ¼ 2f kI ½I M ni A

NA Vw Np

k p ½Mp Np PM 2f kI ½INA Vw

ð43Þ

ð44Þ

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 formulation to increase the number of polymer particles. This is a speciﬁc feature of emulsion polymerization that cannot be achieved in any homogeneous (bulk or solution) free-radical polymerization. For the Smith–Ewart Case 1, k tr; M ½Mp g ka ½Rw and then Eq. (45) applies. M ni A

kp PM k tr; M

ð45Þ

Therefore the molecular weights are controlled by chain-transfer reactions and are independent of Np . Pseudo Bulk System (Smith–Ewart Case 3) For Smith–Ewart Case 3, the number of radicals per particle is large and the kinetics 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 controlled 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). 6.5.1.2

6.5 Molecular Weights n1 X dMmi cc i ¼ k tr; M ½Mp Rmi þ 2cd ði 1ÞRmi þ ði 1Þ Rki Rmk dt iNi k¼1

ðmacromolecules/sÞ 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)]. Rn0 ¼ ð2cd þ cc Þ

m02 m þ k tr; M ½Mp 0 Np V V

Rn1 ¼ ð2cd þ 2cc Þ

m 0 m1 m þ k tr; M ½Mp 1 Np V V

Rn2 ¼ ð2cd þ 2cc Þ

m0m2 m2 m þ 2cc 1 þ k tr; M ½Mp 2 Np V Np V V

X dR m ¼ ka ½Rw Np þ k tr; M ½Mp R m dm¼1 þ kp ½Mp ðRm1 R m Þ dt X ðcc þ cd Þ Rm ¼ 0 k tr; M ½Mp R m 2 Rm Np !1=2 X ka ½Rw Np2 R m ¼ nNp ¼ m0 ¼ 2cd þ 2cc

ð47Þ

ð48Þ

m1 ¼

ka ½Rw Np þ k p ½Mp m 0 þ k tr; M ½Mp m 0 k p ½Mp m 0 A m0 m ð2cd þ 2cc Þ þ k tr; M ½Mp ð2cd þ 2cc Þ 0 þ k tr; M ½Mp Np Np

m2 ¼

ka ½Rw Np þ k p ½Mp ð2m1 þ m 0 Þ þ k tr; M ½Mp m 0 2k p ½Mp m1 A m0 m ð2cd þ 2cc Þ þ k tr; M ½Mp ð2cd þ 2cc Þ 0 þ k tr; M ½Mp Np Np

ð49Þ

The instantaneous number and weight-average molecular weights are then given by Eqs. (50).

Mni ¼

Mwi

k p ½Mp PM m ð2cd þ cc Þ 0 þ k tr; M ½Mp Np

m m m2 ð2cd þ 2cc Þ 0 2 þ 2cc 1 þ k tr; M ½Mp m 2 Np Np ¼ PM m 0 m1 ð2cd þ 2cc Þ þ k tr; M ½Mp m1 Np

ð50Þ

271

272

6 Emulsion Polymerization

If bimolecular termination is predominant, then the instantaneous average molecular weights reduce to Eqs. (51). M ni ¼

k p ½Mp Np ð2cd þ cc Þm 0

PM

M wi

m cc m1 2cd þ cc cc 1þ ¼ 2þ PM A Mni PM m1 ðcd þ cc Þm 0 cd þ cc 2ðcd þ cc Þ

ð51Þ

Therefore: M wi ¼ 2Mni

if bimolecular termination is only by disproportionation; and

3 M wi ¼ M ni 2

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

kp 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 structure (molecular size) during the rest of the process, the molecular weight of the nonlinear polymers evolves throughout the whole process because they may become active and inactive several times during that process. Thus, inactive nonlinear chains formed earlier in the process may become active through either a chain-transfer reaction to polymer or a propagation to pendent double bonds, further polymerizing and modifying their structure. The microstructure of these nonlinear 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 crosslinking 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-

6.6 Particle Morphology

sion polymerization, because of the compartmentalized nature of the system. Nevertheless, some solutions have been reported [37–40, 43–46]. The discussion of these approaches is outside the scope of this chapter. Branches are formed by chain transfer to polymer (either intermolecular or intramolecular) or by propagation to pendent double bonds (PDBs). Terminal double bonds may be formed by chain transfer to monomer, termination by disproportionation, and b-scission at relatively high temperatures. PDBs may also be present if multifunctional monomers are used in the formulation. Intermolecular chain transfer to polymer and propagation to PDBs produces long-chain branches (LCBs), and intramolecular chain transfer to polymer, so-called backbiting, produces short-chain branches (SCBs). Chain transfer to polymer occurs through hydrogen abstraction either from the backbone or from a side group. A crosslink point is a tetrafunctional unit that links two polymer chains, namely an H-shaped structure. The mechanisms leading to tetrafunctional units are: intermolecular chain transfer to polymer followed by bimolecular termination by combination of branched radicals, or propagation of a growing branched radical on a terminal or pendent double bond, or polymerization of formulations including multifunctional monomers (often called crosslinkers).

In the free-radical polymerization of vinylic monomers, gel is formed if the following conditions are fulﬁlled: (1) a mechanism leading to LCBs is operative, for example chain transfer to polymer, and (2) a mechanism that links polymer chains is also operative, for example termination by combination or propagation to pendent double bonds. Obviously, if multifunctional monomers are homopolymerized or copolymerized with vinylic monomers, gel is also formed. In this case, the predominant reaction linking polymer chains and leading to polymeric networks is the propagation to pendent double bonds of the multifunctional monomers. Under these circumstances, polymer chains of increasing density of branching and crosslinking are formed and their molecular weight will rapidly increase, leading to the formation of gel polymer.

6.6

Particle Morphology

Latexes made out of composite polymer particles, that is, particles containing different phases, present deﬁnitive advantages in many applications. Thus, particles formed by an elastic core and a hard shell are used as impact modiﬁers for polymer matrices. Hard core–soft shell particles are particularly useful for paints because they have a low minimum ﬁlm formation temperature and are not sticky at higher temperatures. Hollow particles are eﬃcient opaciﬁers, and hybrid polymer– polymer particles, such as epoxy–acrylic polymer particles, combine the properties

273

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 concentration 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 morphology. During this migration, the size of the clusters may vary because of (1) polymerization 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

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 promoting 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 energy of the system; it depends on the polymer–polymer and polymer–water interfacial tensions [47].

6.7

Living Polymerization in Emulsion

Free-radical polymerization (FRP) is a widespread technique for preparing longchain 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 toward functional monomers, and to the mild reaction conditions required to run the polymerization. Moreover, as discussed in the preceding sections in this chapter, emulsion polymerization (and in general every polymerization reaction carried out using water as the continuous medium) has alleviated all the residual inconveniences of FRP. In fact, the corresponding processes do not involve a large demand for organic solvents, they easily achieve large monomer conversions, and they exhibit reduced problems of heat removal, thus making the process intrinsically 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 functionalization), 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 between the free-radical polymerization technique and other polymerization techniques 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, viscosity, 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 living polymerization, instead, these changes are equally distributed among all the polymer chains, which grow uniformly during the whole reaction. Moreover, as

275

276

6 Emulsion Polymerization

the living chains are still able to restart propagating when the monomer is completely depleted, living polymerization represents a route to the production of block copolymers by successive additions of monomers, an approach clearly not possible in FRP. 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 polymer 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 industrial 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 limitations. It can be applied to a broad range of monomers, nearly all vinyl and vinylidene monomers, it is easily operated in the presence of impurities, such as residual 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 reducing the radical concentration so as to minimize the rate of bimolecular termination (a second-order reaction with respect to radical concentration, the propagation being 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). Using 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 instantaneous 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 indicates the rate constant of the reversible termination). If the rate of this new termination is so large as to be dominant with respect to that of the irreversible termination 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 ‘‘initiator’’ providing the capping species, indicated as RX. This initiator reactivates itself 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 insertion 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

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 monomer), and the total number of polymer chains. Note that some irreversibly terminated 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 enumerated in the following sections. 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 nitroxide radical group), to form dormant polymer chains, R n X: 6.7.1.1

kde

Rn þ X S RnX kac

ðaÞ

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 suffers very limited applicability to monomers unlike styrene, and requires high operating temperatures (about 120–140 C). More recent studies were aimed to reduce the operating temperature and to broaden the monomer applicability so as to enlarge the spectrum of block copolymers accessible by this technique [58]. Despite these eﬀorts, the range of application remains quite limited. 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 halogen. It is clear that, if the radical produced can undergo a few propagation steps 6.7.1.2

277

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 between active and dormant states as is found in NMP. The resulting reversible reaction is represented by Eq. (b), where X indicates the halogen atom, MeðnÞ the metal with the corresponding oxidation state and Li the ligand. kde

R n þ X –Meðnþ1Þ =Li S R n X þ MeðnÞ =Li kac

ðbÞ

ATRP owes most of its success to its high compatibility with many diﬀerent monomers, such as styrene, acrylates, methacrylates, (meth)acrylamides, and acrylonitrile, 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 copper atom and it has been noticed that they not only prepare the copper for the reaction, 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]. 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 mechanism. 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 between 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 dormant species with iodine. 6.7.1.3

kex

Rn þ Rm I ! Rm þ RnI

ðcÞ

Therefore, 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 properly 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 bimolecular 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, provided that the initiator concentration is small compared to that of the transfer agent.

6.7 Living Polymerization in Emulsion

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 process to the polymerization of acrylates (for example, n-butyl acrylate), and for the complete lack of control reported for other monomers [64–66]. 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. 6.7.1.4

kadd

ðþÞ

kfrag

R m YC . ðZÞYR n ! R m þ R n YCðZÞY R n þ R m 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 correct 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 polymerized ﬁrst. Even though the most satisfactory results have been obtained by RAFT polymerization 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 process is also very eﬀective for many other monomers, such as acrylates and methacrylates [67–70]. Moreover, the operating temperatures used to carry out this polymerization are usually close to those typical of conventional radical polymerization

279

280

6 Emulsion Polymerization

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].

Fig. 6.8.

[67–70]. The low temperature, along with the wide range of compatible monomers, 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 expressed 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 process is intended to produce block copolymers, in which case terminated homopolymer chains represent a signiﬁcant drawback with respect to the product quality. 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 termination. It has been exhaustively demonstrated that, in spite of the fact that these bimolecular reactions, deactivation and termination, are both very fast and controlled

6.7 Living Polymerization in Emulsion

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 deactivation simply produces a dormant chain again, termination subtracts polymer chains from the activation/deactivation equilibrium and, according to simple stoichiometric arguments, trapping radicals (nitroxides) are accumulated. The ultimate 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. R¼

kac ðRXÞ0 3kde kt t

1=3 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 corresponding reaction scheme, Eq. (b)], remains roughly constant during the whole process, and therefore the activation process can be approximated as a monomolecular process as in NMP. On the other hand, degenerative transfer and RAFT are characterized by completely 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

ð54Þ

No 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

281

282

6 Emulsion Polymerization

chain concentrations. It can be shown [72] that the time needed to obtain 90% conversion after proper tuning of the process parameters so as to obtain a deﬁned fraction 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

kt ðln 10Þ 2 fkp2 ðRXÞ0

ð55Þ

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 promote smaller dead chain contents [compare Eq (55)], in the case of styrenic copolymers this leads to limited improvements only, since undesired side reactions negatively 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 segregated systems, namely emulsion polymerization. Let us focus on a system characterized 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 average 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. Therefore, the fraction of dead chains in the system can be adjusted by tuning the frequency of entry properly, while the transfer reaction rate independently controls the homogeneity of polymer growth. It is again important to notice that in emulsion polymerization the rate of formation of dead chains by irreversible termination 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

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 involved at all. Accordingly, the average number of active chains per particle remains close to 0.5. 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 immediately. In other words, the principle behind radical segregation cannot hold when NMP is active, since two radicals are generated each time an activation reaction 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 deactivation 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 deactivation, 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 activation 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.

283

284

6 Emulsion Polymerization

6.7.4

Application of LRP in Heterogeneous Systems 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 commercially 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 initially 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 distributed 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 monomer 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 monomer 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 complicated [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 polymerization, where it is possible to exploit the segregation typical of these systems. However, 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 6.7.4.1

6.7 Living Polymerization in Emulsion

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. 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 primary 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 miniemulsion polymerizations by NMP; all show good control of the MWD and the aforementioned problem of slow polymerization rates [76]. Due to the improvements in nitroxide eﬃciency, studies involving polymerization of acrylates have also appeared [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 polymerization with several monomers [82]. Figure 6.10 shows a typical result of a block copolymerization of MMA and styrene, which proved to give good control of polymer growth together with high polymerization rates. Still, some problems remain unsolved. 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]. 6.7.4.2

285

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.

6.8

Emulsion Polymerization Reactors

Emulsion polymers are ‘‘products-by-process’’ whose microstructure and properties are determined during the polymerization. Therefore, the reactor type, the operation mode, and the control strategy play a key role in achieving an eﬃcient, safe, and consistent production of high-quality emulsion polymers. 6.8.1

Reactor Types and Processes

In principle, both stirred-tank reactors and tubular reactors, and combinations thereof, may be employed. Stirred-tank Reactors The stirred-tank reactor is the reactor most commonly used in emulsion polymerization. This reactor may operate in batch, semibatch, or continuous mode. A batch reactor is a closed system in which time is the only independent variable. The batch operation can be used for small-scale production of homopolymers from monomers with a relatively low heat of polymerization. However, the drawbacks associated with this type of operation limit its industrial use. These drawbacks are: 6.8.1.1

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 generation 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

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 continuously over a period of time. Most commercial emulsion products are manufactured 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 polymer 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 morphology, 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 continuously. The main characteristic of a continuous stirred tank reactor (CSTR) is the broad residence time distribution (RTD), which is characterized by a decreasing exponential function. The same behavior describes the age of the particles in the reactor 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 copolymer 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 hightonnage 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. 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 reactor, where reaction time is equivalent to the space time (t) in the tube. For this reactor type, grade transition between diﬀerent polymer grades is instantaneous, but the control of polymer properties is almost impracticable. An important disadvantage of the tubular reactor is the inadequate mixing that can lead to phase separation, reactor plugging, and wall fouling [92]. Several modiﬁcations have been performed to improve radial mixing and minimize the associated problems, but to date tubular reactors have not been widely utilized for industrial production. The 6.8.1.2

287

288

6 Emulsion Polymerization

most important modiﬁed tubular reactors include loop reactors, pulsed-ﬂow reactors, wicker-tube reactors, and Couette–Taylor ﬂow reactors. The continuous-loop reactor is probably the only tubular reactor used in commercial production of emulsion polymers [93, 94] and its use is limited to production 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 residence 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 mechanically 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 eﬀective mixing, minimizing phase segregation, wall fouling, and tube plugging [99, 100]. The introduction of sieved plates, Raschig rings, or baﬄes is reported to improve mixing [101, 102]. However, these internal elements may be sources of coagulation, 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 dispersion 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 hydrodynamically 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]. 6.8.2

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 properties, lowers the attractiveness of tubular reactors for commercial production. Therefore the discussion will be limited to stirred-tank reactors.

6.8 Emulsion Polymerization Reactors

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. 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 dissipated 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 impellers on the same shaft can also be employed. In any case, the agitation requirements in emulsion polymerization are often related to the operation mode. In batch operations, vigorous agitation is required in the initial stages to avoid monomer segregation and promote good phase dispersion. Later on, a lower agitation intensity is needed to avoid shear-induced coagulation. In semicontinuous operations, multiple impellers are normally used to ensure 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 facilitate 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 increases ionic strength and thus promotes coagulation, and the accumulation of emulsiﬁers, which leads to local nucleations and/or ﬂocculations. Continuous operation requires mixing characteristics similar to those of the semibatch process. 6.8.2.1

289

290

6 Emulsion Polymerization

Empirical correlations for the key aspects of mixing, such as power consumption, 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 ﬁeld. Heat Transfer Polymerization reactions are highly exothermic and the heat generated must be removed in order to control reactor temperature. In commercial emulsion polymerization, the heat removal rate from the reactor is often the factor limiting productivity. 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]: 6.8.2.2

improved jacket design to increase turbulence, and hence the heat-transfer coeﬃcient (dimpled, half-pipe); cooled baﬄes (the use of internal coils is not an option because they tend to increase fouling and coagulation); external loop heat exchangers for the reaction medium – in this case, a mechanically 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, extra 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 wellmixed stirred-tank reactors the properties do not vary signiﬁcantly with the position in the reactor, and time is the only independent variable. Therefore, the necessary balances for the reactor design may be made at macroscopic level – that is, for the reactor as a whole.

6.9 Reaction Engineering

6.9.1

Mass Balances

Considering inlet and outlet streams in the reactor, the mass balance [Eq. (56)] for any species i results, dNi ¼ Fie Fis þ ðRi ÞV dt

ð56Þ

where Ni [mol] is the total amount of compound i in the reactor; Fie [mol s1 ] the inlet molar ﬂow rate of component i; Fis [mol s1 ] the outlet molar ﬂow rate of i, (Ri ) [mol s1 m3 ] the net generation rate of i in the reactor, and V [m 3 ] the reactor volume. Equation (56) applies for the particular combination of monomers, initiator, water, and emulsiﬁer, and for amounts of each polymerized monomer. Equation (56) also applies to the number of polymer particles and of precursor particles and the moments of the molecular weight distribution, although in those cases units other than moles should be used. The speciﬁc forms of the net generation rates are discussed below. Usually, the polymerization rate of monomer j is expressed as the rate of monomer consumption (Rpj ), and hence ðRMj Þ ¼ Rpj . For batch processes, Fie ¼ Fis ¼ 0, and for semicontinuous operation Fis ¼ 0. For the continuous operation Fis can be calculated from Eq. (57), where V is constant in a continuous operation and Q s is the volumetric outlet ﬂow rate. Fis ¼ Q s

Ni V

ð57Þ

In emulsion polymerization, the density of the reaction medium does not change signiﬁcantly. Therefore, the volumetric inlet and outlet ﬂow rates can be considered to be the same. The advance of the polymerization is usually given in terms of the conversion Xi of monomer i [Eq. (58)]. Here Polymeri and Monomeri are either in grams or in moles. Xi ¼

Polymeri Polymeri þ Monomeri

ð58Þ in the reactor

When more than one monomer is polymerized, an overall conversion can be calculated from Eq. (59), where XT can be gravimetric or molar depending on the units used for Polymeri and Monomeri (grams or moles, respectively). 2

3 Polymeri 5 X XT ¼ 4 X Polymeri þ Monomeri X

ð59Þ in the reactor

291

292

6 Emulsion Polymerization Tab. 6.3.

Usual conversion deﬁnitions.[a]

Conversion

Fractional conversion of monomer i Overall fractional conversion Overall global conversion

Batch

Ni0 Ni Ni0 P ðNi0 Ni Þ P Ni0

Semicontinuous Ð Ni0 þ Fie dt Ni Ð Ni0 þ Fie dt Ð P ðNi0 þ Fie dt Ni Þ Ð P ðNi0 þ Fie dtÞ Ð P ðNi0 þ Fie dt Ni Þ P NTi

Continuous (steady state) Fie Fis Fie P ðFie Fis Þ P Fie

[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 copolymer 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 Rpi

ð60Þ

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 monomer i polymerized. Npol1 y1cum ¼ X Npoli

ð61Þ

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 enthalpy 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, respectively. Rpi [mol m3 s1 ] is the polymerization rate of monomer i, (DHri ) [J mol1 ]

6.9 Reaction Engineering

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

Ni cpi

X dT X ¼ Rpi ðDHri ÞV Fie cpie ðT Te Þ dt þ Q transfer þ Q loss þ Q stirring

ð62Þ

For 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 heattransfer area and DTml the logarithmic mean temperature diﬀerence between the cooling ﬂuid and the reaction medium. Q transfer ¼ UADTml

ð63Þ

DTml is given by Eq. (64), where Twe and Tws [K] are the inlet and outlet temperatures of the cooling ﬂuid (normally water) in the jacket. If Twe A Tws then DTml A ðT Tw Þ. DTml ¼

ðT Twe Þ ðT Tws Þ ðT Twe Þ ln ðT Tws Þ

ð64Þ

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. Nu ¼ a Re b Pr c

j jw

d ð65Þ

Due to changes in the properties of the reaction media (for example, viscosity) the overall heat-transfer coeﬃcient changes during the process. In semicontinuous operation, the heat-transfer area varies during the operation.

293

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

ð66Þ

Combination of Eqs. (62) and (63) or (66) allows the estimation of the polymerization 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]. 6.9.3

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 occasions 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 depends on the method used to measure it. Thus, histograms are used when the PSD is determined by transmission electron microscopy. However, the mathematical analysis is simpler if the PSD is deﬁned in terms of the number density of 3 ], nðvÞ. The units of nðvÞ are the polymer particles of unswollen volume v [mparticle 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). Np ðv1 ! v2 Þ ¼

ð v2

nðvÞ dv

ð67Þ

v1

Np ¼

ðy

nðvÞ dv

ð68Þ

0

Equation (69) gives the macroscopic population balance for a CSTR, where the lefthand 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 second 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 formation of particles of unswollen volume v by particle coagulation. In Eq. (69) nðvÞ and ne ðvÞ are the reactor and inlet number density of polymer particles, Q s [m 3 s1 ] 3 s1 ] the voluis the volumetric ﬂow rate, V [m 3 ] the reactor volume, rv ðvÞ [mparticle metric growth rate of each particle of volume v, kðv; v 0 Þ the coagulation rate con-

6.9 Reaction Engineering

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 y kðv; v 0 Þnðv 0 Þ dv 0 ne ðvÞ nðvÞ ¼ V V qt qv V v0 ð 1 vv0 þ kðv v 0 ; v 0 Þnðv v 0 Þnðv 0 Þ dv 0 2V v0

ð69Þ

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). nðv0 Þ ¼

R nuc V rv ðv0 Þ

ð70Þ

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 formulations. Equations (69) and (70) are conveniently solved by using orthogonal collocation [119, 120]. 6.9.4

Scaleup

The main objective of scaleup is to reproduce the laboratory results in commercialscale reactors in such a way that the end-use polymer properties can be maintained. 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 desired 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 maintained constant from the smaller scale to the commercial equipment. In theory, this could be done by applying the principle of similarity by means of dimensionless groups characterizing the phenomena of interest. However, when several mechanisms are involved in the process, it is impossible to maintain all the dimensionless 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 considered 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

295

296

6 Emulsion Polymerization Tab. 6.4.

Several scale factors during scaleup.

Parameter [a]

Pilot scale (50 L)

Plant scale of 50 m 3 (scaleup procedure)

P (power) N (impeller rpm) Q (impeller ﬂow) P=V (power/volume) q=V (heat transfer rate/volume) ND (tip speed) Q=Q S (impeller ﬂow/feed rate) Q=H (impeller ﬂow/head) Re (Reynolds number)

1 1 1 1 1 1 1 1 1

10 3 0.21 215 1 0.08 2.1 0.21 46.8 21.5

[a] Units

10 8 10 10 4 10 5 1 100 10 1 10 3

100 0.1 100 0.1 0.05 1 0.1 100 10

10 5 1 10 3 100 0.21 10 1 10 100

0.1 0.01 10 0.0001 0.01 0.1 0.01 10 3 1

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 excessive 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 parameters that have a major eﬀect on the polymer properties. This task is diﬃcult in emulsion polymerization, where mixing aﬀects several polymer properties. Monomer droplet and particle size, mass transfer between the diﬀerent phases, coagulation 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

Production rate: The goal is to maximize the production rate of the available reactors. 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 speciﬁcations of safety, production rate, and product quality, it is necessary to develop suitable 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 reactors 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 properties are produced in the reactor. In polymer latexes the properties that one would like to control during the polymerization include: monomer(s) conversion, copolymer composition, MWD, PSD, particle morphology, branching and crosslinking, gel content, and particle size distribution. Not all of these properties can be monitored 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 selection of the most appropriate technique [124, 126]. For instance, monomer conversion and copolymer composition can be monitored on-line by means of densimetry, 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 monitoring 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 different 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.

297

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 disadvantages of each one, and ﬁnally examples of application for emulsion polymerization reactors will be given. 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 successful implementation of this technique for latexes with a high solids content has been reported [134], the setup is liable to suﬀer clogging. Additional disadvantages of this monitoring technique include the time delays associated with the analysis itself when multicomponent copolymerizations are monitored. This drawback can be partially alleviated by using capillary columns and modern gas chromatography equipment with reduced analysis time. 6.10.1.1

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 uncertainty to the measurements. Note also that formulation ingredients such as CTAs are not measurable by this means. 6.10.1.2

Densimetry This technique is based on the density change that occurs when monomer is converted to polymer. This diﬀerence, which is obviously a maximum in bulk polymerization, allows emulsion polymerizations of relatively high solids content to be accurately 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 thermostated measurement cell and the system is liable to suﬀer clogging. A further dis6.10.1.3

6.10 On-line Monitoring in Emulsion Polymerization Reactors

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 monitored almost continuously. Ultrasound The principle of ultrasound relies on the propagation of the ultrasonic wave pressure. 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 continuous 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 systems 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 velocity and attenuation fully, calibration is required to obtain accurate predictions of monomer conversion or other properties such as particle size because the theoretical 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. 6.10.1.4

Spectroscopic Techniques Since the mid-1990s, there has been plenty of activity regarding the use of spectroscopic 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 insitu measurements in remote and harsh environments (high temperatures, pressures, 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 polymerization reaction monitoring. These can be divided between absorption- and emission-based techniques. IR, NIR, and MIR are absorption-based. 6.10.1.5

299

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 techniques 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 region, 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 monitor emulsion polymerization reactions [147], diﬃculties due both to broad bands associated with water absorption and to the eﬀect of the particle size have been reported [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. However, 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 addition, 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 dispersive 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 region, 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 spectroscopy 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 polymerization in dispersed media. It is very sensitive to CbC bonds. This makes it possible to follow, with high accuracy, 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 a)

Weight Fraction MMA

0.006 0.005 0.004 0.003 0.002 0.001 0 0

0.2

0.4

0.6

0.8

1

Conversion b)

Weight Fraction BA

0.05

0.04

0.03

0.02

0.01

0 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 spectroscopy [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-

301

6 Emulsion Polymerization c)

Solids Content (%)

302

60

50

40

30

20

10 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 necessary because PLS, a multivariate linear calibration, may not be suﬃcient if the degree of nonlinearity between the spectra and the properties of interest is signiﬁcant. 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 sensors 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 estimation techniques [114, 153–155]. Reaction calorimetry is based on the energy balance in the reactor, given by Eq. (71), 6.10.1.6

X

Ni cpi

dT ¼ Q r þ Q feed þ Q transfer þ Q loss þ Q stirring dt

ð71Þ

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

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 reactor (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

ð72Þ

The 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 balance 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. m w cpw

dTws ¼ UAðTws TÞ þ m_ w cpw ðTwe Tws Þ dt

ð73Þ

Heat-balance calorimetry allows the simultaneous estimation of U and Q r , provided that Twe Tws could be accurately measured. Therefore, heat-balance calorimetry is best suited for large-scale industrial reactors because a signiﬁcant diﬀerence 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 information 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 reaction and the overall heat-transfer coeﬃcient from temperature measurements of the reactor and the jacket. This is done by taking advantage of the diﬀerent dynamics 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 oscillatory temperature signals allows calculatation of the UA term and Q r simultaneously [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 sinusoidal 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 uncertain [158]. The amount of free monomer and the copolymer composition in emulsion polymerization reactors can be inferred from measurement of the heat of reaction, Q r

303

6 Emulsion Polymerization

a) 350 Free Monomer (g)

Gravimetry

300

Chromatography Calorimetry

250 200 150 100 50 0 0

50

100

150

200

250

300

350

Time (min)

b) 1.0 ________

Terpolymer Composition

304

Calorimetry

Full Points GC

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 measurement (gravimetry and gas chromatography) for a VAc/BA/AA high solids content semibatch emulsion polymerization. In Figure 6.13(b), the estimation of the terpolymer composition from calorimetric measurements is compared with NMR and GC measurements for a VAc/MMA/BA emulsion terpolymerization [159].

6.11 Control of Emulsion Polymerization Reactors

6.11

Control of Emulsion Polymerization Reactors

The objective of reactor control is to achieve an eﬃcient, safe, and consistent production of high-quality materials. The quality of the emulsion polymers is given by its end-use properties such as mechanical strength; tack, peel, and shear resistances; brightness; impact strength; weatherability; drawability; and elastic modulus. These end-use properties depend on the molecular and morphological properties of the latex, which are the variables that can be controlled in the reactor. Therefore, the relationship between the structural and morphological properties of the polymer latex and the end-use properties is required to establish which are the characteristics of the polymer latex to be produced/controlled in the reactor (Figure 6.1). The development of ﬁrst-principle structure–properties relationships is very challenging as the basis is not yet well established. Thus, the current industrial practice is a trial and error methodology. In order to reduce this time-consuming methodology semiempirical approaches can be used. An example of this type of approach was recently reported to relate the MWD of an acrylic latex of a given composition and the adhesive properties [160, 161]. However, this relationship only provides a way to estimate the end-use properties from a given microstructural property (MWD in the example); in practice, one would like to know the microstructure that will yield a given set of end-use properties (see Figure 6.14). This can be achieved by inverting the model using appropriate techniques of optimization or neural networks [160, 161]. Once the desired molecular and morphological properties of the latex are established, a closed-loop control strategy can be implemented. A typical closed-loop control strategy used to control copolymer composition and molecular weight distribution is shown in Figure 6.14. In the control scheme, x are state variables of the polymerization reactor, y are measurable properties of the polymer latex, ymeas are properties that are monitored on-line, and u are the manipulated variables of the process (basically ﬂow rates of monomers and chain-transfer agents).

Fig. 6.14. Closed-loop control strategy for optimal control of composition and MWD in emulsion polymerization reactors.

305

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 trajectories 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 desired 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. Min

ð 1

½M p

0

1 dXT Rp

ð74Þ

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 comonomer 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 ½M2 p

¼

ð2y1i 1Þ G fð2y1i 1Þ 2 4r1 ð y1i 1Þ y1i r2 g 1=2 2r1 ð y1i 1Þ

ð75Þ

The 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 polymer 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 transfer to a CTA is the main termination event, each of those MWDs can be characterized by the number-average chain length, X ni , according to Eq. (76). X ni ¼

k p1 ½M1 p þ k p2 ½M2 p ktr; CTA ½CTAp

ð76Þ

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

produced in the conversion increment j, and k is the number of increments into which XTf is divided. Wc ðnÞ

k k 1 X DXT X n n ¼ Wj ðnÞDXTj ¼ exp XTf j¼1 XTf j¼1 X nij X nij

! ð77Þ

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 ½Wcd ðnÞ Wc ðnÞ 2 Min X ni

ð78Þ

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 containing 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)]. X ni o

½M1 p þ ½M2 p ½CTAp

¼ f ðXT Þ

ð79Þ

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 present 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

307

308

6 Emulsion Polymerization

process under adiabatic conditions. To assess risk parameters, adiabatic calorimeters 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 criterion. 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 polymerization (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. X DT ¼

i

MiTOT ðDHri Þ X Ni cpi

ð80Þ

i

The adiabatic temperature increase, DT, is calculated for the experiments carried out in the adiabatic calorimeters by subtracting the Tonset from the maximum temperature achieved, Tmax . DT ¼ Tmax Tonset

ð81Þ

Thus, 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 optimal control strategies under safe conditions. A typical result obtained in an adiabatic calorimeter for an emulsion polymerization 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 determined. 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 concentration 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

a) 120

T (°C)

100

80

60

40

20 0

50

100

150

200

250

300

200

250

300

time (min)

b) P (atm)

4 3,5 3 2,5 2 1,5 1 0,5 0

50

100

150

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 maximum 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 (concentration 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-

309

6 Emulsion Polymerization

Temperature (°C)

140 120

Tonset ∆T

100

Tmax

80 60 40 20 0 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.

140 Temperature (°C)

310

T

120

work

+ ∆T T

limit

100 80 T

work

60 SAFE REGION

40 ∆T

20 0 0

2 DANGER

4

6

8

Polym er / M onom er [M ]

p

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.

10

6.11 Control of Emulsion Polymerization Reactors

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 dXT

ð82Þ

d½M2f þ M2pol b0 dXT

ð83Þ

d½CTA f þ CTA pol b0 dXT

ð84Þ

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

ð85Þ

M2f þ M2pol a M2TOT

ð86Þ

CTA f þ CTA pol a CTATOT

ð87Þ

The optimization provides the amounts of monomers and CTAs in the reactor at any overall conversion. These proﬁles are independent of the kinetics of the process 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 implementation of the closed-loop strategy (Figure 6.14) reduces to tracking these proﬁles. 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 estimation 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 manipulated 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 produce 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 controllers (IMCs) [164] being the ones that have gained more attention in the specialized literature.

311

6 Emulsion Polymerization 0.015

2 Butyl acrylate Styrene

1.5

0.012

CTA

0.009 1 0.006 0.5 0.003

0

0

0.2

0.4

0.6

0.8

1

Total amount of CTA (mol/L)

Total amounts of monomers (mol/L)

312

0

Overall Conversion

Optimal trajectories for the amount of monomer and CTA to produce S/n-BA ¼ 50:50 copolymer with a bimodal MWD. Fig. 6.18.

An example of the performance of an on-line control strategy like the one depicted 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. Figure 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 addition of TDM was required to start producing the low molecular-weight mode. Figure 6.19 presents a comparison between the desired and obtained copolymer composition and MWD during the controlled experiment.

Notation

A Ap as c cc

total heat-transfer area [m 2 ] total surface area of the polymer particles [m 2 ] saturated surface of the polymer particles covered by 1 mol of surfactant [m 2 mol1 ] overall termination rate coeﬃcient in the polymer particles [Eq. (9)] [s1 ] rate coeﬃcient for bimolecular termination by combination [s1 ]

Notation

a)

Copolymer Composition

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Conversion

b) Desired

w

df/dlog(M )

0.6

X=0.18

0.5

X=0.47 X=0.60

0.4

X=0.77

0.3

X=0.96

0.2 0.1 0.0 3.0

3.5

4.0

4.5

5.0

5.5 6.0 log (M )

6.5

7.0

w

Experimental results of the on-line controlled emulsion polymerization of S/n-BA: (a) cumulative copolymer composition; (b) MWD produced at diﬀerent conversions. Fig. 6.19.

cd CMC cpi cpie cpw ½CTAp CTA f

rate coeﬃcient for bimolecular termination by disproportionation [s1 ] critical micellar concentration [mol m3 ] heat capacity of compound i in the reactor [J mol1 K1 ] heat capacity of compound i under the entry conditions [J mol1 K1 ] heat capacity of the cooling ﬂuid [J kg1 K1 ] concentration of CTA in polymer particle [mol m3 ] free amount of CTA in the reactor [mol]

313

314

6 Emulsion Polymerization

CTA pol CTATOT D dd dp f fac fc fde Fie Fis H hi (DHri Þ I [I] icrit jcrit j Ki kðv; v 0 Þ ka kac kadd kam kd kd ðnÞ kde kex ðÞ kfrag ðþÞ

kfrag kI kp k pi kpji kt kt

ktr; CTA k tr; M k tw

amount of CTA polymerized [mol] total amount of CTA in the formulation [mol] impeller diameter [m] diameter of the monomer droplets [nm] diameter of the polymer particles [nm] eﬃciency factor of the initiator radicals frequency of activation in living polymerization [s1 ] frequency of bimolecular termination [s1 ] frequency of deactivation in living polymerization [s1 ] inlet molar ﬂow rate of component i [mol s1 ] outlet molar ﬂow rate of component i [mol s1 ] velocity head [J kg1 ] internal heat-transfer coeﬃcient [J m2 s1 K1 ] polymerization heat of monomer i under the reactor conditions [J mol1 ] amount of initiator [mol] concentration of initiator in the aqueous phase [mol m3 ] critical length of the oligoradicals formed from desorbed radicals critical length of the oligoradicals formed from initiator partition coeﬃcient of monomer i between the phase j and the aqueous phase. coagulation rate constant for particles of volumes v and v 0 [m 3 particle1 s1 ] entry rate coeﬃcient [m 3 mol1 s1 ] activation rate coeﬃcient in NMP or ATRP [s1 or m 3 mol1 s1 ] addition rate coeﬃcient in RAFT [m 3 mol1 s1 ] rate coeﬃcient for radical entry into the micelles [m 3 mol1 s1 ] rate coeﬃcient for radical exit [s1 ] rate coeﬃcient of radical exit from particles with n radicals [s1 ] deactivation rate coeﬃcient in NMP or ATRP [m 3 mol1 s1 ] exchange rate coeﬃcient in DT [m 3 mol1 s1 ] fragmentation rate coeﬃcient in RAFT, backward reaction [s1 ] fragmentation rate coeﬃcient in RAFT, forward reaction [s1 ] rate coeﬃcient for initiator decomposition [s1 ] propagation rate constant [m 3 mol1 s1 ] average propagation rate constant of monomer i in copolymerization [m 3 mol1 s1 ] propagation rate constant of radicals with terminal unit j with monomer i [m 3 mol1 s1 ] termination rate coeﬃcient in the polymer particles [m 3 mol1 s1 ] reversible termination rate coeﬃcient in living polymerization [m 3 mol1 s1 ] average chain transfer to CTA rate constant in copolymerization [m 3 mol1 s1 ] chain transfer to monomer rate coeﬃcient [m 3 mol1 s1 ] termination rate coeﬃcient in the aqueous phase [m 3 mol1 s1 ]

Notation

m Micrit Mif Mipol ½Mi p MiTOT Mm Mmi Mn Mni ½Mp mw m_ w Mw Mwi ½Mw N n n NA nðvÞ ne ðvÞ Ni Ni0 nm Nm Nn Np Npoli Npr NTi Nu P Pj PM Pr q Q Q feed Q loss Qr

partition coeﬃcient of small radicals between polymer particles and the aqueous phase [Eq. (11)] number of oligoradicals of critical length formed from desorbed radicals amount of free monomer i in the reactor [mol] amount of monomer i polymerized [mol] concentration of monomer i in the polymer particles [mol m3 ] total amount of monomer i in the formulation [mol] number of inactive chains of length m number of inactive chains of length m in particles with i radicals cumulative number-average molecular weight instantaneous number-average molecular weight monomer concentration in the polymer particles [mol/m 3 ] mass of the cooling ﬂuid in the jacket [Kg] mass ﬂow rate of the cooling ﬂuid [Kg/s] cumulative weight-average molecular weight instantaneous weight-average molecular weight monomer concentration in the aqueous phase [mol/m 3 ] impeller rpm number of radicals in a particle average number of radicals per particle Avogadro’s number number of polymer particles per unit of unswollen volume of particle [particles m3 particle ] inlet number density of polymer particles [particles m3 particle ] total amount of compound i in the reactor [mol] moles of monomer i in the reactor at time zero [mol] aggregation number of surfactant [molecules/micelle] number of micelles number of polymer particles with n radicals number of polymer particles moles of monomer i polymerized [mol] number of precursor particles total amount of monomer i to be feed in a semicontinuous operation [mol] Nusselt dimensionless number impeller power consumption [J s1 ] time-averaged probability of ﬁnding an active chain with ultimate unit of type j average molecular weight of the repeating unit in the polymer chain Prandtl dimensionless number heat transfer rate [J s1 ] impeller ﬂow [m 3 s1 ] sensible heat generation rate due to feeding into the reactor [J s1 ] rate of heat losses to the surroundings [J s1 ] heat generation rate by polymerization reaction [J s1 ]

315

316

6 Emulsion Polymerization

Qs Q stirring Q transfer Re Ri ri R jcrit Rm Rmi R nuc Rp Rpi Rpp RX rv rv ðvÞ Rnk ½Rw ST Sw t T DT Te Tg Tlim Tmax DTml Tonset Tw Twe Tws U V Vi Vd Vw Vp vp Vpol W Wc ðnÞ Wcd ðnÞ

volumetric ﬂow rate [m 3 s1 ] rate of heat production by the agitator [J s1 ] heat removal rate [J s1 ] Reynolds dimensionless number net generation rate of component i [mol m3 s1 ] reactivity ratio of monomer i number of oligoradicals of critical length formed from initiator number of radicals of length m number of radicals of length m in particles with i radicals nucleation rate of polymer particles [particles m3 s1 ] overall polymerization rate [mol m3 s1 ] polymerization rate of monomer i [mol m3 s1 ] polymerization rate per polymer particle [mol particles1 s1 ] amount of the ‘‘capping’’ species in controlled polymerization [mol] volumetric growth rate of one polymer particle [m 3 s1 ] volumetric growth rate of a particle of volume v [m 3 s1 ] generation rate of the kth moment of the distribution of inactive chains concentration of radicals in the aqueous phase [mol m3 ] total amount of surfactant in the reactor [mol] amount of surfactant in the aqueous phase [mol] reaction time [s] reactor temperature [K] adiabatic temperature increase temperature of the feed [K] glass transition temperature [K] maximum temperature achievable for a safe operation [K] maximum temperature obtained during an adiabatic polymerization reaction [K] logarithmic mean temperature diﬀerence temperature at which the polymerization start under adiabatic conditions [K] jacket temperature [K] inlet temperature of the cooling ﬂuid in the jacket [K] outlet temperature of the cooling ﬂuid in the jacket [K] overall heat-transfer coeﬃcient [J m2 s1 K1 ] reactor volume [m 3 ] volume of monomer i [m 3 ] volume of the droplet phase [m 3 ] volume of the aqueous phase [m 3 ] volume of the polymer particles [m 3 ] volume of a swollen polymer particle [m 3 ] volume of polymer [m 3 ] volume of water [m 3 ] cumulative weight MWD desired cumulative MWD

References

Wj ðnÞ [X] Xi X ni X nij XT XTf DXTj y1cum y1i

instantaneous weight MWD of the polymer formed at conversion increment j concentration of ‘‘living agent’’ in controlled polymerization [mol m3 ] conversion of monomer i instantaneous number-average chain length instantaneous number-average chain length of the polymer formed in conversion increment j overall conversion ﬁnal overall conversion jth overall conversion increment cumulative copolymer composition referred to monomer 1 instantaneous copolymer composition referred to monomer 1

Greek a1 a2 g d h l mk nk r t j F P fM j fi P fp fww C jw

parameter of Eq. (29) parameter of Eq. (30) generation rate of small radicals by chain transfer [Eq. (11)] [mol m5 ] critical length for entry of radicals generated from the initiator consumption rate of small radicals generated by chain transfer [Eq. (11)] [m2 ] overall mass-transfer rate coeﬃcient [m 3 s1 ] kth-order moment of the distribution of active chains kth-order moment of the distribution of inactive chains frequency of radical entry [s1 ] space time [s] viscosity of the reaction medium at the reactor temperature [kg m1 s1 ] fraction of the heat of reaction used to heat the reactor walls volume fraction of monomer in the polymer particles volume fraction of monomer i in phase j volume fraction of polymer in the polymer particles volume fraction of water in the aqueous phase parameter of Eq. (17) viscosity of the reaction medium at the wall temperature [kg m1 s1 ]

References 1 J. M. Asua (Ed.), Polymeric Dispersions:

Principles and Applications, Kluwer Academic Press, Dordrecht, 1997. 2 P. A. Lovell, M. S. El-Aasser (Eds.), Emulsion Polymerization and Emulsion Polymers, John Wiley, Chichester, UK, 1997. 3 H. Warson, C. A. Finch, Applications

of Synthetic Resin Latices, 3 vols., John Wiley, Chichester, UK, 2001. 4 D. Urban, K. Takamura (Eds.), Polymer Dispersions and Their Industrial Applications, Wiley-VCH, Weinheim, Germany, 2002. 5 W. D. Harkins, J. Am. Chem. Soc. 1947, 69, 1428–1444.

317

318

6 Emulsion Polymerization 6 W. J. Priest, J. Phys. Chem. 1952, 56, 7

8 9

10

11 12

13 14 15 16

17 18 19

20

21

22

23

24

1077–1082. R. M. Fitch, C. H. Tsai, Polymer Colloids, R. M. Fitch (Ed.), Plenum Press, New York, 1971. J. Ugelstad, F. K. Hansen, Rubber Chem. Technol. 1976, 49, 536–609. P. J. Feeney, D. H. Napper, R. G. Gilbert, Macromolecules 1987, 20, 2922–2930. J. W. Vanderhoff, E. B. Bradford, H. L. Tarkowski, J. B. Shaffer, R. M. Wiley, Adv. Chem. Ser. 1962, 34, 32– 51. J. M. Asua, Prog. Polym. Sci. 2002, 27, 1283–1346. K. E. J. Barret (Ed.), Dispersion Polymerization in Organic Media, John Wiley, London, 1975. C. Larpent, Surfactant Sci. Ser. 2003, 115, 145–187. T. Fukuda, Y. D. Ma, H. Inagaki, Macromolecules 1985, 18, 17–26. F. R. Mayo, F. M. Lewis, J. Am. Chem. Soc. 1994, 66, 1594–1601. J. Forcada, J. M. Asua, J. Polym. Sci.: Polym. Chem. Ed. 1985, 23, 1955– 1962. J. L. Gardon, J. Polym. Sci.: Polym, Chem. Ed. 1968, 6, 2859–2879. J. Guillot, Makromol Chem. Rapid. Commun. 1980, 1, 697–702. P. D. Armitage, J. C. de la Cal, J. M. Asua, J Appl Polym Sci. 1994, 51, 1985–1990. S. Omi, K. Kushibiki, M. Negishi, M. Iso, Zairyo Gijutsu 1985, 3, 426– 441. J. Ugelstad, P. C. Mork, H. R. Mfutakamba, E. Soleimany, I. Nordhuus, K. Nustad, R. Schmid, A. Berge, T. Ellingsen, O. Aune, G. W. Poehlein, R. H. Otewill, J. W. Goodwin (Eds.), NATO ASI Series 1983, 1, 51–99. L. M. Gugliotta, G. Arzamendi, J. M. Asua, J. Appl. Polym. Sci. 1995, 55, 1017–1039. F. K. Hansen, J. Ugelstad, J. Polym. Sci: Polym. Chem. Ed. 1978, 16, 1953– 1979. I. A. Maxwell, B. R. Morrison, D. H. Napper, R. G. Gilbert, Macromolecules 1991, 24, 1629–1640.

25 J. M. Asua, Macromolecules 2003, 36,

6245–6251. 26 J. T. O’Toole, J. Appl. Polym. Sci.

1965, 9, 1291–1297. 27 B. G. Li, B. W. Brooks, J. Polym. Sci.,

28 29

30 31

32

33

34 35

36

37

38 39

40

41 42 43 44 45

Part A: Polym. Chem. 1993, 31, 2397– 2402. W. V. Smith, R. H. Ewart, J. Chem. Phys. 1948, 16, 592–599. M. Nomura, M. Harada, W. Eguchi, S. Nagata, ACS Symp Ser. 1976, 24, 102–121. F. K. Hansen, J. Ugelstad, Makromol Chem. 1979, 180, 2423–2434. J. R. Richards, J. R. Congalidis, R. G. Gilbert, J. Appl. Polym. Sci. 1989, 37, 2727–2756. P. H. H. Araujo, J. C. de la Cal, J. M. Asua, J. C. Pinto, Macromol. Theory Simul. 2001, 10, 769–779. A. Urretabizkaia, G. Arzamendi, J. M. Asua, Chem. Eng. Sci. 1992, 47, 2579–2584. D. Satas, Adhesive Age 1972, 15, 19– 23. O. Elizalde, M. Vicente, J. R. Leiza, J. M. Asua, Polym. React. Eng. 2002, 10, 265–283. G. Lichti, R. G. Gilbert, D. H. Napper, in I. Piirma (Ed.), Emulsion Polymerization, Academic Press, New York, 1982, pp. 93–144. H. Tobita, A. E. Hamielec, Makromol. Chem, Macromol. Symp. 1994, 35–36, 193–212. F. Teymour, J. D. Campbell, Macromolecules 1994, 27, 2460–2469. C. Plessis, G. Arzamendi, J. R. Leiza, D. Charmot, H. A. S. Schoonbrood, D. Charmot, J. M. Asua, Ind. Eng. Chem. Res. 2001, 40, 3883–3894. A. Butte´, G. Storti, M. Morbidelli, Macromol. Theory. Simul. 2002, 11, 37– 52. P. J. Flory, Faraday Discussions of the Chemical Society 1974, 57, 7–18. H. Tobita, Macromolecules 1993, 26, 836–841. D. Charmot, J. Guillot, Polymer 1992, 33, 352–360. G. Arzamendi, J. M. Asua, Macromolecules 1995, 28, 7479–7490. H. Tobita, Acta Polym. 1995, 46, 185– 203.

References 46 A. Ghielmi, S. Fiorentino, G.

47 48 49 50 51 52

53

54 55 56

57

58

59

60 61 62 63 64 65

66 67

68

Storti, M. Morbidelli, J. Polym. Sci. Part A: Polym. Chem. 1998, 36, 1127– 1156. L. J. Gonza´lez-Ortiz, J. M. Asua, Macromolecules 1995, 28, 3135–3145. L. J. Gonza´lez-Ortiz, J. M. Asua, Macromolecules 1996, 29, 383–389. L. J. Gonza´lez-Ortiz, J. M. Asua, Macromolecules 1996, 29, 4520–4527. M. Szwarc, Nature 1956, 178, 1168. M. Szwarc, J. Polym. Sci. A: Polym. Chem. 1998, 36, ix. G. Moad, D. H. Solomon, The Chemistry of Free-Radical Polymerization, Pergamon, Oxford, 1995. R. G. Gilbert, Emulsion Polymerization. A Mechanistic Approach, Academic Press, London, 1995. G. Moad, E. Rizzardo, D. H. Solomon, Polym. Bull. 1982, 6, 589. D. H. Solomon, E. Rizzardo, P. Cacioli, US Patent 4 581 429, 1986. M. K. Georges, R. P. N. Veregin, P. M. Kazmaier, G. K. Hamer, Macromolecules 1993, 26, 2987. M. K. Georges, R. P. N. Veregin, P. M. Kazmaier, G. K. Hamer, M. Saban, Macromolecules 1994, 27, 7228. D. Benoit, V. Chaplinski, R. Braslau, C. J. Hawker, J. Am. Chem. Soc. 1999, 121, 5929. M. Kato, M. Kamigaito, M. Sawamoto, T. Higashimura, Macromolecules 1995, 28, 2093. J. S. Wang, K. Matyjaszewski, J. Am. Chem. Soc. 1995, 117, 5614. J. S. Wang, K. Matyjaszewski, Macromolecules 1995, 28, 7901. V. Coessens, T. Pintauer, K. Matyjaszewski, Prog. Polym. Sci. 2001, 26, 337. T. E. Patten, K. Matyjaszewski, Adv. Mater. 1998, 10, 901. K. Matyjaszewski, S. G. Gaynor, J. S. Wang, Macromolecules 1995, 28, 2093. S. G. Gaynor, J. S. Wang, K. Matyjaszewski, Macromolecules 1995, 28, 8051. A. Goto, K. Ohno, T. Fukuda, Macromolecules 1998, 31, 2809. A. Goto, K. Sato, Y. Tsujii, T. Fukuda, G. Moad, E. Rizzardo, S. H. Thang, Macromolecules 2001, 34, 402. G. Moad, J. Chiefari, B. Y. K.

69

70

71

72 73 74 75 76 77 78

79 80 81 82

83

84

85

86

Chong, J. Krstina, R. T. A. Mayadunne, A. Postma, E. Rizzardo, S. H. Thang, Polym. Int. 2000, 49, 993. R. T. A. Mayadunne, E. Rizzardo, J. Chiefari, J. Krstina, G. Moad, A. Postma, S. H. Thang, Macromolecules 2000, 33, 243. B. Y. K. Chong, T. P. T. Le, G. Moad, E. Rizzardo, S. H. Thang, Macromolecules 1999, 32, 2071. A. Butte´, G. Storti, M. Morbidelli, Modern Styrenic Polymers, Polystyrenes and Styrenic Copolymers, John Wiley, Chichester, UK, 2003. H. Fischer, J. Polym. Sci. A: Polym. Chem. 1999, 37, 1885. H. Fischer, Macromolecules 1997, 30, 5666. A. Butte´, G. Storti, M. Morbidelli, Macromolecules 2001, 34, 5885. B. Charleux, Macromolecules 2000, 33, 5358. M. F. Cunningham, C. R. Chimie 2003, 6, 1351. J. Cao, J. He, C. Li, Y. Yang, Polym. J. 2001, 33, 75. M. F. Cunningham, M. Xie, K. B. McAuley, M. K. Georges, B. Keoshkerian, Macromolecules 2002, 35, 59. M. Li, K. Matyjaszewski, J. Polym. Sci. A: Polym. Chem. 2003, 41, 3606. S. G. Gaynor, J. Qiu, K. Matyjaszewski, Macromolecules 1998, 31, 5951. J. Qiu, S. G. Gaynor, K. Matyjaszewski, Macromolecules 1999, 32, 2872. S. W. Prescott, M. J. Ballard, E. Rizzardo, R. G. Gilbert, Aust. J. Chem. 2002, 55, 415. J. Chiefari, Y. K. Chong, F. Ercole, J. Krstina, J. Jeffery, T. P. T. Le, R. T. A. Mayadunne, G. F. Meijs, C. L. Moad, G. Moad, E. Rizzardo, S. H. Thang, Macromolecules 1998, 31, 5559. W. Smulders, R. G. Gilbert, M. J. Monteiro, Macromolecules 2003, 36, 4309. H. de Brouwer, J. G. Tsavalas, F. J. Schork, M. J. Monteiro, Macromolecules 2000, 33, 9239. Y. Luo, J. G. Tsavalas, F. J. Schork, Macromolecules 2001, 34, 5501.

319

320

6 Emulsion Polymerization 87 C. Farcet, B. Charleux, R. Pirri, 88

89

90

91 92 93 94 95 96

97

98

99 100

101

102

103

104

105

Macromol. Symp. 2002, 192, 249. K. Matyjaszewski, J. Qiu, N. V. Tsarevsky, B. Charleux, J. Polym. Sci. A: Polym. Chem. 2000, 38, 4724. R. Greene, R. A. Gonza´lez, G. W. Poehlein, ACS Symp. Ser. 1976, 24, 341–358. C. Kiparissides, J. F. MacGregor, A. E. Hamielec, Can. J. Chem. Eng. 1980, 58, 48–55. H. C. Lee, G. W. Poehlein, Chem. Eng. Sci. 1986, 41(4), 1023–1030. B. A. Paquet, W. H. Ray, AIChE J. 1994, 40(1), 73–87. K. Geddes, Chemistry & Industry 1983, 6, 223–227. K. Geddes, Surface Coatings Int. 1993, 76, 330–339. C. Abad, J. C. de la Cal, J. M. Asua, Polymer 1995, 36, 4293–4299. P. H. H. Araujo, C. Abad, J. C. de la Cal, J. C. Pinto, J. M. Asua, Polym. React. Eng. 1999, 7, 303–326. C. Abad, J. C. de la Cal, J. M. Asua, J. Appl. Polym. Sci. 1995, 536, 419– 424. C. Abad, J. C. de la Cal, J. M. Asua, in Polymeric Dispersions: Principles and Applications, J. M. Asua (Ed.), Kluwer Academic Press, Dordrecht, 1997, pp. 333–348. J. Meuldijk, A. L. German, Polym. React. Eng, 1999, 7(2), 207–230. F. H. A. H. Van den Boomen, I. Meuldijk, D. Thoenes, Chem. Eng. Sci. 1999, 54(15–16), 3283–3290. C. Sayer, M. Palma, R. Guidici, Ind. Eng. Chem. Res. 2002, 41(7), 1733– 1744. J. Meuldijk, G. J. G. Van Strey, F. A. H. C. Van Doormalen, Chem. Eng. Sci. 1992, 47(9–11), 2603–2608. M. Apostel, W. Pauer, H. U. Moritz, J. Kremestotter, K. D. Hungenberg, Chimia 2001, 55(3), 229–230. C. Herbel, S. Fukhauser, W. Huebinger, W. Kastenhuber, J. Kremeskoetter, WO 99 44737, PCT Int. Appl., 1999. W. Schmidt, S. Kossak, J. Langenbuch, H. U. Moritz, C. Herrmann, D. Kremeskoetter, Dechema Monographs, 1998, 134, 509–519.

106 T. Inamura, K. Sato, S. Ishikura, M.

107

108

109

110

111

112

113

114

115

116

117

118 119 120

121

Nomura, Polym. Int. 1993, 30(2), 203– 206. X. Wei, K. Yoshikawa, I. Yokota, S. Sato, M. Nomura, Polym. Mat. Sci. Eng. 1999, 80, 528–529. X. Wei, H. Takahashi, S. Sato, M. Nomura, J. Appl. Polym. Sci. 2001, 80(11), 1931–1942. G. W. Poehlein, in Polymeric Dispersions: Principles and Applications, J. M. Asua (Ed.), Kluwer Academic Press, Dordrecht, 1997, pp. 305–332. K. J. Karpenter, in Handbook of Batch Process Design, P. Sharrat (Ed.), Chapman and Hall, London, 1997. S. Nagata, Mixing Principles and Applications, John Wiley, New York, 1975. J. B. P. Soares, A. E. Hamielec, in Polymeric Dispersions: Principles and Applications, J. M. Asua (Ed.), Kluwer Academic Press, Dordrecht, 1997, pp. 289–304. H. U. Moritz, Polymer Reaction Engineering, K. H. Reichert, W. Geiseler (Eds.), Verlag Chemie, Weinhem, Germany, 1989. I. Saenz de Buruaga, A. Echevarria, P. D. Armitage, J. C. de la Cal, J. R. Leiza, J. M. Asua, AIChE J. 1997, 43, 1069–1081. A. Guyot, F. Chu, M. Schneider, C. Graillant, T. F. McKenna, Prog. Polym Sci. 2002, 27, 1573–1615. A. C. I. A. Peters, G. C. Overbeck, A. J. P. Buckmann, J. C. Padget, T. Annable, Prog Org Coatings 1996, 29, 183–194. A. C. I. A. Peters, G. C. Overbeck, T. Annable, Prog Org Coatings 2000, 38, 137–150. J. B. Rawlings, Ph.D. Thesis, University of Wisconsin, 1985. J. B. Rawlings, W. H. Ray, Polym. Eng. Sci. 1988, 28, 237–256. P. H. H. Araujo, J. C. de la Cal, J. M. Asua, J. C. Pinto, Macromol Theory Simul. 2001, 10, 769–779. A. Bisio, in Scale-up of Chemical Processes: Conversion from Laboratory Scale Tests to Successful Commercial Size Design, A. Bisio, R. L. Kabel (Eds.), John Wiley, New York, 1985.

References 122 G. B. Tatterson, Scale-up and Design

123

124

125

126

127

128

129

130

131

132

133

134 135

136 137

138

139

of Industrial Mixing Processes, McGrawHill, New York, 1994. H. F. Rase, Chemical Reactor Design for Process Plants: Vol I. Principles & Techniques, John Wiley, New York, 1977. C. H. Chien, A. Penlidis, JMS-Rev., Macromol. Chem. Phys. 1990, 30(1), 1– 42. W. Hergeth in Polymeric Dispersions: Principles and Applications, J. M. Asua (Ed.), Kluwer Academic Press, Dordrecht, 1997, pp. 243–256. O. Kammona, E. G. Ghatzi, C. Kiparissides, JMS-Rev., Macrom. Chem. Phys. 1999, 39(1), 57–134. A. Urretabizkaia, E. D. Sudol, M. S. El-Aasser, J. M. Asua, J. Polym. Sci. Part A: Polym. Chem. 1993, A31, 2907–2913. L. M. Gugliotta, M. Arotcarena, J. R. Leiza, J. M. Asua, Polymer 1995, 36, 2019–2023. P. Canu, S. Canegallo, M. Morbidelli, G. Storti, J. Appl. Polym. Sci. 1994, 54, 1899–1917. A. Echevarria, J. R. Leiza, J. C. de la Cal, J. M. Asua, AIChE J. 1998, 44(7), 1667–1679. M. Vicente, S. Ben Amor, L. M. Gugliotta, J. R. Leiza, J. M. Asua, Ind. Eng. Chem. Res. 2001, 40, 218– 227. A. Guyot, J. Guillot, C. Graillat, M. T. Llauro, J. Macromol. Sci. – Chem. 1984, A21, 683–699. J. R. Leiza, M. Montes, J. C. de la Cal, J. M. Asua, Process Control and Quality 1993, 4, 197–210. A. Urretabizkaia, J. R. Leiza, J. M. Asua, AIChE J. 1994, 40, 1850–1864. M. Alonso, F. Recasens, L. Puigjanser, Chem. Eng. Sci. 1986, 41, 1039–1044. M. J. Ahmed, D. E. Mackey, ACS Symp. Ser. 1989, 404, 296–305. S. Canegallo, G. Storti, M. Morbidelli, S. Carra, J. Appl. Polym. Sci. 1993, 47, 961–979. J. Alig, P. Hauptmann, S. Wateraigi, Teubner Texte zur Physik 1986, 9, 364–373. P. Hauptmann, F. Dinger, R.

140

141

142

143

144 145

146

147

148

149

150

151 152

153

154

155

156

Sa¨uberlich, Acta Polym. 1983, 34, 303–304. S. Canegallo, M. Apostolo, G. Storti, M. Morbidelli, J. Appl. Polym. Sci. 1995, 57, 1333–1346. K. Ho¨rning, W. D. Hergeth, S. Waterwig, Acta Polym. 1989, 40, 257– 260. K. Ho¨rning, W. D. Hergeth, K. G. Ha¨usler, S. Waterwig, Acta Polym. 1991, 42, 174–177. J. B. Callis, M. Eaton, M. L. Callis, C. Wu, J. D. S. Danielson, Process Control and Quality 1995, 8, 1–40. S. Mijovic, J. Andjelic, Polymer 1995, 36, 3783–3786. C. Kiparissides, E. G. Chatzi, O. Kammona, J. Appl. Polym. Sci. 1997, 63, 799–809. M. Agnely, D. Charmot, J. Sawatzki, N. Dupuy, C. Bauer, B. Amran, J. P. Huvenne, Appl. Spectrosc. 2000, 54, 528–535. P. D. Gossen, J. F. MacGregor, R. H. Pelton, Appl. Spectrosc. 1993, 47, 1852–1870. M. M. Reis, P. H. H. Araujo, C. Sayer, R. Giudici, Polymer 2003, 44, 6123–6128. M. M. Reis , P. H. H. Araujo, C. Sayer, R. Giudici, Macromol. Rapid. Commun. 2003, 24, 620–624. O. Elizalde, J. R. Leiza, J. M. Asua, Macromol. Symp. 2004, 206, 135– 149. E. D. Tolin, D. A. Fluegel, US Patent 3 275 809, 1966. B. F. Beekingham, B. N. Heudy, J. V. Simons, GB Patent 1 549 841, 1979. I. Sa´enz de Buruaga, M. Arotc¸arena, P. D. Armitage, L. M. Gugliotta, J. R. Leiza, J. M. Asua, Chem. Eng. Sci. 1996, 51, 2781– 2786. P. Guinot, N. Othman, G. Fevotte, T. F. McKenna, Polym. React. Eng. 2000, 8, 115–134. N. Othman, A. M. Santos, G. Fevotte, T. F. McKenna, Macromol. Symp. 2000, 150, 109–114. R. Carloff, A. Proß, K. H. Reichert, Chem. Eng. Techn. 1994, 17, 406– 413.

321

322

6 Emulsion Polymerization ¨ dtke, K. H. Reichert, 157 A. Tietze, Lu Chem. Eng. Sci. 1996, 51, 3131– 3137. 158 S. Kraemer, R. Gesthuisen, S. Engell, J. R. Leiza, J. M. Asua, Comp. Chem. Eng., in press, 2004. 159 I. Sa´enz de Buruaga, J. R. Leiza, J. M. Asua, Polym. React. Eng. 2000, 8, 39–75. 160 O. Elizalde, M. Vicente, J. R. Leiza, J. M. Asua, Polym. React. Eng. 2002, 10, 265–283.

161 O. Elizalde, M. Vicente, C. Plessis,

162 163 164

165

J. R. Leiza, J. M. Asua, J. Coat. Technol. 2004, 1, 45–51. B. W. Bequette, Ind. Eng. Chem. Res. 1991, 30, 1391–1413. C. Kravavis, M. Soroush, AIChE J. 1990, 30, 249–264. C. G. Economou, M. Morari, B. Plasson, Ind. Eng. Chem. Res. 1986, 25, 403–411. M. Vicente, J. R. Leiza, J. M. Asua, AIChE J. 2001, 47, 1594–1606.

323

7

Ionic Polymerization1 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 polyaddition. 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 radical 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 determined 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 in1) 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

324

7 Ionic Polymerization

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. þ X @@Pnþ X þ M ! @@Pnþ1

ð1Þ

or þ @@Pn Y þ þ M ! @@P nþ1 Y

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 system, 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

ð3Þ

All 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 polymerization. In free-radical polymerization, there are system-immanent, unavoidable termination 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 reactions 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 polymerization, such as in the elimination of LiH or NaH during anionic polymerization 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.

7.2 Anionic Polymerization

@@CH2 Cþ HPh þ PF 6 ! @@CH2 CHPhaF þ PF5

ð4Þ

Many other similar eliminations are not true termination reactions, but are transfer reactions, when the eliminated species can re-initiate a new chain, as in reaction (5). @@CH2 aCþ ðCH3 Þ2 ;

SO3 H ! @@CH2 aCðCH3 ÞbCH2 þ H2 SO4

ð5Þ

So, ionic polymerization oﬀers the possibility, by proper choice of the reaction conditions, of running as a living polymerization. Living polymerization in its pure form is deﬁned not only by the absence of any termination reaction destroying the active center, but also by the absence of any transfer reaction, so that the kinetic chain length and the length of the individual chain are the same and increase linearly with conversion. However, especially in many cationic systems, the active center may be preserved throughout the reaction, but transfer reactions to other molecules such as monomers, polymers, solvent, and so on may stop the individual chain and initiate a new growing chain [Eqs. (6)–(8)]. - - -Pn þ M ! - - -Pnþ1

- - -Pn

þ T ! - - -Pn þ T

T þ M ! - - -P1

propagation

ð6Þ

transfer

ð7Þ

re-initiation

ð8Þ

So, the growth of the individual chains may be terminated by transfer reactions. The active center itself is preserved, but a new chain is started. Thus, transfer reactions change the molecular weight of the polymer, but they do not change the concentration of active centers. The conversion kinetics is only changed by transfer reactions if the re-initiation is much slower than the propagation step. Another diﬀerence between free-radical and ionic polymerization that must also be kept in mind is economic rather than scientiﬁc. Most of the monomers can be polymerized by radical or ionic mechanisms (see Table 7.1), but in most cases the requirement for purity of monomers and solvents in ionic polymerization is much higher, and initiators are more expensive, so ionic polymerization is chosen only if the monomers do not polymerize by a radical pathway, or if ionic polymerization oﬀers other advantages, such as access to molecular structures like block copolymers or specially designed molecular weight distributions.

7.2

Anionic Polymerization

Anionic polymerization involves a wide variety of reactions leading to high molecular weight molecules with the participation of an anionic species. The most important systems are those for the anionic polymerization of diene rubbers [14–

325

326

7 Ionic Polymerization Tab. 7.1.

Monomers with diﬀerent initiation mechanisms.

Monomers

Ethylene a-Oleﬁns 1,1-Dialkyl oleﬁns 1,3-Dienes Styrene, a-methylstyrene Halogenated oleﬁns Vinyl esters Acrylates, methacrylates Acrylonitrile, methacrylonitrile Acrylamide, methacrylamide Vinyl ethers N-Vinylcarbazole N-Vinylpyrrolidone Aldehydes, ketones

Type of initiation Radical

Cationic

Anionic

þ þ þ þ þ þ þ þ þ þ

þ þ þ þ þ þ þ þ

þ þ þ þ þ þ þ

16], 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 especially, 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 monomers 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 rationalize the inﬂuence of polar additives on the microstructure is given in Ref. 25. Generally, the vinyl content increases with the fraction of free ions. 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/ 7.2.1.1

7.2 Anionic Polymerization 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 Na K Rb Cs

35 10 15 7 6

52 25 40 31 35

13 65 45 62 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 depending 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 nonpolar solvents, one of the most important industrial systems, and the determination 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 number, which generally is between 2 and 6, depending on the structure of P (monomer or initiator), solvent, and temperature (see Table 7.4). K ass

ðPaLiÞn ! nPaLi

ð9Þ

Tab. 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 Diethyl ether THF THF THF Diglyme Diglyme Diglyme

10:1 5:1 6:1 3:1 1:1 4:1 2:1 1:1

16 10 43 25 17 87 85 78

327

328

7 Ionic Polymerization Tab. 7.4.

Association number for diﬀerent Li organyls.[a]

Li alkyl

Solvent

n

n-BuLi

benzene cyclohexane THF cyclohexane benzene THF benzene hexane cyclohexane benzene THF benzene cyclohexane hexane benzene hexane cyclohexane cyclohexane benzene hexane

6–6.3 6 2–2.8 4 4 1.1 4 4 2 2 1 2 2 2 2 2 2–4 2–4.3 2–3.7 2

sec-BuLi

t-BuLi Menthyl-Li Benzyl-Li Poly(styryl)-Li

Poly(isoprenyl)-Li

Poly(butadienyl)-Li

[a] Compilation

of data given in Ref. 6, pp. 16, 20, 138.

With the assumption that association is high and the concentration of the monomeric 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. ½PaLi ¼

K ass ½PaLi0 n

1=n ð10Þ

Assuming 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 propagation 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 n

ð11Þ

This 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 polymerization, 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

systems, is used here as an example to point out which aspects may become important 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 simple 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 between initiating and propagating species is likely to occur and must be considered, as well as intermediate equilibria between hexamers, tetramers, dimers, and monomers. In Ref. 35 this problem is addressed for the cross-association of styryland butadienyllithium. There are a number of recent publications reporting much higher association numbers of up to 100 and more [36–39], which are under discussion [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) compared 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 considered 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 ass ð12Þ þ þ ½PaLi0 ½PaLi ¼ 4 4 2 Figure 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

-1

K ass / M

kp / M s

10

10

-1

10000

0.01

0.1

0.00001 0.00000001 0.0025

0.003

0.0035

0.001 0.004

1/T / 1/K

Fig. 7.1.

Association equilibrium constant K ass and propagation rate constant k p as f ðTÞ [44].

329

7 Ionic Polymerization 10

1

α

330

0.1

0.01

0.001 0.000001

0.0001

Kass / M

0.01

1

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). Fig. 7.2.

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, 33]. Table 7.5 gives some examples of the temperature dependence of the propagation 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 temperatures and tackling the problem of temperature dependence of the association behavior. In Ref. 10 an attempt is made to describe the propagation over a wider tem-

Literature values for propagation rate coeﬃcient kp0 ¼ k p ðK=2Þ 0:5 for Li-initiated styrene polymerization in hydrocarbon solvents. Tab. 7.5.

˚

kOp; 0 [MC0:5 sC1 ]

Ea [kJ molC1 ]

Solvent

Temperature [ C]

Ref.

4:5 10 8 2:6 10 9 2:9 10 11 5:2 10 8 5:1 10 7 6:8 10 8

60.3 64.0 78.6 60.3 56.9 63.6

none none cyclohexane benzene toluene toluene

20–50 4–21 30–50 10–30 20–50 30–60

49 50 51 27 52 53

7.2 Anionic Polymerization 1 acc. to [49] acc. to [50] acc. to [51] acc. to [27] acc. to [53] acc. to [52]

0.01

1/2

kp K / M s

-1 -1

0.1

0.001

0.0001 0.003

0.0031

0.0032

0.0033

0.0034

1/T / K

-1

0.0035

0.0036

0.0037

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 ﬁrstorder 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 lithiuminitiated polymerization, the best-investigated ionic system, shows the diﬃculties in kinetic modeling of ionic polymerization. Contrary to free-radical polymerization, the existence of all possible diﬀerent species must be considered for every system under investigation. 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 definition 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 reactions being absent. 7.2.1.2

331

332

7 Ionic Polymerization

I þ M ! P1

initiation

ð13Þ

P1 þ M ! Piþ1

propagation

ð14Þ

In 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 dt

ð15Þ

y X d½M ¼ k i ½I ½M þ k p ½M ½Pi ¼ RP dt i¼1

ð16Þ

d½P1 ¼ k i ½I ½M þ k p ½P1 ½M dt

ð17Þ

d½Pi ¼ k p ½Pi1 ½M þ k p ½Pi ½M dt

ð18Þ

y X

½Pi ¼ ½I 0 ½I

ð19Þ

i½Pi ¼ ½M0 ½M

ð20Þ

i¼1 y X i¼1

For fast initiation (k i g k p ), all initiator is transferred to growing chains immediately, so monomer conversion and the kinetic chain length are given by Eqs. (21)– (23). ½M ¼ ½M0 ek p ½I 0 t

ð21Þ

½M0 ½M ½M0 ½M0 ekP ½I 0 t ½M0 ¼ ¼ xM ½I 0 ½I 0 ½I 0

ð22Þ

½Pi ¼ ½I 0

ð23Þ

n¼ with y X i¼1

7.2 Anionic Polymerization

The resulting frequency distribution hðiÞ of chain lengths i can be derived by solving 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 ¼ hðiÞ ¼ X ¼ ði 1Þ! ½Pi ½I 0

ð24Þ

The weight distribution is given by Eq. (25). wðiÞ ¼

nði1Þ en ði 1Þ!ðn þ 1Þ

ð25Þ

The average number- and weight-average degrees of polymerization and the polydispersity for this Poisson distribution are given by Eqs. (26)–(28). Pn ¼

½M0 ½M ¼1þn ½I 0

Pw ¼ 1 þ n þ D¼

ð26Þ

n A 1 þ Pn 1þn

ð27Þ

Pw A1 Pn

ð28Þ

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 growing 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. ½M ½M0 ¼ ð1 rÞð½I ½I 0 Þ þ r½I 0 ln

½I ½I 0

ð29Þ

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 socalled 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).

hðiÞ ¼

½Ni ¼ y X ½Ni i¼1

i y r qu X uj e R j! j¼i rð1 eu=R Þ

ð30Þ

333

7 Ionic Polymerization

wðiÞ ¼

y i½Ni r i ru=ðr1Þ X uj ru u=R e Þ þ ¼ ð1 rÞð1 e y X r1 R j! j¼i i½Ni

ð31Þ

i¼1

ð1 rÞð1 eu=ðr1Þ Þ þ Pn ¼

ru r1

ð32Þ

ð1 eu=ðr1Þ Þ

ru 1 ru þ 2ru þ ð2r 1Þðr 1Þð1 eu=ðr1Þ Þ r 1 r 1 Pw ¼ ru ð1 rÞð1 eu=ðr1Þ Þ þ r1

ð33Þ

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 distribution 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 polymerization 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

Poisson

0.04

h(i)

334

0.03

Gold

0.02

Schulz-Flory

0.01 0 0

50

100

150

i 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 weightaverages are 102, 64, and 52.

200

7.2 Anionic Polymerization 1.4

1 0.8

1.3

xM

D

0.6 1.2 0.4 1.1

0.2

1

0 0

0.2

0.4

0.6

0.8

1

xI

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. Fig. 7.5.

tion proceeds and more chains will be initiated and propagate. The broadest distribution 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 observed, depending on ½M0 =½I 0 and conversion. The long-term stability of laminar ﬂow tubular reactors is questioned [76], because of the possibility of very long chains growing near the reactor walls, where the residence time approaches inﬁnity, 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 homogeneous continuous stirred tank reactor (HCSTR), the steady-state concentrations of monomer and initiator can be derived from the monomer and initiator mass bal-

335

336

7 Ionic Polymerization

ance, neglecting volume changes, by Eqs. (35) and (36), which must be solved simultaneously. 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 V ½M ¼ k i ½I þ kP k i ½M 2 ½I t 2 þ ½M

ð35Þ

vin; I ½I f t ½I ¼ V k i ½Mt þ 1

ð36Þ

In a CSTR, the narrow Poisson distribution resulting from the chemistry is superimposed 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 kP ½Mt þ 1=t kP ½M þ 1=t wð jÞ ¼ j

1=t kP ½Mt þ 1=t

2

ð37Þ j1

kP ½M kP ½M þ 1=t

ð38Þ

Pn ¼ 1 þ kP ½Mt

ð39Þ

Pw ¼ 1 þ 2kP ½Mt

ð40Þ

Equation (37) is equivalent to Eq. (34) with the propagation probability given by Eq. (41). p¼

kP ½M kP ½M þ 1=t

ð41Þ

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 longchain branching in batch and CSTR reactors is described in Refs. 84 and 85. Side Reactions Pure living polymerization of hydrocarbon monomers usually occurs at low temperatures, where side reactions are not important. At higher temperatures, however, two side reactions do become important, elimination of LiH and transfer to compounds with an acidic hydrogen [5, 6, 31, 86, 87]. 7.2.1.3

7.2 Anionic Polymerization

337

There have been some investigations on the stability of alkyllithium derivatives 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 ¼ 9:5 10 10 expð13407=TÞ s1 for n-BuLi can be evaluated. As with all rate coeﬃcients of ionic polymerization, this elimination also depends on the reaction system. 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 propagate (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 propagation (seconds) at temperatures of 80 C and higher.

Bu

Li

Bu

k tt

+

LiH

i

i

+

Li Bu +

+

PsLi

i Scheme 7.1.

-

Bu

i

Termination in anionic polymerization of styrene.

Polybutadienyllithium is somewhat more stable than polystyryllithium (k tt ¼ 6:7 105 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 usually no eﬀect on the polymerization rate, because the number of active chain ends

PsH

338

7 Ionic Polymerization

Bu

Li

Li

Bu +

k tr

i

+ i Li

Li +

kp

1 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. However, 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. 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 association 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 developed originally for free-radical polymerization, is in many cases suﬃcient to describe anionic copolymerization also. 7.2.1.4

F1 ¼

r1 f12 þ f1 f2 þ 2f1 f2 þ r2 f22

r1 f12

ð42Þ

However, there is one important diﬀerence between free-radical and living anionic polymerization, and this is the lifetime of the growing chain. This diﬀerence becomes 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

feed for incremental conversion. Because of the diﬀerent reactivity of the monomers, 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 called ﬁrst-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 heterogeneity. 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-SBSBSSBS

I-BBBBBSBB 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 copolymerization parameters. In free-radical polymerization there is just one pair of parameters, 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 composition 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

339

7 Ionic Polymerization Tab. 7.6. Copolymerization parameters for Li-initiated copolymerization of styrene (¼ M1 ) and butadiene (¼ M2 ).[a]

˚

T [ C]

Solvent

r1

r2

25 20 30 50 30 78 0 25 25

bulk toluene benzene cyclohexane heptane THF THF THF diethyl ether

0.04 0.004 0.035 0.025 0.1 11 0.2 0.3 0.4

11.2 12.9 10 15.1 7 0.04 5.3 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 feature 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 appreciable extent, and after the butadiene is consumed the overall polymerization rate becomes that of pure styrene.

1 xS = 1 0.8

xS = 0 xS = 0.35

conversion

340

0.6 xS = 0.5 0.4

xS = 0.65

0.2

0 0

5000

10000

15000

20000

t/s

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. Fig. 7.6.

25000

30000

7.2 Anionic Polymerization 0.8

x S = 0.65 0.6

xS inpolymer

x S = 0.5

0.4

x S = 0.35

0.2

0 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 combination 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 consumed. Overall, therefore, there is a block of nearly pure butadiene with some isolated 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 consumed ﬁrst and then butadiene, but again a copolymer with nearly pure blocks of homopolymer will result. 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 distribution. 7.2.1.5

341

342

7 Ionic Polymerization

The structure of copolymers in terms of composition or sequence length distribution 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 sequential 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 byproducts – 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.

Reactivity ratios i for side reactions

R-Li

ki k p,S

+nS

SSSSSS

R-So-L -Li + R-Li -L

k S Bu k p , Bu

+ m Bu

R-So-Bup-L -L + R-Li -Li + R-Buq-Li

k Bu S k p ,S

k tt k p ,i

SSSSBBBBB + BBBBB

k tr k p ,i

+lS

R-So-Bup -Sr ––Li + R-Buq-Ss-Li + R-S - t-Li Scheme 7.4.

Dead chains

SSSSBBB + SSSSBBBSSSS

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

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 obvious 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 0 ½M1 ¼ ½M2 0 ½M2

½M1 þ1 ½M2 ½M2 þ1 r2 ½M1

r1

ð43Þ

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 distributions in a tubular reactor are described in Refs. 135 and 136. Semi-batch operation with a programmed initiator feed [137, 138] and oscillating feeds to homogeneous CSTRs oﬀer the possibility of a wide range of MWDs [139–144]. 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 operated 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 described 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 aliphatic solvents, can be varied by the addition of polar compounds (see Table 7.3) over a wide range. 7.2.1.6

343

344

7 Ionic Polymerization Tab. 7.7. Microstructure of polybutadienes produced in hydrocarbon solvent with diﬀerent catalyst systems.

Catalyst

cis-1,4 [ %]

trans-1,4 [ %]

1,2 [ %]

Nd Co Ni Ti Li

98 96 96 93 36

1 2 3 3 52

1 2 1 4 12

Copolymers from styrene and butadiene (SBRs) with a more or less random comonomer 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 continuous 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 ratio. 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 apportioning 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 researchers toward extending this technique to other vinyl monomers such as acrylates (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

i

ii

iii

H

H

H

H

H

CN

H

COR

iv Scheme 7.5.

v Monomers with heteroatoms for anionic polymerization.

7.2 Anionic Polymerization

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 monomers 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).

A

+

H

CH3

H

COOMe

A CH2

O

ð44Þ

OMe

H3C

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

COOMe

CH3

H

+

R

CH3

CH3 OMe O

H3C

O

O

ð45Þ

CH3

CH2

OMe

O MeO

Me

COR

H

CH3

CH2

+

O

O H3C

+ HC O 3

O OMe

OMe

ð46Þ

Another side reaction is the enolization of acrylate polymer chains [Eq. (47)]. H A

+

CH2 C COOR

CH2 C

+

AH

ð47Þ

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

345

346

7 Ionic Polymerization

7.2.2, but side reactions are more important. Some recent developments in the polymerization 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 heteroatoms at the double bond can also be polymerized via an anionic mechanism. The only monomer with some industrial importance is formaldehyde. It is polymerized [180] via its carbon–oxygen double bond to give polyacetal homopolymer or polyoxymethylene [Eq. (48)]. nCH2 O ! ðaCH2 aOaÞn

ð48Þ

The polymerization is a precipitation polymerization from gaseous formaldehyde in an inert solvent such as cyclohexane at fairly low temperatures to prevent depolymerization 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 sulﬁdes, 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 energy 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 polymers. 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

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 ¼ A kd ½Pi ½Me ½Me

KP ¼

ð51Þ

ln½Me ¼ ðDHP TDSP Þ=RT

ð52Þ

For monomer concentrations below this equilibrium concentration, no polymerization 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 polymerization is thermodynamically forbidden. TC is given by Eq. (53), where the standard state refers to unit concentration. DHP DHP0 ¼ 0 DSP DSP þ R ln½M

TC ¼

ð53Þ

From Table 7.8 it can be seen that for most of the cyclic monomers, the reversibility 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 O

O

Atoms in ring

States [a]

DHp0 [kJ molC1 ]

DSp0 [ J molC1 KC1 ]

[M]e [M]

3

gc

140

174

4 1016

4

lc

82.4

74

3 1011

5

ss

14

13.5

1:6 102

6

lc

7.1

27.6

6

ss

22.9

41.1

7

lc

13.8

4.6

O O P

O

OMe

O

N H

1.58

O O

O

O

N H

1:16 102

O

2:2 103

O

[a] State of monomer (ﬁrst letter) and polymer (second letter): g – gaseous, l – liquid, s – solution, c – condensed.

347

7 Ionic Polymerization

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

348

X

kinter X X

X

X

X

X

kintra +

X

+

X

X

X

X

X

kp X

Scheme 7.6.

X

X

X

X

X

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. Intramolecular 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). ½Mn e ¼

kp ðnÞ A n5=2 k p ðnÞ

ð54Þ

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 Nacyllactam 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 N

R

R

O

N

O

B

O

N

O

+

+

N

O +

N

N

O R

H O N

R

O

O

N

O

O H O N

N

N

O

O

N

+

O

propagation O H O N *

N

O +

N

O *

O H O N

N

n

O H O N

N n

Scheme 7.7.

O

O

N

O

n

N

O +

H O N

O H O N

O NH

N

O +

N

n

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 compared 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 processes [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-

O

7 Ionic Polymerization

350

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 OH + KOH

HO OH

O K

HO

OK

+

+

+

OK

OK

+

+

+

+

OK

O K

+

+

+

+ H2O initiatior formation

+

OH

HO

OK

O

O

O

OK

O

O

+

OK

O

O

+

OK

O

copolymerization EO / PO

+

+

R

R OK

+

H2O

+

OH

R O

O K

HO

+

R

R nO K

+

+

O

Termination

H2 O

mOH

O

R n OH +

O

mO K

+

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 maximum conversion to the potassium alkoxide. Remaining water will terminate growing chains, forming hydroxyl end groups. Chains with alkoxide end groups can undergo chain transfer reaction with hydroxyl-terminated chains. This proton exchange, 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 hydroxyl groups, so the equilibrium constant for the proton exchange is around unity.

7.3 Cationic Polymerization

This equilibrium between active and non-active chains is very fast, so in general a Poisson distribution will result. For the case where the ﬁrst addition is diﬀerent from the subsequent ones, either the Gold distribution [58] in Eq. (30) or a distribution given by Weibull [198] can be used. A recursive formula for a more complex distribution assuming diﬀerent reactivities for each addition of ethylene oxide is given in Ref. 199. However, it has been shown that normally the chain length dependence of the propagation rate coeﬃcient is not important, with the exception of the ﬁrst unit [200, 201]. Both distributions can be used to describe ethoxylation reactions [202]. There are a number of recent publications on the kinetics and copolymerization kinetics of ethylene and propylene oxide initiated with several alcohols and KOH [203–209]. The copolymerization parameters are generally in the ranges of rE ¼ 2–4, rP ¼ 0.15–0.3, showing the higher reactivity of ethylene oxide. One important side reaction, which may occur at higher temperatures, is the isomerization of propylene oxide to allyl alcohol [210, 211], leading to an unsaturated end group (1) which may aﬀect polyurethane production. R O CH2CH CH2

1

The industrial process [23, 212, 213] of ethoxylation and propoxylation is usually a semi-batch process. The starter alcohol and KOH are mixed and water is removed by distillation. In a second step, monomers are fed into the reactor, where the feed rate is chosen so as to be able to remove the heat of polymerization and to keep the latent heat of polymerization of unreacted monomers in a safe state. By this process, homopolymers and random copolymers are accessible. Block copolymers are produced by successive feeds of the respective monomers. Catalyst is removed by addition of acids and subsequent crystallization and ﬁltration of precipitated salts. An optional fourth step is the removal of volatile compounds by distillation.

7.3

Cationic Polymerization

There are many monomers that can be polymerized via a cationic mechanism (see Table 7.1), but the most important polymers from an industrial point of view are homo- and copolymers from isobutene and from some heterocyclic monomers such as trioxane, tetrahydrofuran, and epoxides. There is a detailed discussion on the mechanistic features in Refs. 1–5, 7, and 181–183. 7.3.1

Cationic Polymerization of Vinyl Monomers

Cationic polymerization of vinyl monomers is by electrophilic addition of the monomer to a growing carbenium ion. The reactivity of vinyl monomers in cati-

351

352

7 Ionic Polymerization

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 H2C

CH N

CH

H2C

CH

H2C

Me H2C

CH

H2C Me

OR

ð55Þ

OMe

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 species 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 coeﬃcients the polymerization may become diﬀusion-controlled, which has been addressed [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 equilibrium strongly aﬀects the width of the molecular weight distribution. For the sequence 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 termination. k ass þM H3 CaCHA ! H3 CaCHþ A ! þM

R

k ion

D ¼ Pw =Pn ¼ 1 þ ½I 0 k p =kass

a

a

kp

R

H2C

ð56Þ

ð57Þ

Bimodal 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 alkylation 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

* +

C

*

A

+

+

C

A

+

+

n

ð58Þ

n

* n

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 addressed 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 living 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. Homopolymers 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 polymerized cationically, because of the existence of side reactions leading to dioxane or

353

354

7 Ionic Polymerization

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 ﬂuorosulfonic 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

2

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 discussion on the kinetics of propagation for THF is given in Ref. 182. Because of the low energy gain from ring strain, the equilibrium monomer concentration 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 reactions to polymer are of great importance with regard to the molecular weight distribution (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 SO 3

CCl4

25

CH2 Cl2

25

CH3 NO2

25

ionic covalent ionic covalent ionic covalent covalent ionic covalent ionic

4 102 6 105 3:1 102 1:7 104 2:4 102 5 104 1:4 104 4 104 1:9 103 1:8 105

Oxepane

CF3 SO 3

H3 NO2

25

Oxazoline

CH3 C6 H4 O

CD3 CN

40

7.3 Cationic Polymerization

2.5

ln[M]

2 1.5 1 0.5 0 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 concentration 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 equilibrium between linear and cyclic polymers are found for 1,3-dioxolane and siloxanes [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 monomers [Eq. (59)]: that is, the system is not yet in equilibrium with respect to macrocycles and transfer reactions may still proceed.

+O +

O

+

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 conversion 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 1,3dioxolane, ethylene oxide, and similar comonomers, the comonomer is nearly completely 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

355

356

7 Ionic Polymerization

NMR and stability studies, that the comonomers are randomly distributed along the chains [237, 238]. This is attributed to the fact that intermolecular chain transfer (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. þ RaOCH2 aOCH2 aOCH2 aOaCHþ 2 T RaOCH2 aOCH2 aOaCH2 þ CH2 O

ð60Þ

This unzipping depolymerization occurs during polymerization, but it may also occur 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 oxide, 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 zirconium 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 catalysts 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 descriptions 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

very high rate coeﬃcients for propagation, especially in the case of cationic polymerization, 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 structure 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 parameters 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 monomers and solvents. This may prevent the wider employment of ionic polymerization. Generally, ionic polymerization is used for monomers which cannot be polymerized by free radicals, or for the production of polymer structures which are otherwise not accessible.

Notation

D fi Fi hðiÞ i ½I f kt k tt kAB kass k ion kp kd ki K K ass [M] ½Meq

polydispersity index mole fraction of monomer i in the monomer mixture mole fraction of monomer i in polymer chains being formed frequency distribution of chain length degree of polymerization initiator feed concentration rate coeﬃcient for termination [M1 s1 ] rate coeﬃcient for thermal termination, for example, LiH elimination [s1 ] rate coeﬃcient for chain propagation [M1 s1 ] for monomer B added to chain ending with A rate coeﬃcient of association rate coeﬃcient of ionization, dissociation rate coeﬃcient for chain propagation [M1 s1 ] rate coeﬃcient for decomposition [s1 ] rate coeﬃcient for chain initiation [M1 s1 ] equilibrium constant, general equilibrium constant of association monomer concentration [M] equilibrium monomer concentration

357

358

7 Ionic Polymerization

½Mf Mn Mw ½Ni n Pn Pw r1 ; r2 r Rp R T TC v_ in; I v_ in; M wðiÞ xM xI

monomer feed concentration number-average molecular weight [g mol1 ] weight-average molecular weight [g mol1 ] concentration of polymer chains with degree of polymerization i association number or ring size number average degree of polymerization weight average degree of polymerization copolymerization parameters according to the terminal model ratio of k p =k i for living polymerization rate of propagation [M s1 ] gas constant temperature ceiling temperature initiator feed ﬂow [L s1 ] monomer feed ﬂow [Ls1 ] mass distribution of chain length monomer conversion initiator conversion

Greek DGp DHp DSp t n

Gibbs free energy of polymerization enthalpy of polymerization entropy of polymerization residence time average kinetic chain length

Acronyms s- (t-, n-)Bu CSTR HCSTR M, M T, T THF PaLi PsH MWD Pi HIPS NMR RIM SBR SCSTR

sec.- (tert.-, n-)butyl continuous stirred tank reactor homogeneous continuous stirred tank reactor monomer molecule ( , with active center) transfer agent ( , with active center) tetrahydrofuran growing polymer chain with lithium counter ion polystyrene molecule ending with hydrogen molecular weight distribution polymer chain with degree of polymerization i high-impact polystyrene nuclear magnetic resonance reaction injection molding styrene–butadiene rubber segregated continuous stirred tank reactor

References

References 1 B. L. Erusalimski, Mechanisms of

2

3

4

5

6

7

8 9

10

11 12 13

14

15

Ionic Polymerization, Consultants Bureau, New York, 1986. K. Matyjaszewski (ed.), Cationic Polymerization, Marcel Dekker, New York, 1996. J. P. Kennedy, E. Marechal, Carbocationic Polymerization, WileyInterscience, New York, 1982. J. P. Kennedy, B. Ivan, Designed Polymers by Carbocationic Macromolecular Engineering – Theory and Practice, Hanser Publishers, Mu¨nich, 1992. M. Szwarc, M. Beylen, Ionic Polymerization and Living Polymers, Chapman & Hall, New York, 1993. H. L. Hsieh, R. P. Quirk, Anionic Polymerization, Marcel Dekker, New York, 1996. M. Szwarc, Ionic Polymerization Fundamentals, Hanser Publishers, Mu¨nich, 1996. T. Fukuda, K. Kubo, Y.-D. Ma, Prog. Polym. Sci., 1982, 17, 875. B. R. Morrison, M. C. Piton, M. A. Winnik, R. G. Gilbert, D. H. Napper, Macromolecules, 1993, 26, 4368; M. D. Zammit, T. P. Davis, G. D. Willet, K. F. O’Driscoll, J. Polym. Sci., Part A: Polym. Chem., 1997, 35, 2311. K.-D. Hungenberg, F. Bandermann, L. Janko, K. Knoll, in K. H. Reichert, H. U. Moritz (eds.), DECHEMA Monographs, Vol. 131 (5th Int. Workshop on Polymer Reaction Engineering, 1995), p. 387, VCH, Weinheim, 1995. G. Spach, J. Chem. Soc. (Lond.), 1962, 355. R. A. Finnegan, H. W. Kutta, J. Org. Chem., 1965, 30, 4138. W. J. Kern, J. N. Anderson, H. E. Adams, T. C. Bouton, T. W. Bethea, J. Appl. Polym. Sci., 1972, 16, 3123. W. Cooper, in: W. M. Saltman (ed.), The Stereo Rubbers, Wiley-Interscience, New York, 1977. J. Lal, J. E. Mark, Advances in Elastomers and Rubber Elasticity, Plenum Press, New York, 1986.

16 F. R. Eirich, Science and Technology of

17

18

19

20 21

22

23

24

25

25a

26

27 28 28a

Rubber, Academic Press, New York, 1978. H. Gausepohl, H. Haberkorn, K.-D. Hungenberg, R. Kick, K. Knoll, P. Wittmer, in G. Becker, D. Braun, (eds.), Kunststoﬀ-Handbuch, Vol. 4, Polystyrol, Chapter 7, Hanser Publishers, Mu¨nich, 1996. J. Scheirs, D. B. Priddy (eds.), Modern Styrenic Polymers: Polystyrenes and Styrenic Copolymers, John Wiley, Chichester, 2003. A. Noshay, J. E. McGrath, Block Copolymers, Overview and Critical Survey, Academic Press, New York, 1977. G. Holden, Thermoplastic Elastomers, Hanser Publishers, Mu¨nich, 2000. R. P. Quirk, D. J. Kinning, L. J. Fetters, in G. Allen, J. C. Bevington, S. L. Aggarwal, (eds.), Comprehensive Polymer Science, Pergamon Press, Oxford, 1989. F. E. Bailey, Jr., J. V. Koleske, Alkylene Oxides and Their Polymers, Marcel Dekker, New York, 1991. F. E. Bailey, J. V. Koleske, Polyoxyalkylenes, in Ullmann’s Encyclopedia of Industrial Chemistry, Vol. 28, 6th ed., VCH, Weinheim, 2003. W. Diller, P. Gupta, P. Haas, K. Schauerte, R. Sundermann, K. Uhlig, ´ın Kunststoﬀ-Handbuch, Vol. 7, Polyurethane, Hanser Publishers, Mu¨nich, 1996. D. Hesterwirth, D. Beckelmann, F. Bandermann, J. Appl. Polym. Sci., 1999, 73, 1521, 1533. F. C. Foster, J. L. Binder, Adv. Chem. Ser., 1957, 19, American Chemical Society, Washington DC. H. Hostalka, R. V. Figini, G. V. Schulz, Makromol. Chem., 1964, 71, 71. D. J. Worsfold, S. Bywater, Can. J. Chem. 1960, 38, 1891. D. J. Worsfold, J. Polym. Sci., A-1, 1967, 5, 2783. E. W. Duck, J. M. Locke, J. Inst. Rubber Ind., 1968, 2, 223.

359

360

7 Ionic Polymerization 29 S. Bywater, Macromol. Chem. Phys., 30 31

32

33 34 35 36

37

38

39 40

41 42 43 44

45

46

47

48

49

1998, 199, 1217. S. Bywater, Prog. Polymer. Sci., 1994, 19, 287. R. N. Young, R. P. Quirk, L. J. Fetters, Adv. Polym. Sci., 1984, 56, 1. M. van Beylern, S. Bywater, G. Smets, M. Szwarc, D. J. Worsfold, Adv. Polym. Sci., 1988, 86, 87. D. B. Priddy, M. Pirc, B. J. Meister, Polym. React. Eng., 1992–3, 1, 343. J. E. L. Roovers, S. Bywater, Macromolecules, 1975, 8, 251. R. Ohlinger, F. Bandermann, Makromol. Chem., 1980, 181, 1935. R. N. Young, L. J. Fetters, J. S. Huang, R. Krishnamoorti, Polym. Int., 1994, 33, 217. L. J. Fetters, N. P. Balsara, J. S. Huang, H. S. Jeon, K. Amdal, M. Y. Lin, Macromolecules, 1995, 28, 4996. L. J. Fetters, J. S. Huang, R. N. Young, J. Polym. Sci., Part A: Polym. Chem., 1996, 34, 1517. A. L. Frischknecht, S. T. Milner, J. Chem. Phys., 2001, 114, 1032. A. A. Arest-Yakubovich, J. Polym. Sci., Part A: Polym. Chem., 1997, 35, 3613. S. Bywater, Macromolecules, 1998, 31, 6010. S. Bywater, J. Polym. Sci., Part A: Polym. Chem., 1998, 36, 1065. S. Bywater, Macromol. Chem. Phys., 1998, 199, 1217. T. Tanlak, A. Ahmad, M. N. Treybig, R. G. Anthony, J. Appl. Polym. Sci., 1978, 22, 315. T. D. Edgar, S. Hasan, R. G. Anthony, Chem. Eng. Sci., 1970, 25, 1463. D. C. Timm, J. W. Rachow, ACS Adv. Chem. Ser. 1974, 133, 122; D. C. Timm, L. F. Kubicek, Chem. Eng. Sci., 1974, 29, 2145. S. Auguste, H. G. M. Edwards, A. F. Johnson, Z. G. Meszena, P. Nicol, Polymer, 1997, 38, 4357. S. Auguste, H. G. M. Edwards, A. F. Johnson, Z. G. Meszena, P. Nicol, Polymer, 1996, 37, 3665. J.-J. A. Your, G. D. Karles, J. G. Ekerdt, I. Trachtenberg, J. W.

50

51 52

53

54 55 56 57

58 59 60 61 62

63 64 65

66 67 68 69 70

71 72

Barlow, Ind. Eng. Chem. Res., 1989, 28, 1456. W. H. Christiansen, J. G. Ekerdt, I. Trachtenberg, J. W. Barlow, Ind. Eng. Chem. Res., 1990, 29, 463. T. R. Landon, R. G. Anthony, AIChE J., 1972, 18, 843. E. Ceaucescu, R. Bordeianu, I. Cerchez, M. Balanescu, P. Ghioca, Rev. Roum. Chim., 1981, 26, 509. Y. L. Spirin, A. R. Gantmakher, S. S. Medvedev, Doklady Akad. Nauk S.S.S.R., 1962, 146, 368. M. Szwarc, Nature, 1956, 178, 168. M. Szwarc, M. Levy, R. Milkovich, J. Am. Chem. Soc., 1956, 78, 2656. R. P. Quirk, B. Lee, Polym. Int., 1992, 27, 359. S. Penczek, P. Kubisa, in Encyclopedia of Polymer Science and Engineering, Supplement Volume, 2nd ed., WileyInterscience, New York, 1989. L. Gold, J. Chem. Phys. 1958, 28, 91. V. S. Nanda, R. K. Jain, J. Polym. Sci., Pt. A, 1964, 2, 4583. H. Ho¨cker, Makromol. Chem., 1982, 157, 187. ¨ller, Macromolecules, A. H. E. Mu 1994, 27, 1685. ¨ller, R. Zhuang, G. A. H. E. Mu Litvinenko, Macromolecules, 1995, 28, 4326. R. V. Figini, Makromol. Chem., 1964, 71, 193. R. V. Figini, Makromol. Chem., 1967, 107, 170. R. V. Figini, H. Hostalka, K. Hurm, G. Lohr, G. V. Schulz, Z. Physikal. Chem., 1965, 45, 269. B. D. Coleman, T. G. Fox, J. Am. Chem. Soc., 1963, 85, 1241. B. D. Coleman, T. G. Fox, J. Polym. Sci., 1968, 4, 345. D. C. Pepper, Makromol. Chem., 1995, 196, 963. K. Matyjaszewski, J. Polym. Sci., 1993, 31, 995. K. Matyjaszewski, R. Szymanski, M. Teoderescu, Macromolecules, 1994, 27, 7565. S. Zhu, W.-J. Wang, J. Polym. Sci., Pt. B: Polym. Phys., 1999, 37, 961. G. I. Litvinenko, A. A. Arest-

References

73

74 75 76 77 78 79 80 81

82 83 84

85 86 87

88 89 90 91 92

93 94 95

Yakubovich, Macromol. Chem., Theory Simul., 1992, 1, 321. G. I. Litvinenko, A. A. ArestYakubovich, Macromol. Chem., Theory Simul., 1995, 4, 347. Z. Tadmore, J. A. Biesenberger, I&EC Fundamentals, 1966, 5, 336. D.-M. Kim, E. B. Nauman, I&EC Research, 1999, 38, 1856. D.-M. Kim, E. B. Nauman, I&EC Research, 1997, 36, 1088. W. H. Abrahams, I&EC Fundamentals, 1963, 2, 221. B. D. Coleman, F. Gornick, G. Weiss, J. Chem. Phys. 1963, 39, 3233. M. Szwarc, M. Litt, J. Phys. Chem. 1958, 62, 568. T. A. Oroﬁno, F. Wenger, J. Chem. Phys., 1961, 35, 532. D. A. Couso, L. M. Alassia, G. R. Meira, J. Appl. Polym. Sci., 1985, 30, 3249. R. Tappe, F. Bandermann, Angew. Makromol. Chem., 1988, 160, 117. J. A. Biesenberger, Z. Tadmor, J. Appl. Polym. Sci., 1965, 9, 3409. H. Fathi, A. E. Hamielec, D. J. Davison, Polym. React. Eng., 1996, 4, 243. J. E. Puskas, Macromol. Chem., Theory Simul., 1993, 2, 141. M. D. Glasse, Prog. Polym. Sci., 1983, 9, 133. A. A. Arest-Yakubovich, G. I. Litvinenko, Prog. Polym. Sci., 1996, 21, 335. W. H. Glaze, J. Lin, E. G. Felton, J. Org. Chem., 1965, 30, 1258. W. H. Glaze, J. Lin, E. G. Felton, J. Org. Chem., 1966, 31, 2643. W. H. Glaze, G. M. Adams, J. Am. Chem. Soc., 1966, 88, 4653. R. A. Finnegan, H. W. Kutta, J. Org. Chem., 1965, 30, 4138. B. L. Wakeﬁeld, The Chemistry of Organolithium Compounds, Pergamon Press, Oxford, 1974. J. W. Burley, R. N. Young, J. Chem. Soc. (B), 1971, 1018. G. Spach, M. Levy, M. Szwarc, J. Chem. Soc. (London), 1962, 355. G. T. Viola, M. Bortolotti, A. Zazzetta, J. Polym. Sci., Part A: Polym. Chem., 1996, 34, 13.

96 R. W. Pennisi, L. J. Fetters,

Macromolecules, 1988, 21, 1094. 97 W. Nentwig, H. Sinn, Makromol.

Chem., Rapid Commun., 1980, 1, 59. 98 J. N. Anderson, W. J. Kern, T. W.

99 100 101 102 103

104

105

106

107

108 109 110 111 112 113 114 115 116 117

Bethea, H. E. Adams, J. Appl. Polym. Sci., 1972, 16, 3133. D. B. Priddy, Adv. Polym. Sci., 1994, 111, 67. A. L. Gatzke, E. Vanzzo, Chem. Commun., 1967, 1180. A. L. Gatzke, J. Polym. Sci., A-1, 1969, 7, 2281. I. A. Lifshits, Y. B. Podolnyi, Vysokomolek. Soedin., B, 1971, 13, 149. I. A. Lifshits, Y. B. Podolnyi, Vysokomolek. Soedin., A, 1970, 12, 2655. A. A. Arest-Yakubovich, I. V. Zolotareva, N. I. Pakuro, E. V. Kristal’nyi, B. I. Nakhmanovich, Vysokomolek. Soedineni., A, 1996, 38, 418. L. H. Peebles Jr., Molecular Weight Distributions in Polymers, Interscience Publishers, London, 1971. N. A. Dotson, R. Galvan, R. L. Laurence, M. Tirrel, Polymerization Process Modeling, VCH, Weinheim, 1996. A. F. Johnson, M. A. Moshin, Z. G. Meszena, P. Graves-Morris, J. Macromol. Sci. – Revs. Macromol. Chem. Phys., C, 1999, 39, 527. D.-Y. Yan, J. Chem. Phys., 1984, 80, 3434. D.-Y. Yan, G.-F. Cai, J. Macromol. Sci., Chem., A, 1986, 23, 37. S. C. Jain, V. S. Nanda, Eur. Polym. J., 1977, 13, 137. D.-Y. Yan, C.-M. Yuan, J. Macromol. Sci., Chem., A, 1986, 23, 769. D.-Y. Yan, C.-M. Yuan, J. Macromol. Sci., Chem., A, 1986, 23, 781. C.-M. Yuan, D.-Y. Yan, Makromol. Chem., 1986, 187, 2629. C.-M. Yuan, D.-Y. Yan, Makromol. Chem., 1986, 187, 2641. D.-Y. Yan, C.-M. Yuan, Makromol. Chem., 1987, 188, 341. D.-Y. Yan, C.-M. Yuan, Makromol. Chem., 1987, 188, 333. D.-Y. Yan, C.-M. Yuan, Polymer, 1988, 29, 924.

361

362

7 Ionic Polymerization 118 D.-Y. Yan, C.-M. Yuan, Eur. Polymer 119 120 121 122 123 124 125

126

127

128

129 130

131

132 133 134 135 136 137

138 139 140 141

142

J., 1988, 24, 924. G.-F. Cai, D.-Y. Yan, M. Litt, Macromolecules, 1988, 21, 578. D. J. Worsfold, J. Polym. Sci., A-1, 1967, 5, 2783. R. C. Bening, in Ref. 18. K. Knoll, in Ref. 17. J. E. McGrath, J. Polym. Sci., Polym. Symp., 1977, 60, 29. M. Morton, J. Polym. Sci., Polym. Symp., 1977, 60, 1. N. Hadjichristidis, S. Pispas, G. Floudas, Block Copolymers, WileyInterscience, New-York, 2002. F. J. Balta´ Calleja, Z. Roslaniec (eds.), Block Polymers, Marcel Dekker, New York, 2000. A. Noshay, J. E. McGrath, Block Copolymers, Academic Press, New York, 1977. F. Bandermann, H.-D. Speikamp, L. Weigel, Makromol. Chem., 1985, 186, 2017. L. H. Tung, G. Y. S. Lo, D. E. Beyer, Macromolecules, 1978, 11, 616. D. Hesterwerth, F. Bandermann, Angew. Makromol. Chem., 1991, 84, 111. T. A. Antkowiak, A. E. Oberster, H. G. Halasa, D. P. Tate, J. Polym. Sci., Part A-1, 1972, 10, 1319. C. F. Wofford, H. L. Hsieh, J. Polym. Sci., Part A-1, 1969, 7, 461. BASF AG, DE 19 801 101, 1999. F. Bandermann, Angew. Makromol. Chem., 1971, 18, 137. W. Gebert, F. Bandermann, Angew. Makromol. Chem., 1977, 57, 153. G. R. Meira, A. F. Johnson, Polym. Eng. Sci., 1981, 21, 415. F. Bandermann, R. Dunsing, K. F. Elgert, Chem. Ing. Techn., 1974, 46, 615. F. Bandermann, R. Dunsing, Chem. Ing. Techn., 1974, 46, 617. B. Langner, F. Bandermann, Angew. Makromol. Chem., 1978, 71, 101. J. Klahn, F. Bandermann, Angew. Makromol. Chem., 1979, 83, 1. R. G. Gosden, S. Auguste, H. G. M. Edwards, A. F. Johnson, Polym. React. Eng., 1995, 3, 331. R. G. Gosden, Z. G. Meszena, M. A.

143

144

145 146 147 148 149 150 151 152 153 154 155 156 157

158

159 160

161 162 163 164 165 166 167 168 169 170 171

Moshin, S. Auguste, A. F. Johnson, Polym. React. Eng., 1997, 5, 45, 205. Z. G. Meszena, Z. Viczian, R. G. Gosden, M. A. Moshin, A. F. Johnson, Polym. React. Eng., 1999, 7, 71. R. G. Gosden, K. Sahakara, A. F. Johnson, J. Chen, R. F. Li, X. Z. Wang, Z. G. Meszena, Polym. React. Eng., 2001, 9, 249. Dow Chemical, US 4 883 846, 1989. BASF AG, WO 98/07765, 1998. BASF AG, US 5 990 236, 1999. Dow Chemical, US 4 572 819, 1986. BASF AG, US 5 902 865, 1999. Sulzer Chemtech, US 5 602 216, 1997. D. B. Priddy, M. Pirc, J. Appl. Polym. Sci., 1989, 37, 393, 1079. BASF AG, US 5 717 040, 1998. Bayer AG, US 4 200 713, 1980. BASF AG, US 5 391 655, 1995. Shell Oil, US 5 587 438, 1996. Dow Chemical, US 4 725 654, 1988. D. Threadingham, W. Olbrecht, J. P. Lambert et al., Rubber, 3. Synthetic, in Ullmann’s Encyclopedia of Industrial Chemistry, Vol. 31, 6th ed., Wiley-VCH, Weinheim, 2003. A. Echte, Handbuch der Technischen Polymerchemie, VCH, Weinheim, 1993. Firestone, US 3 317 918, 1956. M. Demirors, Styrene–Butadiene Copolymers, in Encyclopedia of Polymer Science and Technology, Vol. 4, 3rd ed., Wiley-Interscience, New York, 2003. Goodyear Tire & Rubber Co., US 5 100 965, 1992. Goodyear Tire & Rubber Co., US 5 448 003, 1995. Bridgestone, US 5 578 542, 1996. Bridgestone, US 5 625 017, 1997. Bridgestone, EP 0 768 323 A1, 1996. Firestone Tire & Rubber Co., EP 0 171 517, 1985. Phillips Petroleum Company, US 2 975 160, 1961. Phillips Petroleum Company, US 3 294 768, 1966. T. C. Buton, S. Futamara, Rubber Age (N.Y.), 1974, 106, 33. H. L. Hsieh, W. H. Glaze, Rubber Chem. Technol., 1970, 43, 22. G. Holden, N. R. Legge, Thermopastic

References

172 173 174 175 176 177 178

179 180 181

182

183

184

185 186 187 188

189 190

191 192 193 194

Elastomers Based on Polystyrene– Polydiene Block Copolymers, Hanser Publishers, Mu¨nich, 1987. BASF AG, EP 25 942, 1981. BASF AG, US 4 861 742, 1989. L. Fetters, Macromolecules, 1979, 12, 344. P. Guyot, Polymer, 1981, 22, 1724. Laboﬁna, PCT Int. Appl. 89/10 378, 1989. Sumitomo, US 4 845 173, 1989. M. van Beylen, S. Bywater, G. Smets, M. Szwarc, D. J. Worsfolg, Adv. Polym. Sci., 1988, 86, 89. D. Baskaran, Prog. Polym. Sci., 2003, 28, 521. Polyacetal, PERP report Nexant, Chem Systems, New York, 2002. J. E. McGrath (ed.), Ring-Opening Polymerization – Kinetics, Mechanisms, and Synthesis, ACS Symp. Ser. 286, American Chemical Society, Washington DC, 1985. D. J. Brunelle (ed.), Ring-Opening Polymerization – Mechanisms, Catalysis, Structure, Utility, Hanser Publishers, Mu¨nich, 1993. K. J. Ivin, T. Saegusa (eds.), RingOpening Polymerization, Vols. 1–3, Elsevier, New York, 1984. J. S. Riffle, K. Matyjeszwski (eds.), Makromol. Chemie, Macromol. Symp. 42/43, 1991. F. Franta, P. Rempp (eds.), Makromol. Chemie, Macromol. Symp. 32, 1990. A. Miyake, W. H. Stockmayer, Makromol. Chem., 1965, 88, 90. H. Jacobsen, W. H. Stockmayer, J. Chem. Phys., 1950, 18, 1600. F. Fahnler, in Kunststoﬀ-Handbuch, Vol. 3/4, Polyamide, p. 48ﬀ, Hanser Publishers, Mu¨nich, 1998. K. Hashimoto, Prog. Polym. Sci., 2000, 25, 1411. M. J. Kohan, Polyamides, in Ullmann’s Encyclopedia of Industrial Chemistry, Vol. 31, 6th ed., Wiley-VCH, Weinheim, 2003. Atoﬁna, EP 1 165 661, 2000. S. K. Ha, J. L. White, Int. Polym. Processing, 1998, 13, 136. B. J. Kim, J. L. White, Int. Polym. Processing, 2002, 17, 33. BASF, US 6 441 109, 2002.

195 Gh. Rusu, K. Ueda, E. Rusu, M.

Rusu, Polymer, 2002, 42, 5669. 196 P. J. Flory, J. Am. Chem. Soc., 1940,

62, 1561. 197 N. M. van Os (ed.), Nonionic

198 199 200

201 202 203 204 205

206

207

208

209

210 211 212

213

Surfactants, Marcel Dekker, New York, 1998. W. Weibull, B. Nycander, Acta Chem. Scand., 1954, 8, 847. G. Natta, E. Mantica, J. Am. Chem. Soc., 1952, 74, 3152. E. Santacesaria, M. DiSerio, R. Garaffa, G. Addino, Ind. Eng. Chem. Res., 1992, 31, 2413. A. J. Lowe, B. Weibull, J. Polym. Sci., 1954, 12, 493. W. Hreczuch, G. Bekierz, Tenside Surf. Deterg., 1995, 32, 55. F. Heatley, G. Yu, C. Booth, T. G. Blease, Eur. Polym. J., 1991, 27, 573. M. Adal, P. Flodin, Tenside Surf. Deterg., 1994, 31, 1. M. DiSerio, G. Vairo, P. Iengo, F. Felippone, E. Santacesaria, Ind. Eng. Chem. Res., 1996, 35, 3848. E. Santacesaria, M. DiSerio, P. Iengo, in Reaction Kinetics and the Development of Catalytic Processes, G. F. Froment, K. C. Waugh, (eds.), Elsevier, New York, 1999. M. DiSerio, R. Tesser, A. Dimiccoli, E. Santacesaria, Ind. Eng. Chem. Res., 2002, 41, 5196. M. DiSerio, S. Di Martino, E. Santacesaria, Ind. Eng. Chem. Res., 1994, 33, 509. A. de Lucas, L. Rodriguez, M. Perez-Collado, Polym. Int., 2002, 51, 1066. L. E. Pierre, C. C. Price, J. Am. Chem. Soc., 1956, 78, 3432. D. M. Simons, J. J. Verbanc, J. Polym. Sci., 1960, 44, 303. S. D. Gagnon, Polyethers (Propylene Oxide Polymers), in Kirk-Othmer Encyclopedia of Chemical Technology, Vol. 19, 4th. ed., John Wiley, New York, 1996. D. M. Bach, E. M. Clark, R. Ramachandran, Polyethers (Ethylene Oxide Polymers), in Kirk-Othmer Encyclopedia of Chemical Technology, Vol. 19, 4th. ed., John Wiley, New York, 1996.

363

364

7 Ionic Polymerization 214 P. H. Plesch, Prog. Reaction Kinetics, 215 216

217

218

219

220 221 222

223

224

225

226 227 228 229

230 231 232

233

1993, 18, 1. R. B. Taylor, F. Williams, J. Am. Chem. Soc., 1969, 91, 3728. A. A. Berlin, S. A. Vol’fson, Polym. Sci. USSR (Engl. Transl.), 1994, 36, 505. K. Matyjaszewski, D.-H. Lin, Makromol. Chem., Macromol. Symp., 1991, 47, 221. K. Matyjaszewski, R. Szymanski, M. Teodorescu, Macromolecules, 1994, 27, 7565. P. H. Plesch, The Chemistry of Cationic Polymerization, Pergamon Press, Oxford, 1963. P. Sigwalt, Makromol. Chem. Macromol. Symp., 1991, 47, 179. K. Matyjaszewski, P. Sigwalt, Polym. Int., 1994, 35, 1. ¨dos, J. J. P. Kennedy, T. Kelen, F. Tu Polym. Sci., Polym. Chem. Ed., 1975, 13, 2277. ¨ dos, B. Turcsnayi, T. Kelen, F. Tu J. P. Kennedy, J. Polym. Sci., Polym. Chem. Ed., 1977, 15, 3047. K. S. Whiteley, T. G. Higgs, H. Koch, R. L. Mayer, W. Immel, Polyoleﬁns, in Ullmann’s Encyclopedia of Industrial Chemistry, Vol. 28, 6th ed., Wiley-VCH, Weinheim, 2003. S. Penczek, P. Kubisa, K. Matyjaszewski, Adv. Polym. Sci., 1985, 68/69, 35. S. Penczek, P. Kubisa, K. Matyjaszewski, Adv. Polym. Sci., 1980, 37, 1. T. Saegusa, S. Kobayashi, Prog. Polym. Sci., 1973, 6, 107. S. Kobayashi, K. Morikawa, T. Saegusa, Polym. J., 1979, 11, 405. K. Weißermel, E. Fischer, K. Gutweiler, D. D. Hermann, Kunststoﬀe, 1964, 54, 410. J. M. Andrews, J. A. Semlyen, Polymer, 1972, 13, 142. J. A. Semlyen, P. V. Wright, Polymer, 1965, 10, 643. J. M. McKenna, T. K. Wu, G. Pruckmayr, Macromolecules, 1977, 10, 877. T. G. Croucher, R. E. Welton, Polymer, 1976, 17, 205.

234 V. Jaacks, ACS Adv. Chem. Series,

1969, 91, 371. 235 Y. Yamashita, T. Tusda, M. Okada,

236 237 238 239

240

241 242 243

244

245 246

247

248

249

250 251

S. Iwatsuki, J. Polym. Sci., A-1, 1966, 4, 2121. G. Opitz, Plaste Kautschuk, 1973, 20, 597. D. Fleischer, R. C. Schulz, Makromol. Chem., 1975, 176, 677. M. Kucera, J. Pichler, Polymer, 1964, 5, 371. I. Ishigaki, Y. Morita, K. Nishimura, A. Ito, J. Appl. Polym. Sci., 1974, 18, 1927. M. Hasegawa, K. Yamamoto, T. Shiwaku, T. Hashimoto, Macromolecules, 1990, 23, 2629. W. Kern, V. Jaacks, J. Polym. Sci., 1960, 48, 399. W. K. Busﬁeld, D. Merigold, Makromol. Chem., 1970, 138, 65. G. L. Collins, R. K. Greene, F. M. Berardinelli, W. H. Ray, J. Polym. Sci., Polym. Chem. Ed., 1981, 19, 1597. A. Echte, Handbuch der Technischen Polymerchemie, VCH, Weinheim, 1993. DuPont, PCT Int. Appl. 92/14 773, 1992. G. Pruckmayr, P. Dreyfuss, Polyethers, Tetrahydrofuran, in KirkOthmer Encyclopedia of Chemical Technology, Vol. 19, 4th ed., John Wiley, New York, 1996. P. Dreyfuss, M. P. Dreyfuss, G. Pruckmayr, Tetrahydrofuran Polymers, in Encyclopedia of Polymer Science, Vol. 16, 2nd ed., WileyInterscience, New York, 1989. H. Nagahara, K. Hamanaka, K. Yoshida, T. Iwaisako, J. Masamoto, Chem. Lett., 2000, 2. G. Sextro, Polyoxymethylenes, in Ullmann’s Encyclopedia of Industrial Chemistry, Vol. 28, 6th ed., Wiley-VCH, Weinheim, 2003. J. Masamoto, Prog. Polym. Sci., 1993, 18, 1. M. Kucera, E. Spousta, Makromol. Chem., 1964, 76, 183, 190; 1965, 82, 60; 1966, 90, 215.

365

8

Coordination Polymerization1 Joa˜o 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 applications. They are found in products as diverse as prosthetic implants, gas pipelines, 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 dramatically 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 heterogeneous, but some processes also use soluble catalysts. There are many diﬀerent types of catalysts for oleﬁn polymerization and they can be used to synthesize polymer 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 overview of the diﬀerent scientiﬁc and technological aspects related to oleﬁn polymerization with coordination catalysts. 8.1.2

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 coordination 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). Because of backbiting and chain transfer to polymer reactions, LDPE has both shortand 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 predominantly 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

0.915-0.935 g/cm3

0.915-0.94 g/cm3 / 0.88 - 0.912 g/cm3

Fig. 8.1.

Polyethylene microstructures.

HDPE

0.96-0.97 g/cm3

8.1 Polyoleﬁn Properties and Applications

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 disrupts 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 in ﬁlms, and shares the market with LDPE. Polypropylene is the second most important commercial polyoleﬁn. While freeradical polymerization can only produce atactic polypropylene, Ziegler–Natta catalysts 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 polypropylene and some stereoblock elastomeric polypropylenes made with metallocene catalysts 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 composition is required, while reactors in series are employed for the commercial produc-

atactic

Cp2ZrCl2

isotactic

Et(Ind)2ZrCl2

syndiotactic

iPr(Flu)(Cp)HfCl2

isotacticstereoblock

(NMCp)2ZrCl2

isotacticatacticstereoblock

Et(Me4Cp)(Ind)TiCl2

hemi-isotactic

iPr(Cp)(Ind)ZrCl2

Fig. 8.2.

Polypropylene microstructures and some metallocene catalysts used in their synthesis.

367

368

8 Coordination Polymerization

rubber domains

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 distributions. 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 amorphous 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 comprises 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 decreases 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 preferred 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 typical examples of dienes used for the manufacture of EPDM rubbers.

8.1 Polyoleﬁn Properties and Applications

5-ethylidene-2-norbornene

1,4-hexadiene

CH - CH3

CH2 = CH - CH2 - CH = CH - CH3

dicyclopentadiene

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 composition, long-chain branching, and stereoregularity. Therefore, determining the microstructure 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 technique for measuring the molecular weight distribution of polyoleﬁns. Since gel permeation chromatography is a very well-established technique it will not be discussed here. Several publications provide additional discussions on the GPC technique [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 crystallizability of HDPE and LLDPE depends strongly on the fraction of a-oleﬁn comonomer incorporated into the polymer chains, that is, chains with an increased aoleﬁn fraction have a decreased crystallizability. A similar statement can be made for polypropylene and other polyoleﬁn resins that are made with prochiral monomers: resins with high stereoregularity and regioregularity have higher crystallizabilities 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

369

8 Coordination Polymerization

Differential Crystaf Profile

Cumulative Crystaf Profile

Crystaf vessel

dw/dTc

370

w Tc

Tc

Tc Chemical Composition Distribution

w

Calibration Curve

Tc

α-olefin fraction

α-olefin fraction

Estimation of the chemical composition distribution of a polyoleﬁn using a Crystaf proﬁle and a calibration curve.

Fig. 8.5.

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. Temperature 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 generally 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

Weight fraction

Easier to crystallize

More difficult to crystallize

Mol % α-olefin comonomer Crystallization temperature Chemical composition distribution of a polyoleﬁn made with a heterogeneous Ziegler–Natta catalyst. Fig. 8.6.

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 always 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 addition, some catalysts will also produce polypropylene with a distribution of stereoregularities 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 catalysts make HDPE, LLDPE, and polypropylene with narrow distribution of chemical composition and molecular weight with the theoretical polydispersity of 2. The average 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 consequently 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 synthesis 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

371

8 Coordination Polymerization

0.06

2x105

0.04 105

Mw

Normalized response (a.u.)

372

8x104 0.02 6x104

0.00

4x104 0

2

4

6

8

10

x, mole % butene 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].

Fig. 8.7.

very often subject to signiﬁcant experimental error. Nonetheless, very good evidence of the presence of long-chain branches can be found by studying the rheological 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 introduced. More recently, the discovery of soluble metallocene/aluminoxane catalysts opened the doors to a new revolution in the production of polyoleﬁns. These catalyst 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

properties of polyoleﬁns made with these catalysts and the fact that polymerization processes that were originally designed to use Ziegler–Natta catalysts can be converted, with minimal changes, to operate with metallocenes – the so called ‘‘dropin’’ technology. Although traditional heterogeneous and homogeneous Ziegler–Natta catalysts are commonly used as the standard example of coordination polymerization, coordination catalysts include any complex of transition metals and organic ligands: Phillips catalysts are heterogeneous, chromium-based complexes that are not classiﬁed as Ziegler–Natta catalysts; metallocenes are complexes of a transition metal – in most cases an early transition metal – and cyclopentadienyl or cyclopentadienylderivative ligands; late transition metal catalysts may have a variety of ligands containing 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 electronic 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 activators 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 A X L

L R A L

AlR3

C

Al

AlR2X2

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.

373

374

8 Coordination Polymerization

L R A L

L R A L

L A L R

C*

P*1

M

L A L

( L A L

n

)

Pol

n

R P*1 Fig. 8.9.

nM

P*1+n

Monomer coordination and insertion.

Polymerization with coordination catalysts proceeds via two main steps: monomer coordination to the active site, and monomer insertion into the growing polymer 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 monomer 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 between the comonomers to coordinate to the active sites and to be inserted into the growing polymer chains. Diﬀerent rates of coordination and insertion of comonomers 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 polymerization can be regio- and stereoselective, depending on the design of the ligands 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

Regioselectivity 1,2-insertion Pol L A L

L A L

2

1

2

Pol

1

2,1-insertion Pol L A L

L A L

2 1

1 2

Pol

Stereoselectivity

Pol L A L

L A L Pol

Pol L A L

L A 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. Bimolecular 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 catalyst 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.

375

376

8 Coordination Polymerization

chain transfer by β-hydride elimination L A L

L H A L

L H A L

Pol

Pol

Pol P*H

P*r

Dr

chain transfer by β-methyl elimination L A L

CH3 L A L

Pol

P*

P*r

L CH3 A L

Pol

Pol

Dr

chain transfer to hydrogen L A L

Pol P*r

L H A 2 L

H2

P*H

H2

H L A L

Pol

Pol

Dr

chain transfer to monomer L A L

L A L

Pol

P*r

P *r

M

L A L

Pol

Dr

chain transfer to cocatalyst L A L

Pol

P*r Fig. 8.11.

P*A

Al

Pol

D Al

Chain termination mechanisms.

L R A L 2 C* Fig. 8.12.

L A L R

AlR3

L L R A A L R L 2 Cd

Catalyst deactivation by bimolecular reactions.

AlR2

Pol

8.2 Catalysts for Oleﬁn Polymerization

Dr

H2

C

Dr

M

AlR3 C*

Dr

P*r

P*H

P*H

M

P*Me

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 nature of the active site will aﬀect the catalytic mechanism and consequently the activity and the selectivity of the polymerization. Changes in the polymerization reactor 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 coordination catalysts. Catalytic activity is a measurement of how fast the monomer can be polymerized. 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 diﬀerent 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 catalyst activity in units of kgpolymer (mol catalyst h molmonomer )1 . Since the measured polymerization rate does not necessarily have a ﬁrst-order dependence on monomer concentration, this form may also be oversimpliﬁed, but it is certainly preferable to the ﬁrst one. In addition, since coordination catalysts generally deactivate during polymerization, one should be very careful when extrapolating these average 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 polymerization 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.

377

8 Coordination Polymerization

polymerization rate

378

Decay type Build-up type

polymerization time Polymerization kinetics with coordination catalysts showing diﬀerent deactivation rates. Fig. 8.14.

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 catalyst 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 catalysts consisted of particulate titanium chloride crystals. Two major advances in heterogeneous Ziegler–Natta catalysts were the use of supports and internal donors. Inert supports such as magnesium chloride can improve the morphology of the ﬁnal 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 heterogeneous 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

has a very broad molecular weight distribution, with polydispersities ranging from 12 to 24. Interestingly, hydrogen is not an eﬀective chain-transfer agent and generally 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 (alkylaluminum) compounds. Vanadium catalysts are very sensitive to reaction conditions [23]. Changes in the composition of the solvent, polymerization time, and temperature can aﬀect catalyst activity and selectivity. 8.2.2

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 catalysts. 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 currently experiencing in polyoleﬁn manufacture [25]. Other bulky Lewis acids such as trispentaﬂuorophenylborane (B(C6 F5 )3 ) are also very good cocatalysts for metallocenes. 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

Cl

Zr Fig. 8.15.

Cl Structure of a typical metallocene catalyst, Cp2 ZrCl2 .

379

380

8 Coordination Polymerization

X

X

M

X

M

X

X

M

X

C2v

Cs (meso)

C2 (racemic)

(atactic PP)

(atactic PP)

(isotactic PP)

Ex: Cp2ZrCl2

Et(Ind)2ZrCl2

X

M

X

Ex: Et(Ind)2ZrCl2 X

M

X

Cs

C1

(syndiotactic PP)

(no unequivocal prediction)

Ex: iPr(Flu)(Cp)HfCl2

Ex: (Flu)(MeCp)ZrCl2

Symmetry requirements for the production of polypropylenes with diﬀerent stereoregularities. Fig. 8.16.

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 polymerization mechanism. This eﬀect is called the ‘‘biting angle’’. The literature on metallocene catalysts is very rich – one may even say opulent – and thousands of metallocene 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 orientation 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 conditions 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

CH3

H3C

CH3 H3C H3C

Si

H3C

CH3 N

H3C Fig. 8.17.

CH3

Ti

C

CH3 CH3

A half-sandwich metallocene.

Another important class of metallocene catalysts, the ‘‘half-sandwich’’ metallocenes (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 catalysts 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 Section 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 comonomers (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 several 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 catalysts have a well deﬁned structure and in homogeneous systems can behave as single-site catalysts.

381

382

8 Coordination Polymerization Forward Chain Walking P

Forward Chain Walking P

A

Chain Walking followed by Insertion

Monomer Insertion P

P

Backward Chain Walking

A

A

A

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 produce branched polyethylene from pure ethylene without addition of comonomers. The chain walking mechanism involves simultaneous isomerization and polymerization 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 concentration 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 temperatures polymerization of a-oleﬁns with certain diimine–NiCl2 catalysts behave as a living polymerization: that is, the number-average molecular weight increases linearly with polym