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

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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 Joaõ 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

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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

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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

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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

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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

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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 Marıá 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 Sańchez, 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 Sebastiań 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 Sebastiań 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 Sebastiań 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


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 Sebastiań 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

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XXXIV


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. Sańchez 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


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].


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


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


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


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.


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


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,


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


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


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


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


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


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.


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


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


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


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


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-


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


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


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


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


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


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


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


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.


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


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).


1 1 þ ln jî 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


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


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


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


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


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


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


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


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


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

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

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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


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

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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

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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.

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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

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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

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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-


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

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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


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

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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.


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

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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).


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


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


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.


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 Marećhal [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


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


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


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


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


*

...

*

...

*

...

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.


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


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


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


[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.


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.


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.


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


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


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


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


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).


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 Vî 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).


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 Vî 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 Â 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 Vî ðsM 1 X½1Z ; . . . ; sM Z

þMX ½NZ Z

MA

MAN

; sM 1 ; . . . ; sM

A

M þMX z

Þ ¼ Vî 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

!, lÂX 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 lÂ 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 lÂX 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 lÂZliAm

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 lÂ 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 lÂYjiAk þ

NA X

#) MA2 j lÂYji

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


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-


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)].


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


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


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 Vî 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

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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 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:


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.


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


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.


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 ]


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


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


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


+

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


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


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


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


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


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


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


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.


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


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


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).


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


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


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


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


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


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 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.


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


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


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


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.


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


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)].


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 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


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


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


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 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


Drop number

200

A

150 100 50 0 15

35

55

75

95


Drop number

200

B

150 100 50 0 15

35

55

75

95


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


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


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


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-

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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-

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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

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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).

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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

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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

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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]

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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.

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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


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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

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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-

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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-

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256


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.

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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


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


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


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


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).

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264


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


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


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


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


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


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

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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

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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-


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

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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.


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

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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


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

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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.


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.

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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


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

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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


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


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 ]


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


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-


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


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


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


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-


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


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


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


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


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


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


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


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


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


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

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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


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


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


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


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


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


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


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


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


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


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


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.


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


7.2.2, 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)


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


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-


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


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


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.


* +

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


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


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


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


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

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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.

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1969, 91, 371. 235 Y. Yamashita, T. Tusda, M. Okada,

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8

Coordination Polymerization1 Joaõ 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


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


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.


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


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


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


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


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-


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


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


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


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


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


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].


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 polymerization time and the polydispersity index (PDI) is approximately 1.

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts

8.3

Polymerization Kinetics and Mechanism with Coordination Catalysts 8.3.1

Comparison between Coordination and Free-radical Polymerization

One of the main differences between the polymerization kinetics with coordination catalysts and free-radical initiators is that the former depends on the characteristics of the active site as well as on monomer type, while the latter is almost exclusively regulated by monomer type. As we will see, even though this may not constitute a problem for establishing an operative mechanism for coordination polymerization, it creates a significant challenge for model parameter estimation. In free-radical polymerization, the initiator fragment moves away from the growing chain and therefore can influence chain growth and termination during only the first few monomer insertion steps. Consequently, the values of the polymerization kinetic parameters depend mainly on the monomer type, thus permitting the creation of tables of rate constants and activation energies as a function of monomer type, independently of the type of initiator used during polymerization. In contrast, in coordination polymerization chain growth and termination take place by insertion of the monomer or chain-transfer agent into a metal–carbon bond, as proposed by the Cossee mechanism. Consequently, electrical and steric effects around the active site affect polymerization kinetics as much as does the monomer type. The mechanisms of free-radical and coordination polymerization are contrasted in Figure 8.19. In this way, rate constants for coordination polymerization depend not only on the monomer type but also on the nature of the active sites present during polymerization. Since the nature of the active sites is a rather complex (and unfortunately poorly understood) function of polymerization conditions such as temperature, catalyst/cocatalyst ratio and type, presence and concentration of catalyst modifiers, and solvent type, among other factors, this makes the determination of general tables of polymerization rate constants and activation energies for coordination polymerization virtually impossible. On the other hand, the same phenomena, that is to say those that make it difficult to predict the behavior of coordination polymerization a priori, are also responsible for the remarkable flexibility of coordination catalysts, since polymers with completely different properties can be made with only a few monomer types by simply varying the way these monomers are inserted into the polymer chain via active-site design. 8.3.2

Polymerization Kinetics with Single-site Catalysts Homopolymerization Several metallocene, late transition metal, and homogeneous Ziegler–Natta catalysts behave as single-site catalysts, while all heterogeneous Ziegler–Natta and Phillips catalysts have more than one site type. The mechanism for coordination 8.3.2.1

383

384


Free-radical polymerization is monomer-based: R* + M

RM*

RM* + M

RMM* •••

R(M)n* + M

R(M)n+1*

Coordination polymerization is site-based:

CH2 - CH2

Ti

CH2

CH2

Fig. 8.19. Comparison of the monomer propagation step in free-radical and coordination polymerization.

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

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts ka

C þ Al ! C

ð1Þ

After the catalyst is activated by reaction with the cocatalyst, it is ready to polymerize the monomer, M, forming a living polymer chain P1 of length 1 [Eq. (2)]. ki

C þ M ! P1

ð2Þ

A subsequent monomer propagation reaction with a growing polymer of chain length r, Pr , increases its length to r þ 1 [Eq. (3)]. kp

Pr þ M ! Prþ1

ð3Þ

In general, the first monomer insertion step is considered to have a different reaction rate constant, k i , from the subsequent monomer propagation steps, k p . Since in practice these constants are very difficult (if not impossible) to estimate independently, most kinetic models make the reasonable simplifying assumption that k i G k p . The most important transfer reactions in coordination polymerization are: (1) bhydride elimination; (2) transfer to chain-transfer agent; (3) transfer to monomer; and (4) transfer to cocatalyst. b-Hydride elimination is a first-order reaction in which the hydrogen atom attached to the b-carbon in the living chain is abstracted by the active center, forming a metal hydride center, CH , and a dead polymer chain containing a terminal vinyl unsaturation, D¼ r [Eq. (4)]. k tb

Pr ! CH þ D¼ r

ð4Þ

The metal hydride center is available for polymerization and will undergo an initiation reaction in the presence of monomer [Eq. (5)], similarly to that of [Eq. (2)]. k iH

CH þ M ! P1

ð5Þ

The independent estimation of k i and k iH can be very involved; often the simplifying assumption k i G k iH is adopted. For polypropylene polymerization, b-hydride transfer will produce a dead polymer chain with a vinylidine chain end. On the other hand, vinyl-terminated chains are produced when b-methyl-transfer reactions take place [Eq. (6)]. k tbCH3

þ D¼ Pr ! CCH r 3

ð6Þ

Chain-transfer agents are commonly used to control the molecular weight of the polymer. By far the most common chain-transfer agent used in olefin polymerization is hydrogen [Eq. (7)]. k tH

Pr þ H2 ! CH þ Dr

ð7Þ

385

386


Chain transfer to hydrogen leads to the production of a dead chain with a saturated chain end, Dr , and a metal hydride active site that can be initiated with monomer according to Eq. (5). Transfer to monomer also occurs during olefin polymerization, leading to a vinyl-terminated chain for the case of polyethylene (and a vinylidine-terminated chain for the case of polypropylene) [Eq. (8)]. k tM

Pr þ M ! P1 þ D¼ r

ð8Þ

Finally, the cocatalyst may also act as a chain-transfer agent [Eq. (9)]. If the cocatalyst is trimethylaluminum, (CH3 )3 Al, this elementary step produces a methylatedactive center, which can also be initialized by the monomer in a subsequent step: k tAl

Pr þ Al ! CCH3 þ Dr; Al

ð9Þ

Coordination catalysts are very sensitive to polar impurities and may also be deactivated due to mono- or bimolecular mechanisms. Even though these elementary reactions are not very well understood from a quantitative point of view, we will show next some simple reaction mechanisms proposed to describe them. Reactions with polar impurities, I, leading to the formation of a deactivated site, Cd , and a dead polymer chain have been modeled with the very simple equations [Eqs. (10) and (11)]. kdI

Pr þ I ! Cd þ Dr kdI

C þ I ! Cd

ð10Þ ð11Þ

Likewise, monomolecular and bimolecular deactivation reactions have been described with the simple kinetic schemes described by Eqs. (12)–(14). kd

Pr ! Cd þ Dr kd

C ! Cd kd

2C ! 2Cd

ð12Þ ð13Þ ð14Þ

These deactivation equations are not easy to prove, but have been successful in describing several olefin polymerization processes with Ziegler–Natta and metallocene catalysts. The term ‘‘single-site catalyst’’ has a very precise meaning in coordination polymerization. From the point of view of catalyst structure, single-site catalysts are those where all active sites are represented by the same chemical species and have the same polymerization kinetic parameters. In other words, single-site catalysts can be represented with a single set of the elementary reactions described by Eqs. (1)–(14) or any other equivalent set of polymerization mechanism equations. From a polymer microstructure point of view, single-site catalysts will produce linear


polymers that have a theoretical polydispersity of 2 instantaneously. These two conditions are not independent. In fact, the second condition is the consequence of the first. The qualification ‘‘instantaneously’’ used above should not be taken lightly, since this may lead to significant misunderstanding of catalyst behavior. An instantaneous property refers to the property of the polymer made at a given instant in time under a set of invariant polymerization conditions. If these conditions change throughout the polymerization, so will the instantaneous property under examination. Therefore, even though polymer made with a single-site coordination catalyst has a polydispersity of 2 instantaneously, the polydispersity of the accumulated polymer may be much higher (but never lower) than 2 if reaction conditions are allowed to vary during the polymerization, as indicated in Figure 8.20 for the case of decreasing chain-transfer agent concentration as a function of polymerization time. It is only when the polymerization is operated at steady state that the instantaneous properties of the polymer will be equal the accumulated ones. Polymers made by a mechanism where the chain can either grow by propagation or terminate by transfer reactions, as quantified by Eqs. (1)–(14), follow Flory’s most probable chain length distribution [33] [Eq. (15)]. wðrÞ ¼

r r exp rn2 rn

ð15Þ

1.4

chain transfer agent

time

1.2

1

0.8

w (r )

time

0.6

0.4

0.2

0 1

1.5

2

2.5

3

log r Fig. 8.20. Instantaneous chain length distributions of a polymer made with a single-site catalyst when the concentration of chain transfer agent the reactor decreases with increasing polymerization time.

3.5

4

4.5

387

388


In Eq. (15), wðrÞ is the weight distribution of chains of length r and rn is the number-average chain length of the polymer population. Therefore, from a polymer microstructure point of view, a coordination catalyst is considered to have only one site type if it produces polymer with a chain length distribution that follows Eq. (15) instantaneously. Copolymerization The polymerization model most commonly adopted for olefin copolymerization is the terminal model, particularly for studies of polymerization kinetics. In the terminal model, only the last monomer molecule added to the chain end influences polymerization and transfer rates. Besides the fact that it is logically expected, there is also significant experimental evidence supporting the terminal model for olefin polymerization. Since monomer propagation and chain-transfer reactions take place by insertion between the chemical bond formed by the metal in the active site and the polymer chain end, it is certainly reasonable to assume that both the nature of the active site and the type of monomer last added to the chain will affect these reactions. On the other hand, higher-order models such as the penultimate and pen-penultimate models have not found widespread use in coordination polymerization. Copolymerization models are similar to homopolymerization models, with the added complexity that more polymerization rate constants are required. An accepted form of the terminal model for the binary copolymerization of olefins is shown in Table 8.1. Notice that, except for the fact that the polymerization rate constants now depend on the monomer and the chain end type, the mechanism is essentially the same as the one described in Section 8.3.2.1 for homopolymerization. For the case of linear binary copolymers, the instantaneous bivariate chain length and chemical composition distribution, wðr; yÞ, is described by Stockmayer’s distribution [Eqs. (16)–(19)] [34]. 8.3.2.2

rffiffiffiffiffiffiffiffi r r r ry 2 exp exp rn2 rn 2pk 2k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ FA ð1 FA Þ 1 4FA ð1 FA Þð1 rA rB Þ

wðr; yÞ ¼

rA ¼

k p; AA ; k p; AB

rB ¼

k p; BB k p; BA

y ¼ FA FA

ð16Þ ð17Þ ð18Þ ð19Þ

As indicated in Eq. (19), y is the deviation from the average molar fraction of monomer type A in the copolymer, F A . As usual, the instantaneous value of F A can be calculated using the molar fraction of monomer A in the reactor, fA , and the reactivity ratios rA and rB by the Mayo–Lewis equation, Eq. (20) [35]. FA ¼

ðrA 1Þ fA2 þ fA ðrA þ rB 2Þ fA2 þ 2ð1 rB Þ fA þ rB

ð20Þ

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts Tab. 8.1.

Terminal model for binary copolymerization of olefins.[a] ka

Activation

C þ A ! C

Initiation

C þ A ! P1; A k i; B C þ B ! P1; B

Propagation

Pr; A þ A ! Prþ1; A k p; AB Pr; A þ B ! Prþ1; B k p; BA Pr; B þ A ! Prþ1; A k p; BB Pr; B þ B ! Prþ1; B

b-Hydride elimination

Pr; A ! CH þ D¼ r; A k tb; B Pr; B ! CH þ D¼ r; B

Transfer to hydrogen

Pr; A þ H2 ! CH þ Dr; A k tH; B Pr; B þ H2 ! CH þ Dr; B

Transfer to monomer

Pr; A þ A ! P1; A þ D¼ r; A k t; AB Pr; A þ B ! P1; B þ D¼ r; A k t; BA Pr; B þ A ! P1; A þ D¼ r; B k t; BB Pr; B þ B ! P1; B þ D¼ r; B

Transfer to cocatalyst

Pr; A þ Al ! CCH3 þ Dr; Al kAl; B Pr; B þ Al ! CCH3 þ Dr; Al

Deactivation with impurities

Pr; A þ I ! Cd þ Dr; A kdI; B Pr; B þ I ! Cd þ Dr; B

Monomolecular deactivation

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

k i; A

k p; AA

k tb; A

k tH; A

k t; AA

kAl; A

kdI; A

kd; A

[a] A and B represent the two monomer types. All other symbols are the same as those used for homopolymerization, but the reaction rate constants indicate the type of chain end and reacting monomer with the generic nomenclature k i; j where i is the chain end type and j represents the monomer type.

In the same way that Flory’s distribution is the necessary outcome of adopting the homopolymerization model described by Eqs. (1)–(14), Stockmayer’s distribution is the consequence of adopting the binary copolymerization mechanism described in Table 8.1. The attentive reader may have noticed that Stockmayer’s distribution is an extension of Flory’s distribution for the case of copolymers. In fact, upon integration in the interval y a y a y, Stockmayer’s distribution reduces to Flory’s distribution as demonstrated by Eq. (21). rffiffiffiffiffiffiffiffi r r r ry 2 r r exp exp dy ¼ 2 exp wðrÞ ¼ 2 rn 2pk rn rn 2k y rn ðy

ð21Þ

Therefore, the chain length distributions of linear binary (or multicomponent) copolymers also follow Flory’s most probable distribution and have, instantaneously, a polydispersity of 2.

389


Similarly, an expression for the chemical composition distribution alone can be obtained by integrating Eq. (16) over all chain lengths, that is, in the interval 0 a r a y: wð yÞ ¼

ðy 0

rffiffiffiffiffiffiffiffi r r r ry 2 3 dr ¼ rffiffiffiffiffi exp exp 5=2 rn2 rn 2pk 2k 2k y 2 rn 4 1þ rn 2k

ð22Þ

It is difficult to overestimate the explanatory power contained in these few simple expressions. Equation (16) teaches us that longer polymer chains also have narrow chemical composition distributions, as illustrated in Figure 8.21. This result simply reflects the fact that statistical deviations from the average copolymer composition become less likely as the chain increases in length, as would be expected since chains with infinite length should all have exactly the same average copolymer composition. Equation (16) also shows that the tendency to form comonomer blocks (rA rB ! y) will necessarily broaden the chemical composition distribution of copolymers, while the tendency toward monomer alternation (rA rB ! 1) will narrow the chemical composition distribution and in the limit make all chains have the composition F A ¼ 0:5, as illustrated in Figure 8.22. In fact, the chemical composition distribution is best apprehended by plotting Eq. (22) as a function of the lumped parameter rn =k. Figure 8.23 shows that the chemical composition dis-

0.01 0.009

r = 250 r = 500 r = 1000

0.008 0.007 0.006

w (r ,y )

390

0.005 0.004 0.003 0.002 0.001 0 -0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

y Fig. 8.21. Chemical composition distribution for several chain lengths, as described by Stockmayer’s bivariate distribution. Distribution parameters: rn ¼ 1000, FA ¼ 0:5, rA rB ¼ 1.

0.04

0.06

0.08

0.1

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 0.035

r1r2 = 0.01 r1r2 = 1.0 r1r2 = 5

0.03

w (r , y )

0.025

0.02

0.015

0.01

0.005

0 -0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

y Fig. 8.22. Chemical composition distribution for the same chain length and varying rA rB , as described by Stockmayer’s bivariate distribution. Distribution parameters: rn ¼ 1000, FA ¼ 0:5, r ¼ 1000.

40

rn /κ = 1000 rn /κ = 2000 rn /κ = 4000

35 30

w( y)

25 20 15 10 5 0 -0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

y Chemical composition distribution as a function of the lumped parameter rn =k in Eq. (22). Fig. 8.23.

0.04

0.06

0.08

0.1

391

392


tribution becomes broader as the value of rn =k decreases, reflecting the fact that polymers with shorter number-average chain lengths and a tendency to form blocks (see Eq. (17): k increases when rA rB increases) have broader chemical composition distributions. In conclusion, the width of the instantaneous chemical composition distribution of binary copolymers made by coordination polymerization is a function of a single lumped parameter rn =k and, for a given copolymer composition, depends only on the number-average chain length and reactivity ratio product rA rB .

8.3.3

Polymerization Kinetics with Multiple-site Catalysts

All heterogeneous Ziegler–Natta and Phillips catalysts have two or more active-site types and many soluble Ziegler–Natta and metallocene catalysts may also show multiple-site behavior [36, 37]. In addition, several metallocene catalysts, when supported on organic and inorganic carriers, may behave like multiple-site catalysts even if they behaved as single-site catalysts in solution polymerization. Therefore, several of the catalysts used industrially for polyolefin manufacturing have in fact two or more active-site types. The kinetics of polymerization with multiple-site catalysts is generally considered to be the same as with single-site catalysts, as described by Eqs. (1)–(14) for homopolymerization and in Table 8.1 for copolymerization, with different polymerization kinetic parameters assigned to each site type. In some cases, the polymerization mechanism may be extended to include site transformation steps, where sites of one type may change into sites of another type, such as the one described with the reversible reaction in Eq. (23), where D could be a catalyst modifier such as an electron donor, for instance. C1 þ D $ C2

ð23Þ

Since these additional site transformation steps may affect the relative ratio of site types present in the reactor but do not influence the general polymerization behavior with multiple-site catalysts, for the sake of simplicity they will not be further considered in this chapter. It is usually straightforward to detect the presence of multiple-site types on a coordination catalyst because these catalysts will produce polymer with polydispersity higher than 2 even under invariant polymerization conditions. The simplest way to visualize this phenomenon is to assume that every different site type on a multiplesite catalyst produces polymers that follow a distinct Flory’s distribution: that is, those with a distinct number-average chain length, rn; i [38]. In this way, the chain length distribution for the whole polymer is a combination of distinct Flory’s distributions weighted by the mass fraction of polymer made on each site type, m i [Eq. (24)].

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 1 0.9 0.8 0.7

w (r )

0.6 0.5 0.4 0.3 0.2 0.1 0 1

1.5

2

2.5

3

3.5

4

4.5

log r Fig. 8.24. Chain length distribution of a polymer made with a coordination catalyst containing three different active-site types. See Figure 8.25 for the corresponding chemical composition distribution. Distribution parameters: rn; 1 ¼ 500, rn; 2 ¼ 1000, rn; 3 ¼ 2000, m1 ¼ 0:3, m2 ¼ 0:3, m3 ¼ 0:4.

wðrÞ ¼

n X

mi

i¼1

r rn;2 i

r exp rn; i

ð24Þ

The graphical representation of Eq. (24) for a coordination catalyst having three distinct site types is shown in Figure 8.24. It is important to remember that the use of Eq. (24) is a direct consequence of assuming that the mechanism of polymerization for multiple-site catalysts is described with the same set of equations, Eqs. (1)–(14), used to describe single-site catalysts. In other words, Flory’s distribution is the logical consequence of the mechanism adopted for coordination polymerization. For polymers made with multiple-site catalysts, the instantaneous number- and weight-average chain lengths are given by Eqs. (25) and (26).

rn ¼

rw ¼ 2

n X mi

!1

r i¼1 n; i n X i¼1

m i rn; i

ð25Þ

ð26Þ

393

8 Coordination Polymerization 25

20

w (fraction of ethylene)

394

15

10

5

0 0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

fraction of ethylene Fig. 8.25. Chemical composition distribution of polymer made with a coordination catalyst containing three different active-site types. See Figure 8.24 for the corresponding chain

length distribution. Distribution parameters: ðrn =kÞ1 ¼ 2000, ðrn =kÞ2 ¼ 4000, ðrn =kÞ3 ¼ 8000, FA; 1 ¼ 0:88, FA; 2 ¼ 0:90, FA; 3 ¼ 0:93, m1 ¼ 0:3, m2 ¼ 0:3, m3 ¼ 0:4.

Therefore, the polydispersity index of a polymer made with a catalyst having n site types is given by Eq. (27), which is always greater than or equal to 2. n X rw PDI ¼ ¼ 2 m i rn; i rn i¼1

!

n X mi

!

r i¼1 n; i

ð27Þ

Similarly, the bivariate molecular weight and chemical composition distributions of binary copolymers made with multiple-site catalysts can be described as a weighted superposition of Stockmayer’s distributions [39]. If we consider only the chemical composition component of the distribution, as described by Eq. (22), the distribution of polymer made with a multiple-site catalyst becomes Eq. (28).

wð yÞ ¼

n X

3 m i sffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 5=2 k y 2 rn i¼1 1þ i 4 2 rn i 2 k i

ð28Þ

Figure 8.25 shows the chemical composition distribution obtained with Eq. (28) of a polyolefin made with a catalyst having three distinct active-site types, whose chain length distribution has already been presented in Figure 8.24. Notice the characteristic bimodal distribution of ethylene/a-olefin copolymers made with het-

w (r, y)


Fig. 8.26. Bivariate distribution of chain length and chemical composition of a polymer made with a coordination catalyst containing three different active-site types.

Distribution parameters: rn; 1 ¼ 500, rn; 2 ¼ 1000, rn; 3 ¼ 2000, FA; 1 ¼ 0:88, FA; 2 ¼ 0:90, FA; 3 ¼ 0:93, m1 ¼ 0:3, m2 ¼ 0:3, m3 ¼ 0:4, ðr1 r2 Þ1 ¼ ðr1 r2 Þ2 ¼ ðr1 r2 Þ3 ¼ 1.

erogeneous Ziegler–Natta catalysts. These two figures illustrate a very important aspect of olefin polymerization with heterogeneous Ziegler–Natta catalysts, the fact that the sites that make polymer chains with higher number-average chain lengths also have a lower reactivity ratio toward comonomer incorporation. This seems to be a universal property of heterogeneous Ziegler–Natta catalysts and is dramatically illustrated in Figure 8.26 for the bivariate distribution of chain length and chemical composition of a model polymer. Much research has been done to overcome this phenomenon, which can very often be seen as a limitation of heterogeneous Ziegler–Natta catalysts, since low molecular weight chains with high aolefin fractions are easily extracted from the polymer particles and may cause reactor fouling, a high extractable content in films – a serious limitation for food and medical applications – and odor problems during processing. The multiplicity of active sites on heterogeneous Ziegler–Natta and Phillips catalysts is a truly complex phenomenon and other explanations proposed to describe the observed broad chain length and chemical composition distributions have also been proposed, but most rely on similar approaches to the one described herein. The references at the end of the chapter give more information on this topic [40, 41]. 8.3.4

Long-chain Branch Formation

The mechanism of long-chain branch (LCB) formation with coordination polymerization catalysts is terminal branching. In this mechanism, a dead polymer chain containing a terminal unsaturation – generally a vinyl group – is copolymerized with a growing polymer chain to form an LCB (Figure 8.27) [15]. We have already seen that dead polyethylene chains with terminal unsaturations will be formed by

395


Pr+s,4

Pr,2

CH

2

Zr

=

B

2

Zr

CH

B

=

396

Ds,1= Fig. 8.27.

Mechanism of long-chain branch formation with coordination catalysts.

b-hydride elimination and transfer to monomer reactions, as described by Eqs. (4) and (8), respectively. Since these chains can be reincorporated into the growing polymer chains via LCB-formation reactions, they are frequently called macromonomers. Macromonomers can be made in situ via b-hydride elimination and transfer to monomer reactions or they can be added, as an additional comonomer type, to the polymerization reactor. In this way, long-chain branch formation can be seen as a copolymerization reaction with a very long a-olefin and, as expected, catalysts that have a high reactivity ratio toward a-olefin incorporation can also form polymers with high LCB contents. Several metallocene catalysts and some homogeneous Ziegler–Natta catalysts are effective in forming polymer chains with LCBs. In contrast, heterogeneous Ziegler–Natta catalysts make only linear polyolefins. The polymerization mechanism described by Eqs. (1)–(14) for homopolymerization needs to be augmented by only one additional equation, Eq. (29), to include long-chain branch formation. kLCB

Pr; i þ D¼ s; j ! Prþs; iþjþ1

ð29Þ

In Eq. (29), the subscripts i and j indicate the number of LCBs per chain, while the subscripts r and s are their respective chain lengths. Therefore, when a linear growing chain (i ¼ 0) reacts with a linear macromonomer ( j ¼ 0), a chain with one LCB is formed (i þ j þ 1 ¼ 1). Similar equations are easily developed for copolymerization. The weight distribution of chain length for polymer populations containing i LCBs per chain, wðr; iÞ is given by Eq. (30) [42], in which the parameter t is given by Eq. (31). wðr; iÞ ¼ t¼

1 r 2iþ1 t 2iþ2 expðtrÞ ð2i þ 1Þ!

rate of chain transfer þ rate of LCB formation rate of propagation

ð30Þ ð31Þ

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts 0.0004 Linear 0.00035

0.0003

w (r , y )

0.00025 1 LCB 0.0002 2 LCB 3 LCB

0.00015

4 LCB 5 LCB

0.0001

0.00005

0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

r Chain length distributions of the several polymer populations of branched polyolefins made with a single-site coordination catalyst (1=t ¼ 1000). Fig. 8.28.

Therefore, in the absence of LCB formation, t ¼ 1=rn , and Eq. (30) reduces to Flory’s most probable distribution, Eq. (15). Figure 8.28 shows the chain length distribution for several polymer populations with increasing numbers of LCBs per chain. As expected, the chain length average increases with an increasing number of LCBs per chain. An analytical solution for the instantaneous chain length distribution for the whole polymer produced in a CSTR is also available [Eq. (32)] [15]. The function I1 is the modified Bessel function of the first kind of order 1, given by Eq. (33). Bessel functions are easily found in mathematical tables and are readily available is most scientific software applications [43, 44]. wðrÞ ¼

I1 ðxÞ ¼

pffiffiffi ð1 aÞt expðrtÞ rt a pffiffiffi I1 2 1þa ð1 þ aÞ a y X ðx=2Þ 12k k!Gðk þ 2Þ k¼0

ð32Þ

ð33Þ

The parameter a is defined by Eq. (34), where f ¼ is the molar fraction of macromonomer in the reactor, measured with respect to the total concentration of polymer; s is the reciprocal of the average reactor residence time; kLCB is the rate constant for LCB formation; and Y is the concentration of growing polymer chains in the reactor.

397

8 Coordination Polymerization 0.00035

0.0003

α = 0.1 0.00025

0.0002

w (r )

398

0.00015

0.0001

α = 0.3

0.00005

α = 0.5

0 0

5000

10000

15000

20000

25000

30000

35000

40000

r

Fig. 8.29. Overall chain length distribution for branched polyolefins made with a single-site coordination catalyst (1=t ¼ 1000).

a¼

f¼ 1 þ s=ðkLCB YÞ

ð34Þ

Figure 8.29 shows how the chain length distribution of the whole polymer is affected by the value of the parameter a. First, notice that a belongs to the interval [0, 1]. All chains are linear when a ¼ 0 since this implies that either f ¼ ¼ 0 (without macromonomers, no LCBs can be formed) or s=ðkLCB YÞ ! y. The latter condition is obeyed when either s ! y (that is, the reactor residence time tends to zero) or kLCB Y ¼ 0. Both cases imply that no macromonomers are accumulated in the reactor. On the other hand, LCB formation is maximal when a ¼ 1, a condition obeyed only when all dead polymer chains in the reactor contain terminal unsaturations, f ¼ ¼ 1, and the residence time in the reactor is infinite, s ! 0, or the rate of LCB formation is infinite, kLCB Y ! y. Therefore, Eq. (32) captures, in a very elegant way, all the factors determining LCB formation with a coordination catalyst. The parameter a can also be related to the number of LCBs per chain for the whole polymer, B, by Eq. (35). B¼

a 1a

ð35Þ

8.4 Single Particle Models – Mass- and Heat-transfer Resistances

Notice that the average number of LCBs per chain can vary from 0 when a ¼ 0 to infinity when a ¼ 1. Since most long-chain-branched polyolefins made with coordination catalysts are only sparsely branched, with values of B rarely exceeding unity, the upper limit of the paramter a should be considered only as a theoretical possibility never to be reached in practical situations. Chain length averages and polydispersity index for the whole polymer can also be related to the parameters a or B via Eqs. (36)–(38). 1 r n ¼ ð1 þ 2BÞ t

ð36Þ

2 r w ¼ ð1 þ 2BÞð1 þ BÞ t

ð37Þ

PDI ¼ 2ð1 þ BÞ

ð38Þ

These equations demonstrate that the polydispersity index of long-chain-branched polyolefins is always greater than 2 and that the chain length averages increase with an increasing number of LCBs per chain. It is also interesting to calculate the mass fraction of polymer populations containing i LCBs per chain, m i , by Eq. (39). mi ¼

ð2iÞ! a i ð1 aÞð2i þ 1Þ i!ði þ 1Þ! ð1 þ aÞ 2iþ2

ð39Þ

Notice that, for sparsely branched polymers, most of the chains are linear, but the number of more highly branched species increases as a ! 1, as illustrated in Figure 8.30. Similarly, an extension [Eq. (40)] of Stockmayer’s distribution can be derived for binary copolymers containing LCBs formed by terminal branching [45]. wðr; y; iÞ ¼

rffiffiffiffiffiffiffiffi 1 r ry 2 r 2iþ1 t 2iþ2 expðrtÞ exp ð2i þ 1Þ! 2pk 2k

ð40Þ

These equations give a very accurate portrait of the chain microstructure of these polyolefins. They are, in fact, a window into their chain architecture and can be very useful in understanding the constitution of these complex polymers.

8.4

Single Particle Models – Mass- and Heat-transfer Resistances

Most commercial processes for the manufacture of polyolefins use solid catalysts, such as heterogeneous Ziegler–Natta and Phillips catalysts. Many metallocene catalysts have also been supported on inorganic carriers, typically silica, for industrial

399

8 Coordination Polymerization 1 α = 0.05 α = 0.1

0.9

α = 0.2 α = 0.4 α = 0.8

0.8 0.7 0.6

mi

400

0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

i (LCB/chain) Fig. 8.30.

Mass fraction of chains with i LCB per chain, as a function of the parameter a.

use [24]. As in any solid-catalyzed reaction, interparticle and intraparticle massand heat-transfer resistances may become a limiting step in heterogeneous olefin polymerization processes. Olefin polymerization with heterogeneous catalysts has several very distinct characteristics that should be discussed from a qualitative point of view before a more quantitative treatment is presented. Effective heterogeneous catalysts for olefin polymerization are highly porous, as is usual with most heterogeneous catalysts. When a fresh catalyst particle is first fed to the reactor it is evidently free of polymer, but its pores quickly become filled with polymer molecules formed as monomer diffuses from the bulk phase in the reactor to the surface of the active sites on the catalyst pores. At this point, the three alternatives illustrated in Figure 8.31 are possible: (1) the polymer phase clogs the catalyst pores and inhibits any additional polymerization from happening; (2) the structure of the catalyst is not strong enough to resist the expansion forces of the fast-forming polymer chains and ‘‘explodes’’ in several smaller particles, generating fines; (3) the growing polymer chains are capable of fragmenting the catalyst particle in an ordered way and act as a binder between the catalyst fragments, forming an expanding polymer particle composed of polymer chains surrounding catalyst fragments. The last alternative is the only one that has industrial interest; much catalyst research is behind the design of catalyst particles that can be properly fragmented to form uniform polymer particles. This leads to the so-called ‘‘replication phenomenon’’, whereby the size distribution of the catalyst particles is neatly replicated by the size distribution of the polymer particles exiting the reactor, as illustrated in


catalyst particle with pores clogged with polymer

catalyst fragmentation and formation of fines

desired expanding catalyst-polymer particle

Fig. 8.31. Particle growth and fragmentation for polymerization with heterogeneous coordination catalysts.

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

401


Polymerization

Catalyst

Polymer

Cumulative %

402

Particle size Replication phenomenon for polymerization with heterogeneous coordination catalysts. Fig. 8.32.

milder conditions is generally recommended to avoid the formation of intraparticle hot spots and the improper fragmentation of the catalyst particles when subjected to the more severe polymerization conditions existing in industrial polymerization reactors. Several older and some more recent simulation studies deal with these initial instants of polymerization, but they are beyond the scope of this chapter. These initial particle-fragmentation models are more interesting for prepolymerization studies, since the amount of polymer made during this very short initial stage is unlikely to have a marked influence on the overall properties of the

Secondary Particle or Macroparticle

Primary Particle or Microparticle

Catalyst fragment Fig. 8.33.

The multigrain model.

Growing polymer shell


polymer produced in the reactor. A 2001 review covers some of these alternative models [36]. The multigrain model equation for spherical secondary particles is the classic diffusion-reaction equation in a sphere, Eq. (41), where Ms is the monomer concentration in the secondary particle as a function of polymerization time, t, and radial position, rs . qMs 1 q qMs ¼ 2 Deff rs2 RVp rs qrs qt qrs

ð41Þ

The effective diffusivity in the secondary particle, Deff , can be estimated using the conventional expression for effective diffusivity in porous heterogeneous catalysts, Eq. (42), where Db is the monomer bulk diffusivity in the reaction medium, and e and t are the void fraction and tortuosity of the polymer particle, respectively. The fact that both e and t are likely to vary as a function of the degree of fragmentation and expansion of the secondary particle is certainly one of the difficulties in getting a good estimate for Deff . Deff ¼

eDb t

ð42Þ

Finally, RpV is the average volumetric rate of polymerization in the secondary particle at a given radial position. Since, in the multigrain model, the polymerization is assumed to take place only at the surface of the primary particles, this term couples the models for the primary and secondary particles. Equation (41) is subjected to the following boundary conditions given by Eqs. (43) and (44), where Rs is the radius of the secondary particle, ks is the mass-transfer coefficient in the external film surrounding the secondary particle, and Mb is the monomer concentration in the bulk phase. Equations (43) and (44) are the classic boundary conditions for heterogeneously catalyzed chemical reactions, namely symmetry at the center of the particle and stationary convective mass transfer through the mass-transfer boundary layer surrounding the particle, respectively. Finally, the initial condition is given by Eq. (45). qMs ðrs ¼ 0; tÞ ¼ 0 qrs Deff

qMs ðrs ¼ Rs ; tÞ ¼ ks ðMb Ms Þ qrs

Ms ðrs ; t ¼ 0Þ ¼ Ms0

ð43Þ ð44Þ ð45Þ

The initial concentration in the secondary particle, Ms0 , may be set to zero for a monomer-free catalyst condition, but this generally leads to stiff differential equations that may be very hard to solve. It is also common to assume a pseudo-steady-

403

404


state concentration at t ¼ 0 to obtain the initial condition for Eq. (41). Unless one is interested in the monomer profiles for the very first seconds of polymerization, this approximation generally leads to a system of partial differential equations that is simpler to integrate. Spherical primary particles are modeled with a similar equation [Eq. (46)]. qMp 1 q 2 qMp Dp rp ¼ 2 rp qrp qt qrp

ð46Þ

In Eq. (46), the subscript p refers to values in the primary particles. Equation (47) has been suggested to estimate the effective diffusivity of monomer in the primary particle, where Da is the diffusivity of monomer in amorphous polymer and w and i are correction factors to account for the decrease in diffusivity due to chain crystallinity and immobilization of the polymer amorphous phase due to the crystallites. As can be very well imagined, these parameters are also hard to determine and Dp is generally used as an adjustable parameter in the model. Dp ¼

Da wi

ð47Þ

Notice that Eq. (46) does not contain a polymerization reaction term. Because the multigrain model assumes that polymerization takes place at the surface of the catalyst fragment embedded within the primary particle, the reaction term appears as one of the two required boundary conditions [Eq. (48)]. Equation (48) states that the rate of monomer diffusion at the surface of the catalyst fragment, R c , equals the rate of monomer consumption due to polymerization at rate of Rpc , and Eq. (49) imposes the condition that the concentration at the surface of the primary particle equals the equilibrium concentration of monomer absorbed onto the polymer phase, Meq . 4pR 2c Dp

qMp 4 ðrp ¼ R c ; tÞ ¼ pR 3c R pc 3 qrp

Mp ðrp ¼ R p ; tÞ ¼ Meq a Ms

ð48Þ ð49Þ

The equilibrium concentration of monomer in polymer can be related by Eq. (50) to the monomer concentration in the secondary particle at a given radial position and time if a partition coefficient, KMP , between the two phases is known. Meq ¼

Ms KMP

ð50Þ

The polymerization rate at the surface of the catalyst fragment is given by Eq. (51), where C is the concentration of active sites at the surface of the catalyst fragment.

8.4 Single Particle Models – Mass- and Heat-transfer Resistances Tab. 8.2. Temperature profiles in the primary and secondary particles according to the multigrain model.

Secondary particle

Primary article

qTs 1 q qTs ¼ 2 rp Cp ke rs2 þ ðDHp ÞRpV rs qrs qt qrs qTs ðrs ¼ 0; tÞ ¼ 0 qrs qTs ke ðrs ¼ Rs ; tÞ ¼ hðTb Ts Þ qrs Ts ðrs ; t ¼ 0Þ ¼ Ts0

qTp qTp 1 q ¼ 2 ke rp2 rp qrp qt qrp 4 2 qTp 4pRc ke ðrp ¼ R p ; tÞ ¼ ðDHp Þ pRc3 3 qrp rp Cp

Tp ðrp ¼ R p ; tÞ ¼ Ts Tp ðrp ; t ¼ 0Þ ¼ Tp0

R pc ¼ k p C Mðrp ¼ R c ; tÞ

ð51Þ

Finally, the initial condition for the primary particle is stated in Eq. (52). Mp ðrp ; t ¼ 0Þ ¼ Mp0

ð52Þ

Once again, a pseudo-steady-state approximation may be adopted to reduce the stiffness of the system of differential equations for short polymerization times. The multigrain model also includes a set of equations to describe the temperature profiles in the primary and secondary particles. These equations are summarized in Table 8.2. Mathematical models for solving this system of partial differential equations with moving boundaries are involved and have been discussed in the literature [36, 51–60]. This versatile model has been used extensively to describe polymerization with heterogeneous Ziegler–Natta catalysts. Although it is difficult to make general statements for such complex systems, it can be said with confidence that most of the mass- and heat-transfer resistances will take place at the beginning of the polymerization when the concentration of active sites on the secondary particles is at its maximum. As polymer is formed, it pushes apart the active sites in what can be visualized as a dilution effect. In this way, as the secondary particle grows, the catalyst concentration decreases, and naturally intraparticle mass- and heattransfer effects become less prevalent. For the same reason, particle hot spots are more likely to be observed with highly active catalyst particles at the beginning of the polymerization. This effect is highly undesirable since it may lead to softening of the polymer phase and result in particle agglomeration and severe reactor fouling, especially in gas-phase reactors. The equations presented so far for the multigrain model are mass- and energybalance equations in a spherical catalyst particle used for conventional heterogeneously catalyzed reactions subjected to a moving boundary due to polymer formation. To predict polymer properties such as chain length and chemical composition, these monomer and temperature profiles must be coupled with an additional set of equations that describes polymerization and termination mechanisms

405

406


taking place on the surface of the catalyst. The method of moments is generally the preferred technique used in conjunction with the multigrain model, but its discussion will be deferred to Section 8.5. Instead, we will use Stockmayer’s bivariate distribution, Eq. (16), to illustrate how polymer properties can be conveniently predicted from the multigrain model. First, it should be remembered that Stockmayer’s distribution is an instantaneous distribution; that is, it describes the distributions of chain length and chemical composition for polymer made at a given instant in time at a given spatial location in the reactor. Now, consider the polymerization of ethylene and an a-olefin, 1-hexene for instance, taking place in a spherical porous catalyst particle. Hydrogen is used as the chain-transfer agent. Assume also that the catalyst has only one type of active site, as would be observed when supporting a metallocene on a porous silica particle, for instance. The primary and secondary particles at a given instant in time are subject to mass-and heat-transfer resistances that result in radial monomer concentration and temperature profiles. Assuming that ethylene propagates at a much higher rate than 1-hexene and that both have comparable diffusivities, the radial profile for ethylene will be much steeper than for 1-hexene. Similarly, the radial profile for hydrogen can be assumed to be very flat, since the hydrogen diffusivity is high and its reaction rate is low. (Notice that a low reaction rate of the chain-transfer agent as compared to the polymerization rate for the monomers is a requirement for the production of high molecular weight polymers.) Given that the monomer concentration and temperature affect the parameters of the Stockmayer distribution, rn =k, each radial position i in the secondary particle is associated with a unique Stockmayer’s distribution, wi ðr; yÞ, as depicted in Figure 8.34. Polymer richer in the slow reacting comonomer, 1-hexene, is produced near the center of the particle because the molar ratio of 1-hexene/ethylene increases from the surface to the center of the particle. Likewise, chains with lower molecular weight are produced at the center of the particle because the molar ratio of hydrogen/(ethylene þ 1-hexene) increases from the surface to the center of the particle. These concentration gradients will, therefore, broaden the distributions of molecular weight and chemical composition of polymer made with a heterogeneous catalyst, even a single-site catalyst, due to the spatial variations of concentrations within the particle. The summation of all these distributions over the polymer particle, weighted by the amount of polymer made at each radial position, gives the distribution for the whole particle at a given instant in time wp ðr; yÞ, as described by Eq. (53), where Rp; i is the rate of polymerization at radial position i and m is the number of radial positions used in the discretization of the macroparticle. m X

wp ðr; yÞ ¼

Rp; i wi ðr; yÞ

i¼1 m X i¼1

ð53Þ Rp; i


w (fraction of ethylene)

30

407

1.2

25

1

4

15

0.8

w (r )

20

3

10

1

0.2

0 0.7

0.72

0.74

4 3 2

0.4

2

5

0.6

0.76

0.78

0.8

1

0

0.82

1

fraction of ethylene

2

3

4

log r

12 3 4

H2 C6H12 C2H4 Fig. 8.34. Using Stockmayer’s distribution with the multigrain model to predict the distribution of chain length and chemical composition of polyolefins.

These instantaneous distributions can then be integrated in time to obtain the cumulative distribution in the reactor per polymer particle, Wp ðr; yÞ [Eq. (54)]. m ð X Rp; i wi ðr; yÞ dt Wp ðr; yÞ ¼

i¼1

m ð X

ð54Þ Rp; i dt

i¼1

For the case of catalysts containing multiple-site types, such as heterogeneous Ziegler–Natta catalysts, a similar approach applies, by defining one Stockmayer’s distribution for each active-site type. In this case, the overall distribution of chain length and chemical composition in the particle at a given instant equals the summation of the distributions over all site types and all radial positions in the particle [Eq. (55)], where the subscript j indicates the site type of a catalyst containing n site types.

5

408

8 Coordination Polymerization m X n X

wp ðr; yÞ ¼

Rp; i; j wi; j ðr; i¼1 j¼1 m X n X

yÞ ð55Þ

Rp; i; j

i¼1 j¼1

Many other single-particle model formulations exist with lower or higher levels of sophistication. Most of these models generate qualitative results that are similar to the ones described in this section. As usual with any modeling effort, the degree of model sophistication must be justified by the quality of the experimental data available to support the model assumptions. Our recent (2001) review gives coverage of these alternative models [36].

8.5

Macroscopic Reactor Modeling – Population Balances and the Method of Moments

Population balances coupled with the method of moments can be considered the method of choice in most olefin polymerization simulation models. Population balances are molar balances (steady-state or dynamic) of all the important chemical species present in the reactor: living and dead polymer chains, catalyst sites, monomers, and chain-transfer agents. Solving the dynamic population balances for living and dead chains of length r allows the recovery of the complete chain length distribution as a function of polymerization time. Alternatively, instead of solving the whole population balance, one may use the method of moments to solve for only a few moments of the chain length distribution, a technique that requires much less computational effort. In this case, it is possible to model how chain length averages vary as a function of polymerization time, but the information about the complete distribution is irretrievably lost for more complex cases. In this section, we will first illustrate how to use the method of moments for homopolymerization and then show how these equations can be easily adapted to copolymerization using the method of pseudo-kinetic constants. 8.5.1

Homopolymerization

In the following model development, we will use the polymerization kinetics mechanism described by Eqs. (1)–(14) to derive the population balances and moment equations for homopolymerization with a catalyst containing only one site type. Catalysts containing two or more site types are handled similarly by defining a set of equations with distinct polymerization kinetic constants for each different site type. For living polymer chains of length r b 2 made in a CSTR, the dynamic population balance can be derived as Eq. (56).

8.5 Macroscopic Reactor Modeling – Population Balances and the Method of Moments

dPr ¼ Prin þ k p MðPr1 Pr Þ ðk tb þ k tH H2 þ k tAl Al þ k tM M þ kdI I þ kd þ sÞPr dt ð56Þ In Eq. (56) and in the subsequent ones, the notation x in represents the feed flow rate of a given chemical species to the reactor, and s is the reciprocal of the average residence time in the CSTR. Notice that Eq. (56) is simply the molar balance for chains of length r: chains are generated by propagation of a chain of length r 1 or by transfer from a feed stream (coming from a previous reactor in a series of reactors, for instance), and are consumed by either propagation to chains of length r þ 1, by transfer and deactivation reactions leading to dead polymer chains, Dr , or by exiting the reactor in an outlet stream. Similarly, for chains of unit length, Eq. (57) applies. dP1 ¼ P1in þ ðk i C þ k iH CHi ÞM k p P1 M dt ðk tb þ k tH H2 þ k tAl Al þ k tM M þ kdI I þ kd þ sÞP1

ð57Þ

The population balances for C and CH are needed to solve Eqs. (56) and (57). P They are given in Eqs. (58) and (59) respectively, where Y 0 ¼ y r ¼1 Pr . dC ¼ C in þ ka CAl þ ðk tAl Al þ k tM MÞY 0 ðk i M þ kdI I þ kd þ sÞC dt

ð58Þ

dCH in ¼ CH þ ðk tb þ k tH HÞY 0 ðk iH M þ kdI I þ kd þ sÞCH dt

ð59Þ

Often, the reaction between catalyst and cocatalyst is considered instantaneous. Consequently, the term ka CAl can be dropped from Eq. (58) and it is assumed that C ðt 0 Þ ¼ Cðt 0 Þ. The concentration of deactivated catalyst sites at any time is obtained via a molar balance [Eq. (60)]. Cd ðtÞ ¼ C ðt 0 Þ C ðtÞ CH ðtÞ Y 0 ðtÞ

ð60Þ

Population balances for dead polymer chains are also easily derived from the polymerization mechanism equation, Eq. (61). ^ r dDr dD¼ dDr; Al dD ¼ þ r þ dt dt dt dt ^r ^ in þ ½ðk tH H2 þ kdI I þ kd Þ þ ðk tb þ k tM MÞ þ k tAl AlPr sD ¼D r

ð61Þ

409

410


In Eq. (61), dead chains with all types of chain ends (Dr ; D¼ r , and Dr; Al ) have been ^ r for simplicity. Separate population balances for combined in a single variable D dead chains with different chain end types could also be kept, if necessary. Molar balances for monomer, chain-transfer agent, impurities, catalysts, and cocatalyst [Eqs. (62)–(66)] are also required to solve the system of ordinary differential equations defined by Eqs. (56)–(61). dM ¼ M in ðk p Y 0 þ k tM Y 0 þ k i C þ k iH CH þ sÞM G M in ðk p Y 0 þ sÞM dt

ð62Þ

dH2 ¼ H2in ðk tH Y 0 þ sÞH2 dt

ð63Þ

dI ¼ I in ½kdI ðY 0 þ C þ CH Þ þ sI dt

ð64Þ

dC ¼ C in ðka Al þ sÞC dt

ð65Þ

dAl ¼ Al in ðka C þ k tAl Y 0 þ sÞAl dt

ð66Þ

Because of the long-chain approximation, the simplification k p Y 0 g k tM Y 0 þ k i C þ k iH CH in Eq. (62) is very often applied. Elegant ways of solving the population balances defined by Eqs. (56)–(66) have been developed to model how the complete chain length distribution of polyolefins varies as a function of polymerization time in batch, semibatch, or continuous reactors [61, 62]. When only chain length averages are required, the method of moments is the most adequate technique for solving this problem. The ith moment, mðiÞ , of a given distribution, f ðxÞ, is defined by Eq. (67). mðiÞ ¼

X

x i f ðxÞ

ð67Þ

x

Therefore, the zeroth moment is simply the total number (or concentration) of living polymer chains [Eq. (68)].

Y ð0Þ ¼

y X r ¼1

Pr ¼ P1 þ

y X

Pr

ð68Þ

r ¼2

Notice that, for coordination polymerization, Y ð0Þ will be approximately equal to the number of active sites at a given time in the reactor, since initiation reactions tend to be very fast in these systems.


Taking the first derivative of Eq. (68), one obtains Eq. (69). y dY ð0Þ dP1 X dPr ¼ þ dt dt dt r ¼2

ð69Þ

Substituting Eqs. (56) and (57) into Eq. (69) generates the final expression for the zeroth moment of living polymer chains in a CSTR. After some algebraic manipulation, Eq. (70) is obtained. dY ð0Þ ¼ Y ð0Þin þ K i M ðK t þ K d þ sÞY ð0Þ dt

ð70Þ

Here the several kinetic parameters were lumped into the constants K t ; K d , and K i according to Eqs. (71)–(73), to allow for a more concise notation. K t ¼ kbt þ k tH H2 þ k tM M þ k tAl Al

ð71Þ

K d ¼ kd þ kdI I

ð72Þ

K i ¼ k i C þ k iH CH

ð73Þ

Similarly, the first moment of the living chains corresponds to the weight of these chains [Eqs. (74) and (75)].

Y ð1Þ ¼

y X

rPr ¼ 1 P1 þ

y X

rPr

ð74Þ

r ¼2

r ¼1 y dY ð1Þ dP1 X dPr ¼ þ r dt dt dt r ¼2

ð75Þ

Once again, substituting Eqs. (56) and (57) into Eq. (75) leads to the final expression for the first moment of living polymer chains in a CSTR, Eq. (76). dY ð1Þ ¼ Y ð1Þin þ K i M ðK t þ K d þ sÞY ð1Þ þ k p MY ð0Þ dt

ð76Þ

The equations for the second moment of living polymer, Eqs. (77)–(79), are derived in an analogous manner. Y ð2Þ ¼

y X r ¼1

r 2 Pr ¼ 1 2 P1 þ

y X r ¼2

r 2 Pr

ð77Þ

411

412

8 Coordination Polymerization y dY ð2Þ dP1 X dPr ¼ þ r2 dt dt dt r ¼2

ð78Þ

dY ð2Þ ¼ Y ð2Þin þ K i M ðK t þ K d þ sÞY ð2Þ þ k p Mð2Y ð1Þ þ Y ð0Þ Þ dt

ð79Þ

A similar procedure leads to the moment equations for the chain length distribution of dead polymer molecules [Eqs. (80) and (81)].

X ðiÞ ¼

y X

r i Pr

ð80Þ

r ¼1 y dX ðiÞ X dXr ¼ ri dt dt r ¼2

ð81Þ

Substituting Eq. (61) into Eq. (81) produces the general expression for the ith moment of the dead polymer chains, Eq. (82). dX ðiÞ ¼ X ðiÞin þ ðK t þ K d ÞðY ðiÞ P1 Þ sX ðiÞ dt

ð82Þ

Notice that P1 is subtracted from Y ðiÞ for exactness since ‘‘dead chains’’ of length 1 are simply monomer units and should not be counted as dead polymer chains. This correction, however, is negligible for high polymers and can be omitted for most modeling applications. Equation (70), (76), (79), and (82) can be solved with the molar balances for the reactants, Eqs. (62) to (66), to calculate the leading moments of the chain length distribution. The values of the moments, as a function of polymerization time, can then be used to compute the chain length averages with the expressions described below. The number-average chain length, rn , is easily related to the zeroth and first moments of the distributions of chain length of living and dead polymer by Eq. (83). y X

rn ¼

rðDr þ Pr Þ

r ¼1 y X

¼ ðDr þ Pr Þ

X ð1Þ þ Y ð1Þ X ð1Þ G ð0Þ ð0Þ ð0Þ X þY X

ð83Þ

r ¼1

Since, for most coordination polymerizations, the amount of dead polymer far exceeds the amount of living polymer in the reactor, the approximation indicated in Eq. (83) is very commonly used.


The weight-average chain length, rw , is likewise obtained from the ratio of the second to the first moments of living and dead chains, obtained from Eq. (84). y X

rw ¼

r 2 ðDr þ Pr Þ

r ¼1 y X

¼ rðDr þ Pr Þ

X ð2Þ þ Y ð2Þ X ð2Þ G ð1Þ ð1Þ ð1Þ X þY X

ð84Þ

r ¼1

Finally, the polydispersity index, PDI, is given by Eq. (85).

PDI ¼

rw ðX ð2Þ þ Y ð2Þ ÞðX ð0Þ þ Y ð0Þ Þ X ð2Þ X ð0Þ ¼ G 2 rn ðX ð1Þ þ Y ð1Þ Þ ðX ð1Þ Þ 2

ð85Þ

For the case of multiple-site catalysts, population balances are derived for each catalyst site type, and the chain length-averages for the whole polymer are found by averaging the values calculated for each site type [Eqs. (86) and (87)]. n X

rn ¼

ð1Þ

ðXj

j¼1 n X

n X

ð0Þ

j¼1 n X

G ð0Þ

ðXj

rw ¼

ð1Þ

þ Yj Þ þ Yj Þ

j¼1

j¼1

n X ð2Þ ð2Þ ðXj þ Yj Þ

n X

j¼1 n X

j¼1 n X

G ð1Þ

ðXj

ð1Þ

þ Yj Þ

j¼1

ð1Þ

Xj

ð86Þ ð0Þ

Xj

ð2Þ

Xj

ð87Þ ð1Þ

Xj

j¼1

Population balances and the method of moments can also be combined with the multigrain model and other polymer particle growth models. In this case, the population balances are defined for each position in the particle to obtain the radial profiles of chain length averages [36, 51–60]. 8.5.2

Copolymerization

Population balances for copolymerization can be developed using the polymerization kinetics presented in Table 8.1. This approach generates equations that are similar to the ones obtained for homopolymerization but contain more terms to account for the effect of chain ends on the kinetics of propagation and termination.

413

414


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

ð90Þ

Applying the definitions in Eqs. (90)–(93) to Eq. (90), one obtains Eq. (94). fA ¼

Pr; A ; Pr; A þ Pr; B

fA ¼

A ; AþB

M ¼AþB

fB ¼ 1 fA

fB ¼ 1 fA

ð91Þ

ð92Þ ð93Þ


dPr ¼ Prin þ ðk p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB ÞPr1 M dt ðk p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB ÞPr M ðk tb; A fA þ k tb; B fB ÞPr ðk tH; A fA þ k tH; B fB ÞPr H2 ðk tAl; A fA þ k tAl; B fB ÞPr Al ðk t; AA fA fA þ k t; AB fA fB þ k t; BA fB fA þ k t; BB fB fB ÞPr M ðkdI; A fA þ kdI; A fB ÞPr I ðkd; A fA þ kd; B fB ÞPr sPr

ð94Þ

Equation (94) can be expressed in the more compact form of Eq. (95), where the pseudo-kinetic constants are defined by Eqs. (96)–(102). dPr ¼ Prin þ k^p MðPr1 Pr Þ ðk^tb þ k^tH H2 þ k^tAl Al þ k^tM M þ k^dI I þ k^d þ sÞPr dt ð95Þ k^p ¼ k p; AA fA fA þ k p; AB fA fB þ k p; BA fB fA þ k p; BB fB fB

ð96Þ

k^tb ¼ k tb; A fA þ k tb; B fB

ð97Þ

k^tH ¼ k tH; A fA þ k tH; B fB

ð98Þ

k^tAl ¼ k tAl; A fA þ k tAl; B fB

ð99Þ

k^tM ¼ k t; AA fA fA þ k t; AB fA fB þ k t; BA fB fA þ k t; BB fB fB

ð100Þ

k^dI ¼ kdI; A fA þ kdI; A fB

ð101Þ

k^d ¼ kd; A fA þ kd; B fB

ð102Þ

Notice that Eqs. (56) and (95) are equivalent, with the only difference that Eq. (95) uses pseudo-kinetic constants in place of the actual kinetic constants found in Eq. (56). The beauty of this modeling approach is that it is not necessary to develop new equations for copolymerization (binary or higher): the equations developed for homopolymerization, including the moment equations, are equally applicable to copolymerization, provided that pseudo-kinetic constants are used to replace the actual polymerization kinetic constants. To calculate the pseudo-kinetic constants one must know the values of fA and fA at each polymerization time. Values for fA are easily calculated from the molar balance for the monomers, Eq. (103). dA ¼ Ain ðk i C þ k i; H CH ÞA ðk p; AA fA þ k p; BA fB ÞAY 0 dt ðk t; AA fA þ k t; BA fB ÞAY 0 sA

ð103Þ

Since most of the monomer is consumed in polymerization reactions, Eq. (103) is commonly reduced to the simpler form of Eq. (104).

415

416


dA ¼ Ain ðk p; AA fA þ k p; BA fB ÞAY 0 sA dt

ð104Þ

The analogous equation for monomer B is Eq. (105). dB ¼ B in ðk p; BB fB þ k p; AB fA ÞBY 0 sB dt

ð105Þ

The long-chain approximation can be used to calculate the values of fA via Eqs. (106)–(108). k p; AB Pr; A B ¼ k p; BA Pr; B A

ð106Þ

and therefore: k p; AB fA fB ¼ k p; BA fB fA ¼ k p; BA ð1 fA Þ fA fA ¼

k p; BA fA k p; BA fA þ k p; AB fB

ð107Þ ð108Þ

The use of Eq. (108) to calculate fA assumes that this value is not a function of chain length; compare Eq. (91). It has been shown that this hypothesis is valid for high polymers [63, 64]. Finally, the average copolymer composition [Eq. (109)] is easily obtained from Eqs. (104) and (105). FA ¼

A ; AþB

FB ¼ 1 FA

ð109Þ

8.6

Types of Industrial Reactors

The polymerization of olefins with coordination catalysts is performed in a large variety of polymerization processes and reactor configurations that can be classified broadly into solution, gas-phase, or slurry processes. In solution processes, both the catalyst and the polymer are soluble in the reaction medium. These processes are used to produce most of the commercial EPDM rubbers and some polyethylene resins. Solution processes are performed in autoclave, tubular, and loop reactors. In slurry and gas-phase processes, the polymer is formed around heterogeneous catalyst particles in the way described by the multigrain model. Slurry processes can be subdivided into slurry–diluent and slurry–bulk. In slurry–diluent processes, an inert diluent is used to suspend the polymer particles while gaseous (ethylene and propylene) and liquid (higher a-olefins) monomers are fed into the reactor. On the other hand, only liquid monomer is used in the slurry–bulk pro-

8.6 Types of Industrial Reactors

(d) (a) (b) (c)

(e)

Reactor configurations used with olefin polymerization: (a) Autoclave; (b) Loop; (c) Fluidized-bed; (d) Vertical gas-phase; (e) Horizontal gas-phase. Fig. 8.35.

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

417

418

8 Coordination Polymerization unreacted monomer

catalyst cocatalyst

offgas

N2 Reactor 1

Reactor 2 powder silo polymer pellets

propylene

propylene

ethylene

ethylene

hydrogen

hydrogen

Fig. 8.36.

N2 Extruder

Example of a gas-phase tandem reactor process.

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


Product from reactor 1

Product from reactors 1 and 2 (reactor blend)

catalyst and cocatalyst monomers and chain transfer agents monomers and chain transfer agents

Fig. 8.37.

Production of reactor blends in tandem reactors.

catalyst particles is uniform, the fractional polypropylene content in the ethylene– propylene copolymer is exactly the same in each polymer particle exiting Reactor 2. Figure 8.38 illustrates this situation. A very different picture emerges when using two CSTRs in series. Because the residence time distribution of an ideal CSTR, EðtÞ, with average residence time tr , is given by the usual exponential decay equation [Eq. (110)], then some particles will leave Reactor 1 after a short time while others will only leave after spending a considerably longer time in the reactor. 1 t EðtÞ ¼ exp tr tr

419

ð110Þ

Since the same will happen in Reactor 2, in the end the ratio of polypropylene to ethylene–propylene copolymer per particle exiting Reactor 2 will also vary widely, which may be undesirable in some applications. Some of the reactor configurations shown in Figure 8.35 can reduce this phenomenon, particularly the configuration adopted for the gas-phase horizontal reactor, because the residence time distribution of this reactor is the equivalent to about three to four CSTRs in series. (Remember that the residence time of an infinite series of ideal CSTRs is that of a plug-flow reactor.) A more recent solution for this problem, in fact a completely new alternative to tandem reactor technology, is the multizone reactor that will be described in more detail below (see Section 8.6.4).

420


Fig. 8.38.

Reactor blends produced in two plug-flow reactors and two CSTRs in series.

These polymerization processes were originally designed to polymerize olefins with heterogeneous Ziegler–Natta catalysts, with the exception of some solution processes that were designed to work with homogeneous Ziegler–Natta catalysts for the production of EPDM rubbers or some types of polyethylene resins. However, heterogeneous Ziegler–Natta processes can be adapted to the use of metallocene catalysts with minimal changes if the metallocenes are supported on an inert carrier such as silica. Although some differences in particle morphology have been noticed when heterogeneous Ziegler–Natta catalysts are replaced by supported metallocenes, these Ziegler–Natta processes are also currently being used with metallocene catalysts on an industrial scale. In fact, the possibility of using metallocenes in conventional Ziegler–Natta reactors is one of the main reasons for the fast adoption of metallocenes by the polyolefin manufacturing industry. 8.6.1

Gas-phase Reactors

Gas-phase reactors for olefin polymerization are divided into two classes: fluidizedbed reactors and stirred-bed reactors. The stirred-bed reactors can be further classified into vertical and horizontal.


Gas-phase reactors, especially fluidized-bed reactors, are the most common configuration for the polymerization of ethylene to produce HDPE and LLDPE. They are also a very common choice for the second reactor in the production of highimpact polypropylene. Fluidized-bed reactors were developed by Union Carbide (Unipol Process) – currently Univation – and British Petroleum (BP). Some details in their configuration may vary, but their main characteristics are the same. In fluidized-bed reactors, gaseous monomers, chain-transfer agent, inerts, and catalysts are fed continuously into the reactor. The polymerization temperature should be kept well below the melting point of the polymer to avoid particle agglomeration, loss of fluidization, and bed collapse. Since polymer particles are formed in the gas phase in absence of diluent, there is no need for further separation steps when the product is exiting the reactor (except for removal of unconverted monomer), which is a clear advantage of gas-phase processes over slurry and solution processes. The heat of polymerization can be removed by heat exchangers placed on an external recirculation loop. However, low boiling point hydrocarbons and a-olefin comonomers can be introduced into the reactor in the liquid phase to absorb the heat of polymerization by their latent heat of evaporation in an operation procedure called condensed mode. Since most polymerization reactors are limited by their heat removal capability, this technique permits a substantial increase in reactor productivity. There are two principal designs for stirred-bed gas-phase reactors: vertical (originally developed by BASF/Targor) and horizontal (originally developed by Amoco– Chisso). These reactors have several of the advantages of fluidized-bed reactors but the polymer bed is suspended by mechanical agitation. Therefore, impeller design is of the utmost importance in these reactors to avoid reactor fouling and particle agglomeration. Stirred-bed reactors are generally smaller than fluidized-bed reactors, thus permitting grade transition with less production of off-specification material. Both vertical and horizontal designs can be operated in condensed mode to increase productivity. As mentioned above, the narrower residence time distribution of horizontal gas-phase stirred-bed reactors is advantageous for the production of reactor blends with a narrow distribution of their different components. Narrow residence time distributions are also useful if frequent grade transitions are required. Gas-phase reactors have several advantages, notably:

Good temperature control due to high turbulence and heat-transfer coefficients, and heat removal by the latent heat of vaporization of inerts and monomers. Lower operational costs due to lack of diluent recovery operations and, in the case of fluidized-bed reactors, due to the absence of moving parts. Grade flexibility for molecular weight control, since hydrogen concentration is regulated by varying the partial pressure of hydrogen in the reactor. In slurry and solution reactors, the low solubility of hydrogen in the diluent may reduce the range of possible molecular weights for a given catalyst. Higher comonomer incorporation (that is, production of copolymer with lower

421

422


crystallinity). This is possible since there is no risk of copolymer dissolution in the reaction medium. However, care should be taken not to form sticky polymer particles that can lead to particle agglomeration or reactor fouling. The narrower residence time distribution of horizontal stirred-bed reactors leads to higher yields per pass, formation of less off-specification material, and more uniform impact copolymers and reactors blends. Some disadvantages of gas-phase reactors are:

For the particular case of fluidized-bed reactors, fluidization is not a trivial process and therefore demands better process control and the design of catalyst particles that fluidize well. Severe fouling and polymer particle agglomeration can occur, with occasional reactor shutdown. Fluidized-bed reactors, in particular, are prone to the formation of polymer sheets on the walls (‘‘sheeting’’) or polymer ‘‘chunks’’ that may lead periodic interruption of reactor operation. Fluidized-bed reactors are generally very large, which makes grade transition more time-consuming and might lead to the production of significant amounts of off-specification products. 8.6.2

Slurry Reactors

Slurry processes for olefin polymerization are performed in autoclave or loop reactors. Both reactor configurations are rather old and date from the beginning of commercial olefin polymerization. Most first-generation olefin polymerization processes used autoclave reactors, while the Phillips process employed a loop reactor. Slurry–diluent processes use an inert diluent to suspend the polymer particles. Although the diluent does not directly affect the polymerization, it has been shown that different diluents might change catalyst behavior, probably due to electronic interaction with the active sites. Gaseous monomers and hydrogen are continuously bubbled through the diluent. Liquid a-olefin comonomers, diluent, catalysts, and cocatalyst are continuously fed into the reactor. Alternatively, liquefied propylene can be fed into the reactor (slurry–bulk process). Except from this difference, all other conditions are similar to the slurry–diluent process. Both autoclave and loop reactors have a residence time distribution of CSTRs (loop reactors are operated at very high recirculation ratios), so they share several of the same characteristics. There is a tendency nowadays to favor the loop reactor configuration for the production of polypropylene in slurry–bulk mode. Some advantages of slurry reactors are:

Their simplicity of design and low cost make them a common choice for laboratory-scale studies for screening catalysts and investigation of polymerization kinetics, particularly autoclave configurations. The large amount of diluent used (or liquid monomer) acts as a heat sink, per-


mits very good temperature control, and minimizes the risk of runaway polymerizations. In loop reactors, the high recirculation rate leads to high turbulence and high heat-transfer coefficients. Additionally, the high heat-transfer area available in these reactors permits efficient removal of heat of polymerization and therefore high polymer yields. However, slurry processes also have several disadvantages, such as:

It is necessary to remove the diluent from the polymer formed and recycle it back to the polymerization reactor after purification, thus increasing operational costs and environmental hazards. With Ziegler–Natta catalysts, molecular weight is generally controlled by transfer to hydrogen. Since the solubility of hydrogen in the diluent is not very high, there is less flexibility for controlling molecular weight with this type of reactor. This is not as important for Phillips catalysts, where molecular weight control is achieved via support treatment, but can become a limiting factor with Ziegler– Natta catalysts. Metallocenes are generally very sensitive to the presence of hydrogen and therefore less influenced by this reduced solubility that Ziegler–Natta catalysts. Less crystalline copolymer chains can dissolve in the diluent – particularly the ethylene–propylene copolymer fraction in high-impact polypropylene – causing fouling and increasing the viscosity of the diluent. Therefore, certain low-crystallinity grades cannot be produced with these reactors. It has been speculated that, for the case of metallocene catalysts, some of the limitations encountered with slurry reactors (both CSTR and loop) when producing copolymers of lower crystallinity can be minimized or completely eliminated. As discussed above, heterogeneous Ziegler–Natta catalysts produce LLDPE with very broad chemical composition distributions containing low-crystallinity tails. Such low-crystallinity tails are absent in most polyolefins made with metallocene catalysts, thus minimizing the risk of copolymer dissolution during polymerization in slurry reactors. 8.6.3

Solution Reactors

Solution processes use autoclave, tubular, or loop reactors. As compared to slurry and gas-phase polymerization, solution processes are commonly operated at a much higher temperature to keep the polymer dissolved in the reaction medium, and at much lower average residence times (5–20 min, as opposed to 1–4 h). Since polymerization conditions are more uniform in solutions reactors – there are no inter- and intraparticle heat- and mass-transfer resistances, for instance – this configuration is commonly used for the production of EPDM rubbers with soluble Ziegler–Natta vanadium-based catalysts. Composition homogeneity is a require-

423

424


ment of most EPDM rubbers since the formation of populations with higher crystallinity is generally not acceptable in the rubber industry. Solution reactors can be also used to produce polyethylene resins with soluble Ziegler–Natta or metallocene catalysts. The short residence time used in solution reactors because of their high operation temperature is often an advantage during grade transition. The fact that the polymer is in solution is also beneficial for process control, since solution viscosity can be used as a measure of polymer molecular weight, for instance. However, high solution viscosities are also a limiting factor for these reactors, reducing the achievable polymer concentration in solution, especially for high molecular weight resins. Solution reactors can also operate in a wider range of temperatures than slurry and gas-phase reactors, for which the temperature should be high enough

Internal gas/solid separator

RISER upward pneumatic transport

DOWNCOMER packed bed moving downward

Gas fan Product discharge Catalyst inlet

Heat exchanger Fig. 8.39.

Schematic of a multizone circulating reactor [50].

Notation

to permit high polymerization rates but not so high as to soften the polymer and cause particle agglomeration and reactor fouling. This wider temperature range allows for more flexibility in terms of catalyst types and polymer structural control. In addition, solution reactors can be used to produce polymer from very high to very low crystallinity, since there are no problems with reactor fouling caused by sticky low-crystallinity polymers. 8.6.4

Multizone Reactors

The multizone circulating reactor (see Figure 8.39) is a novel concept for propylene polymerization, developed by Basell. This reactor combines a fast fluidization reactor with a moving packed-bed reactor and can produce reactor blends in a single reactor instead of in a series of reactors [65]. The reactor operates in a cycle: polymer is transferred from the bottom of the fixed-bed section (downcomer) to the bottom of the fluidized-bed section (riser). A gas stream, containing monomers and inerts, conveys the polymer particles to the top of the riser and a centrifugal separator settles them at the top of the downer. Finally, the particles flow by gravity from the top to the bottom of the downer, where the cycle is repeated again. The polymer particles can pass through these two sections of the reactor several times before exiting the reactor. If these two (or more) zones are kept in difference polymerization conditions, a multimodal reactor blend polymer can be produced. It is claimed that because the polymer particles can be made to circulate between the different reactor zones several times before exiting the reactor, a more uniform distribution of blend components will result than in an equivalent resin made on two reactors in series. Of course, this is what would be expected from a reactor blend made in several reactors in series.

Notation

A Al B C C CCH 3 Cd CH D Da Db Deff Dp Dr

monomer type A cocatalyst or activator number of LCBs per chain for the whole polymer; monomer type B catalyst active center methylated active center deactivated site metal hydride center catalyst modifier (electron donor) diffusivity of monomer in amorphous polymer [cm 2 s1 ] monomer bulk diffusivity in the reaction medium [cm 2 s1 ] effective diffusivity in the secondary particle, [cm 2 s1 ] effective diffusivity in the primary particle, [cm 2 s1 ] dead chain with a saturated chain end

425

426


D¼ r ^r D Dr; Al EðtÞ f¼ fA FA H2 i I I1 ka kd Kd kdI ki Ki k iH kLCB KMP kp k p; ij ks Kt k tb k tb-CH3 k tAl k tH k tM m mi M Mb Meq Mp Mp0 Ms Ms0 n

dead polymer chain containing a terminal vinyl unsaturation dead polymer chains with all possible types of chain ends dead polymer chain formed via a transfer to cocatalyst reaction reactor residence time distribution molar fraction of macromonomer in the reactor, measured with respect to the total concentration of polymer molar fraction of monomer A in the reactor average fraction of comonomer A in the copolymer hydrogen number of long-chain branches per polymer chain polar impurity modified Bessel function of the first kind of order 1 rate constant for catalyst activation [s1 ] rate constant for monomolecular or bimolecular deactivation [s1 ] lumped deactivation constant, defined in Eq. (72) rate constant for deactivation by impurity [L mol1 s1 ] initiation rate constant [L mol1 s1 ] lumped initiation constant, defined in Eq. (73) initiation rate constant for a metal-hydride center [L mol1 s1 ] rate constant for LCB formation [L mol1 s1 ] monomer partition coefficient between bulk and polymer phases propagation rate constant [L mol1 s1 ] propagation rate constant for monomer type i and chain end type j [L mol1 s1 ] mass-transfer coefficient in the external film surrounding the secondary particle [cm s1 ] lumped chain-transfer constant, defined in Eq. (71) b-hydride elimination rate constant [s1 ] b-methyl elimination rate constant [s1 ] rate constant for transfer to cocatalysts [L mol1 s1 ] rate constant for transfer to hydrogen [L mol1 s1 ] rate constant for transfer to monomer [L mol1 s1 ] number of radial positions used in the discretization of the macroparticle mass fraction of polymer made on site type i, mass fraction of polymer with i long-chain branches per chain monomer monomer concentration in the bulk phase [mol L1 ] equilibrium concentration of monomer absorbed onto the polymer phase [mol L1 ] monomer concentration in the primary particle [mol L1 ] initial monomer concentration in the primary particle [mol L1 ] monomer concentration in the secondary particle [mol L1 ] initial concentration of monomer in the secondary particle [mol L1 ] number of active-site types in a multiple-site catalyst

Notation

PDI PDI Pr r rA ; rB Rc rn rn Rp; i Rpc RpV

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

defined in Eq. (34) void fraction of the polymer particle

427

428


correction factor to account for the decrease in diffusivity due to chain immobilization of the polymer amorphous phase due to the crystallites defined in Eq. (17) ith moment of a chain length distribution defined in Eq. (31); tortuosity of the polymer particle fraction of chains terminated in monomer of type i correction factor to account for the decrease in diffusivity due to chain crystallinity

i k mðiÞ t fI w

Subscripts i; j r; s

site type; number of long-chain branches chain length

Superscripts feed flow rate of a given chemical species to the reactor [mol L1 s1 ] pseudo-kinetic constant

in ^ Acronyms Cp Crystaf CSTR EPDM GPC HDPE LCB LDPE LLDPE MAO Tref

cyclopentadienyl crystallization analysis fractionation continuous stirred-tank reactor ethylene–propylene–diene rubber gel permeation chromatography high-density polyethylene long-chain branch low-density polyethylene linear low-density polyethylene methylaluminoxane temperature rising elution fractionation

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Encyclopedia of Polymer Science and Technology, 3rd Edition, Wiley-VCH, Weinheim, 2004, pp. 75–131. D. Campbell, R. A. Pethrick, J. R. White, Polymer Characterization, 2nd edition, Stanley Thornes Publishers, Cheltenham, 2000. W. W. Yau, J. J. Kirkland, D. D. Bly, Modern Size-Exclusion Liquid Chromatography: Practice of Gel Permeation and Gel Filtration Chromatography, John Wiley & Sons, New York, 1979. S. Anantawaraskul, J. B. P. Soares, P. M. Wood-Adams, Fractionation of semi-crystalline polymers by crystallization analysis fractionation (Crystaf ) and temperature rising elution fractionation (Tref ). Adv. Polym. Sci., in press. J. B. P. Soares, A. E. Hamielec, in Experimental Methods in Polymer Characterization, R. A. Pethrick (Ed.), John Wiley & Sons, Chichester, 1999, p. 15. A. Faldi, J. B. P. Soares, Polymer, 2001, 42, 3057. J. B. P. Soares, A. E. Hamielec, Polymer, 1996, 37, 4606. A. E. Hamielec, J. B. P. Soares, Prog. Polym. Sci., 1996, 21, 651. J. B. P. Soares, Macromol. Mater. Eng., 2004, 289, 70. E. Y.-X. Chen, T. J. Marks, Chem. Rev., 2000, 100, 1391. G. Henrici-Olive, S. Olive, Angew. Chem. Int. Ed. Engl., 1971, 10(2), 10. H. H. Britzinger, D. Fischer, R. ¨lhaupt, B. Rieger, R. M. Mu Waymouth, Angew. Chem. Int. Ed. Engl., 1995, 34, 1143. K. Soga, T. Shiono, Prog. Polym. Sci., 1997, 22, 1503. R. Mulhaupt, Macromol. Chem. Phys., 2003, 204, 289. H. L. Krauss, in Transition Metals and Organometallics as Catalysts for Olefin Polymerization, W. Kaminsky, H. Sinn (Eds.), Springer-Verlag, Berlin, 1988, pp. 163–168. P. C. Thune, J. Loos, A. M. de Jonga, P. J. Lemstrab, J. W. Niemantsverdriet, Topics in Catalysis, 2000, 13, 67.

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9

Mathematical Methods P. D. Iedema and N. H. Kolhapure 9.1

Introduction

In this chapter some mathematical methods to solve kinetic modeling problems are explained. A very sound basis for this was already laid many years ago by Flory [1]. Here, we want to present modern mathematical tools that have recently been developed through the use of computers. The focus is on the link between kinetic rate data and reactor type on one hand, and distributive properties – in one or more dimensions – on the other. These distributive or microstructural properties are concerned not only with countable quantities, such as the number of monomer units in a polymer molecule, but also with structure in the case of branched polymer molecules. With structure, we discuss the connectivity of branch points and the lengths of the segments between them. In the greater part of the text, reactors are treated in a simplified manner. We consider continuous and batch reactors, but all of them ideally mixed. The effect of incomplete mixing (segregation, macroand micromixing) is addressed in classical textbooks such as Biesenberger [2] and Dotson et al. [3]. Some recent attempts to include the impact of micromixing on distributions are available in the literature [57–59], but this field is still in its infancy. Nevertheless, to still provide a sound basis for issues of mixing, we devote one section to the use of computational fluid dynamics in polymer reaction engineering problems. The microstructural properties that we address are chain length, number of monomer units of one kind (copolymer), number of branch points, number of unsaturated bonds, and number of reactive monomer units (end groups in polycondensation). Problems may require solution of one or more of these properties simultaneously. Here, we will denote this as the dimensionality of the problem at hand. For instance, growth in addition polymerization can be described by a simple 1D (chain length) reaction equation and population balance. kp

R n þ M ! R nþ1


ð1Þ

432

9 Mathematical Methods

dR n ¼ k p MðR n1 R n Þ dt

ð2Þ

In contrast, growth in polycondensation of a trifunctional monomer A with a bifunctional monomer B requires a 3D description. The 3D distribution R n; i; k , where subscripts denote chain length, number of A end groups, number of B end groups, respectively, obeys: kc

R n; i; k þ R m; j; l ! R nþm; iþj1; kþl1

ð3Þ

n1 X i X k X dR n; i; k jðk l þ 1Þlði j þ 1ÞR m; j; l R nm; ijþ1; klþ1 ¼ kc dt m¼1 j¼0 l ¼0

ð4Þ

Note that this describes a reaction between single end groups of two different molecules; end group combinations within one (longer) molecule require a similar approach. These examples illustrate the importance of the dimensionality of the problem at hand. In general, low dimensionality can be dealt with using analytical or differential methods, while higher dimensionality soon requires a Monte Carlo sampling approach. Note that all of these methods, except MC, start with a population balance. Here we will mainly discuss ways to solve such balances of lower or higher dimensionality. This chapter will start with the description of lower-dimensional problems and show to what extent these can be successful. This often involves the reduction of the problem to lower dimensionality, inevitably leading to averaging over one or more dimensions. The most well-known is the method of moments, which, however, does not solve full distributions. This is followed by the introduction of a fairly recent mathematical method based on differential equations that does solve full distributions: the Galerkin h–p finite element method (FEM). Subsequently, we will present advanced applications of the Galerkin-FEM (G-FEM) method, being classes and pseudo-distribution modeling. A completely different approach – originating from polymer network modeling – is then discussed: probability generating functions. To conclude the methods for countable properties, we give an overview of full Monte Carlo simulation methods as introduced by Tobita [11–15, 48–51], mostly for cases where analytical or differential equation approaches fail. A separate section then is spent on very recently developed conditional Monte Carlo methods to synthesize branched architectures. Finally, a section is devoted on applications of computational fluid dynamics in polymer reaction engineering.

9.2

Discrete Galerkin h–p Finite Element Method

The discrete Galerkin h–p finite element method (FEM) is a powerful numerical method to solve chain length distributions for a wide set of polymerization prob-

9.2 Discrete Galerkin h–p Finite Element Method

lems. It has been implemented in the commercially available package PREDICI. A great proportion of the problems discussed in this chapter are solved with this approach. A detailed description of the mathematics of the method is given by Wulkow [4]. Below the main mathematical features are given following the description in Ref. 4. Any set of population balance equations (see, for example, Table 9.2, below) can be written as a set of countable ordinary differential equations [Eqs. (5)]. us0 ðtÞ ¼ fs fu1 ðtÞ; . . . ; us tot ðtÞg s ¼ 1; . . . ; stot

ð5Þ

Here, the us ðtÞ are the concentrations of all the macromolecules with length n at time t, represented by vector us ðtÞ; stot is the dimension (chain length) of the system, for polymer systems typically very large, up to 10 6 . For an approximation u1 of the solution uðt þ tÞ after a time step from t to t þ t a semi-(linear)-implicit Euler scheme is applied [Eq. (6), where j ¼ uðtÞ, A is the derivative f u ðjÞ, and I the identity matrix]. ðI tAÞDu ¼ tf ðjÞ

ð6Þ

u1 ¼ j þ Du The solution of this equation is approximated by a finite element Galerkin method. This is realized by a multilevel algorithm, according to which a subdivision of the j s-axis is constructed (see Figure 9.1). On each interval h a local expansion us of us is used, where j is the level number and l is the interval number: j

usj jh j

¼

l

( h1j , p1j )

pl X

j

j

a kl t k; l ðsÞ

ð7Þ

k¼0

( hmj , pmj )

Fig. 9.1. Division of chain length axis into intervals of length h, where distribution is approximated by Chebyshev polynomials of order p.

433

434


(h1,p1)

(h2,p2)

(h3,p3)

(h4,p4)

(h5,p5) smax

1

(h1,p1+1)

(h2,p2)

(h3L,1) (h3R,1) (h4,p4+1)

(h5,p5)

Fig. 9.2. Refinement strategy. Orders of intervals (h1 ; p1 ) and (h4 ; p4 ) increased by 1. Interval (h3 ; p3 ) is split into two intervals with order 1 (order may be higher according to optimization strategy). Other intervals remain unchanged.

j

The polynomials t k; l are discrete Chebyshev polynomials of degree k. The number j of expansion coefficients pl may differ from interval to interval, such that fitting it to a concentration distribution can be solved by varying grid and order. Note that this feature gives the name to the method: Galerkin h (varying intervals) – p (varying order). The node-order-distribution on the final grid is chosen in such a way that the work necessary to compute the whole distribution is minimal: DF ¼ fðhl ; pl Þ . . . ðhm ; pm Þg

ð8Þ

The construction is started with an initial grid D 0 on the interval [0; s max ], where s max may be very large. It proceeds from level to level by refinements or increases of the order, using information from the previous level. An example is shown in Figure 9.2. The Galerkin h–p method is very flexible, being able to solve simple distributions with few broad intervals, but also complicated, including multimodal distributions with steep flanks requiring intensive local adaptation. Such adaptations are managed automatically with the method. The algorithm as implemented in PREDICI utilizes chain length truncation. A truncation index is calculated, being the maximum chain length up to which a distribution is calculated. From the index for a old , a new index is calculated from the moments of the calculated distribution, s max distribution: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi m m m1 2 new k ¼ 10 ð9Þ ¼ 1 þk s max m0 m0 m0 When higher accuracy is required in tail calculations weight- or wðlog sÞ-based truncation indices can be calculated, where moments m 0 ; m1 , and m 2 are replaced by mk ; mkþ1, and mkþ2 , with k ¼ 1 or 2, respectively. For example, a Flory distribution with average n n ¼ 100 is represented until s max ¼ 1100. A first demonstration of the capabilities of the Galerkin-FEM method will be given in Section 9.4 in a comparison with the method of moments. Further and extensive use is made of the

9.3 Method of Moments

method in the classes and pseudo-distributions approach, to be discussed in Sections 9.5 and 9.6.

9.3

Method of Moments 9.3.1

Introduction

The method of moments is the most well-known method for solving polymerization problems [1–3]. The equations are derived from the population balances. This is realized in a straightforward way for the radical polymerization system of Table 9.1, a 1D problem. Table 9.2 presents the original population balances and Table 9.3 the resulting moment equations, up to the 4th moment. The linear part of this problem can be solved without additional assumptions, but the nonlinear part leads to a closure problem. This will be discussed next. Some results and a discussion on the validity of the method will be given in Section 9.4, in a comparison with the Galerkin-FEM method. 9.3.2

Linear Polymerization

Suppose we want to describe a recombination reaction of two living copolymer chains with one dead chain [Eq. (10)], which is a 2D problem. p

kc; ps

s R n; i þ Rm; j ! Pnþm; iþj

ð10Þ

The first subscripts (n; m) denote chain length, the second (i; j) the number of monomer units of one type per chain; superscripts indicate terminal units, identified as either monomer 1 or monomer 2, so p ¼ 1 or 2 and s ¼ 1 or 2; kc; ps is the

Tab. 9.1. Reaction equations for radical polymerization with transfer to polymer and random scission by H-abstraction.

Mechanism

Reaction equation

Rate factor

Initiation Propagation Disproportionation termination Recombination termination Transfer to monomer Transfer to chain transfer agent S Transfer to polymer Random scission by H-abstraction

I2 ! 2I I þ M ! R1 R n þ M ! R nþ1 R n þ R m ! Pn þ Pm R n þ R m ! Pnþm R n þ M ! Pn þ R1 R n þ S ! Pn þ R1 R n þ Pm ! Pn þ R m þ LCB R n þ Pm ! Pn þ R ml þ Pl

kd ; k i f kp k td k tc km kS k trp m k rs m

435

436

9 Mathematical Methods Tab. 9.2. (Population) balances for radical polymerization with transfer to polymer and random scission by H-abstraction. y X dM ¼ M0 M k p tM R n ¼ M0 M k p tMl 0 dt n¼1 dI2 ¼ I20 I2 kd tI2 t dt dI t ¼ I þ 2kd tI2 k i tIM dt dS t ¼ S0 S ks tSl 0 dt dR1 ¼ R1 þ k i tIM þ ks tSl 0 k p tR1 M þ k rs tl 0 m 0 t dt

(a)

t

(b) (c) (d) (e)

y X dR n ¼ k p MR n þ k p MR n1 k tp R n m1 þ k tp l 0 Pn þ k rs l 0 Pk k rs l 0 m1 þ dt k¼nþ1

ðk tc þ k td Þl 0 R n ks SR n k m MR n R n =t y X dPn ¼ k tp l 0 nPn þ k tp m1 R n þ k rs l 0 Pk þ k rs m1 R n k rs l 0 ðn 1ÞPn dt k¼nþ1

(f )

(g)

n1 1 X þ k td l 0 R n þ k tc R mn R m þ ks SPn þ k m MPn Pn =t 2 m ¼1

rate coefficient for combination between chains with terminal units p and s. The full population balance reads as [5] (the ‘‘þ¼’’ duet means that this is one out of possibly more contributions to the population balance of Pn; i ): 2 X 2 n1 X i X X dPn; i p kc; ps R m; j R snm; ij þ¼ dt r ¼1 q¼1 m¼1 j¼0

ð11Þ

The moments of distributions R and P are defined as: p

lab ¼

y X y X n¼1 j¼1

p

na i b R n; i

mab ¼

y X y X

na i b Pn; i ;

p ¼ 1; 2

ð12Þ

n¼1 j¼1

The population balance of Eq. (11) is expressed in moments by performing the summations as in Eq. (12) and collecting all the terms. In its most general form the result can be expressed as [5]: b 2 X 2 a X X dmab 1X a b r r l l kc; ps þ¼ 2 p¼1 s¼1 g d g; d ag; bd dt g¼0 d¼0

ð13Þ

Usually, when number- and weight-average molecular weights are to be calculated,

9.3 Method of Moments Tab. 9.3. Moment equations for radical polymerization with transfer to polymer and random scission by H-abstraction.

dl 0 1 ¼ k p MR1 ðk tc þ k td Þl 20 ks Sl 0 k m Ml 0 l 0 t dt dl1 1 ¼ k p Ml 0 k tp m1 l1 þ k tp m 2 l 0 þ k rs l 0 ðm 2 m1 Þ k rs l 0 ðm1 m 0 Þ 2 dt 1 ðk tc þ k td Þl 0 l1 ks Sl1 1 k m Ml1 l1 t dl 2 1 1 1 ¼ 2k p Ml1 k tp m1 l 2 þ k tp m3 l 0 þ k rs l 0 m3 m 2 þ m1 þ k rs l 2 ðm1 m 0 Þ 3 2 6 dt 1 ðk tc þ k td Þl 0 l 2 ks Sl 2 k m Ml 2 l 2 t dl3 1 1 1 ¼ 3k p Ml 2 k tp m1 l3 þ k tp m4 l 0 þ k rs l 0 m4 m3 þ m 2 þ k rs l3 ðm1 m 0 Þ 4 2 4 dt 1 ðk tc þ k td Þl 0 l3 ks Sl3 k m Ml3 l3 t dl4 1 1 1 1 ¼ 4k p Ml3 k tp m1 l4 þ k tp m5 l 0 þ k rs l 0 m5 m4 þ m3 m1 5 2 3 30 dt 1 þ k rs l4 ðm1 m 0 Þ ðk tc þ k td Þl 0 l4 ks Sl4 k m Ml4 l4 t dm 0 1 1 k tc þ k td l 20 þ ks Sl 0 þ k m Ml 0 m 0 ¼ k rs l 0 ðm1 2m 0 Þ þ 2 t dt dm1 1 1 ¼ k rs l 0 m 2 m1 m 0 þ k rs l1 ðm1 m 0 Þ k rs l 0 ðm 2 m1 Þ þ k tp l1 m1 2 2 dt 1 k tp l 0 m 2 þ ðk tc þ k td Þl 0 l1 þ ks Sl1 þ k m Ml1 m1 t dm 2 1 1 1 ¼ k rs l 0 m3 m 2 þ m1 m 0 þ k rs l 2 ðm1 m 0 Þ k rs l 0 ðm3 m 2 Þ þ k tp l 2 m1 3 2 6 dt 1 k tp l 0 m3 þ k tc ðl 0 l 2 þ l12 Þ þ k td l 0 l 2 þ ks Sl 2 þ k m Ml 2 m 2 t dm3 1 1 1 ¼ k rs l 0 m4 m3 þ m 2 m 0 þ k rs l3 ðm1 m 0 Þ k rs l 0 ðm4 m3 Þ þ k tp l3 m1 4 2 4 dt 1 k tp l 0 m4 þ k t ðl 0 l3 þ 3l1 l 2 Þ þ k td l 0 l3 þ ks Sl3 þ k m Ml3 m3 t dm4 1 1 1 1 ¼ k rs l 0 m5 m4 þ m3 m1 m 0 þ k rs l4 ðm1 m 0 Þ k rs l 0 ðm5 m4 Þ 5 2 3 30 dt

(a) (b)

(c)

(v)

(e)

(f ) (g)

(h)

(i)

( j)

1 þ k tp l4 m1 k tp l 0 m5 þ k t ðl 0 l4 þ 4l1 l3 þ 3l 22 Þ þ k td l 0 l4 þ ks Sl4 þ k m Ml4 m4 t

it is sufficient to have a; b ¼ 0; 1; 2. Although these forms seem to be complex, they can be derived straightforwardly and solved without any additional assumption or model for linear polymerization – that is, when rate factors are independent of chain length, comonomer content, and so on. In fact, contributions to the population balances from all other kinetic mechanisms – propagation, transfer to solvent or monomer, disproportionation – can be treated in similar but simpler ways. For linear polymerization the method of moments provides exact solutions for number

437

438


and weight averages of molecular weight and copolymer composition. Most well known are the averages for homopolymerization expressed as: Mn ¼ m 0

m1 m0

Mw ¼ m 0

m2 m1

ð14Þ

9.3.3

Nonlinear Polymerization

In nonlinear polymerization rate factors depend on chain length or other microstructural properties, which leads to special problems when utilizing the method of moments [6]. We here address transfer to polymer in radical polymerization leading to long-chain branching (LCB) and random scission. The reaction equation for transfer to polymer in a 1D formulation with the rate factor proportional to chain length, k tp m is Eq. (15). k tp m

R n þ Pm ! Pn þ R m

ð15Þ

The population balance contribution for dead chains is given by: y y X X dPn nPn ; l 0 ¼ Rn þ¼ k tp ðm1 R n l 0 nPn Þ m1 ¼ dt n¼1 n¼1

ð16Þ

y P na Pn , and constructing the moment Defining moments as in Eq. (12), ma ¼ n¼1 equations from Eq. (16) yields:

dma þ¼ k tp ðm1 la l 0 maþ1 Þ dt

ð17Þ

This illustrates a typical closure problem. In linear polymerization, differential equations of the ath moment contain RHS terms with ath or lower moments, which allows direct solution of the set. In contrast, as shown by Eq. (17), nonlinear polymerization leads to higher moments on the RHS. This implies that the set cannot readily be solved without additional assumptions. Another example is random scission of linear dead chains into two living chains (macroradicals): k rs n

Pn ! R m þ R nm

ð18Þ

Again, the rate factor is proportional to chain length. The 1D population balance reads: y X dR n nPn þ¼ k rs dt m¼nþ1

ð19Þ

9.3 Method of Moments

Constructing equations for 0th through 4th moments by summing over chain length leads to: dl 0 þ¼ 0; dt

ð20Þ

dl1 1 þ¼ k rs ðm 2 m1 Þ; 2 dt dl 2 1 1 1 þ¼ k rs m3 m 2 þ m1 ; 3 2 6 dt dl3 1 1 1 þ¼ k rs m4 m3 þ m 2 ; dt 4 2 4 dl4 1 1 1 1 þ¼ k rs m5 m4 þ m3 m1 : 5 2 3 30 dt

ð21Þ ð22Þ ð23Þ ð24Þ

Hulburt and Katz [7] have developed a method to obtain estimates of the higher moments in terms of lower ones using a distribution approximation method. Most previous work has been based on the 3rd-moment closure. A general expression is constructed for moment mi , where l is the highest order of the series of Laguerre polynomials in the approximation, while a and g are parameters to be specified: mi ¼

ði þ g 1Þ! m 0 ðg 1Þ! ðg=aÞ i " # l m X km X j m!ðm þ g 1Þ!ðm þ i þ g 1 jÞ! þ ð1Þ i j!ðm jÞ!ðm þ g 1 jÞ! m¼3 ðg=aÞ j¼0

ð25Þ

The series is truncated at l ¼ i 1, so that the computation of mi requires the coefficient k i1 . This implies that ml and higher moments equal zero. Coefficients k i follow in their turn from the lower moments m0...i1 by:

ki ¼

i X ð1Þ j j¼0

ðg 1Þ! ðg=aÞ ij m ; j!ði þ g 1 jÞ! ði jÞ! ij

ð26Þ

m a2 , which leads to so k 0 ¼ m 0 . It is further assumed that: a ¼ 1 ; g ¼ m0 m 2 =m 0 a 2 k1 ¼ k2 ¼ 0. Thus, closure expressions have been obtained for m3 ; m4, and m5 : m3 ¼

m2 ð2m 2 m 0 m 21 Þ m 0 m1

ð27Þ

439

440


m4 ¼

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

m5 ¼

ð12m14 m 2 42m12 m 22 m 0 þ 36m 20 m 32 þ 20m13 m 0 m3 30m1 m 20 m 2 m3 þ 5m12 m 20 m4 Þð3m12 4m 2 m 0 Þ m 20 m13

ð28Þ

ð29Þ

9.4

Comparison of Galerkin-FEM and Method of Moments

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

9.4 Comparison of Galerkin-FEM and Method of Moments 1.8

x 10-5 kmole/m3

n2Rn

Concentration

1.6 1.4 1.2 1 0.8 0.6

Computation limit P n

0.4 0.2 0 0 10

1.8

Concentration

1.6

2

4

10

10

6

10

Chain Length

8

10

10

10

12

10

kmole/m3

n2Pn

1.4 1.2

Computation limit

1 0.8 0.6 0.4 0.2 0 0 10

2

10

4

10

Chain Length

Radical polymerization in CSTR with transfer to polymer. Reactor and kinetic data: initiator feed: I2; f ¼ 5 103 kmol m3 ; monomer feed Mf ¼ 16:75 kmol m3 ; residence time: t ¼ 30 s; kd ¼ 0:5 1 s1 ; Fig. 9.3.

6

10

8

10

k p ¼ 1:4 10 5 m 3 (kmol s)1 ; k tc ¼ 5 10 10 m 3 (kmol s)1 ; k tp ¼ 180 m 3 (kmol s)1 ; conversion x ¼ 0:249. The discontinuity in the living CLD is due to a vanishing dead chain concentration.

trations increase more rapidly within the computation limit of 10 9 and reach the same magnitude as the dead chain tail concentrations; see Figure 9.3. Thus, living and dead CLDs nicely overlap over the whole range and are correctly calculated, even while a considerable proportion of the monomer units are now polymerized in living chains.

441

442


105

10

4

kg/mole

Living chains, recombination termination only

Mw

Galerkin-FEM

103 102

1010

Moments method (closure µ 3) 20

40

60

80

ktp ( 140

100

120

m3/(kmole.s)

kg/mole

140

160

180

)

Moments method (closure µ 3)

120

Mw100 80 60

Galerkin-FEM

40 Dead chains, recombination termination only

20 0

0

20

40

60

80

100

ktp (m3/(kmole.s))

120

140

160

180

Radical polymerization; for data, see Figure 9.3. Comparison of Galerkin-FEM and method of moments. The deviation is strongest for living chains.

Fig. 9.4.

We conclude that the method of moments yields accurate solutions for all moments (hence Mn ; Mw ; Mz ) in the case of linear polymerization. In those cases where Flory distributions form the solution of instantaneous or steady-state polymerizations, full CLDs can be constructed as simple combinations of Flory distributions. This is not possible when distributions are resulting from combination reactions between other distributions, such as recombination termination. In nonlinear polymerization the method of moments requires estimation of higher moments, by which errors are introduced unavoidably. In those situations the

9.4 Comparison of Galerkin-FEM and Method of Moments 7

10

kg/mole Living chains

6

10

Galerkin-FEM

5

Mw 10

Moments method (closure µ 3)

4

10

3

Dead chains

Living chains

10

2

10

Dead chains

1

10

Disproportionation termination only

0

10

0

Concentration

0.9

500

1000

1500

ktp (m3/(kmole.s))

2000

2500

3000

kmole/m3 0.01

0.8

0.008

0.7

0.006

0.6

0.004 0.002

0.5

0 5 10

0.4

Dead

Living 10

6

10

7

10

8

10

9

Close-up

0.3 0.2 0.1 0 0 10

2

10

4

10

6

10

Chain Length

Fig. 9.5. Radical polymerization; see Figure 9.3. Termination is through disproportionation only. The living chain concentration is the same order of magnitude as the dead chain concentration at high chain length.

8

10

10

10

443

444


Galerkin-FEM method is preferred, although care must be taken when extremely long chain lengths have to be calculated.

9.5

Classes Approach 9.5.1

Introduction

The classes approach is applicable to multidimensional problems where the range of the second and higher dimensions is restricted to a small range. In the examples of this approach we will discuss presently, these second dimensions are the numbers of terminal double bonds (TDBs) on a chain in the case of poly(vinyl acetate) (PVAc) and the numbers of radical sites on a chain in the case of low-density polyethylene (LDPE). The idea simply is to solve the problem rigorously in the first dimension, chain length, for separate classes with a fixed value for the second dimension. Thus, a 2D problem is reduced to a set of 1D problems, where the size of the set is determined by the number of values of the second variable – the number of classes – for which the solution is desired. Obviously, this is only feasible when the number of classes is limited, since the equations have to be implemented separately for each class. A different classes approach is followed by Pladis and Kipparissides [8], who combined it with a method of moments. 9.5.2

Computing the CLD of Poly(vinyl acetate) for a Maximum of One TDB per Chain

The most general set of reaction equations is given in Table 9.4 in terms of all three concentration distribution variables – chain length, number of TDBs, num-

Tab. 9.4.

Reactions for radical polymerization of vinyl acetate. kd

Initiator dissociation

I2 ! 2I

Initiation

I þ M ! R1; 0; 0

Propagation Termination by disproportionation (without TDB creation) Termination by disproportionation (with TDB creation)

R n; i; k þ M ! R nþ1; i; k k td R n; i; k þ R m; j; l ! Pn; i; k þ Pm; j; l

Termination by recombination

R n; i; k þ R m; j; l ! Pnþm; iþ j; kþl

ki

kp

k td

R n; i; k þ R m; j; l ! Pn; iþ1; k þ Pm; j; l k tc

km

Transfer to monomer

R n; i; k þ M ! Pn; i; k þ R1; 1; 0

Transfer to polymer

R n; i; k þ Pm; j; l ! Pn; i; k þ R m; j; lþ1

Terminal double bond propagation

R n; i; k þ Pm; j; l ! R nþm; iþ j1; kþlþ1

k tp m

Notation: m; n: chain length; i; j: number of terminal double bonds (TDB) per chain; k; l: number of branches per chain.

kdb j

9.5 Classes Approach

ber of branches; hence it is a 3D problem [9]. Most of the equations in the table are self-evident except the TDB propagation reaction, which deserves closer examination. It refers to the incorporation of dead chains with a TDB in the growing living chain. The reaction equation is: kdb j

R n; i; k þ Pm; j; l ! R nþm; iþj1; kþlþ1

ð30Þ

In fact, this represents a combination reaction that sums chain lengths, while numbers of branches are summed along with the addition of one extra branch created by the reaction; TDBs are summed as well, except for one TDB consumed in the reaction. It is evident that the reactivity of the dead chain inserted is proportional to the number of TDBs it carries. Now, TDB propagation introduces a nonlinearity of a different kind than that caused by the step involving transfer to polymer discussed previously. In that case the nonlinearity is due to the dependence of the transfer rate on chain length, a variable already present in a 1D description of chain length alone. In contrast, here the rate depends on a second distribution variable, the number of TDBs per chain, which is not dealt with explicitly in the 1D description of chain length. Mathematically speaking, in order to solve the problem in one dimension, chain length, one has to solve it in the second dimension, TDB number, as well. Two methods exist to deal with the TDB-dependent reactivity. The first is the classes approach, to be discussed here, and the second is the pseudodistribution approach, to be presented in Section 9.6. The full 3D molecular weight–terminal double bond-branching population balance equations belonging to the reactions as given in Table 9.4 are listed in Table 9.5. Note that this system is an exact representation of the reactions in Table 9.4, no assumptions have yet been made, and it is valid for cases with an arbitrary number of TDBs per chain. In the present case of a CSTR, the complete population balances of all distributions Q n contain an outflow term of the general shape Q n =t, where t represents the average residence time (equal to volume/volumetric flow). The overall moments l 0 and m1 are defined as usual and in the 3D case follow as:

l0 ¼

y X y X y X i¼0 k¼0 n¼1

R n; i; k

m1 ¼

y X y X y X

nPn; i; k

ð31Þ

i¼0 k¼0 n¼1

Since we will not address branching here, the set has to be reduced to 2D, by summation over the branching index k. The second dimension, the number of TDBs per chain, is concerned with the manner in which they are created. One of two possible mechanisms is transfer to monomer, producing a monoradical with a TDB according to the reaction equation shown in Table 9.4. Subsequent propagation of this monoradical leads to a chain with a TDB. The second mechanism is disproportionation, directly leading to chain with a TDB. Now, chains with more than one terminal double bond can be created in two ways: insertion of chains with a TDB created by disproportionation termination, or by termination by recombination. If

445

446


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

Initiator dissociation Initiation Propagation

dI2 I2 ¼ kd I2 dt t dI I dR1; 0; 0 ¼ 2kd I2 k i MI ; þ¼ k i MI dt t dt dR n; i; k þ¼ k p MðR n1; i; k R n; i; k Þ dt [a]

Termination by disproportionation (TDB creation)

dR n; i; k þ¼ k td l 0 R n; i; k ; dt

dPn; i; k 1 þ¼ k td l 0 ðR n; i1; k þ R n; i; k Þ 2 dt


dR n; i; k þ¼ k tc l 0 R n; i; k ; dt

n1 X i X k dPn; i; k 1 X þ¼ k tc R m; j; l R nm; ij; kl 2 m ¼1 j¼0 l ¼0 dt

Transfer to monomer Transfer to polymer Terminal double bond propagation

dR n; i; k dPn; i; k dR1; 1; 0 þ¼ k m MR n; i; k ; þ¼ k m l 0 M; þ¼ k m MR n; i; k dt dt dt dR n; i; k dPn; i; k þ¼ k tp ðl 0 nPn; i; k1 m1 R n; i; k Þ; þ¼ k tp ðl 0 nPn; i; k þ m1 R n; i; k Þ dt dt y X y X y n1 X iþ1 X k X X dR n; i; k þ¼ kdb R n; i; k iPn; i; k þ jPm; j; l R nm; ijþ1; kl1 dt n¼1 i¼0 k¼0 m ¼1 j¼0 l ¼0

!

dPn; i; k þ¼ kdb il 0 Pn; i; k dt [a] The first term between brackets equals zero in the case in which no account is taken of TDB creation by disproportionation.

disproportionation does not lead to reactive TDBs and recombination is absent, chains may carry one TDB as a maximum, which is the case being generally dealt with in modeling studies of PVAc [10–18]. The specific situation of a maximum of one TDB per chain here serves as an example of the classes approach, while the more general case of more than one TDB will be addressed in the context of pseudo-distribution modeling (Section 9.6). Only two TDB classes are required to fully describe the problem of one TDB as a maximum: 0 and 1 TDB per chain. Consequently, population balances for only four distributions have to be solved – ¼ R n ; Pn ; R¼ n ; Pn – and this can be done in an exact way without additional assumptions. The equations are shown in Table 9.6. The resulting CLDs for a realistic set of kinetic data are shown in Figure 9.6 for the total dead chain concentrations, with and without TDB. 9.5.3

Multiradicals in Radical Polymerization

In most models of radical polymerization, living chains are explicitly taken into account, except for instance in the Monte Carlo simulation approach by Tobita [11–15]. Usually, macroradicals are assumed to possess only one radical site

9.5 Classes Approach Tab. 9.6. Full set of population balance equations for the PVAc problem in the case of a maximum of one terminal double bond per chain.

Propagation Termination by disproportionation

Transfer to monomer

Transfer to polymer


1.2

dR n dR¼n þ¼ k p MðR n1 R n Þ þ¼ k p MðR¼n1 R¼n Þ dt dt dR n dPn þ¼ k td l 0 R n ; þ¼ k td l 0 R n dt dt ¼ ¼ dR n dPn þ¼ k td l 0 R¼n ; þ¼ k td l 0 R¼n dt dt dR¼ dR¼n dPn¼ 1 þ¼ k m l 0 M; þ¼ k m MR¼n ; þ¼ k m MR¼n dt dt dt dR n dPn þ¼ k m MR n þ¼ k m MR n dt dt dR n dPn þ¼ k tp ðl 0 nPn m1 R n Þ; þ¼ k tp ðl 0 nPn þ m1 R n Þ dt dt ¼ ¼ dR n dPn þ¼ k tp ðl 0 nPn¼ m1 R¼n Þ; þ¼ k tp ðl 0 nPn¼ þ m1 R¼n Þ dt dt ! y n1 X X dR n ¼ ¼ þ¼ kdb R n Pn þ Pm R nm dt n¼1 m ¼1 dPn¼ þ¼ kdb l 0 Pn¼ dt

kmole/m3

n 2 ( Pn + Pn= )

1 0.8

kdb = 0 kdb = 2500 m3/(kmole.s) kdb = 6200 m3/(kmole.s)

0.6 0.4

kdb = 31000 m3/(kmole.s)

0.2 0 Radical polymerization of vinyl acetate in a CSTR: sensitivity of chain length distribution to rate of TDB propagation. Reactor and kinetic data: initiator feed: I2; f ¼ 3:85 103 kmol m3 ; monomer feed Fig. 9.6.

Mf ¼ 9 kmol m3 ; residence time: t ¼ 4000 s; kd ¼ 9 106 s1 ; k p ¼ 9500 m 3 (kmol s)1 ; k m ¼ 2:38 m 3 (kmol s)1 ; k td ¼ 10 8 m 3 (kmole s)1 ; k tp ¼ 5 m 3 (kmol s)1 ; conversion x ¼ 0:5.

447

448

9 Mathematical Methods Tab. 9.7. Reaction equations for radical polymerization of ethylene: 2D chain length– multiradical approach. kd

Initiator dissociation Initiation

I2 ! 2I ki I þ M ! R1; 1

Propagation

R n; i þ M ! R nþ1; i

Termination by disproportionation

R n; i þ R m; j ! R n; i1 þ R m; j1


R n; i þ R m; j ! R nþm; iþj2

Transfer to monomer Transfer to solvent

R n; i þ M ! R n; i1 þ R1; 1 km i R n; i þ S ! R n; i1 þ R1; 1

Transfer to polymer

R n; i þ R m; j ! R n; i1 þ R m; jþ1

kpi

k td ij k tc ij

km i

k tp jm

Notation: m; n: chain length; i; j: number of radical sites per chain.

(monoradical assumption). In principle, this is only correct for linear polymerization. In nonlinear polymerization, for example, transfer to polymer or TDB propagation, multiradicals can easily be created. For instance, transfer to polymer is frequently assumed to occur only in dead chains, but obviously it also takes place in living chains. In order to investigate the effects of taking multiradicals into account, we apply the classes approach. This anticipates the expectation that transition regimes exist, where the monoradical assumption no longer holds, while the numbers of radical sites per chain is still low. We take the example of radical polymerization with transfer to polymer; reaction and population balance equations are listed in Tables 9.7 and 9.8. Note that the sec-

Tab. 9.8. (Population) balance equations for radical polymerization of ethylene: 2D chain length–multiradical approach.

Initiator dissociation Initiation Propagation Termination by disproportionation Termination by recombination Transfer to monomer Transfer to solvent Transfer to polymer la ¼

y X y X n¼1 i¼0

na R n; i

dI2 I2 ¼ kd I2 dt t dI I dR1; 1 ¼ 2kd I2 k i MI ; þ¼ k i MI dt t dt dR n; i þ¼ k p iMðR n1; i R n; i Þ dt dR n; i þ¼ k td l 0 fði þ 1ÞR n; iþ1 iR n; i g dt n1 iþ1 P P dR n; i 1 þ¼ k tc jði j þ 2ÞR m; j R nm; ijþ2 2 m ¼1 j¼1 dt dR n; i dR1; 1 þ¼ k m Mfði þ 1ÞR n; iþ1 iR n; i g; þ¼ k m l 0 M dt dt dR n; i dR1; 1 þ¼ ks Sfði þ 1ÞR n; iþ1 iR n; i g; þ¼ k m l 0 S dt dt dR n; i þ¼ k tp ½l 0 nfR n; i1 R n; i g þ l1 fði 1ÞR n; i1 þ iR n; i g dt

9.6 Pseudo-distribution Approach

ond subscript of R n; i denotes the number of radical sites per chain; hence chains with i ¼ 0 represent what is usually called ‘‘dead’’ chains. The transfer-to-polymer reaction is obviously responsible for the creation of multiradicals, as appears from the term R n; i1 on the RHS. Two classes models are formulated, one with two classes (the usual living and dead chains) and one with five classes (up to four radical sites per chain). Note that the implementation of the termination reactions requires the implementation of 4 2 pairs of reaction equations of the form: k tc IJ

IþJ2

J R nI þ R m ! R nþm ;

k td IJ

J J1 R nI þ R m ! R nI1 þ R m

ð32Þ

where RnI denotes the 1D distribution of the class with I radical sites per chain. Thus we see that many more classes rapidly lead to a huge implementation task and the classes method is not appropriate for such problems. We calculated CLDs with the two classes models for various transfer-to-polymer rates. Up to k tp ¼ 500 m 3 (kmol s)1 results are identical, proving the validity of the monoradical-assumption in this range. For k tp ¼ 5000 m 3 (kmol s)1 , on the contrary, we see large differences, as shown in Figure 9.7. According to the fiveclasses model, chains are much shorter and also conversion is lower. This is due to differences in an increase in the reactivity of multiradical chains in different reactions. From the reaction equations we see that propagation increases linearly with the number of radical sites, but termination with the product of radical sites. Hence, accounting for multiradicals leads to an increased termination/propagation ratio and thus to shorter chains and lower conversion under the conditions simulated. Note that the relative heights of the distributions in Figure 9.7 suggest that for k tp ¼ 5000 m 3 (kmol s)1 even more than five classes are required for a proper description. We conclude, however, that the classes approach in this case produces useful information on the multiradical issue, even when employing a limited number of classes.

9.6

Pseudo-distribution Approach 9.6.1

Introduction

This strategy is based on the Galerkin-FEM method. The simultaneous description of CLD and DBD leads to a two-dimensional problem with the property coordinates chain length and number of branches. In order to avoid the computationally expensive 2D problem, a mathematical reduction technique is applied to the original model, which allows the computation of average branching degrees per chain length (and the respective variances). For that reason branching moment distributions are introduced. The population balances for these distributions can be solved

449

9 Mathematical Methods 10-3 kmole/m3

2-classes model 5-classes model

Concentration

0.3

0.2

n2Rn,0 n2Rn,0

0.1

n2Rn,1 n2Rn,1

0 0 10

5

4

Concentration

450

2

10

4

6

10

8

10

10

Chain Length n

10

10

12

10

10-3 kmole/m3

2-classes model 5-classes model

n2Rn,4

n2Rn,1

n2Rn,3

3

n2Rn,2 2

n2Rn,1

1

0

4

10

6

10

8

10

10

10

Chain Length n Radical polymerization with transfer to polymer: multiradical problem. CLDs of chains with 0 . . . 4 radical sites per chain. Comparison of two- and five-classes model. Accounting for multiradicals leads to lower Fig. 9.7.

conversion and shorter chains. Kinetic data: kd ¼ 0:57 s1 ; k p ¼ 1:38 10 5 m 3 (kmol s)1 ; k m ¼ 138 m 3 (kmol s)1 ; ks ¼ 790 m 3 (kmol s)1 ; k tc ¼ k td ¼ 2:82 10 8 m 3 (kmol s)1 ; k tp ¼ 5000 m 3 (kmol s)1 .


as a 1D problem. Thus the original 2D problem is replaced by a series of 1D problems, getting exact information on averages concerning the second (number of branches) distribution. The distributions of the original kinetic model will be called ‘‘real distributions’’ whereas the additional branching moment distributions are named ‘‘pseudo-distributions’’. We will introduce this approach in the case of the 2D CLD/DBD computation for mixed-metallocene polymerization of ethylene. Subsequently, we present applications of the approach to the 3D problems of radical polymerization of vinyl acetate (CLD/DBD/number of terminal double bonds distribution), AB radical copolymerization (CLD/comonomer composition distribution/sequence length distribution), and finally the 2D problem of radical polymerization of polyethylene, where random scission is a complicating factor. 9.6.2

CLD/DBD for Mixed-metallocene Polymerization of Ethylene Formulation of Pseudo-distribution Problem Long branches are created in this system by incorporation of chains with a terminal double bond (TDB) at the catalyst site (constrained-geometry catalyst, CGC) [19–30]. The reaction equations are listed in Table 9.9 and the definitions of the growing and dead chain branching moment distributions are elucidated in Table 9.10. Note that the 0th branching moment distributions represent the original normal chain length distributions. From the branching moment distributions F and C certain averages may be derived. For example, the average number of branches of chains of length n; Bn , as well as the branching density, rn , can be computed from the ratio (for each n) of the 1st and 0th branching moment distribution: 9.6.2.1

Tab. 9.9.

Reaction mechanisms and rate coefficients.

Reaction

Reaction equation initiation[a]

CCGC ;

CCGC ! CCGC þ M ! R1;b 0 Linear cat. activation and initiation[a] Clin ! Clin ; Clin þ M ! R1;l 0 b b Branching cat. propagation R n; i þ M ! R nþ1; i l l Linear cat. propagation R n; þ M ! R i nþ1; i b Branching cat. b-hydride elimination R n; ! C þ Pn;b i CGC i l l¼ [b] Linear cat. transfer to monomer R n; i þ M ! Clin þ Pn; i b ¼ b [b] branching cat. terminal double bond propagation R n; þ P ! R m; j i nþm; iþjþ1

Branching cat. activation and

[a] Catalyst [b] Note

activation and initiation are taken as one step. that Pn;¼ i ¼ Pn;b¼i þ Pn;l¼i .

Rate coefficient ka; CGC k i; CGC ka; lin k i; lin k p; CGC k p; lin kw; CGC km; lin k p; TDB

451

452

9 Mathematical Methods Notation for mixed-metallocene systems.

Tab. 9.10.

b R n; i R nl Pn;b¼i ; Pn;l¼0 ; Pn;¼ i

Pn;b i ; Pn;l 0 ; Pn; i m¼0 ¼ l 0b ¼

y X y X

Pn;¼ i

n¼1 i¼0

n¼1 i¼0

y X y X

y X

b R n; i

n¼1 i¼0

R nb ¼

polymer growing on branching catalyst polymer growing on linear catalyst (0 branches) dead polymer with terminal double bond from branching/linear cat. and their sum, resp. dead polymer without terminal double bond from branching/linear cat and their sum, resp. y X y X m0 ¼ Pn; i

y X

l 0l ¼

R nl

n¼1

b R n; i

i¼0

Pnb¼ ¼

y X

Pn;b¼i

Pnl¼ ¼ Pn;l¼0

Pn¼ ¼

i¼0

Pnb ¼

y X

Pn;¼ i

i¼0

Pn;b i

Pnl ¼ Pn;l 0

Pn ¼

i¼0

Fnb1 ¼

y X

y X

Pn; i

i¼0

y X

b iR n; i

i¼0

Cn¼1 ¼

y X

iPn;¼ i

Cn1 ¼

i¼0

Fnb2 ¼

y X

y X

iPn; i

i¼0 b i 2 R n; i

i¼0

Cn¼2 ¼

y X

i 2 Pn;¼ i

i¼0

Bn ¼ Cn1 =Pn ;

Cn2 ¼

y X

i 2 Pn; i

i¼0

rn ¼ Cn1 =ðnPn Þ

ð33Þ

In fact, F and C represent the moments of the branching distributions of living and dead chains at the given chain length, n. Thus, the branching polydispersity Dn follows from the second branching moment according to: Dn ¼

Cn2 =Cn1 Cn1 =Pn

ð34Þ

In order to derive the balance equations for the branching moment distributions, the following steps have to be performed [31]:

For each reaction step of the kinetic model, the two-dimensional population balance is derived.


The two-dimensional balance is reduced to a one-dimensional balance by applying the above summations over the branching index (in the same type of operation as obtaining moments from distributions). The resulting terms have to be expressed in terms of the branching moment distributions and are added to the system of equations. All equations are solved simultaneously. This procedure can in principle be applied to any polyreaction model where additional properties have to be considered. The key is how a certain reaction step changes the additional property. The total set of balance equations for all the reaction mechanisms involved in the mixed-metallocene problem is: dM ¼ fk i; lin Clin þ k i; CGC CCGC þ ðk p; lin þ km; lin Þl 0b þ ðk p; CGC þ km; CGC Þl 0l gM dt þ

M0 M t

ð35Þ

l dR n; 0 l l ¼ k i; lin MClin dðn 1Þ þ k p; lin ð½R n1; 0 R n; 0 ÞM dt l ðkb; lin þ km; lin MÞR n; 0 b dR n; i

dt

l R n; 0 t

ð36Þ

b dðn 1Þ þ k p; CGC ðR n1 R nb ÞM ¼ k i; CGC MCCGC b þ ðkb; CGC þ km; CGC MÞR n; i

þ k p; TDB

b m¼0 R n; i

þ

n1 X i X

! ¼ b Pm; j R nm; ij1

m¼1 j¼0

dPn;l¼0 Pn;l¼0 l l¼ b ¼ ðkm; lin M þ kb; lin ÞR n; 0 k p; TDB Pn; 0 l 0 dt t dPn;b¼i dt

b b¼ b ¼ ðkm; CGC M þ kb; CGC ÞR n; i k p; TDB Pn; i l 0

dB B ¼ k p; TDB m¼0 l 0b dt t

b R n; i

t

ð37Þ

ð38Þ

Pn;b¼i t

ð39Þ

ð40Þ

Here dðiÞ is the Kronecker’s delta function which has value of 1 for i ¼ 0 and 0 for i 0 0. Note that growing chains at the linear catalyst do not carry branches; hence the second index is always zero. Now, we will derive the equations for the pseudodistributions in a stepwise manner. Reactions not affecting the number of branches per chain, that is, those describing propagation, transfer to monomer,

453

454


and b-hydride elimination, will be presented firstly. The reaction creating branching, TDB propagation, is more complex and will be discussed subsequently. Propagation step In this case the degree of branching is not altered, so this is a simple step from this point of view. The contribution to the balance of ath branching moment distributions of living chains in reduced or pseudo-distribution notation follows, by multiplication with ia and taking the summation over the branching index i:

i y dFan X dR n ¼ þ¼ k p MðFan1 þ Fan Þ ia dt dt i¼0

ð41Þ

Transfer to monomer, b-hydride elimination Again, the degree of branching is not affected, hence the contributions to the pseudo-distribution balances of the living and dead chains are given by:

dFan þ¼ k m MFan ; dt dFan þ¼ kb Fan ; dt

dCan þ¼ k m MFan ; dt

dCan þ¼ kb Fan ; dt

ð42Þ

ð43Þ

TDB propagation The one-dimensional contributions for this mechanism describing chain length are obtained from the TDB propagation terms in the 2D equations by summing over the number of branches, i:

dR nb þ¼ k p; TDB R nb m¼0 dt

ð44Þ

dPn¼ þ¼ k p; TDB Pn¼ l 0b dt

ð45Þ

n1 X dR nb ¼ þ¼ k p; TDB R nb Pnm dt m¼1

ð46Þ

The first branching moment distribution follows by multiplication of Eqs. (44)–(46) with the number of branches i, and subsequent summation: dFnb1 þ¼ k p; TDB Fnb1 m¼0 dt

ð47Þ

dCn¼1 þ¼ k p; TDB Cn¼1 l 0b dt

ð48Þ


The production term requires some more extensive algebraic manipulation: y n1 X i1 X X dFnb1 b ¼ þ¼ k p; TDB i R m; j Pnm; ij1 dt m ¼1 j¼0 i¼0

¼ k p; TDB

n1 X y X

0 y X

b @ R m; j

m¼1 j¼0

¼ k p; TDB

n1 X y X

¼ k p; TDB

b R m; j

y X ¼ ði þ j þ 1ÞPnm; i

!

i¼0

n1 X

y X

m¼1

j¼0

n1 X

1 ¼ A iPnm; ij1

i¼ j

m¼1 j¼0

¼ k p; TDB

!

b jR m; j

y X

¼ Pnm; i

þ

y X

b R m; j

y X ¼ ði þ 1ÞPnm; i

j¼0

i¼0

!

i¼0

¼ b ¼1 b ¼ ðFmb1 Pnm þ Rm Cnm þ Rm Pnm Þ

ð49Þ

m¼1

The second branching moment distribution follows by multiplication of the TDB propagation terms in the 2D equations with the squared number of branches i 2 and subsequent summation: dFnb2 þ¼ k p; TDB Fnb2 m¼0 dt

ð50Þ

dCn¼2 þ¼ k p; TDB Cn¼2 l 0b dt

ð51Þ

The production term again requires some more extensive algebraic manipulation: y n1 X i1 X X dFnb2 b ¼ i2 R m; þ¼ k p; TDB j Pnm; ij1 dt m¼1 j¼0 i¼0

¼ k p; TDB

n1 X y X

b R m; j

m¼1 j¼0

¼ k p; TDB

n1 X y X m¼1 j¼0

0 y X @ i2P ¼

!

nm; ij1

1 A

i¼ j

b R m; j

y X ¼ ði þ j þ 1Þ 2 Pnm; i i¼0

!

455

456


0 B B B B n1 B X B ¼ k p; TDB B B m¼1B B B @

y X j¼0

j

2

b R m; j

y X

¼ k p; TDB

m¼1

þ

y X

b R m; j

y X

1 ¼ i 2 Pnm; i

C C C C y y y y C X X X X b ¼ b ¼ þ2 jR m; j iPnm; i þ 2 jR m; j Pnm; i C C C j¼0 j¼0 i¼0 i¼0 C C y y y y X X X X C b ¼ b ¼ A þ2 R m; j iPnm; i þ R m; j Pnm; i j¼0

i¼0

j¼0 n1 X

¼ Pnm; i

i¼0

i¼0

j¼0

i¼0

¼2 ¼1 ¼ b Fmb2 Pnm þ Rm Cnm þ 2Fmb1 Cnm ¼1 b1 ¼1 b b ¼ þ 2Fm Pnm þ 2R m Cnm þ R m Pnm

! ð52Þ

The principles of the pseudo-distribution model are applicable to any reactor. Results are presented for a semi-batch reactor using kinetic data from a realistic mixed-metallocene system in Figure 9.8. Construction of the Full 2D Distribution The pseudo-distribution approach thus allows us to calculate the 1D CLD and the average number of branches versus chain length. The question arises of how to obtain the full bivariate chain length–number of branches distribution from the pseudo-distribution model. The issue is that construction of branching distributions at each chain length requires an assumption to be made about the shape of the branching distribution, or more precisely for chains originating from the branching catalyst (Pnb¼ ). For instance, if we suppose that each monomer unit in a chain of given length has equal probability of being branched, then the branching distribution can be described by a binomial distribution. However, from the branching polydispersity as calculated from Eq. (34) we observe that the distribution for dead chains from the branching catalyst is narrower than a binomial distribution. An alternative approximation method is required, generating branching distributions at a given chain length with correct values for both the average number of branch points N n and the branching polydispersity Dn. A three-parameter binomial distribution modified by raising of it to a power an (the third parameter) varying with chain length, the variable power binomial distribution [VPBD] can satisfy these requirements: 9.6.2.2

pn; N ¼

n N

ðrn Þ N ð1 rn ÞðnNÞ

an ð53Þ

Note that in the VPBD the parameter a permits the distribution width to be changed: a > 1 leads to a narrower distribution. The fitting procedure is described in more detail by Iedema et al. [20]. The VPBD method turns out to yield identical solutions to the classes (Section 9.5) and pgf (Section 9.7) methods as applied to the mixed-metallocene problem. The resulting full bivariate CLD/DBD for the case of a CSTR is shown in Figure 9.9, together with the VPBD fit.

9.6 Pseudo-distribution Approach 1.4

Semibatch, rCGC = 0.2

dw/d{log(MW)}

1.2 1 0.8 0.6

600 s

0.4 0.2

t = 100 s

0 3 10

4

5

10

6

10

10

7

10

Molecular Weight x 10-3

0.8

0.6

600 s 500 s 400 s

Batch time

Branching density ρb

1

300 s 200 s

0.4

100 s

0.2 0 2 10

Semibatch, rCGC = 0.2 3

10

4

Chain Length

10

Fig. 9.8. Mixed-metallocene polymerization of ethylene in a semibatch reactor; branching (constrained geometry) catalyst: CGC-Ti; linear catalyst: Et[Ind]2 ZrCl2 . Reactor and kinetic data: initial concentration CGC-Ti: 8 107 kmol m3 ; initial concentration Et[Ind]2 ZrCl2 : 3:2 106 kmol m3 ; monomer molar feed

5

10

flow: 8 107 kmol s1 ; batch time: 600 s; k i; CGC =k p; CGC ¼ k i; lin =k p; lin ¼ 1; k p; CGC ½M ¼ k p; lin ½M ¼ 500 s1 ; kb; CGC ¼ 0:3 s1 ; kb; lin ¼ 0:7 s1 ; km; CGC =k p; CGC ¼ 0; km; lin =k p; lin ¼ 0:0014; kH; CGC ¼ 250 m 3 kmol1 s1 ; kH; lin ¼ 0; k p; TDB ¼ 1750 m 3 kmol1 s1 .

We conclude that the pseudo-distribution approach can be applied successfully, provided a good approximation can be made for the branching distribution at given chain length from the branching moments. The method is valid for batch reactors as well, in contrast to, for example, the pgf–cascade method (Section 9.7), which is restricted to steady state reactors. It is to be preferred over classes methods in cases, like the metallocene one, where the second distribution dimension may assume high values as well.

457


an

3

2

10-2

Fraction

458

10-4

VPBD power

10 1

10 2

n4

10 3

10

10 5

10-6 CSTR rCGC = 0.2 [H2] = 1.13 10-3 kmole/m3

10-8 10-10

102

10-12 0

103 10

20

30

Number of Branche s N

104

40

Mixed-metallocene polymerization of ethylene in a CSTR. Kinetic data are the same as in Figure 9.8. Residence time CSTR: 300 s; feed concentrations identical to initial concentrations in Figure 9.8. Bivariate chain length/number of branches distribution. Hydrogen present. Based on molecular weight distribution and branching distribution from

Fig. 9.9.

50

60

105

n ai Ch

100 101

th ng e L

n

the Galerkin-FEM model. Assumption: variable power binomial distribution of number of branch points at each chain length of dead chains from CGC-Ti with power an as shown in the insert. The discontinuity at N ¼ 1 indicates the existence of a ‘‘ridge’’ at N ¼ 0, due to the contribution of branchless chains from Et[Ind]2 ZrCl2 .

9.6.3

CLD/Number of Terminal Double Bonds (TDB) Distribution for Poly(vinyl acetate) – More than one TDB per Chain General Case This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6. Here, we address the more general problem of more than one TDB per chain [9]. This occurs as a consequence of insertion of TDB chains created by disproportionation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a 1D formulation by developing the TDB and branching moment expressions. The (N; M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by: 9.6.3.1

N; M Fn;:;:

¼

y X i¼0

i

N

y X k¼0

! M

k R n; i; k

N; M Cn;:;:

¼

y X i¼0

i

N

y X k¼0

! M

k Pn; i; k

ð54Þ


459

Tab. 9.11. The general (N; M)th double moment formulation of the population balance equation set of Table 9.10, obtained by multiplying by the TDB number and branching number indices, i N and k M , and subsequent summation over these indices.

Propagation Termination by disproportionation


dFnN; M

d

y X i¼0

iN

y X

k M R n; i; k

k¼0

N; M þ¼ k p MðFn1 FnN; M Þ

¼ dt dt N; M dFn þ¼ k td l 0 FnN; M dt

y y X X dCnN; M 1 þ¼ k td l 0 iN k M R n; i1; k þ FnN; M 2 dt i¼0 k¼0

! [a]

dFnN; M þ¼ k tc l 0 FnN; M dt y y n1 X i X k X X dCnN; M 1 X þ¼ k tc iN kM R m; j; l R nm; ij; kl 2 dt m ¼1 j¼0 l ¼0 i¼0 k¼0

Transfer to monomer

Transfer to polymer

dFnN; M þ¼ k m MFnN; M ; dt dCnN; M þ¼ k m MFnN; M dt

dF1N; M þ¼ k m l 0 M dt

y y X X dFnN; M þ¼ k tp l 0 n iN k M Pn; i; k1 m1 FnN; M dt i¼0 k¼0

!

y y X X dCnN; M þ¼ k tp l 0 n iN k M Pn; i; k1 þ m1 FnN; M dt i¼0 k¼0

!

0


!

1 y X y X y X iPn; j; l B R n; i; k C y y B C n¼1 j¼0 l ¼0 X X dFnN; M C N MB þ¼ kdb i k B C n1 X iþ1 X k B C dt X i¼0 k¼0 @ A þ jPm; j; l R nm; ijþ1; kl1 m ¼1 j¼0 l ¼0

dCnN; M þ¼ kdb l 0 CnðNþ1Þ; M dt [a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.

Performing the corresponding summations on the equations in Table 9.6, one obtains the (N; M)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N; M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M ¼ 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N ¼ 0; 1, and 2. Solving the model thus essentially means solving the population balances of the real concentration distributions R n and Pn and the pseudo-distributions FnN; M and CnN; M .

460

9 Mathematical Methods Tab. 9.12. The (0; 0)th moment distribution formulation of the population balance equation set of Table 9.11 (taking N ¼ 0 and M ¼ 0); this set is solved by the TDB moment distribution model.

Propagation Termination by disproportionation Termination by recombination Transfer to monomer Transfer to polymer Terminal double bond propagation

dR n þ¼ k p MðR n1 R n Þ dt dR n dPn þ¼ k td l 0 R n ; þ¼ k td l 0 R n dt dt n1 dR n dPn 1 X þ¼ k tc l 0 R n ; þ¼ k tc R m R nm 2 m ¼1 dt dt dR n dR1 dPn þ¼ k m MR n ; þ¼ k m l 0 M þ¼ k m MR n dt dt dt dR n dPn þ¼ k tp ðl 0 nPn m1 R n Þ; þ¼ k tp ðl 0 nPn þ m1 R n Þ dt dt ! y n1 X X dR n dPn þ¼ kdb R n þ¼ kdb l 0 Cn1; 0 Cn1; 0 þ Cm1; 0 R nm ; dt dt n¼1 m ¼1

The resulting equations for N ¼ 0 and M ¼ 0 are listed in Table 9.12. Here, the (0; 0)th moments are the usual 1D chain length distribution variables, defined by:

Rn ¼

y X y X

R n; i; k ð¼ Fn0; 0 Þ Pn ¼

i¼0 k¼0

y X y X

Pn; i; k ð¼ Cn0; 0 Þ

ð55Þ

i¼0 k¼0

All of the derivations are straightforward, but the TDB propagation deserves closer examination. From Table 9.11 we have the general (N; M) formulation in: y y X X dFnN; M iN kM þ¼ kdb dt i¼0 k¼0

R n; i; k

y X y X y X

iPn; j; l þ

n¼1 j¼0 l¼0

n1 X iþ1 X k X

! jPm; j; l R nm; ijþ1; kl1

m¼1 j¼0 l ¼0

ð56Þ Taking M ¼ 0 and N ¼ 0, the first term between the brackets can be rewritten as: y X y X

R n; i; k

y X y X y X

! iPn; i; k

n¼1 i¼0 k¼0

i¼0 k¼0

y X y y X X ¼ ðR n; i; k Cn1; 0 Þ ¼ i¼0 k¼0

n¼1

y X n¼1

! Cn1; 0

y X y X i¼0 k¼0

R n; i; k ¼ R n

y X n¼1

Cn1; 0

ð57Þ


461

The second term between the brackets can be rewritten as: y X y X n1 X iþ1 X k X

jPm; j; l R nm; ijþ1; kl1

i¼0 k¼0 m¼1 j¼0 l ¼0

8 0 19 = X n1 <X y X k X y y n1 X X @ jPm; j; l ¼ R nm; ijþ1; kl1 A ¼ Cm1; 0 R nm ð58Þ ; : m¼1 k¼0 l¼0 j¼0 m¼1 i¼j1 Thus, we find the TDB propagation term for the living chains to be given by: y n1 X X dR n 1; 0 þ¼ kdb R n; i; k Cn1; 0 þ Cm R nm dt n¼1 m¼1

! ð59Þ

Applying M ¼ 0 and N ¼ 0 to the dead chains formulation yields: dPn þ¼ kdb l 0 Cn1; 0 dt

ð60Þ

The RHS expressions of Eqs. (59) and (50) contain higher TDB moment (1; 0) distributions (Cn1; 0 ) than the (0; 0) moment distributions, R n and Pn , they describe.

Tab. 9.13. The (1; 0)th moment distribution formulation of the population balance equation set of Table 9.11 (taking N ¼ 1 and M ¼ 0); this set is solved by the TDB moment distribution model.


dFn1; 0 1; 0 þ¼ k p MðFn1 Fn1; 0 Þ dt dFn1; 0 þ¼ k td l 0 Fn1; 0 ; dt


dFn1; 0 þ¼ k tc l 0 Fn1; 0 ; dt

Propagation

Transfer to monomer Transfer to polymer Terminal double bond propagation

! y X dCn1; 0 1 1 þ¼ k td l 0 iR n; i1 þ Fn1; 0 ¼ k td l 0 Fn1; 0 þ R n [a] 2 2 dt i¼0 n1 X dCn1; 0 þ¼ k tc Fm1; 0 R nm dt m ¼1

dFn1; 0 dF11; 0 dCn1; 0 þ¼ k m MFn1; 0 ; þ¼ k m l 0 M; þ¼ k m MFn1; 0 dt dt dt dFn1; 0 dCn1; 0 þ¼ k tp ðl 0 nCn1; 0 m1 Fn1; 0 Þ; þ¼ k tp ðl 0 nCn1; 0 þ m1 Fn1; 0 Þ dt dt ( ) y n1 X X dFn1; 0 ; 1; 0 1; 0 2; 0 1; 0 1; 0 1; 0 þ¼ kdb Fn Cn þ ðCm R nm þ Cm Fnm Cm R nm Þ dt n¼1 m ¼1 dCn1; 0 þ¼ kdb l 0 Cn2; 0 dt

[a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.

462


This is a direct consequence of the fact that the reactivity of the dead chains depends on the number of TDBs they carry. Hence, at this point we are already confronted with the closure problem present in the TDB moment model, anticipating that solving these higher moments leads to even higher moments in the equations. Next, we will derive the higher moment equations. For N ¼ 1 and M ¼ 0 we obtain the set of population balance equations for the pseudo-distributions of order (1; 0) as listed in Table 9.13. The termination by recombination equation is derived by developing the summation term as: ! y n1 X i X 1X i R m; j R nm; ij 2 i¼0 m¼1 j¼0 0 1 n1 X y y X 1X ¼ R m; j @ iR nm; ij A 2 m¼1 j¼0 i¼j n1 X y y X 1X R m; j ði þ jÞR nm; i ¼ 2 m¼1 j¼0 i¼0

!

n1 y y y y X X X X 1X jR m; j R nm; i þ R m; j iR nm; i ¼ 2 m¼1 j¼0 j¼0 i¼0 i¼0

¼

!

n1 n1 X 1X 1; 0 ðFm1; 0 R nm þ R m Fnm Þ¼ Fm1; 0 R nm 2 m¼1 m¼1

ð61Þ

This yields for the recombination contribution to the population balance: n1 X dCn1; 0 Fm1; 0 R nm þ¼ k tc dt m¼1

ð62Þ

With respect to TDB propagation, with M ¼ 0 and N ¼ 1 the first term between brackets becomes that in Eq. (64) y X y X

iR n; i; k

! iPn; i; k

n¼1 i¼0 k¼0

i¼0 k¼0

¼

y X y X y X

y X y X i¼0 k¼0

iR n; i; k

y X n¼1

! Cn1; 0

¼

y X n¼1

! Cn1; 0

y X y X

iR n; i; k ¼ Fn1; 0

i¼0 k¼0

y X

Cn1; 0

n¼1

ð63Þ and the second term, somewhat more complicated, becomes

9.6 Pseudo-distribution Approach y X y X i

n1 X iþ1 X k X

i¼0

m¼1 j¼0 l¼0

k¼0

¼

y X i¼0

( i

n1 X i X

! jPm; j; l R nm; ijþ1; kl1 )

ð jPm; j R nm; ijþ1 Þ

m¼1 j¼0

8 0 19 n1 X y y <X = X @ jPm; j iR nm; ijþ1 A ¼ :m ¼1 j¼0 ; i¼ jþ1 ( ¼

n1 X y X

y X jPm; j ði þ j 1ÞR nm; i

m¼1 j¼0

¼

n1 X y X m¼1

¼

n1 X

!)

i¼0

2

j Pm; j

j¼0

y X

R nm; i þ

y X j¼0

i¼0

y X jPm; j ði 1ÞR nm; i

!

i¼0

2; 0 1; 0 ðCm R nm þ Cm1; 0 Fnm Cm1; 0 R nm Þ

ð64Þ

m¼1

Thus, we find the TDB propagation terms for the living and dead chains to be: ( ) y n1 X X dFn1; 0 1; 0 1; 0 2; 0 1; 0 1; 0 1; 0 Cn þ ðCm R nm þ Cm Fnm Cm R nm Þ ð65Þ þ¼ kdb Fn dt n¼1 m¼1 dCn1; 0 þ¼ kdb l 0 Cn2; 0 dt

ð66Þ

The higher moments observed earlier on the RHS are seen again. We finally present the result for one higher moment distribution, N ¼ 2, M ¼ 0, for which we obtain the set of population balance equations for the pseudodistributions of order (2; 0) as listed in Table 9.14. First, the termination by recombination term is developed: y n1 X i X 1X i2 R m; j R nm; ij 2 i¼0 m¼1 j¼0

!

0 1 ! n1 X y y n1 X y y X X 1X 1X 2 2 @ A ¼ R m; j i R nm; ij ¼ R m; j ði þ jÞ R nm; i 2 m¼1 j¼0 2 m¼1 j¼0 i¼ j i¼0

463

464


Tab. 9.14. The (2; 0)th moment distribution formulation of the population balance equation set of Table 9.11 (taking N ¼ 2 and M ¼ 0); this set is solved by the highest TDB moment version of the TDB moment distribution model.


dFn2; 0 2; 0 þ¼ k p MðFn1 Fn2; 0 Þ dt dFn2; 0 þ¼ k td l 0 Fn2; 0 ; dt


dFn2; 0 þ¼ k tc l 0 Fn2; 0 ; dt

Propagation

Transfer to monomer

Transfer to polymer


! y X dCn2; 0 1 1 þ¼ k td l 0 i 2 R n; i1 þ Fn2; 0 ¼ k td l 0 Fn2; 0 þ Fn1; 0 þ R n [a] 2 2 dt i¼0 n1 X dCn2; 0 1; 0 þ¼ k tc ðFm2 R nm þ Fm1; 0 Fnm Þ dt m ¼1

dFn2; 0 dF12; 0 þ¼ k m MFn2; 0 ; þ¼ k m l 0 M dt dt 2; 0 dCn þ¼ k m MFn2; 0 dt dFn2; 0 dCn2; 0 þ¼ k tp ðl 0 nCn2; 0 m1 Fn2; 0 Þ; þ¼ k tp ðl 0 nCn2; 0 þ m1 Fn2; 0 Þ dt dt 8 y X > > Fn2; 0 Cn1; 0 > > < 2; 0 dFn n¼1 þ¼ kdb n1 3; 0 2; 0 1; 0 > dt X > Cm R nm þ Cm1; 0 Fnm þ 2Cm2; 0 Fnm > > þ : 2; 0 1; 0 1; 0 2C R 2C F þ Cm1; 0 R nm nm m m nm m ¼1

9 > > > > = >; > > > ;

dCn2; 0 þ¼ kdb l 0 Cn3; 0 dt [a] The first term between brackets equals zero in the case where TDB creation by disproportionation is not accounted for.

n1 X y y y y y y X X X X X 1X ¼ j 2 R m; j R nm; i þ R m; j i 2 R nm; i þ jR m; j iR nm; i 2 m¼1 j¼0 j¼0 j¼0 i¼0 i¼0 i¼0

¼

!

n1 n1 X 1X 2; 0 2; 0 1; 0 1; 0 ðFm R nm þ R m Fnm þ 2Fm1; 0 Fnm Þ¼ ðFm2 R nm þ Fm1; 0 Fnm Þ 2 m¼1 m¼1

ð67Þ This yields for the recombination contribution to the population balance: n1 X dCn2; 0 2 1; 0 þ¼ k tc ðFm R nm þ Fm1; 0 Fnm Þ dt m¼1

ð68Þ

The first term in the equation describing TDB propagation is derived in a similar way to Eq. (62), while the second term follows as:


X y X y y y X X dR n; i; k i i2 ¼ dt i¼0 k¼0 i¼0 k¼0 ¼

y X

( i

2

n1 X iþ1 X k X

! jPm; j; l R nm; ijþ1; kl1

m¼1 j¼0 l¼0

n1 X i X ð jPm; j R nm; ijþ1 Þ

)

m¼1 j¼0

i¼0

8 0 19 n1 X y y <X = X @ jPm; j i 2 R nm; ijþ1 A ¼ :m¼1 j¼0 ; i¼ jþ1 ( ¼

n1 X y X m¼1 j¼0

¼

n1 X

y X jPm; j ði þ j 1Þ 2 R nm; i

!)

i¼0

2; 0 1; 0 2; 0 ðCm3; 0 R nm þ Cm1; 0 Fnm þ 2Cm2; 0 Fnm 2Cm R nm

m ¼1 1; 0 2Cm1; 0 Fnm þ Cm1; 0 R nm Þ

ð69Þ

The total contribution of TDB propagation to the population balance is thus given by: ( y X dFn2; 0 Cn1; 0 þ¼ kdb Fn2; 0 dt n¼1 n1 3; 0 2; 0 2; 0 1; 0 X Cm R nm þ Cm1; 0 Fnm þ 2Cm2; 0 Fnm 2Cm R nm þ 1; 0 2Cm1; 0 Fnm þ Cm1; 0 R nm m¼1

)

ð70Þ For the dead chains we have: dCn2; 0 þ¼ kdb l 0 Cn3; 0 dt

ð71Þ

It is obvious that even higher TDB moment distribution balances can be constructed, but we will restrict ourselves to the ones developed for up to the second moments. When applying the TDB moment model, in principle two solution strategies are possible: one using the zeroth and first TDB moments only and the second with zeroth, first, and second TDB moments. In the first case we have to find a closure relationship for the second moment, and in the second case, one for the third moment. Below, we show that the system becomes simpler in the case of a maximum of one TDB per chain.

465

466


Closure relations

We have adopted a simple form for these relationships:

Cn2; 0 ¼ Dn

Cn1; 0 1; 0 C Pn n

ð72Þ

Cn3; 0 ¼ Dn0

Cn2; 0 2; 0 C Cn1; 0 n

ð73Þ

Here the functions Dn and Dn0 are in fact polydispersities of the branching moment distributions and in principle are to be determined as functions of chain length n. Inserting these closure relationships in Eqs. (65), (66), and (70), (71), reduces them to: ( ) y n1 X X dFn1; 0 Cm1; 0 1; 0 1; 0 1; 0 1; 0 1; 0 þ¼ kdb Fn Cn þ Cm Fnm þ Dm 1 Cm R nm dt Pm n¼1 m¼1 ð65aÞ dCn1; 0 dt

þ¼ kdb l 0 Dn

Cm1; 0 Pn

Cn1; 0

ð66aÞ

( y X dFn2; 0 þ¼ kdb Fn2; 0 Cn1; 0 dt n¼1 2 39 Cm2; 0 > 2; 0 n1 C 1; 0 F 2; 0 þ 2C 2; 0 F 1; 0 þ D 0 X 1 Cm R nm 7= m nm m 6 m nm 1; 0 C þ 4 5 m > ; m ¼1 1; 0 þ 2Cm1; 0 Fnm þ Cm1; 0 R nm ð70aÞ dCn2; 0 C 2; 0 þ¼ kdb l 0 Dn0 n1; 0 Cn1; 0 dt Cn

ð71aÞ

An exact determination of Dn and Dn0 is not possible, but they can be estimated from results of other methods, for instance a TDB classes model in regions with few TDBs per chain. TDB Pseudo-distribution Approach for a Maximum of one TDB per Chain This case allows a few simplifications. Firstly, only the first TDB moment distributions, Fn1; 0 and Cn1; 0 , have to be solved in addition to the real concentration distributions, since all higher TDB moment distributions are identical to these. Second, the TDB propagation contributions to the population balances become greatly simplified and their closure problem vanishes, since with Cn2; 0 ¼ Cn1; 0 Eqs. (65) and (66) become: 9.6.3.2

9.6 Pseudo-distribution Approach y n1 X X dFn1; 0 1; 0 Cn1; 0 þ Cm1; 0 Fnm þ¼ kdb Fn1; 0 dt n¼1 m¼1

dCn1; 0 þ¼ kdb l 0 Cn1; 0 dt

! ð74Þ

ð75Þ

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

467

468


Fig. 9.10. Radical polymerization of vinyl acetate in a CSTR. The reactor and kinetic data are the same as in Figure 9.6. Left: chain length distributions for dead and living chains.

The part of the living chain CLD from the TDB moment model beyond 10 10 is estimated. Right: numbers of TDBs per chain for various kdb .

9.6 Pseudo-distribution Approach Tab. 9.15. Reaction mechanisms for LDPE (first subscript: chain length; second subscript: number of branch points).

Mechanism

Reaction equation

Rate factor

Initiation Propagation Disproportionation termination Recombination termination Transfer to monomer Transfer to chain-transfer agent S Transfer to polymer Pre-scission (formation of secondary macroradicals) Scission of macroradicals

I2 ! 2I I þ M ! R1; 0 R n; i þ M ! R nþ1; i R n; i þ R m; j ! Pn; i þ Pm; j R n; i þ R m; j ! Pnþm; iþj R n; i þ M ! Pn; i þ R1; 0 R n; i þ S ! Pn; i þ R1; 0 R n; i þ Pm; j ! Pn; i þ R m; jþ1 þ LCB R n; i þ Pm; j ! Pn; i þ Rm; j

kd ; k i f kp k td k tc km kS k trp k rs

Rn; i ! R nm; ij þ Pm; j

ksec

9.6.4

Radical Polymerization of Ethylene to Low-density Polyethylene (LDPE) Introduction This problem has received considerable attention for a long time. Modeling is complicated, since branching is involved, and the importance of random scission for LDPE has now been recognized [31, 32, 54]. We address the 2D problem of CLD/ DBD calculation here. The full set of reaction equations is given in Table 9.15, notation in Table 9.16, and population balance equations in Table 9.17. For merely 9.6.4.1

Tab. 9.16.

Notation for LDPE system.

R n; i Rn; i Pn; i m0 ¼

Primary radical chains Secondary radical chains Dead polymer chains y X y X

Pn; i

l0 ¼

n¼1 i¼0

Rn ¼

y X

R n; i

y X y X

y X

Rn ¼

y X

y X

Rn; i

y X y X

iR n; i

F1 n ¼

y X

Pn ¼

i¼0

Bn ¼ Cn =Pn rn ¼ Cn =ðPn nÞ.

F2 n ¼

y X

y X

Pn; i

i¼0

iRn; i

Cn1 ¼

i¼0

i 2 R n; i

Rn; i

n¼1 i¼0

i¼0

i¼0

Fn2 ¼

l0 ¼

n¼1 i¼0

i¼0

Fn1 ¼

R n; i

y X

iPn; i

i¼0

i 2 Rn; i

i¼0

number of branches per chain branching density

Cn2 ¼

y X i¼0

i 2 Pn; i

469

470

9 Mathematical Methods Tab. 9.17.

Population balance equations for LDPE.

2D population balance equations dR n; i ¼ k p MðR n1; i R n; i Þ ðk td þ k tc Þl 0 R n; i þ k trp ðnl 0 Pn; i1 m1 R n; i Þ dt ks SÞR n; i k rs m1 R n; i þ ðk m M þ8 9 = y < y X X þ ksec gðm; nÞ ½bðm; j; njiÞR m; j : ; m ¼nþ1 j¼i

(a)

þ ðk m M þ ks SÞl 0 dðn 1ÞdðiÞ þ k i f MIdðn 1ÞdðiÞ n1 X i dPn; i 1 X ¼ k td l 0 R n; i þ k tc ðR m; j R nm; ij Þ þ k trp ðnl 0 Pn; i þ m1 R n; i Þ 2 m ¼1 j¼0 dt

(b)

þ ðk m M þ k8s SÞR n; i þ k rs ðnl 0 Pn; i þ m1 9 R n; i Þ y < y = X X þ ksec gðl; nÞ ½bðm; j; njiÞR m; j : ; j¼i l ¼nþ1 dR n; i

¼ nk rs l 0 Pn; i ksec R n; i dt 1D concentration distribution equations dR n ¼ k p MðR n1 R n Þ ðk td þ k tc Þl 0 R n þ k trp ðnl 0 Pn m1 R n Þ ðk m M þ ks SÞR n dt

k rs m1 R n þ ksec

y X

(c)

(d)

gðm; nÞRl þ ðk m M þ ks SÞl 0 dðn 1Þ þ k i f MIdðn 1Þ

m ¼nþ1 n1 dPn 1 X ¼ k td l 0 R n þ k tc R m R nm þ ðk trp þ k rs Þðm1 R n nl 0 Pn Þ þ ðk m M þ ks SÞR n 2 m ¼1 dt

þ ksec

y X

(e)

gðm; nÞR m

m ¼nþ1

dR n ¼ nk rs l 0 Pn ksec R n dt 1D pseudo-distribution equations, 1st branching moment y dFn1 X dR n; i 1 ¼ k p MðFn1 i þ Fn1 Þ ðkM M þ kS SÞFn1 ðk td þ k tc Þl 0 Fn1 dt dt i¼0

(f )

(g)

þ k trp fm1 Fn1 þ l 0 nðCn1 þ Pn Þg k rs m1 Fn1 ( " j #) y y X X X þ ksec gðm; nÞ R m; ibðm; j; njiÞ þ ðkM M þ kS SÞl 0 dðn 1Þ j m ¼nþ1

j¼0

i¼0

þ k i f Mdðn 1Þ y n1 1 X dCn X dPn; N ¼ ¼ ðkM M þ kS SÞFn1 þ k td l 0 Fn1 þ k tc i Fm1 R nm dt dt m ¼1 i¼0 ( " j #) y y X X X þ ðk trp þ k rs Þðm1 Fn1 l 0 nCn1 Þ þ ksec gðn; mÞ R m; ibðm; j; njiÞ j n¼mþ1 y dR X n; i ¼ ¼ nk rs l 0 Cn1 ksec F1 i n dt dt i¼0

dF1 n

j¼0

(h)

i¼0

(i)

9.6 Pseudo-distribution Approach Tab. 9.17. (continued)

1D pseudo-distribution equations, 2nd branching moment y dFn2 X dR n; i 2 ¼ ¼ k p MðFn1 i2 þ Fn2 Þ ðkM M þ kS SÞFn2 ðk td þ k tc Þl 0 Fn2 dt dt i¼0

( j)

þ k trp fm1 Fn2 þ l 0 nðCn2 þ 2Cn1 þ Pn Þg ( " j #) y y X X X gðm; nÞ R m; i 2 bðm; j; njiÞ þ ksec j m ¼nþ1

j¼0

i¼0

k rs m1 Fn2 þ ðkM M þ kS SÞl 0 dðn 1Þ þ k i f Mdðn 1Þ y n1 X dCn2 X dPn; i 1 ¼ ¼ ðkM M þ kS SÞFn2 þ k td l 0 Fn2 þ k tc i2 ðFm2 R nm þ Fm1 Fnm Þ dt dt m ¼1 i¼0 ( " j #) y y X X X 2 2 2 gðn; mÞ R m; j i bðm; j; njiÞ þ ðk trp þ k rs Þðm1 Fn l 0 nCn Þ þ ksec n¼mþ1

j¼0

y X dR n; i ¼ ¼ nk rs l 0 Cn2 ksec F2 i2 n dt dt i¼0

dF2 n

(k)

i¼0

(l)

computational reasons it is assumed that the scission reaction proceeds in two steps, a pre-scission reaction yielding a secondary macroradical Rn; i , and a subsequent breakage of this radical: k rs ðn1Þ

Pn; i þ R m; j ! Rn; i þ Pm; j

ð76Þ

ksec Rn; i !

ð77Þ

R nm; ij þ Pm; j

The term (n 1) in the pre-scission equation [Eq. (76)] is due to the fact that we are dealing with random (pre-)scission here: that is, every CaC bond has equal probability of being attacked to form a secondary radical. The population balance contribution from this reaction is similar to that from a transfer-to-polymer reaction (see Table 9.17). The 2D population balance contribution from the actual scission step reads as (Table 9.17): 9 8 = y < y X X dR n; i gðm; nÞ ½bðm; j; njiÞRm; þ¼ ksec j ; : dt m¼nþ1 j¼i

ð78Þ

Here, the functions g and b are the most general form of probability functions, expressing the way in which the fragment lengths are distributed (g) and how the branch points are redistributed on these fragments (b). The function gðm; j; nÞ describes the probability that a chain of length n with j branch points breaks into a fragment with length m, while bðm; j; nj jÞ is the conditional probability that a fragment of length m created by scission of a chain n=i carries exactly j branch points. The fragment length distribution function reflects the scission mechanism acting.

471

472


The Galerkin-FEM approach allows us to choose any model describing such mechanisms, either chemical or mechanical, if they are expressed in terms of fragment lengths as functions of overall chain length or numbers of branch points [54]. An overview of such models is given elsewhere [32]. In the most realistic models the effect of branching and architectures is accounted for; this gives rise to predominantly long and short fragments. The fundamental problem here, to start with, is that the second dimension of the 2D population balance problem, the branching distribution, has to be solved to calculate the first dimension, CLD. Moreover, not only the branching distribution, but also the character of the branched architectures, determine the scission function. One way of addressing this problem is employing an empirical approximation relationship for the fragment length function gðn; mÞ found for ‘‘topological scission’’ of LDPE architectures [33]: 1 ð2s þ mÞ 2 gðn; mÞ ¼ n1 ( X 1 m¼1

þ

ð2s þ mÞ

1 ð2s þ n mÞ 2

2þ

1

)

ð79Þ

ð2s þ n mÞ 2

with s the average segment length, according to: s¼

nn 1 þ 2 LCB=m 0

ð80Þ

In Eq. (80), LCB is the overall concentration of long chain branches, following from a balance coupled with the transfer-to-polymer reaction (see Table 9.17). The fragment length distribution gðn; mÞ for topological scission is a function of chain and fragment length only, and independent of the number i of branch points on the original chain. Note that this is an approximation method that fits to the differential equation approach. Obviously, it accounts for branching architectures in an averaged manner. Fully accounting for architectures is not possible with the differential equation method, since it does not describe connectivity in molecules. Doing this in more rigorous ways, full [11–15] or conditional [33–35] Monte Carlo (MC) simulations can be used (see Section 9.8). We stress here that the strength of the differential method discussed here is its ability to implement any function describing scission in overall molecular dimensions, such as chain length. This is not readily possible in MC simulations. In order to solve the one-dimensional chain length distribution problem, the scission contribution of Eq. (78) is summed over the number of branch points on i P bðn; i; mj jÞ ¼ 1, so this population balance assumes fragments, j. By definition a one-dimensional form: j¼0 y X dR m gðn; mÞR n þ¼ ksec dt n¼mþ1

ð81Þ


The 1D population balance for the pseudo-distribution or first branching moment distribution Fn follows by taking the first branching moment of Eq. (78):

d

y X

jR m; j

j¼0

dt

( " #) y y i X X X dFm gðn; mÞ R n; i jbðn; i; mj jÞ ¼ þ¼ ksec dt n¼mþ1 j¼0 i¼0

ð82Þ

Depending on the branch point redistribution function, the term in square brackets (in fact the expectation value of the number of branches on a fragment) can be evaluated. Similarly, we find for the second branching moment expression:

d

y X

j 2 R m; j

j¼0

dt

( " #) y y i X X X dFm2 2 ¼ þ¼ ksec gðn; mÞ R n; i j bðn; i; mj jÞ dt n¼mþ1 j¼0 i¼0

ð83Þ

The task is now to find empirical expressions for the first and second branching moment summation terms between brackets in Eqs. (82) and (83) similar to Eq. (79) for the fragment lengths. The shape of these is strongly dependent on the scission mechanism [32]. Note that finding solutions for this problem is important for calculation of the branching density and the branching polydispersity as functions of chain length. From this the shape of the branching distribution at constant chain length can be estimated, which then produces an estimation of the full CLD/DBD. Finally, in Figure 9.11 we show MWDs for two different scission models. The ‘‘linear scission’’ case assumes scission of unbranched chains. The topological scission case employs the fragment length function of Eq. (79). A marked difference is observed. 9.6.5

Radical Copolymerization Introduction We now address the problem of finding the 3D distribution of chain lengths, copolymer composition, and sequence length distribution in radical copolymerization. We have several options here. One could be explicitly solving the 3D conseq centration variable at the sequence level, Pn; i; s , denoting the concentration of sequences of length s on chains with a total number of monomer units n (usually called chain length), and number of monomer units of one kind i. Note that i=n is then the fractional copolymer composition. We will not do this, but instead choose a simpler option and solve a different 3D concentration variable at the seq chain level, Pn; i; s . Subscripts n and i have the same meaning as in Pn; i; s , but s here denotes the number of monomer sequences of one kind. Solving this prob9.6.5.1

473


0.6 0.6 0.5 0.5

dW/d{log(n)}

474

0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.0 0 10

1e+0

101

1e+1

102

1e+2

103

1e+3

104

1e+4

105

1e+5

Chain Length

106

1e+6

107

1e+7

108

1e+8

Radical polymerization of ethylene in a CSTR. Linear and topological scission with the same polydispersity of 26 as experimental MWD from SEC-MALLS [35]. Solid line, experimental MWD; dash-dot line, topological scission result; dash-dot-dot line, linear scission result. Fig. 9.11.

lem in the full three dimensions would yield the average sequence length on chains Pn; i; s , simply following as i=s. We will show next how to solve the problem using pseudo-distributions and thus find the average copolymer composition and sequence length as a function of chain length, n. Reaction equations are listed in Table 9.18, notation and definitions in Table 9.19. Note that macroradicals are specified according to terminal unit, with an upper index A or B. Balance Equations All the reactions listed in Table 9.18 have contributions to the set of balance equations. We will give a few examples of the generation of these contributions from some of the reaction equations. 9.6.5.2

Tab. 9.18.

Reaction equations for radical copolymerization.

Initiation kd

Propagation kAA

Termination kc

A A R n; i; s þ A ! R nþ1; iþ1; s

A A R n; i; s þ R m; j; t ! Pnþm; iþj; sþt

I þ A ! R1;A 1; 0

B B R n; i; s þ B ! R nþ1; i; s

B B R n; i; s þ R m; j; t ! Pnþm; iþj; sþt

I þ B ! R1;B 0; 0

A B R n; i; s þ B ! R nþ1; i; s

A B R n; i; s þ R m; j; s ! Pnþm; iþj; sþt

I2 ! 2I ki

ki

kBB

kAB

kBA

B A R n; i; s þ A ! R nþ1; iþ1; sþ1

kc

kc

9.6 Pseudo-distribution Approach Tab. 9.19.

Notation and definitions. l 0A ¼

Overall moments

y X y X y X

A R n; i; s

l 0B ¼

n¼1 s¼1 i¼1

Pn; i ¼

2D concentrations

y X

Pn; i; s

R nA ¼

1D concentrations

Pn; s ¼

GnA ¼

1st moment distributions number of A sequences

FnA ¼

y X y X

y X y X

A R n; i; s

R nB ¼

m0 ¼

y X y X y X

y X

Pn; i; s

A iR n; i; s

GnB ¼

y X y X A s R n; i; s

R nA ¼

i¼1

Number of A units per chain as a function of n Number of A sequences per chain as a function of n Average sequence length of A as a function of n Average sequence length of B as a function of n

y X y X

y X y X

B R n; i; s

Pn ¼

B iR n; i; s

Ln ¼ Cn ¼

!

s¼1 i¼1

þ¼ kBA A

dt

y X y X

B B Rn1; i1; s1 ¼ kBA AR n1

ð84Þ

s¼2 i¼2

For the first moment distribution of the number of A units we have:

d

y X y X s¼1 i¼1

dt

! A iR n; i; s

¼

y X y y X X dGnA B B þ¼ kBA A iR n1; iR n1; i1; s1 ¼ kBA A i1 dt s¼2 i¼2 i¼2

ð85Þ The last term here contains the two-dimensional distribution multiplied by i: B i.R n1; i1 , which upon summation over i, due to its index (i 1), produces the B B and Gn1 . Hence, we get: sum of two one-dimensional distributions: R n1 dGnA B B þ¼ kBA AðR n1 þ Gn1 Þ dt

y X y X

Pn; i; s

iPn; i; s

y X y X s Pn; i; s s¼1

i¼1

nnA ¼ Ln =Pn snA ¼ Cn =Pn lnA ¼ nnA =snA lnB ¼ ðn nnA Þ=snA ¼ nnB =snA

kBA

Rn;A i; s

y X y X

s¼1 i¼1

A BxA-propagation: Rn,B i, s B A m RnB1, In this reaction all the indices iniB1, sB1 crease by 1. The contribution to the balance equations from which the length distribution is calculated simply follows by taking the double sum over i and s:

y X y X

A R n; i; s

s¼1 i¼1

y X y X B s R n; i; s s¼1

y X y X s¼1 i¼1

s¼1 i¼1

FnB ¼

Pn; i; s

n¼1 s¼1 i¼1

s¼1 i¼1

s¼1 i¼1

s¼1

B R n; i; s

i¼1

s¼1 i¼1

1st moment distributions of number of A units

y X y X y X n¼1 s¼1 i¼1

s¼1

d

475

ð86Þ

i¼1

476


Obviously, in this case the problem in obtaining the first moment distribution of the number of A sequences is exactly identical to that for the A units, so we have: dFnA B B þ Fn1 Þ þ¼ kBA AðR n1 dt

ð87Þ

k AA

A AxA-propagation: Rn,A i, s B A m RnB1, Here, only two of the three indices iniB1, s crease by 1. The length distribution is again calculated simply by taking the double sum over i and s which is identical to the length distribution term found in the previous propagation case, Eq. (84):

d

y X y X

! A R n; i; s

s¼1 i¼1

þ¼ kAA A

dt

y X y X

A A R n1; i1; s ¼ kAA AR n1 ;

ð88Þ

s¼2 i¼2

For the first moment distribution of the number of A units, for this propagation step we have:

d

y X y X

! iRn;A i; s

s¼1 i¼1

¼

dt

y X y y X X dGnA A A iRn1; iRn1; þ¼ kAA A i1; s ¼ kAA A i1 ; ð89Þ dt s¼2 i¼2 i¼2

which again is equal to the one found before, Eq. (85). Hence we end up with an expression containing two one-dimensional distributions on the right-hand side: dGnA A A þ Gn1 Þ þ¼ kAA AðR n1 dt

ð90Þ

For this propagation step the situation for the first moment distribution of A sequences is different, since we now have

d

y X y X i¼1 s¼1

dt

! A sR n; i; s

¼

y X y y X X dFnA A A sR n1; sR n1; þ¼ kAA A i1; s ¼ kAA A s; dt s¼2 i¼2 s¼2

ð91Þ

where in the last term the summation over s proceeds with a two-dimensional disA tribution Rn1; s simply having s as the index. This yields: dFnA A ; þ¼ kAA AFn1 dt containing only one one-dimensional distribution.

ð92Þ

9.6 Pseudo-distribution Approach kBB

B BxB propagation Rn,B i, s m RnB1, Only the chain length index is increased. Using i, s a similar argument to previously, we can say that the right-hand side expressions for both the number of A units and the number of A sequences moment distributions will produce two-dimensional distributions as intermediate results having i and s as indices. Summation of these will finally produce expressions like Eq. (92):

dGnB B þ¼ kBB BGn1 dt

ð93Þ

dFnB B þ¼ kBB BFn1 dt

ð94Þ kc

A Termination by combination: Rn,A i, s B Rm, This is the reaction j, t m P nBm, iBj, sBt equation for termination between macroradicals with identical terminal groups. The balance equation describing the production of dead chains Pn; i; s is:

n1 X i1 X s1 X dPn; i; s 1 A A ðR m; ¼ k tAA j; t R nm; ij; st Þ 2 dt m¼1 j¼1 t¼1

ð95Þ

The first moment distribution of the number of A units leads to:

d

y X y X s¼1 i¼1

dt

! iPn; i; s

( ) n1 X y i1 X y X s1 X X dLn 1 A A ¼ ¼ k tAA i ðR m; j; t R nm; ij; st Þ 2 dt m¼1 i¼1 j¼1 s¼1 t¼1 ( ) n1 X y i1 X X 1 A A i ðR m; j R nm; ij Þ ¼ k tAA 2 m¼1 i¼1 j¼1

ð96Þ

The last term can be rearranged to give for the first moment of the number of A units: 0 1 ( ) n1 X y i1 n1 X y y X X X X dLn 1 1 A A A @R A A ¼ k tAA i ðR m; iR nm; j R nm; ij Þ ¼ k tAA m; j ij 2 2 dt m¼1 i¼1 m¼1 j¼1 j¼1 i¼ j n1 X y y X X 1 A A R m; ði þ jÞR nm; ¼ k tAA j i 2 m¼1 j¼1 i¼0

!

n1 y y y y X X X X X 1 A A A A ¼ k tAA jR m; R nm; R m; iR nm; j iþ j i 2 m¼1 j¼1 j¼1 i¼1 i¼1 n1 n1 X X 1 A A A A ¼ k tAA ðGmA R nm þ Rm Gnm Þ ¼ k tAA GmA R nm 2 m¼1 m¼1

!

ð97Þ

477

478


An expression for the contribution from the termination reaction to the first moment distribution of the number of A sequences is obtained in an identical manner: n1 X dCn A ¼ k tAA CmA R nm dt m¼1

ð98Þ

For the termination between B-terminated macroradicals the expressions obtained are exactly the same as Eqs. (97) and (98). For the reaction between A- and Bterminated macroradicals we find: n1 X dLn B A B þ¼ k tAB ðGmA R nm þ Rm Gnm Þ dt m¼1

ð99Þ

n1 X dCn B A B ðCmA R nm þ Rm Cnm Þ þ¼ k tAB dt m¼1

ð100Þ

The consumption terms for the macroradicals corresponding to the termination reactions have simpler forms like the balance equation in three dimensions: dRn;A i; s dt

þ¼ k tAA Rn;A i; s

n1 X i1 X s1 X

A A Rm; j; t k tAB Rn; i; s

m¼1 j¼1 t¼1

¼ Rn;A i; s ðk tAA l0A þ k tAB l0B Þ

n1 X i1 X s1 X

B Rm; j; t

m¼1 j¼1 t¼1

ð101Þ

From this the moment distribution expressions for the number of A units and for the number of A sequences easily follow as: dFnA þ¼ FnA ðk tAA l 0A þ k tAB l 0B Þ dt

ð102Þ

dGnA þ¼ GnA ðk tAA l 0A þ k tAB l 0B Þ dt

ð103Þ

Once the pseudo-distributions have been solved, the average copolymer composition nnA =n and sequence lengths lnA can be calculated with the expressions given in Table 9.19. Note that the average number of B units nnB can simply be derived from nnA , since nnB ¼ n nnA . For large numbers of sequences per chain we may further assume that these are equal for both monomers: snB ¼ snA . Thus, the average sequence length for B; lnB, can also be inferred easily. Some illustrative calculations have been carried out for a batch copolymerization and the results (together with kinetic and reactor data) are shown in Figure 9.12.


Copolymer composition

1 0.8

B

0.6 0.4

A

0.2 0 101

102

103 104 Chain length

105

106

Sequence length

104 103 102

B

101 A 100 10-1 101

102

103 104 Chain length

Fig. 9.12. Average copolymer composition and sequence length as functions of chain length for batch copolymerization. Kinetic data: kd ¼ 6:25 103 s1 ; k i ¼ 3:75 104 m 3 (kmol s)1 ; k pAA ¼ 1000 m 3 (kmol s1 ); k pAB ¼ 100 m 3 (kmol s)1 ; k pAA ¼ 1000 m 3 (kmol s)1 ; k pBA ¼ 10 6 m 3 (kmol s)1 ; k pBB ¼ 10 5 m 3 (kmol s)1 ; kc ¼ 3 10 5 m 3

105

106

(kmole s)1 ; cA0 ¼ 2 kmol m3 ; cB0 ¼ 2 kmol m3 ; cI0 ¼ 102 kmol m3 ; batch time: 400 s. Shorter chains (produced early in the batch) contain equal amounts of A and B with sequence lengths of 2.5 on the average. Long chains (late in the batch) possess more B, with sequences of up to 2000.

479

480


9.7

Probability Generating Functions 9.7.1

Introduction

Probability generating functions (pgf ) are defined as polynomials in the transformation variable z [where the coefficients p i represent a probability distribution (that is, a length distribution)]:

GðzÞ ¼

y X

pi z i

ð104Þ

i¼0

Distribution moments and coefficients can be obtained from GðzÞ according to: qG ; m 0 ¼ Gð1Þ; m1 ¼ qz z¼1 ! 1 q iG pi ¼ i! qz i

m2 ¼

q2G qz 2

! ð105Þ z¼1

ð106Þ

z¼0

We will discuss pgfs as used in a transformation procedure to solve population balance equations [3], and as employed in the cascade theory of polymer networks [37–44]. 9.7.2

Probability Generating Functions in a Transformation Method

According to the transformation procedure, the population balance equations in terms of discrete variables (chain length, number of branch points) are transformed into a set of equations in z. The transformed set is solved and subsequently inverted to the original discrete variable domain. This process makes use of some interesting properties of certain mathematical expressions as transformed into the z-domain. Transformations and inversions are tabulated in textbooks [3]. As an example we take linear AB step polymerization in a batch reactor with equal initial end group concentration P0 : k

Pn þ Pm ! Pnþm

ð107Þ

n1 y X X dPn Pm Pnm 2kPn Pn ¼k dt m¼1 n¼1

ð108Þ

The total chain concentration is m 0 ¼ Eq. (106) we obtain Eq. (109), yielding:

y P n¼1

Pn , so by taking the zeroth moment of

9.7 Probability Generating Functions

dm 0 ¼ kðm 0 Þ 2 dt

ð109Þ

m 0 ¼ P0 =ð1 þ ktP0 Þ

ð110Þ

Since by definition [Eq. (104)] Gð1Þ ¼ m 0 , while the convolution property prescribes:

G

n1 X

! Pm Pnm

¼ fGðPn Þg 2

ð111Þ

m¼1

Eq. (108) becomes: dGðzÞ ¼ kfG 2 ðzÞ 2GðzÞGð1Þg dt

ð112Þ

By putting z ¼ 1, Eq. (110) for the total concentration m 0 is reproduced. To solve Eq. (112) we first realize that under initial conditions the pgf is given as: Gðz; 0Þ ¼ P0 z; integration then leads to: GðzÞ ¼ P0 fGð1Þ=P0 g 2 z=½1 zf1 Gð1Þ=P0 g

ð113Þ

This expression in the z-domain has a standard inverse form in the chain length domain, which represents a Flory distribution as the well-known solution to this problem: Pn ¼ P0 fGð1Þ=P0 g 2 f1 Gð1Þ=P0 g n1

ð114Þ

This pgf transformation procedure is an elegant method that has found many applications [3]. However, its ability to solve complete distribution problems is restricted to cases where explicit inversion is possible. In other cases only the main moments can be calculated, which then often can also be realized directly by the method of moments. 9.7.3

Probability Generating Functions and Cascade Theory

The cascade theory has been developed to deal with problems concerning polymer network formation [37–44]. Consider the polymer network made by polycondensation of f -functional monomers, represented as a rooted tree in Figure 9.13. Let end group conversion be given as a; then for a three-functional monomer the pgf of the connectivity between the zeroth and first generation is given by: F0 ðzÞ ¼ ð1 aÞ 3 þ 3að1 aÞ 2 z þ 3a 2 ð1 aÞz 2 þ a 3 z 3 ¼ ð1 a þ azÞ 3

ð115Þ

481


(1-α) 2z0

2α(1-α)z 1 α2z2 2 F1(z) =

(1-α+ αz) 2

1

(1-α) 3z0

3α(1-α) 2z1

3α 2(1-α)z 2

α3z3

Generation

482

0

F0(z) = (1-α+ αz)3 Fig. 9.13. Principle of probability generating functions and cascade theory for a three-functional polymer network.

Likewise, the pgf for connectivity between first and second (and all subsequent) generations is given by: F1 ðzÞ ¼ ð1 a þ azÞ 2

ð116Þ

In general, for f -functional monomers we thus have: F0 ðzÞ ¼ ð1 a þ azÞ f F1 ðzÞ ¼ ð1 a þ azÞ f 1

ð117Þ

From the connectivity pgfs at subsequent generation levels the branch point probability distributions at a certain level can be inferred. For instance, in the example with f ¼ 3 at level 2 the branch point pgf reads as: F0 ðzÞ ¼ ð1 a þ aF1 Þ 3

ð118Þ

where F1 is given by Eq. (116); coefficients of z i in Eq. (118) represent the probabilities of finding i branch points at level 2. Thus, at an arbitrary level we have: G0 ðzÞ ¼ F0 ðF1 ðF1 ð. . . F1 ðzÞÞÞÞ

ð119Þ

From Eqs. (117), number- and weight-average chain lengths can be derived using Eqs. (105) [37], but a more elegant method is to construct self-consistent equations [45, 46] to solve the pgf. In the three-functional example above, the probabilities of a given node on an arbitrary generation level being connected to zero, one, or two

9.7 Probability Generating Functions

further nodes are expressed by the coefficients for z 0 ; z 1, and z 2 , respectively, in pgf GðzÞ. In addition, being connected to a further node implies the possibility of being connected to even further nodes. This possibility is described by the identical pgf GðzÞ. The connectivity to further nodes if connected to two nodes is expressed by G 2 ðzÞ, since statistics of these connectivities are identical, but independent. Note that the contributions of further connectivities are thus correctly described by the coefficients of the z terms. The self-consistent equation becomes: G ¼ ð1 aÞ 2 þ að1 aÞzG þ a 2 z 2 G 2

ð120Þ

We will illustrate this procedure on a simple linear step-polymerization ( f ¼ 2) to obtain the CLD and on single metallocene ethylene polymerization to find the 2D distribution of chain lengths and numbers of branch points. Linear step-polymerization is treated as above with f ¼ 2, yielding the selfconsistent pgf equation: G ¼ ð1 aÞ þ azG

ð121Þ

and hence: G ¼ ð1 aÞ=ð1 azÞ

ð122Þ

Applying Eqs. (105) to Eq. (121) we find the number- and weight-average for the number of links between monomer units, which after adding 1 yields the familiar expressions for n n and nw already found by Flory [1]: qG a 1 þ1¼ ¼ qz z¼1 1 a 1a 2 q G qG a 2a 2 1þa nw ¼ ¼ þ1¼ 2 qz z¼1 qz z¼1 1 a ð1 aÞ 2 1a nn ¼

ð123Þ

Series expansion in z of Eq. (122) yields for G: G¼

y X ð1 aÞa i z i

ð124Þ

i¼0

of which the coefficients represent a Flory distribution ð1 aÞa i1 , realizing again that chain length is one more than the number of links. The derivation of the CLD/DBD 2D distribution for the metallocene system is based on the fact that all segments between branch points possess the same statistics. The pgf method is then applied to the connectivity of segments rather than monomer units as in the polycondensation problem. A pgf is constructed for the probability of finding connectivity points (branch points) on a segment going in a direction opposite to the growth direction. The self-consistent pgf equation follows from an argument in

483

484


which the probability of a segment having a branch point plays a central role. Assuming steady state and taking the (0; 0)-moment of the population balance equations (35)–(40), we find this branching probability to equal: b¼

kb k p; TDB tl 0 B ¼ ¼ B þ m 0 þ l 0 ð2kb t þ 1Þk p; TDB l 0 þ kb þ 1=t

ð125Þ

A given segment is connected to an initiation point (hence not to a branch point) with probability 1 b, and it is connected to a branch point with probability b. In the case of a branch point the segment is connected to two further segments obeying the same statistics and hence pgf. Thus we have: G ¼ ð1 bÞ þ bzG 2

ð126Þ

which has a quadratic form this time, having Eq. (127) as the solution [45]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4bð1 bÞz ð127Þ G¼ 2b This can be expanded to give: G¼

y X

ð2iÞ! b i ð1 bÞ iþ1 z i i!ði þ 1Þ! i¼0

ð128Þ

where the coefficients for z i represent the probability distribution of the numbers of branch points per molecule. Here, the factor ð2iÞ!=ði þ 1Þ! is called the Catalan number, which has already been derived by Flory [1] and represents the number of different ways a molecule with i branch points can be constructed. Since the number of segments between branch points plus terminal segments per molecule is 2i þ 1, obtaining the length distribution further requires the length distribution of the segments to be taken into account. This should obey a Flory distribution, being dictated by the competition of propagation on the one hand, and termination by b-hydride elimination and TDB incorporation on the other. Hence the numberaverage segment length and probability distribution of segments with length ni follow as: kpM 1 ni nns ¼ ¼ exp ð129Þ p n nns nns kb þ k p; TDB m¼ 0 þ 1=t From Eq. (129) the probability of finding polymers with i branch points (2i þ 1 segments) is easily inferred. The conditional probability of polymers with 2i þ 1 segments, all with a Flory distribution with average nns , of having total length n is known to be: pðnjiÞ ¼

n exp nns ðnns Þ 2iþ1 ð2iÞ! n 2i

ð130Þ

9.8 Monte Carlo Simulations

This result can be understood as the product of 2i þ 1 segmental probability distributions [Eq. (129)], leading to n in the exponential, multiplied by a prefactor representing the number of possibilities of distributing n monomer units among 2i þ 1 segments. Combining Eqs. (129) and (130) gives the full expression for the 2D distribution of CLD and DBD: pðn; iÞ ¼

n 2i ðnns Þ 2iþ1

1 n b i ð1 bÞ iþ1 exp s i!ði þ 1Þ! nn

ð131Þ

As regards a successful application of the pgf cascade method, we conclude that:

connectivity statistics should be identical for all units (monomer units or segments), and a convenient series expansion of the pgf from the self-consistent equation should be possible.

9.8

Monte Carlo Simulations 9.8.1

Introduction

This method is employed for problems where analytical or differential equations approaches are not feasible in view of high dimensionality, or only lead to approximate solutions or averages for one or more dimensions. Examples of such limitations have been discussed above. The Galerkin-FEM method in the pseudodistribution mode was confronted with closure problems in the case of PVAc with more than one TDB per chain. The pgf-method is applicable only to systems with identical statistics of the constituting elements. In principle, Monte Carlo simulation does not suffer from such limitations. Here, we will introduce classical MC, but subsequently mainly discuss some successful applications of advanced MC methods. Classical MC simply describes the reactions of single monomer units. In the case of branching, a reaction event can be either a propagation, a termination or a branching step, according to the relative probabilities of these reactions. This method has, for instance, been applied to a single metallocene system for branched polyethylene to provide the bivariate CLD/DBD [47]. The disadvantage of this approach is that it requires billions of reaction steps to generate sufficiently large populations of molecules, typically 10 6 , to derive accurate statistics. Computationally less demanding is application of MC to larger constituent units than molecules: segments or primary polymers [11–15, 47]. In the aforementioned metallocene example this could be realized for a continuous reactor in a simple manner because of the identical statistical properties of all segments. This increased computation speed by a factor of 15 [47]. However, when the tails of very broad CLDs

485

486


are to be described, typically involving concentrations more than six decades lower than the highest chain concentrations in the system, the numbers of molecules to be generated become excessive, even with primary polymer sampling. Tobita has achieved a great improvement in MC sampling in this respect by applying sampling based on weight-fraction instead of number-fraction distributions [11–15]. We will now discuss this method as applied on mostly branched radical polymerization systems involving transfer to polymer, terminal double bond incorporation, recombination termination, and random scission for both continuous and batch reactors. 9.8.2

Weight-fraction Sampling of Primary Polymers: Batch Reactor, Transfer to Polymer [48]

A branched molecule is thought to be composed of primary polymers (pps) in which growth of linear chains has started from monoradicals or from secondary radical sites at other pps, created by transfer to polymer, and stopped by a certain termination mechanism. Primary polymers can possess one or more branch points at which other pps have been growing. Primary polymer sampling is generally based on the assumption of instantaneous growth of pps. In a batch reactor the length distribution of such pps is then determined by the kinetic conditions (conversion) at the ‘‘birth’’ time, y. The MC process starts with sampling such a birth time y, and under number-fraction sampling its length is sampled from the (usually Flory) number-fraction distribution at y. Since molecules may start growing at any moment, we may (but must not necessarily) consider this first sampled pp as the first one of the whole molecule. The first pp may, by transfer to polymer reactions, receive one or several branch points in the remaining time between y and the batch end time c, and thus become attached to pps created at later birth times. These first, second, third and further pps may in their turn receive branch points and become attached to more pps at later stages. The MC process stops when the last pps no longer obtain branch points. A slightly modified approach does not regard the first sampled pp as the first one in the molecule. Instead it accounts for the possibility that the first sampled pp has started growing from a branch point created on a previously formed pp (between 0 and y), rather than being initiated from a monoradical. Since in all cases the connectivity is based on the same values for the branching probability at the various birth times, the results are identical. Since most pps are present in short chains, this sampling procedure generates mostly short molecules; hence a very great many molecules have to be created to predict long CLD tails accurately. If, instead, one sampled monomer units on pps and was able to predict their connectivity to other parts of the molecule, then most molecules generated would have sizes around the maximum of the weight-fraction distribution. In other words, such a method is significantly more effective at finding long CLD tails. This is the rationale of weight-fraction sampling, since the probability of selecting a monomer unit at random from a pp population is propor-


tional to the length n of the pps. Since in that method a monomer unit is sampled at random in a molecule, it cannot always be on the first pp of the molecule. The MC process consequently proceeds in the second manner described above. The first monomer unit sampled at y forms part of a pp in the zeroth generation, as do the pps to which it (eventually) is connected by branch points from it, on the one hand, and (eventually) to the pp on which it started growing on, on the other. The MC algorithm determines the birth times and lengths of pps in subsequent generations. We will demonstrate the MC procedure on a very simple radical polymerization process with transfer to polymer, while disproportionation is the only termination mechanism (see Figure 9.14). The algorithm employs conversion x, rather than time, as the independent variable. Consequently, it starts with sampling the birth conversion x ¼ y of a zeroth-generation pp by sampling a random number between 0 and end conversion c. Its length is determined by sampling from the weight-fraction distribution (usually according to Flory): wn ðyÞ ¼

n exp nðyÞ nðyÞ 2 n

ð132Þ

The average chain length nðyÞ is given by: nðyÞ ¼ k p MðyÞl 0 ðyÞ=k td l 0 ðyÞ 2 ¼ k p MðyÞ=k td l 0 ðyÞ

pp: brith conversion z(1), 0 < z(1) < z(0)

pp with u(1) z(0) < u(1) < ψ

branch points sampled with Eq. (138) with ρ (ψ,θ) from Eq. (137)

pp: brith conversion z(0), 0 < z(0) < θ

pp with u(1) z(0) < u(1) < ψ

ð133Þ

initial unit

pp with u(0) θ < u(0) < ψ

pp: brith conversion θ, length from wn(θ), Eq. (132)

pp with u(0) θ < u(0) < ψ Fig. 9.14. Monte Carlo sampling of primary polymers and their connectivity for radical polymerization with transfer to polymer in a batch reactor. Conversion c, first pp sampled at birth conversion x ¼ y.

pp with u(1) u(0) < u(1) < ψ

487

488


Sampling can be realized by using the cumulative distribution of wn ðyÞ, a function of y; cwn ðyÞ, with values between 0 and 1, selecting a random number between 0 and 1 (randð1Þ), and finding y by requiring that cwn ðyÞ ¼ randð1Þ. A faster method utilizes the property that sampling twice from the number-fraction distribution and adding results exactly reproduces a weight-fraction distribution [49]. Sampling from nn can be performed rapidly using the simple formula: n ¼ ceil½nðyÞ lnf1=randð1Þg

ð134Þ

where ceil denotes the value obtained when rounding a real number to the nearest higher integer, a standard operation available in most mathematical packages such as MATLAB. Doing this twice and adding the values generates a weight-fraction sample value. The probability of receiving branch points for this pp depends on its average branching density between y and c and is proportional to its now known length, n. The former can be derived from the monomer and branch points balance (Table 9.1): dM dx ¼ k p l 0 ðtÞM ! ¼ k p l 0 ðtÞð1 xÞ dt dt

ð135Þ

dr ¼ k tp l 0 ðtÞ dt

ð136Þ

Combination and integration between y and c yields the average branching density rðy; cÞ: dr 1 1y ¼ C tp ! rðy; cÞ ¼ C tp ln dt 1r 1c

ð137Þ

The number of branch points m on this pp is sampled from a binomial distribution containing average branching density rðy; cÞ and pp length n: pðmÞ ¼

n m

r m rðnmÞ

ð138Þ

Sampling can be performed by employing standard random number generators for a binomial distribution (for example, in MATLAB: m ¼ binorndðn; rÞ). Next, to the pps attached at each of these branch points a birth conversion u, y < u < c, and a length must be assigned. The former should follow from the formation intensity distribution of branching density over the conversion interval y–c as given by Eq. (137). This implies that we should sample u from the conditional (given that this branch point exists at a pp formed at y) probability distribution, as expressed by:


1y 1y ln CPa ðujyÞ ¼ ln 1u 1c

ð139Þ

The sampling can be performed by selecting a random number between 0 and 1 (randð1Þ) and inferring u from it by requiring that CPa (a function of u between 0 and 1) equals randð1Þ. The shape of Eq. (139) is such that on average u is chosen closer to c than to y. This agrees with the fact that branch formation intensity is higher at high conversion. The length of the pp grown at x ¼ u is sampled from the number-fraction distribution:

nn ðyÞ ¼

1 n exp ; nðyÞ nðyÞ

ð140Þ

since a single chain end (instead of an arbitrary unit on the chain) is chosen at random as its starting point. Sampling is easily realized by employing Eq. (133). This eventually leads to a number of pps in generation 0 with specified birth conversions and lengths. Included in the procedure for this generation is the accounting for the eventuality of the first pp [pp(y)] being attached to a pp created earlier, at z, 0 < z < y. This follows from the relative probability of the first pp of being initiated by a transfer-to-polymer reaction [Eq. (141)]. Pb ¼ k tp l 0 m1 =ðk tp l 0 m1 þ 2kd I2 Þ ¼ k tp m1 =ðk tp m1 þ k td l 0 Þ

ð141Þ

The latter equality follows from the quasi-steady-state-assumption. Note that Pb in a batch reactor is a function of conversion. If other transfer mechanisms are present, the denominator is extended with the corresponding contributions to the initiation process. Whether or not the pp is attached to another pp indeed follows by selecting a random number between 0 and 1 and determining whether the inequality randð1Þ < Pb is false or true. If connected (true) then the birth conversion of the earlier pp simply follows from the conditional distribution (given that the first sampled pp is created at x ¼ y and grows from an earlier one): CPi ðzjyÞ ¼ z=y

ð142Þ

CPi is linear with z since from the perspective of the pp created at x ¼ z its probability of undergoing transfer to polymer is simply linearly proportional to conversion. This implies that all values for the birth conversion between 0 and y are equally probable. Sampling can be performed in the same way as described for CPa [Eq. (139)]. Finally, the length of the pp grown at x ¼ u is sampled from the weight-fraction distribution wn ðzÞ, Eq. (132), since any of the monomer units in this pp can undergo branching. This then concludes the MC process at generation 0. If this generation has generated new pps attached in either way to the first one, the generation number is increased by one. Note that pps attached alongside the first pp can only become attached to pps created at higher conversion. In contrast,

489

490


the pp (created at x ¼ z) to which (eventually) the pp sampled first is attached by its chain end can have further branch points created at birth conversions x > z, but in addition it can be itself attached to a pp pp (x < z) created earlier; see Figure 9.14. The number of branch points on the pp formed at x ¼ z also follows from the binomial distribution, Eq. (138), but now with an average branching density rðz; cÞ and a length one less than its sampled length, since one of its monomer units already possesses a branch point (by which its is connected to the first pp sampled). 9.8.3

Example

We demonstrate the algorithm with the synthesis of a molecule with 10 branch points as shown in Figure 9.15. Alongside all the pps between brackets are listed the generation number, birth conversion, and length, respectively. The first pp is sampled at x ¼ y ¼ 0:4 and has length 300 [wn distribution, Eq. (132)]. It possesses three branch points [binomial distribution, Eq. (138)]. Birth conversions of these three pps are sampled using Eq. (139) to be: u ¼ 0:65; 0:45, and 0.62. The lengths of the three pps are sampled from the distribution nn at these conversions [Eq. (140)]: 60, 130, and 80, respectively. The pp sampled first turns out [using probability Pb from Eq. (141)] to be connected to an earlier pp. Birth conversion of this earlier pp equals z ¼ 0:3, as sampled from CPi , Eq. (142). Its length is 230 [wn distribution, Eq. (132)]. This finishes generation 0, which has generated four pps in total. In the first generation two of these pps according to Eq. (138) turn out to

(1, 0.22, 180)

(2, 0.42, 80) (0, 0.3, 230)

(1, 0.37, 140)

(1, 0.6, 110)

(0, 0.4, 300)

(0, 0.65, 60)

(0, 0.45, 130)

(0, 0.62, 80)

(1, 0.51, 60)

(2, 0.65, 100) Fig. 9.15. Example of full Monte Carlo sampling of a branched molecule for radical polymerization with transfer to polymer.


posses branch points: the one with u ¼ 0:45 (one branch point) and the one with z ¼ 0:3 (three additional branch points). Primary polymer (0, 0.45, 130) can have only branch points at later (> 0.45) birth conversion: at u ¼ 0:51, length 60. Primary polymer (0, 0.3, 230) can have branch points at later (>0.3) birth conversion: u ¼ 0:6, length 110, and u ¼ 0:37, length 140). In addition, Primary polymer (0, 0.3, 230) turns out to be attached [Eq. (142)] to an even earlier ( 10 8 took a time of around 10 min (1.5 GHz Athlon CPU Processor), and the whole population of 290,000 took several hours. Note that we applied a cut-off limit: molecules getting beyond 2 10 5 branch points in one generation were stopped, which happened 40 times in the whole population and led to a maximum number of branches per molecule of around 2:6 10 6 . The Galerkin-FEM fiveclasses model converged within 30 s. Obviously, the Galerkin-FEM method is


0.5

Monte Carlo Galerkin-FEM, 5-classes multiradical

dw/d{log(MW)}

0.4

0.3

0.2

0.1

0 0 10

2

10

4

10

6

10

8

10

10

10

Chain Length Good agreement between MC simulations (290,000 molecules) and GalerkinFEM five-classes model (all living and dead chains). Reactor and kinetic data: initiator feed I2; f ¼ 5 103 kmol m3 ; monomer feed Mf ¼ 16:75 kmol m3 ; residence time: t ¼ 30 Fig. 9.16.

s; kd ¼ 0:5 s1 ; k p ¼ 1:4 10 5 m 3 (kmol s)1 ; k td ¼ 5 10 10 m 3 (kmol s)1 ; k tp ¼ 2000 m 3 (kmol s)1 ; conversion x ¼ 0:249; average branching density r ¼ 0:00463; average pp length n n ¼ 144; branching probability Pb ¼ 0:679.

much faster to calculate CLD only. On the other hand, MC simulations provide the full bivariate CLD/DBD. However, it must be noted that we did not extract the full architectural information. This would require construction of incidence matrices [33], which probably limits calculations to around 10,000 branch points in a molecule. 9.8.6

Batch Reactor, Terminal Double Bond Incorporation [15]

The problem of incorporation of chains with a terminal double bond (TDB) exists in polymerizations discussed above, such as radical polymerization of vinyl acetate and olefin polymerization with a constrained-geometry metallocene catalyst (CGC). Tobita [15] has developed an MC algorithm for this problem for the PVAc case. It is assumed that TDBs are created by transfer to monomer only, while recombination is absent, which results in a maximum of one TDB per chain. We largely follow Tobita’s explanation, but differ in that we will assume that disproportionation is the termination mechanism, while transfer to solvent and to polymer are not yet being accounted for. Later we will address the real PVAc problem, which in fact has two branching mechanisms: TDB propagation and transfer to polymer.

493


0.015

dw/d{log(MW)}

494

Monte Carlo

0.01 GF-5

0.005

GF-5, dead GF-2, dead

GF-2

GF-5, living 0 6 10

GF-2, living 7

10

8

10

Chain Length

Tail of the CLD of Figure 9.16. At chain lengths > 10 7 living chain concentrations become of the same order of magnitude as dead chains. The Galerkin-FEM two-classes Fig. 9.17.

9

10

model (no multiradicals) underestimates the tail. The rapid decline of the MC curve at 3 10 8 is due to the cut-off procedure.

Similarly to the MC algorithm for transfer to polymer in a batch reactor, conversion x is employed as the independent variable. The procedure starts in generation 0 by sampling a birth conversion y for the first pp: 0 < y < c, where c is the end conversion, while its length follows by sampling from the weight-fraction distribution wn , Eq. (132), then the average chain length nðyÞ is given by: nðyÞ ¼

k p MðyÞl 0 ðyÞ k m MðyÞl 0 ðyÞ þ k td l 0 ðyÞ 2

¼

k p MðyÞ k m MðyÞ þ k td l 0 ðyÞ

ð149Þ

Now, we consider the possibility that this pp will be incorporated in a pp that grows later, at u: y < u < c. This depends, in the first place, on the probability that the first pp, grown at x ¼ y, possesses a TDB: PTDB ðyÞ ¼

k m MðyÞl 0 ðyÞ k m MðyÞl 0 ðyÞ þ k td l 0 ðyÞ

2

¼

k m MðyÞ k m MðyÞ þ k td l 0 ðyÞ

ð150Þ

In the second place, we have to know which fraction of the pps with a TDB grown at x ¼ y will actually be incorporated as conversion increases, since together with


PTDB ðyÞ this determines the probability of the randomly chosen pp at x ¼ y becoming connected to a later pp. We should consider what happens to pps with a TDB after their creation at x ¼ y. A fraction of them is incorporated, but as the rate at which this happens depends on their concentration, this rate will decrease. In fact, the fractional decrease of these pps exactly follows the decrease in the fraction of TDBs of pps created at x ¼ y. The TDB mole-fraction, FTDB ðy; uÞ, starts at Cm ¼ k m =kp for all birth conversions, so also at x ¼ y, while it decreases according to the balance describing the TDB consumption starting from y: dFTDB ðy; tÞ ¼ k p; TDB l 0 ðtÞFTDB ðy; tÞ dt

ð151Þ

With Eq. (135) this is transformed in terms of birth conversion u (y < u < c), giving: Cp; TDB FTDB ðy; uÞ dFTDB ðy; uÞ ; ¼ 1u du

ð152Þ

where Cp; TDB ¼ k p; TDB =k p . By integration between y and c one obtains: FTDB ðy; cÞ ¼ Cm

1 c Cp; TDB 1y

ð153Þ

1 c Cp; TDB of TDB chains is still present. 1y Or, the probability that such pps created at x ¼ y have reacted at x ¼ c to produce a branch point equals one minus this fraction. The overall probability Pb; TDB ðy; cÞ of a randomly chosen pp(y) to be incorporated in a later pp(c) thus becomes that given by: Thus, we see that at x ¼ c a fraction

(

) 1 c Cp; TDB Pb; TDB ðy; cÞ ¼ PTDB ðyÞ 1 1y

ð154Þ

Whether or not it is connected follows by the checking of the inequality randð1Þ < Pb; TDB ðy; cÞ. Now, the birth conversion u at which incorporation takes place has to be determined. This is realized by using the conditional probability distribution in the pp has reacted u, CPa; TDB ðujyÞ, over the interval y to c, namely givenCthat 1 c p; TDB during the interval – represented by the fraction : 1y Pb; TDB ðy; uÞ ¼ CPb; TDB ðujyÞ ¼ Pb; TDB ðy; cÞ

"(

),( )# 1 u Cp; TDB 1 c Cp; TDB 1 1 1y 1y ð155Þ

495

496


Sampling proceeds in the same manner as discussed in the cases with transfer to polymer [see Eq. (139)]. Connectivity in generation 0 can also occur, when the pp sampled first itself incorporates pp chains with a TDB during its growth at x ¼ y. Obviously, such chains should have been created at birth conversions z before y; hence 0 < z < y. The probability of receiving branch points in this way in fact equals the instantaneous branching density rTDB ðyÞ, which is given by the ratio of TDB propagation to monomer propagation rate: rTDB ðyÞ ¼

k p; TDB l 0 ðyÞnTDB ðyÞ Cp; TDB nTDB ðyÞ ¼ k p ð1 yÞM0 l 0 ðyÞ ð1 yÞM0

ð156Þ

Here, nTDB ðyÞ is the average concentration of TDBs in the reactor, which follows from a TDB balance (production and consumption): dnTDB ðtÞ ¼ k m MðtÞ k p; TDB l 0 ðtÞnTDB ðtÞ dt

ð157Þ

With Eq. (135) and M ¼ M0 ð1 yÞ, it follows that: Cp; TDB nTDB ðyÞ dnTDB ðyÞ ¼ Cm M0 ; dy 1y

ð158Þ

which yields Eq. (159) by integration: nTDB ¼

Cm M0 f1 y ð1 yÞ Cp; TDB g Cp; TDB 1

ð159Þ

With this Eq. (133) becomes: rTDB ðyÞ ¼

Cp; TDB Cm f1 ð1 yÞ Cp; TDB 1 g ðCp; TDB 1Þ

ð160Þ

Given the length of the pp first sampled and rTDB ðyÞ, the number of branch points can be sampled from a binomial distribution, Eq. (138), using a standard binomial distribution random number generator. This yields a certain number of branch points connecting the pp to the same number of pps formed earlier, of which birth conversion and lengths have to be determined. The former follows from the conditional probability that a pp created between 0 and z (0 < z < yÞ is connected to the pp growing at x ¼ y, CPi ðzjyÞ. This probability is proportional to the mole fraction of TDBs on pps created between 0 and z still present at x ¼ y. The average TDB mole fraction of these pps at x ¼ y follows from Eq. (153) by integration between 0 and z: F TDB ðz; yÞ ¼

1 y

ðz 0

FTDB ðz; yÞ dz

ð161Þ


Thus, the normalized CPi ðzjyÞ becomes: ð 1 z FTDB ðz; yÞ dz 1 ð1 zÞ 1Cp; TDB y CPi ðzjyÞ ¼ ð0y ¼ 1 1 ð1 yÞ 1Cp; TDB FTDB ðz; yÞ dz y 0

ð162Þ

Conversion births z are determined by sampling as described previously. The shape of Eq. (162) prescribes that on average z is found to be closer to y than to 0, reflecting the fact that pps generated early have a high chance of being incorporated by pps earlier than y. The lengths of the pps are sampled from the number-fraction distribution in the usual manner [Eq. (133)]. This then concludes generation 0, and the procedure is repeated for higher generations until no more connections are found. 9.8.7

CSTR, Terminal Double Bond Incorporation

The greatest difference from the batch reactor is again the taking into account of the RTD according to the exponential form of Eq. (143). The residence time t (or reduced RT x ¼ t=t) is taken as the independent variable. The algorithm starts with the sampling of x and the determination of the length from wn with nðyÞ after Eq. (149) and the double sampling after Eq. (134). The pp sampled first may through its eventual TDB become connected to a pp that starts growing after the first one, hence having an RT shorter than x. Similarly to the batch reactor, the probability of this connectivity is the product of PTDB; c , the probability of a randomly chosen pp having a TDB, and a factor denoting the decrease of TDB fraction FTDB due to TDB incorporation. PTDB; c follows from Eq. (150), but is a constant in the CSTR case. The TDB fraction must be considered as a function of residence time, FTDB ðtÞ, with starting value FTDB ð0Þ ¼ Cm , as transfer to monomer is the only source of TDBs. The balance equation for FTDB ðtÞ: dFTDB ðtÞ ¼ k p; TDB l 0 FTDB ðtÞ, dt

ð163Þ

which is similar to Eq. (151), with the starting condition, the steady-state equality: l0 ¼

x ; k p tð1 xÞ

ð164Þ

and the definition of the reduced RT leads to: x x ; FTDB ðxÞ ¼ Cm exp Cp; TDB 1x

ð165Þ

497

498


This time, the exponential term describes the decline of unreacted TDB with residence time x; hence one minus this term denotes the fraction of pps having generated a connection as a function of x. Thus, the probability of an arbitrary pp being connected as a function of x becomes: PTDB ðxÞ ¼ PTDB; c

FTDB ðxÞ 1 Cm

ð166Þ

The residence time u, 0 < u < x, of the pp connected has to determined next. It simply follows from the conditional (given that its TDB has reacted) probability CPa ðujxÞ: CPa ðujxÞ ¼

PTDB ðuÞ PTDB ðxÞ

ð167Þ

The chain length of this pp follows from wn . 9.8.8

Incorporation of Recombination Termination [14]

Recombination termination is implemented in the same way in the batch reactor and in the CSTR. First to note is that termination through recombination happens to two pps growing simultaneously, which implies that birth conversions or residence times are identical for the two. The algorithm starts with a check on which of the new pps created in a certain generation by some mechanism is connected to another one by recombination. This probability is obtained from the relative reaction rates; for example, in the case of transfer to polymer only: Ptc ¼ k tc l 20 =ðk tc l 20 þ k td l 20 þ k tp m1 l 0 Þ ¼ k tc l 20 =ðk tc l 0 þ k td l 0 þ k tp m1 Þ

ð168Þ

When recombination is at hand (randð1Þ < Ptc ), determination of the birth conversion or residence time is performed in no other way, as before. The total length of the two pps connected can be found by sampling once from wn and once from nn , and addition. The correctness of this can be understood by realizing that the second pp can be connected to the first one only by one chain end. 9.8.9

Incorporation of Random Scission, Linear Chains, Batch Reactor [50]

Here we will follow Tobita’s explanation of the MC algorithm for radical polymerization with random scission for nonbranched systems, though it includes recombination termination. Later, the link to branching by transfer to polymer will be elucidated. Random scission is assumed to happen to dead chains breaking into a living and a dead fragment. The living fragment starts growing again and will be terminated by some mechanism such as recombination. This implies that a scis-


sion point acts as an initiation point for a new pp, which further obeys the same growth statistics as pps growing from initiator radicals or secondary radical sites on other pps. Scission and subsequent growth may occur several times to pp chains, so finally they may be constructed of various segments, created at various times. The construction process of pps undergoing scission and growth steps is depicted schematically in Figure 9.18 for the case of a batch reactor. The algorithm starts by randomly sampling a birth conversion y0 of Seg-0, the first pp sampled, between 0 and end conversion c. For this first pp we may arbitrarily choose its growth direction: to the right in Figure 9.18. Now in principle, the length of Seg-0 is sampled from wn by sampling twice from nn , but this is not definitively the value it finally will get, since scission may occur. Therefore, the connecting unit between these two is explicitly considered as the initially selected monomer unit. Next, the part to the right of this unit is examined and a check is made on whether scission has taken place. To this end the scission density h (or scission probability of mono-

initially selected unit

Ptc

Prs

scission

θ2 Seg-2”

birth conversion θ

ψ

θ1

Seg-2’

θ1

Seg-0

Seg-1

end conversion

initiation from scission

initiation from scission

θ0 recombination θ2

θ0

CPa ,rs (θ1 | θ0 )

CPi ,rs (θ 2 | θ 0 )

scission at θ1

scission at θ 0 Chain length

Example of the construction of a linear pp chain undergoing scission and recombination in radical polymerization. The square on the LHS marks the (non-scission) initiation point of this chain at birth conversion y2 . Fig. 9.18.

499

500


mer units) is defined; its derivation exactly follows that of the branching density in the case of transfer to polymer. Realizing that dh ¼ k rs l 0 ðtÞ dt

ð169Þ

and applying Eq. (135), we find 1 y0 hðy0 ; cÞ ¼ Crs ln 1c

ð170Þ

where Crs ¼ k rs =k p , which is similar to Eq. (137). It has been demonstrated [51] that for equal scission probability hðy0 ; cÞ of all monomer units in a chain the number average of fragments equals 1=hðy0 ; cÞ, while its length distribution is Flory [Eq. (171)]. nns ðy0 ; cÞ ¼ hðy0 ; cÞ expfhðy0 ; cÞng

ð171Þ

The scission check can now be performed by comparing the RHS of Seg-0, nn ðy0 ; cÞ from Eq. (140) with nns ðy0 ; cÞ: if nn ðy0 Þ > nns ðy0 ; cÞ, scission has taken place, otherwise it has not. Given that scission has taken place, the birth conversion y1 ð> y0 ) at which scission takes place is determined by the expression for the conditional probability distribution, Eq. (139) – in view of the similarity between branching and scission. If scission has taken place, then the probability is 12 that the chain end forms the initiation site for a new pp Seg-1 with growth direction to the right, while its length is sampled from nn ðy1 Þ. Seg-1 is checked for scission using Eqs. (139) and (171). If scission did not happen, termination has taken place to this segment. It is connected to a further pp by recombination with probability Ptc [Eq. (168)]; but in Figure 9.18 it is not connected, so Seg-1 is a terminal segment at the RHS of the initially selected unit. Next, the chain part in the direction left of the initial unit on Seg-0 is considered. Its length is again determined by checking whether scission has taken place [comparing lengths from Eqs. (139) and (172)]. If not, then the chain end represents an initiation site. The probability Prs that this is a site following a scission at y0 is given by the rate of scission relative to the other reaction rates (compare Eq. (141) for the branching probability Pb ): Prs ðy0 Þ ¼

k rs l 0 ðy0 Þm1 ðy0 Þ k rs m1 ðy0 Þ ¼ k rs l 0 ðy0 Þm1 ðy0 Þ þ 2kd I2 ðy0 Þ k rs m1 ðy0 Þ þ ðk td þ k tc Þl 0 ðy0 Þ

ð172Þ

Figure 9.18 shows the situation where Seg-0 has grown at a scission point on Seg2 0 after scission of the latter at birth conversion y2 < y0 , to be selected as a random number between 0 and y0 . The growth direction of Seg-2 0 has then to be selected. There is a probability of 12 that it is in the direction indicated, to the LHS. Its length is determined by comparing lengths with Eqs. (139) and (171). In this case no scis-


sion took place, and hence the possibility of termination by recombination must be checked, using Eq. (168). Recombination happened to Seg-2 00 , also created at y2 . On checking whether scission occurred, this turned out to be not the case, and thus Seg-2 00 becomes the other terminal end, at the LHS of the initial unit. Note that the RHS of the pp first sampled, Seg-0, if it had not undergone scission, might have been connected to another segment by recombination. Then, the algorithm would have followed the same lines as for Seg-2. 9.8.10

Combined Scission/Branching

The combined scission/branching algorithm starts in generation 0 with the scission procedure. Once the linear chain eventually consisting of several segments with different birth conversions has been constructed, the numbers of branch points on these various segments are determined using the binomial distribution [Eq. (138)] with the proper branching densities and lengths. To determine whether scission has taken place on the pps connected via these branch points, the RHS part of the scission algorithm has to be employed. Eventually, scission did not take place, and then the check for recombination should be performed. Now, the terminal at the RHS of the initially selected unit (Seg-1 in Figure 9.18) is always a free end, so it is never connected. In contrast, the terminal at the LHS has been assessed as a initiation site not being created by scission. Hence, this site may have been created either by an initiator radical or at a dead pp backbone by transfer to polymer, with probability Pb from Eq. (141). If it is connected to a pp at a smaller birth conversion (u < y2 in Figure 9.18), then the scission algorithm is fully repeated with the connection point as the initially selected unit. This concludes generation 0. 9.8.11

Scission in a CSTR

As usual in a CSTR, the independent variable is residence time: in the example of Figure 9.18, y0 for Seg-0. Now, the average scission density h is a constant; following by a similar argument to that for branching [Eq. (146)] gives: h ¼ Crs x=ð1 xÞ

ð173Þ

For individual pps the scission density is a function of residence time (compare Eq. (144) for branching density): hðyÞ ¼ hy

ð174Þ

The RTD itself follows Eq. (143). Knowing hðy0 Þ, the scission check can be made by comparing lengths from Eqs. (139) and (171). Given that scission has taken place on the RHS of Seg-0, the shorter residence time y1 ð0 < y1 < y0 Þ at which scission

501

502


takes place is determined by a similar expression (Eq. (147)) to that for branching, giving the conditional scission probability distribution: CPa; rs ðy1 jy0 Þ ¼ y1 =y0

ð175Þ

The LHS part of the branching algorithm is also slightly modified. The probability Prs of Seg-0 being started on a scission point at Seg-2 0 again follows from Eq. (172), but it is constant in a CSTR. The residence time y2 , now longer than y0 , is sampled from a conditional probability distribution similar to Eq. (148): CPi; rs ðy2 jy0 Þ ¼

1 Fðy2 Þ 1 Fðy0 Þ

ð176Þ

9.9

Prediction of Branched Architectures by Conditional Monte Carlo Sampling 9.9.1

Introduction

Except for the full Monte Carlo simulations, all of the previously described mathematical methods were meant to compute microstructural properties in terms of countable quantities: number of monomer units (chain length), number of branch points, and so on. However, when dealing with branched polymer molecules, the connectivity structure, or topology, is a highly important issue for both characterization and properties of branched polymers. It is obvious that, given the number of monomer units and branch points in a molecule, a high variability exists in topology. This section is devoted to prediction methods of branched architectures. We present two methods, both based on conditional Monte Carlo sampling, the first applicable to radical polymerization with transfer to polymer, the second to metallocene-catalyzed polymerization of polyolefins. Both are applicable to CSTRs only. Note that the full Monte Carlo method described previously also generates information on connectivity. It has as such been utilized to predict radii of gyration [11–15, 48–51] for radical polymerization systems. In principle, this method can be extended to other polymerization systems as well. However, conditional MC methods take full advantage of the fact that it is often much easier to find the CLD/DBD, which then makes it possible to focus directly on the more interesting large molecules with many branch points. Although topology is less easy to quantify than countable microstructural properties, highly interesting properties can still be inferred that provide direct characterizations of branched structures. One of these is the well-known radius of gyration, and another is a recently introduced topological characterization originating from rheology: the bivariate seniority/priority distribution [52]. Here, methods will be described to obtain radius of gyration and seniority/priority values from architectures as synthesized by the algorithms to be described. In all cases architectures will be represented in descriptive matrices from graph theory [53].

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

9.9.2

Branched Architectures from Radical Polymerization in a CSTR

For a given combination of chain length and number of branch points (n; N) a great number of molecular topologies is possible, but the specific chemistry of radical polymerization leads to a specific probability distribution of topologies. The synthesis algorithm [33] generates this distribution. Like the full MC method it is based on the primary polymers (pps) being the linear constitutive elements of branched molecules (Figure 9.19). A molecule with N branch points is composed of N þ 1 pps. The length distribution of pps (Flory) follows in much the same way

pp 2 pp 1 pp 3

Start: Seletion of n, N combinations from 3-D n-Nconcentration distribution (Galerkin-FEM) Selection of N+1 primary polymers from Flory distribution Sampling of monomers on pp’s to determine time order: pp1, pp2, pp3, ..., ppN+1. Start with pp1

pp 4 Connection algorithm

Connection algorithm: connects ppi to structure containing pp1...i-1

N

i < N+1 N

Y End: Architecture

Architecture

Scission fragment length distribution

Rheology: Seniority/ Priority

Synthesis algorithm for branched architectures in radical polymerization. Right: algorithm flow diagram. Upper left: connection of primary polymers. Lower left: a resulting architecture with specified connectivity structure and lengths between segments. Fig. 9.19.

Radius of gyration

503

504


as in the full MC method [Eq. (140)]. Note that pps are attachable to other ones by only one terminal. The mechanism making pps attachable at two ends is termination by recombination, but for the time being this will not be considered. The method allows for the fact that pps may undergo scission. Note finally that a primary polymer may carry several branch points. The population of pps thus produced possesses a length distribution that is controlled by the chemistry of the system. The first step of the algorithm is sampling N þ 1 pps from the calculated distribution, which is realized by selecting N numbers ni at random in the interval 1 < ni < n. This generates N þ 1 intervals 1 n1 , ðn1 þ 1Þ n2 ; . . . ; ðnN1 Þ nN ; ðnN þ 1Þ n, representing the desired pp lengths. Thus, a length distribution is obtained that approximates to a Flory distribution for large N and n [33]. The next step is finding the growth time sequence of pps in the molecule to be composed. This determines which pp may have grown from which other already existing pp. Notice that pp growth may be regarded as instantaneous in this radical system. Now, the time order in which pps are sampled is not independent of their length, since more possibilities exist of attaching smaller pps to a longer one than of attaching longer pps to a smaller one. This implies that the longer pps of a molecule, on average, are created before the short ones. In the algorithm, the early-time preference of long pps is realized by sampling of monomer units on pps. This is performed by selecting a random number between 1 and n and determining its location on the sequence of N þ 1 segments used for pp length sampling. This is the pp first in time order. It is removed from the sequence and the sampling is repeated for the remaining segments until the complete time order is determined. The connection part of the algorithm is simple after the time order has been determined; see Figure 9.19. The second pp selected is attached to pp1 ; the third may be attached to pp1 and pp2 , and so on. The algorithm accounts for the fact that a pp’s probability of receiving a branch point is proportional to its length, which reflects branch formation by transfer to polymer. Furthermore, it accounts for the pps’ differences in residence time. A pp early in time order – meaning a pp with a long residence time – has a higher chance of receiving branch points than a later pp. This implies that branch point distribution is heterogeneous, since pps with a longer residence time possess a higher branching density. However, this is counteracted by the circumstance that within a molecule the pps early in time order are longer on average. Note further that within a molecule, longer pps are associated with longer residence times, although obviously within the whole pp population in a CSTR no such relationship exists. We should realize here that the situation for a batch reactor might be different. When conditions change in a batch reactor, the pp length distribution might also change, which should be accounted for in the sampling procedure. The present algorithm is strictly applicable to a CSTR, since it assumes one (steady-state) pp length distribution. The algorithm generates topologies accounting for all the important chemical and reactor conditions influencing them: the specific n; N combination, pp length distribution, and coupling procedure representing transfer to polymer. The specific population of


architectures this produces indeed has properties different from those exhibited by the population of all possible architectures [33]. 9.9.3

Branched Architectures from Polymerization of Olefins with Single and Mixed Branch-forming Metallocene Catalysts in a CSTR [35] Introduction A polymer with length n and N branches may assume a large number of architectures. Due to the statistical nature of chemistry the architectures may feature large differences. We developed a Monte Carlo method virtually synthesizing the polymer according to the proper kinetic rules, which reflects the chemistry of the process. We first discuss a method for a single-catalyst system, which is essentially simpler than the method for a mixed system that will be presented next. Flow diagrams are shown in Figure 9.20. 9.9.3.1

Single-catalyst System In this case the method is based on the separation of the synthesis activities concerning topology and segment lengths. It will be shown next that the topology is created by a series of insertion events, during which structures with certain numbers of branch points are coupled. The probabilities of existence of these structures are only determined by the numbers of branch points they carry, not by the lengths of these segments. We can see this by realizing that in the single-catalyst system all chain segments in a molecule of given n and N should obey the same statistics, since all of them have grown under the same kinetic conditions. This fact implies that interchanging any pair of segments does not affect the probability of existence of a structure as long as it retains the same topology. For a complete architecture this argument is equally valid. As long as the topology is the same and as long as the total number of monomer units is constant, all molecules are equally probable. This implies that we can determine first the topology and afterward the length of the segments. 9.9.3.2

Synthesis of Topology In order to determine the topology of a molecule, as a first step of the algorithm we consider the first insertion of a terminally double-bonded polymer structure into a growing chain attached to the branching catalyst. The number of branch points on the inserted chain, N1 , and the number of branch points on the growing chain (to be formed after the first insertion) must add up to N 1. The situation is depicted in Figure 9.21. Note that with a single catalyst, at steady state in a CSTR, the statistical properties of the inserted chains and growing chains are identical. Insertions are possible in N 1 different ways, each way having its own probability. These probabilities can be calculated from the two-dimensional chain length/number of branch points distribution using the following arguments. The frequency of insertion of a species with a terminal double bond is proportional to its concentration. Therefore, the insertion probability of a polymer chain containing a certain num9.9.3.3

505

506

9 Mathematical Methods Start N

Sampling N1

Start N

N –1 from

Sampling N1

b

( N1 ) P N (eq 178) N2 = N – N1

N = N1

N2 = N – N1

Update of topology (connectivity matrix)

N = N2

N –1 from

b

( N1 ) P n , N (eq 183)

Sampling n1

n –1 ffrom o

b

(n1 , N1 ) P n , N (eq 181, for N1 selected) y

n2 = n – n1

N1 > 0 n

y

Sampling k n – n1 -1 from m

Start N, n

N2 > 0 n

b

N = N1

Segment length algorithm

Final topology (adjacency matrix)

n1 , N N1 1

(eq 184)

Update of architecture (connectivity matrix)

N = N2 2N + 1 segment lengths

Full incidence matrix

(k ) P n

y

N1 > 0 n

y

Full architecture

N2 > 0 n

Rheology

Full architecture

Radius of gyration

Rheology

Radius of gyration

Fig. 9.20. Flow diagrams of synthesis algorithm for branched architectures from metallocene-based polymerization. Left: single-metallocene system. Right: two-catalyst system.

ow

Structure being inserted Structure to grow on branching catalyst

th

N- N 1-1 -

Linear segment on branching catalyst

k

Dir ect ion

of

N 1, n 1

=

==

of gr

?

Branching catalyst

Branching catalyst

Dir ect ion

N1

gro wt h

Structure to grow on branching catalyst

N- N 1 -1, n -n 1 -k -1

Synthesis algorithm for branched architectures from metallocene-based polymerization. Left: single-metallocene system. Right: two-catalyst system. Fig. 9.21.

Structure being inserted, from either catalyst


ber of branches is proportional to that structure’s concentration Therefore, the insertion probability of a polymer chain containing a certain number of branch points, say N1, should be proportional to the concentration of chains with N1 branch points. However, the probability of insertion of a chain with N1 branch points is not the only factor playing a role. Additionally we need to know the probability of the growing chain, after the first insertion, obtaining precisely that number of branch points to arrive at N branch points in total: N N1 1. Now, during the growth process of a polymer chain, the probability that after the first insertion it will receive this particular additional number of branch points is independent of its growth history. This means that it is independent of whether it grows after the first, second, or any other insertion, and it also is independent of the number of branch points on chains previously inserted. Therefore, the probability that the polymer will grow to obtain the N N1 1 additional branch points is simply proportional to the concentration of such chains. Hence, under the condition that a polymer with N branch points is formed, the probability of a first insertion with N1 branch points equals the normalized product of the concentrations of polymers with a terminal double bond and N1 branch points and polymers with N N1 1 branch points. For the formation of polymers with a terminal double bond this leads to the conditional branch point probability density function (PDF): Tg . Considerable effort has therefore been made to describe yield in semicrystalline polymers in terms of crystal plasticity theory. In isotropic polycrystalline materials, five independent crystal slip systems are generally necessary for global plasticity, but this requirement is relaxed in polymers owing to the presence of the amorphous phase. Indeed, yield in polymers is thought to involve a relatively limited number of slip sys-

14.4 Yield and Fracture

tems with slip planes parallel to the chain axes (slip in planes perpendicular to the chain axes would require rupture of covalent bonds). The classical approach of Young to yield in lamellar semicrystalline polymers is based on the nucleation of dislocations by thermal fluctuations [26]. The activation energy is calculated from the shear strain energy for a screw dislocation of width u [Eq. (67), where b is the magnitude of the Burgers vector (usually the repeat distance along the chain), ro is the size of the dislocation core (about 4b according to computer simulations), G is the shear modulus, lc is the length of the dislocation in the direction of the Burgers vector (taken to equal the lamellar thickness) and s is the shear stress]. DU ¼

Gb 2 lc u sbul ln 2p ro

ð67Þ

The activation energy for the formation of the dislocation, DU , corresponds to the maximum of DUðuÞ, which leads to Eqs. (68) and (69). u ¼

Gb 2ps

DU ¼

Gb 2 lc u ln 1 2p ro

ð68Þ ð69Þ

Given DU , one can estimate the critical stress for the formation of a dislocation sc . Usually, it is assumed that DU A 50kT, which gives a sc comparable with experimentally determined values of the shear yield stress, ty . The main objection to the dislocation model is that the predicted temperature dependence of the yield stress is much weaker than that observed experimentally, particularly at higher T, and more recent efforts to describe yield have been based on thermally activated helical motions of polymer chains within their crystals, believed to be associated with the a relaxation, but which may occur below Tm [27]. In an isotropic polycrystalline polymer whose microstructure consists of stacked lamellae arranged in the form of spherolites, the slip systems activated depend on the local orientation of the lamellae with respect to the applied stress and, as deformation proceeds, these orientations are modified. To calculate the evolution of the crystalline texture, one can consider the polymer to behave as a crystalline aggregate. Although the entropic contribution of chain orientation in the amorphous regions may also need to be considered, the major contribution to work hardening in tension is rotation of the slip planes toward the tensile axis, so that the resolved shear stress in the slip direction diminishes. This results in a fiber texture in the limit of large deformations, such that the crystallites are oriented with their c axis (the chain axis) parallel to the stretch direction. Despite the relative success of such models, they do not explicitly address the micro-mechanisms involved in the transformation of the spherulitic texture into a fiber texture. One possibility is that the

747

748

14 Polymer Mechanical Properties

repeated shearing of lamellar fragments renders them thermodynamically unstable because of their small size. Models for the development of crystalline textures during drawing, based on this idea, thus explain the establishment of a new lamellar long period in highly deformed regions, which depends only on the deformation T and not on the initial lamellar thickness [27]. 14.4.4

Crazing and Fracture

Rather than undergo homogeneous (non-cavitational) ductile necking, glassy amorphous polymers often show brittle behavior in tension, in that they fail abruptly prior to yielding and hence at a relatively low strain. In uncrosslinked or lightly crosslinked polymers, this phenomenon is usually associated with crazing [28, 29]. As shown in Figure 14.14, crazes are crack-like defects spanned by highly drawn fibrils with lateral spacings of the order of 10 nm and a constant draw ratio, close to lmax , which implies that fibril extension takes place by ductile drawing at the craze–bulk interface. The presence of the fibrils means that crazes are capable of bearing loads normal to their faces. Indeed, the stress normal to the faces of an isolated growing craze, sc (the fibril drawing stress), is approximately equal to the global applied stress along most of the craze length. Hence, isotropic bulk samples deformed in uniaxial tension may contain a high density of crazes whose planes are parallel and perpendicular to the tensile axis. Even so, plastic deformation associated with crazing remains highly localized and contributes relatively little to the overall deformation. Given that crazes may act as preferential sites for crack initia-

Fig. 14.14. Craze microstructure in a thin film of amorphous polycarbonate deformed in tension at 100 C.


tion, this accounts for their association with brittle macroscopic behavior as defined above. Crazing and ductile necking may nevertheless co-exist, depending on the relative stability of the crazes with respect to crack initiation. Empirical criteria for the formation of crazes in multiaxial stress states, analogous to the von Mises criterion for yield [Eq. (59)], are based on the observation that crazing is absent in both compression and simple shear, which is reasonable given that one would expect cavitation to be favored by large values of p. A critical strain to craze, ec ¼ A þ B/p, is generally found to provide a reasonable description of craze nucleation, so that in terms of the principal stresses the criterion is given by Eq. (70), where A; B; C, and D are constants. s1 ns2 ns3 ¼ C þ

D s1 þ s2 þ s3

ð70Þ

Given that highly triaxial stress states are unfavorable to yielding [compare Eq. (59)], crazing tends to dominate in highly constrained geometries, such as notch tips, or in the vicinity of local stress concentrations in bulk specimens. This is not inconsistent with fibrillation by ductile drawing, because the void formation associated with crazing releases constraints on plastic deformation at the microscopic level. The very small size of craze fibrils means that their surface energy is expected to play an important role in craze formation. Hence, although the large hydrostatic stress gradients associated with closely separated voids favor fibril drawing, they are offset by stress gradients arising from the surface tension at the void tips, G. In kinetic models for fibril formation, the fibrils are therefore argued to adopt the characteristic spacing Do that maximizes the overall stress gradient and hence the rate of fibril extension. It has been shown on this basis that the craze stress sc m G 1/2 , for a fixed deformation rate [28]. There is evidence that polymers with high entanglement densities, ne , craze less readily than polymers with low ne. One possible explanation for this is in terms of the creation of surface associated with fibrillation. For a fixed entanglement network, the creation of free surface must involve chain scission, as shown in Figure 14.15. The higher ne, the more entanglements are lost during fibrillation, the higher the energy cost in creating the fibrils and hence the higher sc. This may account for the ductility of amorphous, high ne ‘‘engineering thermoplastics’’ such as PC, in which crazing tends to be suppressed in favor of homogeneous deformation (sy is not directly dependent on ne ). It follows that highly crosslinked polymers, in which the influence of ne and nx may be considered additive, do not show crazing, although their ductility, and hence their toughness, is limited by the limit extensibility of the network [28]. The competition between homogeneous deformation and crazing is also likely to be strongly influenced by secondary relaxations, so that the relatively low b relaxation temperature of PC (see Section 14.4.2) may also contribute to its ductility. Indeed, shifting the secondary relaxation to higher T by chemical modification at constant ne has been found to lead to embrittlement in PC [30]. Similarly, polymers that craze easily, such as aPS and PMMA, typically have relatively high secondary relaxation temperatures.

749

750


Fig. 14.15. Schematic of craze widening at the craze–bulk interface, illustrating the geometrically necessary entanglement loss.

The macroscopic property of immediate practical concern is often not so much craze formation as fracture resistance. In a cracked specimen, stress is concentrated at the crack tip, and the local stress state depends on the crack length. For small global deformations, it may be expressed in terms of a stress intensity factor, K, given by Eq. (71), where a is the crack length and F is a dimensionless function of the specimen geometry [31, 32]. pffiffiffiffiffi K ¼ F pas

ð71Þ

In many fracture mechanics-based approaches, crack advance is taken to occur when K reaches some critical value K c (equivalent energy-based criteria are also widely used). K c may then be measured using a pre-cracked specimen, in which a and hence K are well defined. In some cases (PMMA, for example) crack advance is observed to proceed via breakdown of a single craze at the crack tip. By modeling a craze as an orthotropic linear elastic body it has been shown that K c is given by Eq. (72), where n is Poisson’s ratio, a is related to the craze anisotropy, sc is the draw stress normal to the craze-bulk interface, vf is the fibril volume fraction in the craze, and sf is the stress to break a craze fibril [33]. 1/4

Kc A a

1 vf E 1 n 2 sc

1/2

pffiffiffiffiffiffiffiffi pDo sf

ð72Þ

This approach incorporates the stress-concentrating effect of cross-tie fibrils, widely observed in crazes in glassy polymers (compare Figure 14.14). In the absence of any stress-concentrating effect, that is, for a ! 0, a time-independent fibril failure criterion sf implies crack advance can never occur, because the stress in a given fibril can never exceed sc . This result has been confirmed by more detailed micromechanical modeling, and is important in that it provides a direct link between the


Fig. 14.16. Fibrillar deformation zone at a crack tip in a bulk specimen of semicrystalline high-density polyethylene deformed at room temperature (thin section stained with RuO4 ).

macroscopic fracture behavior and microscopic quantities, which can be estimated by direct observation of craze microstructures. Although Eq. (72) is not strictly applicable to the more usual case of multiple crazing, or mixed crazing and shear at the crack tip, the guiding principle remains valid and serves to underline the importance of entanglement, not only for craze formation but also for ultimate failure. Bulk semicrystalline polymers deformed in tension also show considerable localized micro-necking at T between Tg and Tm , in the form of either crazes or craze-like deformation zones, such as that shown in Figure 14.16. As with amorphous polymers, these are thought to have a strong influence on the ultimate failure properties. Although the details of the lamellar structure may vary widely, qualitative models for micro-necking in melt-crystallized semicrystalline polymers above Tg show many features in common. Deformation is usually assumed to initiate in interlamellar amorphous regions, which stretch and cavitate, so that cavity sizes are commensurate with the lamellar separation. The intervening material is then drawn down to form the mature fibrillar structure. Again, a high degree of entanglement and strong anchoring of the chains by the lamellae (that is, high M) are expected to promote toughness in semicrystalline polymers [5, 34, 35]. To improve the fracture resistance of polymers that craze readily, but which may have other advantages, such as cheapness in the case of aPS, a common strategy is to increase the density and extent of localized damage (crazing, for example) by creating local stress concentrations [36]. The greater the extent of local damage, the greater the energy dissipation during crack advance, and hence the greater the crack resistance. The presence of rubber inclusions in a glassy or semicrystalline matrix results in considerable local mismatch in stiffness. This increases their stress-concentrating effect and provides local concentrations in the von Mises stress or the hydrostatic stress, which will favor shear and craze yielding respectively. Cavitation of the modifier particles is also thought to be important in tough-

751

752


ening and may in some cases change the local stress state sufficiently to promote a change in mechanism (for example, shear to craze transitions in glassy polymers).

14.5

Conclusion

The aim of this contribution has been to link the basic macroscopic phenomena associated with polymers with the unique features of their structure, the most obvious being the presence of long, flexible molecular chains. The important role of the conformational entropy of flexible chains, not only for rubber elasticity but for polymer dynamics in general, has been demonstrated. Moreover, the concept of an entanglement network, which underpins much of the theory of polymer dynamics in the melt, has also been shown to have important repercussions for the high strain behavior of solid polymers, namely plastic deformation, crazing, and fracture.

Notation

A Ao a a aT b b Cl g C1 g C2 Cy D Do DR E E0 E 00 E Eijkl f fg fv G G0

Helmholtz free energy [J] cross-sectional area of a tensile specimen [m2 ] Helmholtz free energy per unit volume [J m3 ] statistical segment length of the primitive path (tube model) [m] WLF shift factor equivalent bond length (Kuhn length) [m] magnitude of Burgers vector [m] specific heat at fixed specimen length [J kg1 ] WLF constant WLF constant [K] characteristic ratio tensile compliance [Pa1 ] craze fibril spacing [m] Rouse self-diffusion coefficient [m 2 s1 ] Young’s (tensile) modulus [Pa] tensile storage modulus [Pa] tensile loss modulus [Pa] complex tensile modulus [Pa] stiffness tensor [Pa] force [N] fractional free volume at Tg (dimensionless) fractional free volume (dimensionless) shear modulus [Pa] shear storage modulus [Pa]

Notation

G 00 G Ge J K K Kc k L l lc lo M Mb Mc Me m N N0

shear loss modulus [Pa] complex shear modulus [Pa] plateau modulus [Pa] shear compliance [Pa1 ] bulk modulus [Pa] stress intensity factor [Pa m 1/2 ] critical stress intensity for crack advance [Pa m 1/2 ] Boltzmann constant ¼ 1:38066 1023 J K1 tube contour length (tube model) [m] length [m] lamellar thickness [m] mean length of skeletal bonds in a real macromolecular chain [m] molar mass [g mol1 ] molar mass per statistical segment [g mol1 ] critical molar mass [g mol1 ] entanglement molar mass [kg mol1 ] number of beads in Rouse model (dimensionless) number of statistical segments in a freely jointed chain (dimensionless) number of statistical segments between beads (Rouse model) (dimensionless) number of skeletal bonds in a real macromolecular chain (dimensionNl less) first normal stress difference (dimensionless) N1 NA Avogadro’s number ¼ 6:022 10 23 Ne number of statistical segments in a chain of molar mass Me (dimensionless) number of statistical segments in the primitive path (tube model) (diNL mensionless) n number of chains per unit volume [m3 ] entanglement density [m3 ] ne number of crosslinks per unit volume [m3 ] nx PðR; NÞ probability distribution of R in a chain with N links p hydrostatic pressure [Pa] Q activation energy [J] Q heat transfer (into a thermodynamic system) [J] R end-to-end vector of a linear chain [m] maximum end-to-end distance of a real macromolecule R max vector corresponding to the nth bond in a chain [m] rn dislocation core radius [m] ro S entropy [J K1 ] T temperature [K, C] glass transition temperature [K, C] Tg melting point [K, C] Tm U internal energy [J] u width of screw dislocation [m]

753

754


u V vf W

critical screw dislocation width [m] activation volume (Eyring model) [m3 ] craze fibril volume fraction (dimensionless) work (done on a thermodynamic system) [J]

Greek a af G g DE DU DU d dc ec eij eij z zo h ho hg hy Y l m n n n n0 r sc sf sij sij sy t tc tD tR ty WðRÞ

craze anisotropy factor coefficient of thermal expansion of fractional free volume [K1 ] surface tension at craze void tips [N m1 ] shear strain energy difference (Robertson model) [J] shear strain energy of screw dislocation [J] activation energy for screw dislocation [J] phase angle critical crack opening displacement [m] strain to craze engineering strain components of the engineering strain friction coefficient [N s m1 ] monomeric friction coefficient [N s m1 ] viscosity [N s] zero shear viscosity [N s] viscosity at Tg [N s] limiting high shear rate viscosity [N s] structure temperature (Robertson model) [K] deformation ratio (dimensionless) pressure coefficient (stress-dependent von Mises criterion) (dimensionless) jump rate (Eyring model) (dimensionless) Poisson’s ratio (dimensionless) scaling exponent (Zimm model) (dimensionless) attempt frequency (Eyring model) (dimensionless) density [kg m3 ] craze fibril draw stress [Pa] stress to break a craze fibril [Pa] components of the engineering stress [Pa] engineering stress [Pa] tensile yield stress [Pa] relaxation time [s] critical stress for the formation of a screw dislocation [Pa] reptation time [s] Rouse relaxation time [s] shear yield stress [Pa] number of conformations at fixed R (dimensionless)

References

angular frequency (dimensionless) proportion of high energy states (Robertson model)

o w

Acronyms aPS iPP K-BKZ PC PE PMMA PVC WLF

atactic polystyrene isotactic polypropylene Kaye-Bernstein–Kearsley–Zapas bisphenol A polycarbonate polyethylene poly(methyl methacrylate) poly(vinyl chloride) Williams–Landel–Ferry

References 1 P. J. Flory, Statistical Mechanics of

2

3

4

5

6 7

8

9

10

11

Chain Molecules, John Wiley, New York, 1969. W. L. Mattice, U. W. Suter, Conformational Theory of Long Chain Molecules, John Wiley, New York, 1994. L. J. Fetters, D. J. Lohse, W. W. Graessley, J. Polym. Sci., Part B: Polym. Phys., 1999, 37, 1023. M. Rubenstein, R. H. Colby, Polymer Physics, Oxford University Press, Oxford, 2003. I. M. Ward, D. W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, John Wiley, New York, 1993. U. W. Gedde, Polymer Physics, Chapman and Hall, London, 1995. L. R. G. Treolar, The Physics of Rubber Elasticity, Clarendon Press, Oxford, 1958. J. E. Mark, B. Erman, Rubber-Like Elasticity, a Molecular Primer, John Wiley, New York, 1988. B. Erman, J. E. Mark, Structures and Properties of Rubberlike Networks, Oxford University Press, Oxford, 1997. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., John Wiley, New York, 1980. N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behaviour, Springer Verlag, Berlin, 1989.

12 K. L. Ngai, D. J. Plazek, in Physical

13

14

15

16

17

18

19

20

21

Properties of Polymers Handbook (J. E. Mark, editor) AIP Press, Woodbury, NY, 1996. M. Doi, Introduction to Polymer Physics, Oxford University Press, Oxford, 1996. M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986. R. G. Larsen, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1998. R. B. Bird, C. F. Curtiss, R. C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nd ed., John Wiley, New York, 1987. J. des Cloizeaux, G. Jannink, Polymers in Solution: Their Modelling and Structure, Clarendon Press, Oxford, 1990. P. G. DeGennes, Scaling in Polymer Physics, Cornell University Press, Ithaca, 1979. L. J. Fetters, D. J. Lohse, R. H. Colby, in Physical Properties of Polymers Handbook (J. E. Mark, editor) AIP Press, Woodbury, NY, 1996. H. A. Barnes, J. F. Hutton, K. Walters, An Introduction to Rheology, Elsevier, Amsterdam, 1989. R. G. Larson, Constitutive Equations

755

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22

23

24 25 26

27 28

for Polymer Melts and Solutions, Butterworths, Boston, MA, 1988. B. Crist, in Materials Science and Technology, Vol. 12 (E. L. Thomas, editor), VCH, New York, 1993. J. C. Bauwens, C. Bauwens-Crowet, G. Home`s, J. Polym. Sci. A2, 1969, 7, 735. A. S. Argon, Phil. Mag., 1973, 28, 839. R. E. Robertson, J. Chem. Phys., 1966, 44, 3950. R. J. Young, Introduction to Polymers, Chapman and Hall, London, 1981. A. Galeski, Prog. Polym. Sci., 2003, 28, 1643. E. J. Kramer, Adv. Polym. Sci., 1983, 52, 1.

29 E. J. Kramer, L. L. Berger, Adv.

Polym. Sci., 1990, 91/92, 1. 30 L. P. Chen, A. F. Yee, E. J. Moskala,

Macromolecules, 1999, 32, 5944. 31 J. G. Williams, Fracture Mechanics of

32 33 34 35

36

Polymers, Ellis Horwood, Chichister, 1984. H.-H. Kausch, Polymer Fracture, Springer Verlag, Berlin, 1985. H. R. Brown, Macromolecules, 1991, 24, 2752. C. J. G. Plummer, H.-H. Kausch, J. Macromol. Sci. – Phys., 1996, B35, 637. J. J. Benkoski, P. Flores, E. J. Kramer, Macromolecules, 2003, 36, 3289. C. B. Bucknall, Toughened Plastics, Applied Science, London, 1977.

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15

Polymer Degradation and Stabilization1 Tuan Quoc Nguyen

15.1

Introduction

Most synthetic plastics and natural polymers are principally constituted of the elements C, H, N, and O, all of which are primary components of organic molecules. As with smaller organic molecules, these atoms are attached by relatively weak covalent bonds that are susceptible to chemical attack, particularly to oxidative degradation. During their potential lifetime, engineering plastics are exposed to diverse environmental factors which can act alone or in combination to adversely affect the initial material properties. These changes generally occur over several years, but can be accelerated or retarded in the presence of internal and external elements temperature, and temperature variations. Environmental factors that contribute to degradation include mechanical stresses, temperature variations, humidity, sunlight, oxygen, atmospheric pollutants, and microbial enzymes. Depending on the structural level at which material changes occur, it is usual to distinguish between physical, physicochemical, and chemical degradation.

Physical degradation, commonly known as physical aging, results from changes in the thermodynamic state of the system (temperature, pressure, molar volume) during the daily use of the material. During processing, chains may be oriented and the sample inhomogeneously cooled, resulting in internal stress. With time, the chains have the opportunity to relax from their nonequilibrium conformations to a state of lower free energy. Partly immiscible blends, for instance, can remain in a homogeneous metastable state for an extended period of time. The presence of stress or heat can accelerate the phase separation process. The resultant changes in orientation, free volume, internal stress, crystalline content, and


the end of the text, under ‘‘Notation’’.


758

15 Polymer Degradation and Stabilization

phase transitions are known globally as physical aging. In principle, physically aged samples do not affect chemical structure and can be rejuvenated by proper thermal treatment. Physicochemical degradation involves diffusion of small to medium size molecules either from the polymer (plastifier or other additives) or into the polymer (water absorption, solvent, surface-active liquids), resulting in changes in polymer properties. Exudation of plastifier and surface migration of additives followed by washing are among the common problems encountered in natural weathering. In the presence of surface-active agents, environmental stress cracking can also develop. Although preserving the structural integrity of the polymer, physicochemical degradation is in general irreversible as a result of local changes in chemical composition, and can result in severe alterations of material properties. Chemical degradation concerns irreversible changes in the chemical structure of the polymer initiated by a number of activation agents frequently encountered in outdoor weathering: – oxygen (causing oxidation), reactive pollutants (ozone, nitric oxides), enzymes (causing biodegradation); – heat (causing thermal degradation); – photons (from UV or g-radiation), electrons and high energy particles (causing photodegradation and radiative degradation); – mechanical stress (causing mechanochemical degradation). In this review, we will focus exclusively on chemical degradation, with particular emphasis on environmental degradation. It should be kept in mind, however, that under practical weathering conditions, physical, physicochemical and chemical degradation may happen simultaneously and in a synergetic way, in some cases resulting in premature material failure. Although the term ‘‘degradation’’ usually has a negative connotation, some polymer applications require degradability as an asset or even a prerequisite as, for instance, in biodegradable detergents in some biomaterials, and in the manufacture of microelectronic circuitry. In fact, the growing importance of degradation and stability assessment in plastic materials is largely motivated by two apparent contradictary interests:

on one hand, there is an increased consumer demand for quality and durability to avoid the expensive labor costs of replacing deteriorated components; on the other hand, for some applications, the plastic should be able to degrade spontaneously and safely when disposed of, to reduce the solid waste burden. Obviously, this ‘‘degradability-on-demand’’ can be realized only with an adequate understanding of the degradation and stabilization mechanisms.

15.2 General Features of Polymer Degradation

15.2

General Features of Polymer Degradation

Over 95 wt.% of polymers are synthesized from eight vinyl monomers, and most of the remainder are from another couple of dozen monomers with diverse chemical structures. Despite this relative simple chemistry, the problem of degradation and stabilization of synthetic polymers is exceedingly complex. This complexity stems from the structural organization of plastic materials, which occurs at different levels:

At the molecular level, most commercial plastics are mixtures of macromolecules with diverse substances, which can be traces of solvent or catalyst residues, pigments, fillers, and a variety of additives. The macromolecules themselves can be blends of homopolymers or copolymers. As to the differences in small organic molecules, even a simple homopolymer presents a diversity in chain length and configuration. The presence of chain branching, tacticity, monomer linking, or chain defects may all have important consequences for the material’s stability. Molecular chains can arrange themselves into amorphous, oriented, and crystalline domains of different sizes, shapes, and distributions. This intricate morphology is globally known as supramolecular organization. Studying degradation requires not only identification of the diverse chemical reactions which induce structural changes in the polymer, but also the interactions between the numerous chemical species initially present in the sample or formed in the course of degradation. Interdependence of the change in material morphology that occurs upon aging, and its influence on the rate of degradation, should be considered additionally. 15.2.1

Degradative Reactions

From the complexity of chemical processes, and the intricate relationship between chemical, morphological, and environmental factors which control degradation, it is safe to state that there are as many degradation mechanisms as commercial plastic formulations. Obviously, examining in detail all the effects mentioned for a given polymer system requires considerable effort. Fortunately, a large group of polymer degradation problems share a number of common features. Most degradative reactions of polyolefins, for instance, can be envisioned as the reverse process of vinyl polymerization, with four steps: radical initiation, depropagation (the reverse of propagation), chain branching under oxidative conditions (produced by intermolecular and intramolecular hydrogen transfer followed by b-scission), and radical termination. Except for the mode of initiation, the thermal, photolysis, and radiolytic degradation processes are similar in many respects and can be described

759

760


Initiation

PH þ X ! P þ XH

ð1Þ

Radical conversion

P þ O2 ! PO2

ð2Þ

Propagation

PO2 þ PH ! POOH þ P

ð3Þ

Chain branching

POOH ! PO þ OH

ð4Þ

PO þ PH ! POH þ P

ð5Þ

HO þ PH ! H2 O þ P

ð6Þ

PO ! various chain-scission reactions

ð7Þ

P þ P ! inactive products

ð8Þ

PO2 þ PO2 ! inactive products

(9)

Termination

General oxidative degradation mechanism for polyolefins (PH designates the polymer, P a macroradical, and X a radical of unspecified nature). Scheme 15.1.

by the same reaction scheme, a simplified version of which, based on polyolefin degradation but applicable to other types of polymers with a carbon backbone, is given as Eqs. (1)–(9) in Scheme 15.1 [1]. Initiation Covalent bond scission, with formation of free radicals or charged species, is the starting process in any type of chemical degradation. This primary event requires energy, which can be supplied by heat (thermal degradation), UV photons (photodegradation), high-energy photons or particles (radiolysis), mechanical forces (mechanochemical degradation), or reactive chemicals (solvolysis, enzymatic degradation). Heterolytic bond scission is restricted to degradation by reactive chemicals and by enzymes in biodegradation. For the other modes of degradation, bond scission is mainly homolytic, except in some particular situations described in Section 15.2.2. Apart from its free-radical nature, homolytic initiation is generally a complex and less well-known process (radical X of reaction (1)). The exact mechanisms of initiation depend on the polymer structure, the presence of internal and external impurities, and the degradation conditions. Also, the mode of initiation may change during the course of reaction. Under oxidative conditions, for instance, formation of hydroperoxides followed by their decomposition rapidly becomes the prevalent mode of initiation. 15.2.1.1

Propagation Once initiated, the alkyl macroradical can proceed through a number of possible reaction pathways. In the absence of oxygen, some macroradicals can depolymerize by unzipping, or they can abstract a hydrogen atom from a nearby molecule or from another group inside the same molecule. In oxidative environment, the free 15.2.1.2

15.2 General Features of Polymer Degradation

radical combines readily with molecular oxygen to produce peroxy radicals, which can subsequently abstract H atoms to re-form an alkyl radical. Radical conversion reaction is extremely rapid and is diffusion-controlled, owing to the diradical nature of molecular oxygen, with k2 @ 10 8 –10 9 L mol1 s1 . Hydrogen abstraction reactions, on the other hand, require a high activation energy and are the ratedetermining step with k2 /k3 > 10 6 . The activation energy for k3 depends on the enthalpy of reaction DH ¼ De ðPaHÞ De ðPOOaHÞ, where De is the bond dissociation energy. With De ðPOOaHÞ A 360 kJ mol1 , peroxide abstraction reactions are endothermic with a rate constant decreasing rapidly with increasing CaH bond dissociation energy, according to the sequence: tertiary CaH (De ¼ 385 kJ mol1 ) > secondary CaH (400 kJ mol1 ) > primary CaH (410 kJ mol1 ). This behavior has been verified in the case of polyolefins, for which it is found that HDPE is more resistant to oxidative degradation than LDPE, which is itself more resistant than isotactic polypropylene (iPP), owing to the increasing presence of tertiary CaH bonds in the polymer [2]. Chain Branching Hydroperoxide decomposition is the key step in polymer oxidative degradation and has been extensively studied in a number of systems, using low MW analogs of the polymers. Corresponding rate constants and activation energies can be found in Ref. 2, p. 383. The fate of the hydroperoxides formed is diverse. The OaO bond energy is only approximately 170–195 kJ mol1 and is easily cleaved by heat or UV light below 340 nm according to Eq. (10). 15.2.1.3

D;hn 400 C and an estimated melting point of 1400 C, PPP constitutes a reference for heat-resistant polymer. Poly(para-phenylene) can be synthesized by a number of routes, with the best heat-resistant material obtained by polymerization of 1,3-cyclohexadiene, followed by dehydrogenation of the polymer formed according to Eq. (35) [16]. -H

n n

ð35Þ n

Moldable polymers can be obtained by increasing chain flexibility with substitution of lateral groups, at the expense of decreasing thermal stability. Polyimides and polyetherimides (V), with a semi-rigid structure and electronically stabilized aromatic rings, form an important class of melt-processible polymers with outstanding heat resistance (Table 15.4). Apart from the decomposition temperature, the propensity for char formation is another polymer thermal characteristic of practical importance. The char-forming tendency generally shows a negative correlation with flammability: a sample with the greatest char residue is also the least flammable polymer and the presence of flame retardant additives also increases char formation. In the case of fire, the amorphous crust of char material that forms on the fluid surface reduces the heat flux to the burning fluid, which in turn reduces the combustion rate. 15.4.3

Computer Simulation

Thermal degradation involves scission of covalent bonds which can be formally written as Eq. (36). k

A!BþC

ð36Þ

15.4 Thermal Degradation Tab. 15.4. Chemical structure and thermal stability by thermogravimetry of some representative polyetherimides O O

N

N

O

O

O

O

n

Ar

(V) xAr x

˚

Thermal stability [ C]

˚

4,4 0 -biphenyl 4,4 0 -benzophenone 4,4 0 -diphenyl ether 4,4 0 -diphenyl sulfide

Tg [ C]

1 wt.% loss (air)

1 wt.% loss (N2 )

5 wt.% loss (air)

5 wt.% loss (N2 )

247 239 227 209

500 490 480 470

515 485 480 470

530 530 513 504

543 523 523 505

According to Eyring’s transition-state theory, the rate constant k is given by Eq. (37), with kB ¼ Boltzmann’s constant, h ¼ Planck’s constant, DE0 ¼ molecular energy difference at absolute zero between the activated complex and the reactant, and qz and qA the molecular partition functions of the transition state and reactant, respectively. The transition state is distinguished by a single negative vibration, a characteristic which has been used to identify true transition states in computer modeling. kðTÞ ¼ ðkB T/hÞðqz /qA Þ expðDE0 /kB TÞ

ð37Þ

With the increasing capabilities of computers and development of new numerical methods, it is now possible to predict polymer properties computationally. In addition to saving time, computer-aided chemistry can sometimes provide new insights into some decomposition mechanisms which are difficult to obtain by experimental techniques. Computer modeling has been used in an increasing number of ways to simulate thermal degradation. A few representative examples are described below.

Ab-initio calculations were performed on a series of model structures to predict the effect of lateral alkyl substituents on the thermal stability of degradable polycarbonate. The optimized transition structures revealed that the Ca aO bond dissociates first, followed by abstraction of the b-hydrogen atom, developing a carbocation character in the transition state on the Ca atom. Substituents which stabilize the transition state will also accelerate the degradation rate [17].

781


Molecular dynamics simulation has been conducted to understand the significant reduction in flammability of polymer nanocomposites. Using graphite as a model compound, it was found that polymer in an intercalated structure loses fewer fragments generated by the degradation and retains its shape longer as a result of repulsive nonbonding interactions between the polymer and the graphite layers. The fragments generated by thermal activation collide with the graphite and bounce back into the central unit cell, where they undergo recombination reactions with other free radicals. The decrease in polymer fragments which can escape as combustible fuel accounts for the observed improvement in fire resistance [18]. Force field calculations with the MOPAC PM3 package [19] allow the prediction of thermal degradation mechanisms. First, the smallest representative section of the polymer to be investigated was selected. The program then calculated the heat of formation and Gibbs free energies at different temperatures of the polymer and for different radical fragments. The kinetic barrier for bond dissociation was estimated by stretching each individual bond to its breaking point, and plotting the heat of formation (enthalpy) as a function of bond extension. Calculations on the PA-6,6 structure identify the aCOaCH2 a bond as the one having the lowest transition state energy, and the carbonyl carbon as the most susceptible site for radical attack. Upon dissociation, the methyl radical created folds back and attacks the aCbO bond, forming cyclopentanone, in accord with experimental findings [20]. a

a

782

15.4.4

Thermal Oxidative Degradation of Polypropylene

Isotactic polypropylene (iPP) is a major commodity plastic material which cannot be utilized without thermal stabilizers. With a moderately complex structure, iPP is frequently used as a ‘‘model system’’ to test the different theoretical and experimental approaches to macromolecular degradation. Initiation The homogeneous oxidation of PP follows the free-radical auto-oxidation mechanism depicted in Scheme 15.1. Under isothermal conditions, the oxygen uptake curves display a pseudo-induction period during which oxidation is autoaccelerated. The duration for the induction period t i depends on the sample purity and preoxidation history. For clean samples, t i is reasonably reproducible and is of the same order of magnitude as the POOH lifetime, determined by iodometric titration. The temperature dependence of the induction time obeys an Arrhenius law with an apparent activation energy of 105 G 15 kJ mol1 , which is the same as for the decomposition of hydroperoxides. The corresponding rate constants are much lower than for the other degradation processes and account for the induction period during which hydroperoxides accumulate before reaching a maximum. It has long been recognized that residual Ziegler–Natta polymerization catalysts, gener15.4.4.1

15.4 Thermal Degradation

0.3

1.2 oxygen uptake (left scale)

1

mole O 2/kg

0.2

0.6

mole POOH/kg

peroxide content (right scale)

0.8

0.1

0.4 0.2 0

0

10

20

30

40

0 50

degradation time [min] Fig. 15.7. Oxygen uptake (left-hand scale) and hydroperoxide formation (right-hand scale) in the thermo-oxidative degradation of iPP films at 130 C, with small (—) and large (----) spherulites.

ally at the 1–20 ppm level, accelerate the solid-state degradation of PP. The negative influence of polymerization catalyst residues depends not only on the type of catalyst, but also on its concentration. The induction period stage is complicated by the morphology of the sample. In iPP, for instance, a sample with small spherulite sizes (< 100 mm), obtained by rapid quenching, has a short induction time (Figure 15.7). A large spherulite sample (350–500 mm), obtained by prolonged annealing, results in a significant increase in the induction period. The difference was explained by a difference in the oxygen diffusion rate in the oriented amorphous regions, which are more strained in large spherulite structures [21]. It was shown from decomposition kinetics and by treatment with dimethyl sulfide that peroxides consist of two types: a fast-decomposing one composed of peracids, and a slowly decomposing one consisting of hydroperoxides and hydroperesters. During the induction period, the slowly decomposing hydroperoxides accumulate and the oxidation rate is controlled by the rate of decomposition, which may be finally catalyzed by metal ion residues. The autoacceleration stage is controlled by the fast-decomposing peracids [22]. Regardless of the exact mechanism of bond scission, the mechanistic step for initiation in PP can be described schematically by Eq. (38).

783

784


CH3 CH2 CH

+X

CH2

.

.

CH3

CH CH

CH2 ð38Þ

CH3

.

CH2

CH2 C

Propagation For PP, the radical conversion step with oxygen is given by Eq. (39). 15.4.4.2

CH3

.

CH2

CH2 C

CH3 C

+ O2

CH3 CH2

C

.

H

CH2 C

O

H

O

CH3 ð39Þ

Hydrogen abstraction from tertiary carbon has an activation energy lower by approximately 15 kJ mol1 than an abstraction reaction on secondary carbon (Ref. 2, p. 386), and is the predominant mode of formation of hydroperoxides in PP. In addition to energy considerations, the hydrogen abstraction rate constant is dependent on steric factors and polymer conformation. It is found, for instance, that k3 (Scheme 15.1) in solution is lower in a theta solvent than in a good solvent, owing to increased steric repulsion in a contracted molecular coil [7]. Abstraction reaction in polymers can occur intramolecularly or intermolecularly. The former possibility is important in polymers which possess lateral groups (PP and PS), whereas it is nonexistent in linear polymers (HDPE). The possibility of intramolecular H abstraction has been advanced as one of the reasons for the high sensitivity of PP (in comparison to HDPE) toward oxidative degradation. Infrared studies have revealed that more than 90% of hydroperoxides in PP are hydrogen-bonded in sequences of two or more. This result supports an intramolecular hydrogen abstraction reaction, facilitated by a six-membered ring stereochemical arrangement [Eq. (40)]. CH3 CH2 C

CH3

CH3

CH2 C

O O

.

.

CH2 C

CH2 C O

H

CH3

OH + O2

CH3 CH2 C

O

CH3

CH3

CH2 C CH2 C O H O OH

.

ð40Þ


Chain Branching Chain branching by homolytic decomposition of polymer hydroperoxides results in formation of highly reactive polymer alkoxy (PO) and hydroxyl (OH) radicals. The small and mobile OH can readily abstract hydrogen from a nearby polymer chain, creating a secondary or tertiary macroalkyl radical. The tertiary alkoxy radical can abstract hydrogen intramolecularly, forming a primary alkyl radical as in Eq. (41). 15.4.4.3

CH3

C

CH2 CH3 C CH O CH2 H

CH2 CH

O H

CH2

ð41Þ

The polyalkoxy radicals can decompose further by b-scission, yielding ketones, aldehydes, and alkyl radicals, depending on the initial position of the radical [Eqs. (42a), (42b), and (43)].

CH3 CH2

C

CH3 CH2

CH

O CH3 CH 2

C O

β-scission

CH 3 +

CH 2

CH

ð42aÞ

CH3 CH2

C

CH2

CH

+

CH3

ð42bÞ

O

CH3 CH

CH3 CH O

CH

β-scission

CH3 CH

CH 3

H +

C

CH

ð43Þ

O

Similarly, the cleavage of peroxy radicals results in the formation of double bonds along with aldehydes and ketones [Eqs. (44) and (45)].

785


CH3

CH3

CH

CH

β-scission

CH

O O CH3

ð44Þ

CH 3

H

CH

+

C

HC

CH

OH

+

O CH3 CH2 C

CH3

CH2

β-scission

CH

O O

ð45Þ

CH 3

CH3 C

CH 2

CH2

+

O

+

CH

OH

Termination In an oxygen-deficient atmosphere, vinylidene and vinyl compounds may be formed from the disproportionation of polypropylene radicals [Eqs. (46) and (47)]. 15.4.4.4

2PaCH2 aCH ! PaCH2 aCHbCH2 þ PaCH2 aCH2 aCH3

ð47Þ

a

ð46Þ

a

2PaCHaCH2 ! PaCbCH2 þ PaCHaCH3 CH3 CH3 CH3 a

786

In the presence of oxygen, the majority of recombinations occur between tertiary peroxy radicals, to give dialkyl peroxides and oxygen [Eq. (48)]. CH 3 CH2

CH3

CH2

C

+

CH2

C

O

CH2

O O

O

CH3

CH3

CH2 C

C

O

O O

O

CH2

CH2

CH3

CH3

C

C O

ð48Þ CH2

+

O2

O

H = -300 kJ.mol-1

Secondary Reactions In oxidative degradation, it is convenient to make a distinction between primary chemicals formed by reactions of the polymer peroxy radicals, and secondary products created in subsequent reactions of these primary compounds. Ketones and al15.4.4.5


dehydes formed by b-scissions may react further with hydroperoxides to form a variety of oxidation products. The formation of peracids, for example, proceeds via a multistep oxidation of aldehydes [Eq. (49)].

H

CH3 CH2

CH

POO.

POOH

CH3 CH2

C

CH

+ O2, PH

C.

O

O OOH

CH3 CH2

CH

C O

ð49Þ

It should be emphasized that the reaction scheme of Eq. (49) is unlikely if hydroperoxides are distributed homogeneously over the whole sample. Owing to restricted mobility below Tm, the oxidation products cannot diffuse away. Locally, hydroperoxides can reach sufficiently high concentrations for secondary reactions to occur at a significant rate. In a similar fashion, the formation of carboxylic acids, esters and g-lactones proceeds through a complex series of oxidation reactions of alcohols and ketones. The aldehydes and ketones formed by b-scission can undergo Norrish type I [Eq. (50a)] and Norrish type II reactions [Eq. (50b)] during photodegradation (see Section 15.5.3).

CH 3 CH 2

C O

I rish Nor Nor rish

CH 3 CH 2

CH

+

CH3CO

ð50aÞ

CH3 II

CH

CH

CH

CH 3 + CH3COCH3

ð50bÞ

For a long time, the similarity between thermal and photolytic products was a mystery because Norrish photoprocesses are absent in the former situation. By using low MW model compounds, chemical derivatization, and 13 C-NMR techniques, it has been assessed that a large fraction of the carbonyl-containing compounds formed during oxidative degradation of PP are a-methylated acids. Such a carboxylic structure can only originate from the oxidation of macroalkyl radicals [Eq. (51)], which can be formed either by a Norrish I photoprocess or by b-scission of alkoxy radicals (Ref. 10, p. 583).

CH3 CH2

CH

+ O2, PH

∆ or hv

CH3 HOO– CH2

CH 3

O HO

CH

C

CH

ð51Þ

787

788


Once the structure of the carboxylic acids had been elucidated, the IR absorption band which appears as a shoulder at 1740 cm1 in thermal or photo-oxidation of PP, and was initially attributed to ester functions, was reassigned to an acidic group hydrogen-bonded to a vicinal hydroperoxide (VI). CH3

CH3 CH2

CH

CH

O

C O

OH

OH

(VI)

Formation of Volatile Compounds At least 40 different volatile compounds have been identified during the oxidative degradation of iPP, some of which are important for spreading the degradation according to the ‘‘infection’’ mechanism (Section 15.4.5). The mode of formation of the principal volatile products is indicated in Eqs. (52)–(55) [23]. It is proposed that CO and CO2 are formed by the decomposition of carbonyl and carboxyl radicals following hydrogen abstraction from the parent compounds [Eqs. (52) and (53)]. Peracids formed by oxidation of formaldehyde can decompose into formic acid, water, carbon dioxide, and triplet oxygen according to Eqs. (54) and (55). 15.4.4.6

H

CH3 CH2

CH

C

POO.

POOH

CH3 CH2

O

C.

CH

O

ð52Þ

CH3 CH2

OH

CH3 CH2

CH

POO.

C.

CO

+

H

POOH

CH3 CH2

C O

CH

C.

O O

ð53Þ

CH3 CH2

C.

+

CO2

H O H

C O

OH CO2

+

H 2O

ð54Þ


O 2

H

OH

OH 2 H

C O

C

+ O

O2

ð55Þ

15.4.5

Homogeneous versus Heterogeneous Kinetics

Degradation reactions in the solid state are spatially heterogeneous, not only across the sample but also within the sample. At least two factors can account for the heterogeneity of the degradation:

Nonuniform distribution of the reactants (oxygen, additives, impurities): uptake of oxygen and formation of reactive chemicals are the highest near the surface. Additives distribution can also be inhomogeneous as a result of inadequate blending or migration. Heterogeneous structure of the sample: oxygen (or any other chemical reactant) can diffuse more readily into amorphous than into crystalline domains, thus favoring oxidative degradation of the former regions. Superimposed on this macroheterogeneity, which occurs on a micron scale, is microheterogeneity resulting from the low diffusivity of macromolecular species. In solid polymer, it is expected that most macromolecular constituents formed from radical reactions are localized in microdomains of size not exceeding 10–30 nm. The preceding considerations indicate that degradation kinetics of solid polymers cannot be studied solely within the framework of homogeneous free-radical kinetics. In spite of these complications, it has been found in many instances that the homogeneous kinetics treatment of thermal oxidation gives a surprisingly good fit with experimental data [24]. It has been argued that the homogeneous kinetic model may be applied to an inhomogeneous system by considering the parameters as average values. It should be kept in mind, however, that the real concentrations of degraded species may be much higher locally than the mean values averaged over the bulk sample, and that extrapolating microscopic kinetics from macroscopic average data is a nontrivial problem. The evidence for heterogeneous oxidation of solid polypropylene was revealed by chemiluminescence measurements. Chemiluminescence from single polymer powder particles revealed a large spread in the induction periods. The oxidation behavior changed with particle separation, and the result was interpreted as a result of physical ‘‘infection’’ from a particle with the shortest induction period to its nearest neighbors, causing rapid oxidation of the complete sample. The origin of this infection was traced back to the formation of volatile compounds, more specifically acetic acid and formaldehyde, which have the propensity to accelerate oxidative degradation [25]. Highly mobile radicals, such as OH, can also diffuse far and spread the degradation.

789

790


Devising a chemical kinetics model which encompasses all sources of heterogeneity present in a solid polymer is a challenge. The starting point in most heterogeneous kinetics models is to consider that oxidation is initiated nonuniformly at a few sites, presumably around catalyst residues or other impurity centers in the amorphous region of the polymer. One approach to heterogeneous modeling was by Monte Carlo simulation, based on the random-walk diffusion of low MW reactive radicals. Another concept, called the ‘‘epidemic model’’, utilizes the analogy between the spreading of oxidation through the amorphous region of solid polymer, and the infectious contamination of a disease through a population. In this model, three distinct polymer populations are assigned: the ‘‘infectious’’ oxidizing fraction p i , the remaining unoxidized fraction p r , and the ‘‘dead’’ or oxidized fraction pd . These populations may be described by a series of coupled differential equations, Eqs. (56)–(58), which are developed by noting that oxidation can spread only if there is unoxidized material within contact distance of the infected one. dp r /dt ¼ bp r p i

ð56Þ

dpd /dt ¼ ap i

ð57Þ

dp i /dt ¼ dp r /dt dpd /dt

ð58Þ

Integration of the above system of differential equations using experimental b and a parameters give an excellent fit with the chemiluminescence curve when dealing with single particles, but the quality of the fit deteriorated with groups of particles, as a result of multiple-spreading processes [26]. By considering only the oxidized and the unoxidized fractions, a simplified spreading model was developed to explain the time dependence of hydroperoxide concentration during oxidative degradation of LDPE [27], expressed as Eq. (59), where n is the number of oxidized domains, N the total number of amorphous domains, and r the rate coefficient for spreading. dn/dt ¼ r n r ðn 2 /NÞ

ð59Þ

15.4.6

Applications of Thermal Degradation

Although the word ‘‘degradation’’ usually has a pejorative connotation, a number of important industrial applications depend on this class of reactions to develop new products or processes. Analytical Pyrolysis Pyrolysis of a polymer means thermal degradation in the complete absence of any external reactant. Analytical pyrolysis, defined as pyrolysis conducted in combina15.4.6.1


tion with some physicochemical separation technique (pyrolysis–GC–MS, for instance), has found a wide range of applications in biopolymers and synthetic polymers. Polymers pyrolyze by different combinations of three general mechanisms: random scission (PE), depolymerization (POM, PMMA), or elimination of small molecules other than monomer (for instance, HCl in PVC). Each polymer under strictly controlled pyrolytic conditions provides a typical pattern of degradation products, which can be used for fingerprint identification, composition analysis, structure determination, or mechanistic studies [28]. Introduction of New Chemical Functionalities Controlled thermal degradation of polyolefins in the presence of peroxides has been carried out in extruders or in continuous mixers, to reduce polymer MW and sample polydispersity. The resulting material has more processing advantages than the undegraded one. Based on the same principle, random chemical functionalization of polyolefins with polar groups has been achieved by extruding the polymer in the presence of peroxy ketals or peroxy esters [29]. 15.4.6.2

Chemical Modification of Polymer Structure Carbon fiber is used in a variety of structural and electrical applications, and is probably the most well-known example of high-performance material produced by thermal degradation. Although carbon fiber can be produced in several ways, including alignment of molecules of pitch in its mesophase state, the most economical way consists of ‘‘hot-stretching’’ high MW polyacrylonitrile (PAN) fibers and while heating under controlled conditions. The filaments are first oxidized under tension at around 260 C in air, to convert the PAN precursor to a thermally stable structure by an exothermic reaction (E a ¼ 3:5 kJ g1 , DH ¼ 2:5 kJ g1 ), according to Eq. (60) [30]. 15.4.6.3

260°C, air C

C

C

N

N

C N

C

C N

N

N

O

O

ð60Þ

O

O

C OH

C C N

N

N H

N

N

stabilized PAN

N

N

NH2

H

The carbonization step is carried out in an inert atmosphere (high-purity N2 ) above 1000 C to yield carbon fibers containing 93–95% carbon [Eq. (61)] and a tensile modulus (E) of 140–200 GPa. This grade is mostly used in sporting goods and in composites.

791

792

15 Polymer Degradation and Stabilization N

N

N

N

N

N

1000°C, N2

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

ð61Þ

To achieve fibers in which the carbon crystals are further stretched and aligned, graphitization takes place around 2000–3000 C. These graphite fibers have a carbon content greater than 99% and a tensile modulus of 400–1000 GPa. For comparison, the tensile modulus of carbon nanotubes is 1000 GPa, and of diamond 1200 GPa. Metal Injection Molding (MIM) Polyacetals have a low ceiling temperature, and are readily depolymerized by unzipping at low temperature (0.4–0.8% min1 at 222 C for POM). Owing to this low thermal stability, polyacetals can be used only if end-capped with stable groups (acetate or ether). This inherent thermal instability is exploited in an industrial method known as ‘‘metal injection molding’’, which allows fine metal powder mixed with a polymer binder to be processed by injection molding, in much the same way as thermoplastic materials [31]. In a procedure based on POM, the binder is removed by thermal devolatilization according to Eq. (62) 15.4.6.4

HNO3 /150 C

RaðCH2 aOaÞn H ! RaCH2 OH þ ðn 1ÞH2 CbO

ð62Þ

Recycling Polymer can be recycled by reuse of existing material (primary recycling), by regranulating the waste by mechanical means so that it can be melted and formed again (secondary recycling), or by transforming the waste into new chemical compounds (tertiary recycling) through chemical reactions. The last of these methods presents several economic advantages in comparison to primary or secondary recycling, since revalorizing steps can be avoided. Recently, thermolysis has been viewed as a viable alternative to recovery for polymer recycling and numerous studies have been directed toward this objective [32]. The term ‘‘feedstock recycling’’ has been used to describe this new class of plastics recycling technology, which breaks down solid polymers into a spectrum of ba15.4.6.5

15.5 Photodegradation

sic chemical compounds that can be reused as raw materials for the chemical industry. Vinyl polymers, when pyrolyzed at temperatures from 200 to 500 C in the total absence of air, usually degrade to yield monomers (poly(methyl methacrylate), poly(a-methylstyrene), polystyrene, polyisobutylene) or a wide distribution of molecular fragments (polyethylene and polypropylene). In order to reduce the process temperature and to limit the range of products, particularly in the case of polyethylene, several catalyst cracking systems based on zeolites or clays have been developed. A newly developed technique in the recycling field is thermolysis coupled with reactive distillation.

15.5

Photodegradation 15.5.1

Absorption of UV Radiation by Polymers

Photo-oxidation is the most common mode of weathering for industrial polymers, and differs from thermal degradation principally in the mode of chemical activation. In place of vibrational energy, the energy of the photon provides the driving mechanism for free-radical generation. Photochemical processes are based on two fundamental principles: the Grotthus–Draper law, which states that only radiant energy that is effectively absorbed can activate molecules, and the Stark–Einstein law, which asserts that one absorbed photon can induce photochemical reaction of only a single molecule [33]. The first law implies the presence of an appropriate chromophore in the polymer, whereas the second law indicates that the formation of photoproducts is linearly dependent on the light intensity, and that a photoreaction can occur only if the energy of the photon is sufficient to overcome the corresponding activation energy. For most organic molecules, absorption in the near-UV (190–400 nm) involves electronic transition from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). Commercial polymers can be broadly divided into two categories, depending on whether or not they contain delocalized p-electrons, sometimes combined with heteroatoms with nonbonding valence shell electron pairs (O, N, S) in their chemical structure. Polymers which contain carbonyls ( aCbO), conjugated polyenes ( aCbCaCbC a), ketenes ( aCbCaCbO), and sulfones (aFaSO2 aFa) belong to the first group of chromophores and are capable of absorbing light in the near UV (190 nm < l max < 400 nm); see Table 15.5. Those from the second group possess only single covalent bonds, such as CaC, CaH, CaO, CaCl, and CaF, and can in principle absorb light only in the far UV (l max < 190 nm). Owing to vibrational and electronic couplings, UV absorption bands of polyatomic molecules in the condensed state are generally quite broad, with width at half-maximum commonly exceeding 60–80 nm. Even with this broadness, polymers such as polyolefins, polyacetals, poly(vinyl chloride), or polyacrylonitrile

793

a

a

a

a

15 Polymer Degradation and Stabilization Tab. 15.5.

Principal chromophore groups in synthetic polymers[a].

Polymer

Chromophore

lmax [nm]

emax [L molC1 cmC1 ]

Polyesters Polyaromatics Poly(aryl ketone)s Polydienes Conjugated polyenes

a

aCbO aFa aFaCOaFa aCbC a aCbCaCbC a a(CbC)3 a a(CbC)10 a aCbCaCbO aFaSO2 aFa

188 (279) 200 (256) 250 (350) 185 (230) 217 263 432 220 (350)

900 (15) 4 400 (226) 18 000 (120) 8 000 (2) 20 900 @5 10 4 @2 10 5 2 10 4 (30)

a a a

[a] Values

a

Ketenes Sulfones

a

in brackets refer to the secondary absorption band.

should neither absorb nor degrade when exposed to light with wavelength above @230 nm. In practice, however, the UV absorption spectra of commercial samples of all the cited polymers show a broad absorbance band which extends well above the expected limits (Figure 15.8). Generally, these extraneous absorbances are weak and can be determined only after proper signal processing. This situation is illustrated in Figure 15.9 for commercial additive-free films of HDPE and iPP.

1.6 PSU, 1µm PET, 2µm PC, 2µm PS, 5µm PMMA, 20µm PVC, 100µm

1.4 1.2

absorbance

794

1 0.8 0.6 0.4 0.2 0 200

250

300 wavelength [nm]

UV absorption spectra of some common industrial polymer films, at the thickness indicated. Fig. 15.8.

350

15.5 Photodegradation

3.0

2.0 absorbance

HDPE

iPP

1.0

0.0

160

180

200

220

240 260 280 wavelength [nm]

300

320

340

Fig. 15.9. UV absorption spectra of 50 mm-thick films of additive-free commercial HDPE (—) and iPP (---).

Far-UV spectroscopy indicates that the ‘‘absorption edge’’ of long-chain alkanes starts at about 155 nm. Absorption at much longer wavelengths observed in commercial polyolefins could be explained only by the presence of chromophore impurities and chemical defects formed during the synthesis, storage, and processing of the polymer. The absorption peak at 180 nm recorded in HDPE, for instance, has been attributed to vinylidene end groups, as revealed by independent FTIR measurements (Figure 15.9). Impurities in polymers can be classified as internal or external [7]. Internal impurities These are part of the molecular structure, situated either along the chain or at chain end(s), and may consist of:

Anomalous structural units, which result from the kinetics of polymerization. In-chain peroxides, formed during polymer synthesis. Free-radical polymerization is generally accomplished without strict exclusion of air and small amounts of oxygen dissolved in the monomer will be scavenged by the macroradicals and included in the polymer chain as peroxide linkages. Photolysis of these peroxide groups gives alkoxy radicals, leading ultimately to hydroperoxides. Alternatively, if polymerization has been carried at elevated temperatures, the peroxides incorporated will undergo thermal fragmentation and disproportionation to form phenyl alkyl ketone end-group chromophores. Carbonyl containing groups formed during material transformation. Processing, with elevated temperature and high shearing stresses, provides ample opportunities for thermal oxidation. Even at ambient temperature, some polymers such

795

796


as isotactic polypropylene or cis-polybutadiene, can be readily oxidized if unstabilized. External impurities Such impurities are contained in the sample but not incorporated in the polymer structure. During synthesis, processing and storage, the polymer can be contaminated or blended with a variety of external chemical species which may contain chromophore and photoreactive groups. Some typical external impurities found in commercial plastics are:

residual catalysts and initiators, traces of solvents (aromatic solvents, in minute amounts, can act as photosensitizers (Ref. 26, p. 125), pigments, dyes, and additives, traces of metal, metal oxides, or metal salts from the reactor, processing equipment, and containers.

15.5.2

The Solar Spectrum

The majority of polymer bond dissociation energies are within the 290–420 kJ mol1 bracket (allylic CaC being the lowest, and aCH2 aH the highest). The activation energy of photochemical reactions in the gas phase usually lies just a few percent above the corresponding bond dissociation energy, while it can reach 40 kJ mol1 for diffusion-controlled reactions in the condensed state. Therefore, depending on the type of bond to be broken, any absorbed photons with wavelength shorter than approximately 420 nm for the weakest bonds, to 290 nm for the strongest ones, could promote chain scission. The importance of photodegradation in outdoor weathering depends much on the sunlight spectrum. The solar spectral irradiance at the top of the atmosphere is well-approximated by black-body radiation with a temperature of 5770 K. The Earth’s atmosphere, on the other hand, is practically opaque to any wavelength shorter than about 290 nm as a result of ozone absorption. The UV range of solar irradiation is commonly divided into three spectral regions of decreasing wavelengths denoted by UV-A, UV-B and UV-C. Although UV-B light is the most efficient at initiating photodegradation, its intensity at the Earth’s surface is fortunately very limited, owing to the screening effect of ozone, and accounts for 240 nm and oxygen, the phenyl groups in PS, PPO, and PC can undergo ring-opening oxidation with formation of mucondialdehyde (Scheme 15.4) [36].

15.6

Radiolytic Degradation 15.6.1

Interaction of High-energy Radiation with Matter

Understanding the effects of ionizing radiation on the properties of plastics materials is important in nuclear engineering, space research, radiation processing, and radiation sterilization. Radiation composed of photons or particles with energy much higher than those encountered in electron bonding orbitals is referred as high-energy radiation (X- or g-rays, high-energy electrons, protons, and charged particles). Owing to this excess

805


806

(A)

CH3

–O

O

P

.

.CH2

PH

– O– C–

C

–O

O

– O – C–

C CH3

CH3

O

–O –O

CH2

O

.C

O

hv/O2

– O – C–

– C – CH3

–O

– C – OH

–O

– OH

CH3

+ HCOOH + CH3COOH

(B)

.OH 1O

+

– O–

– C–

→

– O–

– C–

→

acids, esters

2

O Scheme 15.4.

O

Chain scission (A) and ring-opening (B) reactions in bisphenol-A polycarbonate.

energy, high-energy radiation can penetrate much deeper into material to create ions, superexcited states, and hot radicals. The degradative effects are also much more extensive than with UV. The principal sources of high-energy radiation are electron beam accelerators, which account for 90% of commercial radiation capacity; the remainder consist of 60 Co installations. 60 Co is unstable and decays to the stable 60 Ni according to Eq. (73). 60

Coðt1/2 ¼ 5:27 yÞ ¼

60

Ni þ b

ð73Þ

The nuclei of Ni atoms that result from this decay are in an excited state and immediately emit two g-rays of energies 1.332 and 1.173 MeV. The low-energy b are absorbed by the 60 Co housing and all the radiolytic effects result from the g-ray emission. Electron accelerators can deliver higher dose rates, whereas 60 Co sources are characterized by a greater depth of penetration. A fast electron loses most of its kinetic energy by inelastic collisions with electrons from the medium, producing energetic secondary electrons. Depending on the energy of the radiation, many secondary electrons of decreasing energy will be

15.6 Radiolytic Degradation

blobs 100-500 eV

spurs 1:1). Either strong acids or strong bases can be used as catalysts. Acids are most widely used, but with bases a higher content of ortho–ortho linkages can be obtained if desired for mechanical properties. Unlike resols, which are mostly methylolated monomer species with some occasional polymeric species, novolacs immediately form true polymer chains of predictable molecular weight. In principle methylene bridges, being more thermodynamically stable than ether bridges, predominate in the polymer chains. So the P/F ratio (P being in excess) determines ideally the (number-averaged) molecular mass Mn . Novolacs cannot form extensive networks by themselves because of the formaldehyde deficiency. A gel could possibly be formed, however, because of branches in the polymer chains when a P/F ratio close to 1:1 is chosen with a three-functional phenolic monomer. 16.2.2.2

16.2 Phenolic Resins

The higher the Mn of the resin and the higher the functionality of the phenolic monomers, the higher the risk of gelation during production. This can be calculated with the Flory–Stockmayer and related mathematical theories. One easy way out is not to use phenol (three-functional) but, for example, p-cresol (twofunctional) as the monomer: only linear polymeric chains can then be formed. Curing of novolacs requires addition of extra formaldehyde; mostly HMTA, typically 8–15%, is used as a solid formaldehyde source. Epoxy-novolacs Without extra formaldehyde, novolac resins are hardly reactive with other resins, as the aromatic hydroxyl groups are relatively inert. They can be used to introduce other reactive functionalities, however. The most important example of a functionalized novolac resin is in the reaction with epichlorohydrin (Scheme 16.3). The resulting resins contain glycidyl groups and can be used as epoxy resins (see Section 16.4). The advantage of these epoxy resins over classical bisphenol-A glycidyl resins is that they contain more epoxy groups per unit of weight and have a higher functionality as a resin (one glycidyl group per phenolic unit). Both phenol- and cresol based novolacs are used as epoxy resin precursors. Besides reaction with epichlorohydrin, phenolic resins are sometimes also reacted with alkyl chlorides or alcohols to etherify the aromatic hydroxy groups. In this way, their compatibility/ solubility in combination with other resins (or solvents) is enhanced. 16.2.2.3

O

OH

OH

OH

O Cl

O

O

O

O

O

(n+2) n Scheme 16.3.

n

Epoxy-novolac resins formed with epichlorohydrin (ECH).

Discoloration The major disadvantage of phenolic resins, both resols and novolacs, is the strong yellow to black color of the resins, or the cured materials. The discoloration (the monomers and sometimes the resins are colorless in principle) takes place when the material comes into contact with oxygen, which is virtually impossible to prevent. By oxidation of methylene bridges, for example, p-quinone puffered structures are formed which have a high extinction coefficient for visible light (Scheme 16.4 top). Azomethine structures also can be formed when oxidation takes place in a resin which was cured in the presence of ammonia (for example, from ‘‘hexa’’); see Scheme 16.4 bottom. 16.2.2.4

841

842

16 Thermosets

1/2 O2 HO

O

CH

O

O

-H2O

OH

OH NH

OH 1/2 O2

OH N

-H2O Scheme 16.4.

Oxidation of phenolic resins leading to discoloration.

16.2.3

Production

Phenol resins, both resols and novolacs, are exclusively made in batch processes. In the production of novolacs, molten phenol is charged into the heated and stirred reaction tank together with the weak acidic catalyst (for example, oxalic acid). At 95 C, the aqueous formalin solution is slowly added, while the reaction heat is removed through distillation of water at slightly diminished pressure. When all the formaldehyde has reacted, the water is distilled off and the reaction with the excess phenol is continued at higher temperature (140–160 C). When the desired viscosity is reached, the excess of unreacted phenol is removed by distillation as well. The molten resin is let down onto a cooling belt, and broken into small glassy lumps. Resol resins are made at temperatures from room temperature to a maximum of 60 C. As in novolacs, molten phenol is charged first into the reactor together with the basic catalyst (for example, calcium oxide), after which the formalin solution is slowly added with cooling of the mixture through water distillation. A higher vacuum (40–50 mbar) must be used than with resol resins because of the lower reaction temperature. When the desired viscosity has been reached, the reaction can be quenched by addition of a small amount of acid (to a pH of approximately 4–5). Then water is removed through distillation, either to dryness or to a concentrated aqueous solution. If the resol resin has sufficient molecular weight to be a stable solid at room temperature, the dry resin melt is processed to glassy lumps over a cooling belt, like a novolac resin. Very low molecular weight resol resins (Mn < 200 g mol1 ) are dispensed as aqueous solutions; resins having higher molecular weight (500–800 g mol1 ) are often diluted with alcohol solvents (for better solubility). 16.2.4

Properties and Applications

Phenol resins have found their ways to numerous applications in varying markets: wood glues, bonding agents for mineral and glass fibers, laminates, molded parts, coatings, inks, rubbers, matrix materials for grinding and brake composites. Their

16.3 Amino Resins

success is explained by their relatively low price and high degree of chemical and thermal inertness. Thermal decomposition sets in only between 220 and 250 C, depending on the composition. Phenol resins are quite stable to hydrolysis, even at high pH, and are therefore suitable for making particle board for contact with concrete castings. Once cured, phenolic materials are hard but brittle. To overcome their brittleness, they can be combined with other resins in thermoset formulations. Their compatibility with other resins can be tuned through the choice of phenolic monomer (resins based on nonylphenol can be dissolved even in hydrocarbons), their molecular weight, and the degree of etherification. Due to their discoloration they are mostly used in non-surface applications. For example, coatings based on phenol/epoxy systems are used as primers but not as topcoats. High-pressure laminates are composed of a pile of papers impregnated with resol resin, on top of which a dećor paper impregnated with a colorless melamine–formaldehyde resin (next paragraph) is placed, the stack being cured together in a press. Resols can also be used in combination with unsaturated rubbers such as EPDM (see Section 16.9) which are cured cationically.

16.3

Amino Resins 16.3.1

Introduction

Amino resins are based on reactions of compounds containing amino, imino, or amide groups with aldehydes [3]. As in phenolic resins, the most widely used aldehyde is formaldehyde. In an aqueous environment a condensation reaction between formalin and the amino, imino, or amide compound leads to highly branched, low molecular weight polymers. Introducing an acid catalyst is usually necessary to initiate further condensation reactions that lead to curing. The commercial history of amino resins goes back to 1924 with the use of urea as the amino source. In 1935 melamine (2,4,6-triamino-1,3,5-triazine)–formaldehyde (MF) resins were introduced, as more expensive but higher quality resins than the urea–formaldehyde (UF) resins. These two resins are still the main amino resins used to date. Applications exist of pure UF or MF, but also of physical or chemical mixtures of melamine and urea resins, leading to melamine–urea–formaldehyde (MUF) resins. 16.3.2

Chemistry

In principle the condensation reaction between formalin and the amino group, referred to as methylolation and somewhat similar to the reaction of phenol and formaldehyde (see Section 16.2.2), can take place at four positions on a urea mole-

843

844

16 Thermosets

NH 2 N

N

2HN

N H

N

OH

Monomethylomelamine H N N

(Mono) NH2

OH N

N N

2HN

N OH H N,N'-Dimethylolmelamine H N N HO

N N H

OH

H N

N(

N H

N(

N

N N(

N

N OH)2

OH)2

2HN

N N

N(

OH )2

N,N,N',N'-Tetramethylolmelamine

N N

N(

N( N )N

OH)2 (Penta)

OH)2 N

N

N(

OH )2

Hexamethylolmelamine Scheme 16.5.

(Tetra)

OH)2

Pentamethylomelamine

2(HO

(Tri)

OH)2

N(

N

(Di)

OH

N,N',N'-Trimethylolmelamine

N,N,N',N"-Tetramethylolmelamine

HO

2HN

OH

N

N H

OH )2

N',N'-Dimethylolmelamine

N

N

N(

N

H N

N,N',N"-Trimethylolmelamine

HO

2 HN

OH

N

N H

N

The nine possible ‘‘methylol’’ adducts of melamine and formaldehyde.

(Hexa)

16.3 Amino Resins

cule and at six positions on a melamine molecule. The methylolation reactions of melamine and urea are fast and reversible at temperatures over 70 C. Melamine actually dissolves in the reaction mixture due to this methylolation reaction. For melamine nine different methylolated melamine molecules have been identified (Scheme 16.5), each with its own molecular equilibrium constant. Both primary (HOCH2 NH) and secondary methylols (HOCH2 NCH2 OH) are formed. In particular the tri- (primary) and hexa- (secondary) methylolated species are the most stable. In the polymerization reactions, depending on the molar ratio of melamine to formaldehyde, any substitution state of melamine can be found. Therefore, MF resins are usually highly branched. For urea four different methylolated ureas have been identified (Scheme 16.6). The existence of a fifth, tetramethylolurea, is uncertain. The primary dimethylolated species is the most stable one and in polymerization reactions urea tends to react primarily as a two-functional monomer. In accordance, UF resins with a low urea/formaldehyde ratio are only slightly branched. O HO

N H

NH2

Monomethyllurea O HO

N H

N H

O OH

2(HO

)N

NH2

N,N-Dimethylolurea

N,N'-Dimethylolurea O )N

2 (HO

N H

OH

Trimethylolurea Scheme 16.6.

The four possible ‘‘methylol’’ adducts of urea and formaldehyde.

Polymerization Chemistry In MF resins, melamine methylol groups condense with melamine amino groups into methylene bridges linking two triazine moieties. Two melamine methylol groups condense to a methylene ether, in short ether bridges (see Scheme 16.7). The main reaction parameters are pH and the molar ratio of formaldehyde to melamine (F/M) and/or urea. Typical evolution of the chemical composition of a melamine formaldehyde resin during synthesis is shown in Figure 16.2. In UF resins urea homocondensation results in methylene and ether bridges as well, as shown in Scheme 16.8. In MUF resins, next to the homocondensations of melamine and urea species, co-condensations can also take place, again via both ether and methylene bridges (see Scheme 16.9) [4]. The critical parameters influ16.3.2.1

845

16 Thermosets

846

NH2

NH2 N H2N

N N

N

+ HO

NH2

N H

NH2

N N

N NH2

NH2 N

2 2HN

Scheme 16.7.

N

N

N N N H H "Methylene bridge"

NH2

N N

N N

H2N

NH2

N N H

OH

H2N

NH2

N N

NH2

N N H

O

N

N N H "Ether bridge" Condensation reactions of amino resins: homocondensation of melamine.

NH2

0.8 0.7 Di Mono

0.6 mol/kg resin

Tri 0.5 Ether

0.4 Tetra 0.3 0.2

Methylene

Penta 0.1 Hexa 0 0

20

40

60

80 Time (min.)

100

Fig. 16.2. The concentration of the methylol groups (solid lines) and the methylene and ether bridges (broken lines) as a function of synthesis time. (F/M ¼ 1:65, 95 C, pH 9.5). For the labels on the methylol lines, see Scheme 16.5.

120

140

16.3 Amino Resins

O

O

O + H2N

N H

HO

NH2

H2N

NH2

O 2

N H

HO

O

N N NH2 H H "Methylene bridge"

O H2N

NH2

847

O N H

O

N H "Ether bridge" Scheme 16.8. Condensation reactions of amino resins: homocondensation of urea.

NH2

encing homo- and co-condensation, as well as homo- and co-condensation, are the molar amounts of M, U, and F during each phase of the resin synthesis, temperature, and pH. Changes in the pH have a direct and substantial influence on the condensation reaction kinetics and, thus, on the number of methylene and ether linkages formed. Changes in the pH of 0.3 can double the rate of an individual reaction. In moderately alkaline media (pH 7–9) methylene bridges are formed predominantly, while in stronger alkaline conditions (pH 9–10) primarily ether bridges are produced. Unlike methylene bridges, ether bridges are susceptible to hydrolysis. Equilibrium is reached after short reaction times. Therefore, a fairly constant concentration of ether bridges is often observed, caused by similar rates of formation and hydrolysis. Methylene bridges do not undergo this hydrolysis and thereby increase in concentration during synthesis; see also Figure 16.2.

NH2 N H2N

O

N

+ NH2

N

HO

N H

NH2

NH2 N

NH2 N 2 HN

H2N

N

2HN

NH2

NH2 O

N N

N H

NH2

NH2 N

N H

"Methylene bridge" H2N

OH

N

O

+ N H

N

O

N

N

+ N H

OH

HO

N H

NH2

H2N

O

N N

N H

O

N H

"Ether bridge" Scheme 16.9.

Condensation reactions of amino resins: co-condensation of melamine and urea.

NH2

848

16 Thermosets

16.3.3

Production

Amino resins are produced in batch processes at temperatures of around 90 C in aqueous solution under alkaline conditions. The solid content (amount of dry substance: melamine, urea, and formaldehyde) in the reactor is around 50–70% by weight. For MF resins the pH during production is around 9–9.5 at the beginning of the process when all the raw materials are added to the reactor, and will stay around that value during resin preparation. For UF resins the first phase of the reaction is similar, alkaline methylolation, but after some hours the pH is brought down to around 5 to promote the bridge formation reactions. When producing a MUF resin, different scenarios for adding the raw materials to the reactor are possible regarding the moment of dosing. One should choose different pH settings accordingly. Numerous recipes are described for synthesizing a MUF resin. It is possible to simply use a mixture of urea–formaldehyde and melamine–formaldehyde resins, but usually melamine and urea are introduced into the reactor at different times. Two major concepts have found general industrial acceptance. One starts with the synthesis of a UF resin, followed by a cocondensation step of melamine (UFþM); the other starts with a MF resin and co-condenses urea in the second stage (MFþU). Mostly, in each case a final urea dosage is used to lower the molar ratio. The UFþM concept is characterized by a backbone of UF chains and a lower branching efficiency of the melamine molecule. In the MFþU concept the branching degree of the melamine molecule is increased and the melamine is more efficiently incorporated into the polymeric structure, resulting in a reduction of the dominant character of the urea backbone. Both methods of production have their own merits in resin stability and in the performance in the different fields of application. The desired degree of condensation of the resins is determined in the case of UF and MUF by the viscosity of the reaction mixture. Typically resins have viscosities in the range of 50–400 mPa s. UF resins are milky white due to small crystallites of linear UF segments; MUF resins can be milky white as well, but also clear. In MF resins the degree of condensation is determined via the so-called ‘‘water tolerance’’. This value is expressed as the relative amount of water that can be added to the resin while remaining clear, that is, not turning turbid because of phase separation of water-insoluble higher oligomers. Increasing condensation times give rise to increasing molecular weight and, hence, lower water tolerance. Because of their reactivity and intrinsic gelating potential, amino resins are characterized by a limited storage stability. In general an aged resin (more than one to three months old) has a very high viscosity or is gelled and cannot be applied any more. Typically acceptable storage times are one month, depending on storage conditions but also on initial viscosity, and/or water tolerance, and/or the chemical structure of the resin. For some applications the resins are spray-dried and applied as dry solid or redispersed solutions. Solid amino resins do not show this fast ageing behavior because the polymerization reactions are halted by the strong decrease in molecular mobility (vitrification) of the polymeric system.

16.4 Epoxy Resins

16.3.4


Curing of amino resins proceeds readily in an acidic environment; an acidic catalyst is usually added to the amino resin just prior to use. As catalysts, ammonium sulfate for UF and MUF and p-toluenesulfonic acid for MF are typical examples. A cured amino resin is hard and brittle. Some degree of flexibilization can be achieved by adding glycols or polymers containing hydroxyl groups to the resin. UF resins are much more sensitive to moisture than melamine resins. Applications which require a certain moisture resistance, either in swelling or hydrolysis, necessitate the use of melamine (MUF or MF). The majority of amino resins are used in combination with wood or paper. The interaction with these materials is very good; amino resins are used as wood glues in wood composites like particle board, plywood, and engineered lumber products. Amino resins can also be the binder material in paper. Most money papers are reinforced by melamine resins. A larger application in combination with paper, especially for melamine–formaldehyde, is as impregnation resins. Paper is impregnated with resin and hot-pressed, giving decorative films for products like worktops and laminate flooring with a very durable surface (hard, scratch- and chemical-resistant). Laminate flooring provides a very good illustrative example of the application of melamine in combination with mainly paper and wood fibers. This product has acquired its own place among other floor coverings such as carpets, tiles and parquet. Going from the bottom to the surface of a laminate floor, one finds: a backing paper impregnated with melamine resin; a substrate such as particle board where the wood particles are glued together with MUF resin; a wood dećor printed paper impregnated with melamine resin, and a transparent overlay paper impregnated with melamine resins. This pile of materials is then hot-pressed (typically at 150– 200 C and 40–60 bar) for between 30 s and 2 min. After sawing into pieces and sanding, the laminate decorative flooring is ready for application in homes and commercial remises, being very durable, scratch- and liquid-resistant, and easy to clean. Other applications include binders for molding materials such as electrical plugs or dinnerware, binders for mineral and glass wool insulation and roofing, flameretarding agents, and leather auxiliaries.

16.4

Epoxy Resins 16.4.1

Introduction

Epoxy resins also can look back at a long history of development, as the first ones became commercial in the early 1940s after their invention in 1938. The term

849

850

16 Thermosets

O

O

O Cl

ECH Scheme 16.10.

O

Glycidyl ether

O O

Glycidyl ester

Epichlorohydrin (ECH) and glycidyl compounds.

‘‘epoxy’’ refers to the presence of an epoxide group in the compound (monomer or resin). This group is a three-membered cyclic ether (two carbon atoms and one oxygen atom in a ring) and is called an oxirane in official IUPAC nomenclature. Epoxy compounds can, for example, be made though oxidation of a carbon–carbon double bond. By far the largest industrial source of epoxy groups is epichlorohydrin (ECH). This compound contains a highly reactive, highly toxic, epoxy group that can easily be attached to other molecules, including polymers, by a substitution reaction at the carbon–chloride bond. Epoxy groups originating from reaction with ECH are called glycidyl groups; examples are given in Scheme 16.10. The epoxy novolac resins, polymeric and polyfunctional glycidyl ethers, have already been discussed (see Section 16.2.2). The most dominant class of expoxy resins comprises glycidyl ethers of bisphenol-A. 16.4.2

Chemistry

The synthesis of epoxy resins from bisphenol-A and epichlorohydrin (ECH) is, in simplified form, a net substitution reaction of the chlorine group of ECH by the aromatic hydroxy groups of bisphenol-A to give a diglycidyl ether with evolution of hydrochloric acid; 1 mol of base is required per hydroxy group to neutralize this acid. In more chemical detail, the aromatic hydroxy group is first deprotonated by the base and then reacts with the epoxy group of ECH. Then the chloride group is eliminated by an internal rearrangement by which a new glycidyl group is formed. The idealized reaction product is the diglycidyl ether of bisphenol-A (DGEBPA), a crystalline solid, as shown in Scheme 16.11. In industrial practice the compound is a liquid which contains a small amount of higher oligomers (n ¼ 0:13), and is called ‘‘standard liquid’’. These higher oligomers are formed by reaction of aromatic hydroxy groups with the glycidyl group of DGEBPA. It is also possible to purposely synthesize higher oligomers (n ¼ 2–20) by catalyzing this reaction with, for example, quaternary ammonium salts or triethanolamine as nucleophilic catalysts. The different ways of making higher molecular weight polymers is described below (see Section 16.4.3). The closer the molar ratio of ECH/bisphenol-A in the resin to 1:1, the higher the molecular weight. A higher molecular weight results in a higher glass transition temperature of the polymer and a lower number of epoxy groups per unit weight (see Table 16.2). In contrast to epoxy-novolac resins, an epoxy resin of this kind is always two-functional, with

16.4 Epoxy Resins

O

Cl

OH

HO

O

- 2 NaCl

2 NaOH

O

Cl

851

O

O

O

DGEBPA

O

OH O

O

O

O

n Higher oligomers Scheme 16.11. Synthesis of the diglycidyl ether of bisphenol-A (DGEBPA), and higher oligomers.

glycidyl groups on both ends only. Some slightly higher functionalities can be the result of side reactions (branching as a result of the aliphatic hydroxyl groups reacting with glycidyl groups during the polymer synthesis). This branching can be minimized by using a selective catalyst such as triethanolamine. Cure For the chemistry of curing epoxy resins both the glycidyl groups and the aliphatic hydroxyl groups are of importance. The higher the molecular weight of the resin, 16.4.2.1

Tab. 16.2.

Epoxy resin overview.

˚

Resin

n

ECH/BPA[a]

DGE-BPA/BPA[b]

Mn

Tg [ C]

Standard liquid 1001 type 1004 type 1007 type 1009 type

0.16 2 5.5 14 16

1.33 1.15 1.07 1.05

2 1.4 1.14 1.12

360 900 1900 4300 5000

90%) even numbers of repeating units (n ¼ 0; 2; 4; 6; 8; 10 . . .) as a result of the consecutive reactions between DGEBPA and bisphenol-A. The presence of about 10% n ¼ 1 dimer in the standard liquid resin causes 10% of the Advancement product to have odd numbers of n. 16.4.3.3

O

O

O

O

HO

(n)

(n + 1)

O

OH

OH O

O

O

O

n Scheme 16.14.

Epoxy resin synthesis via the Advancement process.

O

16.5 Alkyd Resins

16.4.4


Epoxy thermosets have excellent mechanical properties: they are characterized by a high hardness but relatively low brittleness (in comparison with phenolic and amino resins). This is caused by the b-transitions in the bisphenol-A moieties along the polymer chain. The higher the molecular weight, the higher the flexibility of the network (the longer the elastically active network (EAN) chain) and the lower the crosslink density when the cure reaction involves only the glycidyl groups. If a high flexibility is desired with a higher crosslink density, rubbery resins can be used to modify the lower molecular weight resins. During cure they separate as small rubber domains inside the thermoset matrix. The lower molecular weight epoxy resins, when cured with, for example, amines or anhydrides, are used as 2K adhesives and in liquid coatings. When cured with phenolic resins, excellent hydrolysis resistance is obtained. Beverage and food packaging cans are coated on the inside with such systems for good sterilization performance. Epoxy resins are popular for their properties in many other coating applications also, but predominantly in primers and coatings for indoor use. The high Tg of the higher oligomers makes them very suitable for powder coatings (see Section 16.6.5), especially when cured with dicyanodiamide or acidic polyester resins. The presence of many aliphatic hydroxyl groups along the chain provides good chemical interaction with, for example, metal surfaces. This results in excellent adhesion to metal substrates, and a high resistance to corrosion. The disadvantage of epoxy resins based on bisphenol-A or novolac is that they are very prone to photooxidation at the aromatic ether bonds and are, thus, poorly resistant to UV light during outdoor weathering. An example of epoxy resins that cure when a strong acid is liberated is given in Scheme 16.35 (see Section 16.8.5, on UV curing). Press moldings from epoxy resins, catalyzed by tertiary amines for cure, are used in automotive and electronic components. As they only react with their glycidyl end groups, they are relatively insensitive to over-cure. They are furthermore dimension-stable, and are very resistant to heat and chemical influences.

16.5

Alkyd Resins 16.5.1

Introduction

Alkyd resins can be described as polyesters containing unsaturated fatty acids. Many of the chemical and technological aspects coincide with those of saturated polyesters (see Secion 16.6). We discuss them here first because the alkyd technology, itself originating from early in the 20th century, stemmed from the use of vegetable oils in the painting and graphic arts industry. In comparison with simple

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vegetable drying oils, alkyds provide harder films and much faster drying. As vegetable oils, they are crosslinked through an autoxidative drying mechanism by ambient oxygen from the air. They are compatible with a wide range of solvents and other resins. They currently constitute a versatile and economic group of binders for the paint industry, with a world annual volume of approximately 1.5 Mt. 16.5.2

Chemistry

Alkyd resins are essentially built up of three groups of components: polyhydric alcohols, polyacids and (vegetable) monoacids. The polymerization process is by esterification (see Section 16.6.2). Examples of typical alkyd monomers are pentaerythritol (tetramethylolmethane), trimethylolpropane, and glycerol as polyhydric alcohols (Scheme 16.21, below), and mainly phthalic anhydride but occasionally isophthalic acid, maleic anhydride, and adipic acid as polyacids (Scheme 16.22, below). The most commonly used monoacids are polyunsaturated fatty acids (the main chemical species of which are oleic, linoleic, and linolenic acids; Scheme 16.15) that impart to the alkyd its drying properties, and benzoic acid which is sometimes used to raise the Tg . The main vegetable oils of industrial relevance are linseed oil, soybean oil, safflower oil, and sunflower oil. They all contain differ-

O OH

Stearic acid

OH

Palmitic acid

O

O OH

Oleic acid

O OH

Linoleic acid

O

OH

OH

Linolenic acid

OH

Ricinic acid

OH

α-eleostearic acid

O

O

Scheme 16.15.

Some vegetable fatty acids used in alkyd resins.

16.5 Alkyd Resins

857

Unsaturated fatty acid content of vegetable oils.

Tab. 16.3.

Fatty acid

Linseed oil

Palmitic Stearic Oleic Linoleic Linolenic Ricinoleic Eleostearic Iodine number

6 4 22 16 52

Soy oil

Safflower oil

11 4 25 51 9

Sunflower oil

8 3 13 75 1

Tung oil

11 6 29 52 2

4 2 4 8

Castor oil

9 83 8

180

130

145

130

82 175

155

ent amounts of fatty acids (Table 16.3). The numbers given are merely indications, as seasonal influences affect the relative fatty acid contents. The concentration of unsaturated groups in the oil is expressed by the iodine number (g I2 per 100 g oil), based on the known addition reaction of iodine to unsaturated carbon bonds. The fatty acids can be incorporated, along with the other polyester monomers, into the resin as such. This is called the fatty acid process, but, if glycerol also is desired to be present in the resin, it can be more economical for the vegetable oil, that is, a triester (triglyceride) composed of glycerol and fatty acids, to react directly with the polyester ingredients. In that case the oil cannot be introduced in the polyesterification stage, as the oil itself is not compatible with the growing polyester. A kind of dispersion of oil-free polyester particles would then be formed in the oil. These undesired systems are called glyptals. To avoid the formation of glyptals, the fatty acids in the oil have to be randomly distributed over the glycerol of the oil and the polyhydric alcohols present in the formulation in a first transesterification step. The resulting hydroxyesters can intermix in a second step with the acids and then result in a homogeneous product. This ‘‘alcoholysis process’’ is depicted in Scheme 16.16.

C 17 H31 O

O O O O

O

O

O

C 17 H31

HO

OH

HO

OH

O

HO

OH

2

+2

C 17 H31

Scheme 16.16.

HO

C 17 H31

C17 H31

O

+

OH OH

The alcoholysis process for alkyd resins.

In a similar process, called acidolysis (Scheme 16.17), the glycerol from the oil is distributed over the fatty acids and isophthalic acid. This procedure is not possible

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O CO2H

O O

C17 H31 O O O

O

O 2 HO2CC17H31

+2

C17 H31

+

C17 H31 O CO2H

O O

CO2H O

C17 H31

CO2H Scheme 16.17.

The acidolysis process for alkyd resins.

with phthalic anhydride, since free carboxylic acid groups are required for this transesterification reaction. Orthophthalic acid is not stable at the acidolysis temperatures (260 C) and will form mainly the more stable cyclic anhydride. The process is therefore primarily used in the production of printing ink resins, where isophthalic acid is the diacid of choice. If a higher Tg of the resin is desired, for example to obtain harder coatings, the chemistry of the alkyd resin can be modified after the polycondensation process by incorporation of phenolic (see Section 16.2) and/or colophonium resins. The Alkyd Constant In contrast to radical-type polymerization, in the making of branched polycondensates like alkyds the build-up of molecular weight can be controlled to a large degree. The starting point in these calculations is the understanding that an infinite molecular weight (gelation) can in principle only be obtained if the average functionality F of the monomers is equal to or higher than 2. This average functionality is defined as: F ¼ ðEalcohol þ Eacid Þ/M0 , in which Ealcohol denotes the number of equivalents of alcohol groups, and Eacid the number of equivalents of acid groups, while M0 denotes the total number of moles of monomers in the system. In practice, alkyd resins are cooked with a surplus of hydroxyl groups. This means that not all the hydroxyl groups will esterify and consequently the number of effective alcohol equivalents is equal to the number of acid equivalents; in other words, Ealcohol þ Eacid ¼ 2Eacid . Thus, F ¼ 2Eacid /M0 . The alkyd constant K is defined as 2/F, equalling M0 /Eacid , and should not be lower than 1. In practice, an alkyd constant of 1.02–1.06 is recommended to avoid gelation regimes at all times. 16.5.2.1

Autoxidative Drying A basic reaction scheme of the autoxidative drying process is depicted in Scheme 16.18. A catalyst is needed to give this reaction sufficient speed; usually a cobalt(II) salt is used. The diene moieties of the linolenic acids, especially, are involved in this process. After formation of a conjugated hydroperoxide with molecular oxygen from the ambient air, the hydroperoxide decomposes into oxy-radicals that can fur16.5.2.2

16.5 Alkyd Resins

O

Polyester

O Oxygen, Cobalt (II) Polyester

O O

O

OH

Cobalt (II) Polyester

O

Polyester

O

O

O

O

.

O

Polyester

Scheme 16.18.

O

O Autoxidative drying of alkyds and oils.

OH

ther react with unsaturations of other fatty acids. Mainly by addition and recombination reactions of these radicals the final network structure is formed. Side reactions during the autoxidation process can lead to yellowing. 16.5.3

Production

The alcoholysis process has lower raw material costs than the fatty acid process but higher production costs, since it is a two-step process. The decision whether to produce from oil or from fatty acids is thus largely based on the amount of oil/fatty acids in the formulation (see short oil/long oil resins, discussed in Section 16.5.4). Regardless of the way fatty acids are incorporated, the main step of alkyd synthesis is the polyesterification. This process is quite similar to the process for saturated polyester resin production, and actually more or less the same equipment can be used (Figure 16.3). The polyesterification of the alkyd raw materials normally proceeds at 240 C with the aid of xylene as a mode of transportation of the water, by forming a binary azeotrope with it. The xylene and water vapors rise through the column on top of the reactor and are condensed on top of a separator. The xylene is pumped back into the reactor and the water is removed. Because of

859

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16 Thermosets

vent vacuum

distillation column inert gas

FC oil out

T condensate tank

TC T oil in

Fig. 16.3. Reactor design for production of polyester and alkyd resins. FC, flow control; T, temperature measurement; TC, temperature control.

undesirable volatile by-products that often form ternary azeotropes with water, the reaction water cannot be disposed of in the sewage system without incineration. Variables measured at regular intervals to control and, if necessary, adjust the process are:

Acid value: a measure for the amount of acid groups still present in the reaction mixture, expressed in mg potassium hydroxide needed to neutralize 1 g of solid resin. This value drops throughout the process as the esterification proceeds. Viscosity: an indication of the weight-averaged molecular weight, usually measured by the torque of the reactor stirring motor or that of a separate rotor/stator measuring device. The viscosity increases with time as a result of increasing molecular weight of the resin. If desired, a graph can be drawn plotting log viscosity against acid value. This is a more or less straight line and enables the operator to judge whether the resin will be within the specifications, which show up on the graph as a rectangle. After reaching these specifications, potential modifications can be made, for example with silicones for better outdoor durability, or polyamides to impart thixotropy. Finally the resin is cooled, diluted in solvent and/or emulsified, and if necessary filtered.

16.5 Alkyd Resins

16.5.4

Properties and Applications Short Oil Alkyds Short oil alkyds contain less than 35% oil by weight, calculated as triglyceride. These resins have a relatively high Tg and therefore will ‘‘feel’’ dry as soon as the solvent has evaporated from the paint. Chemical drying through the unsaturated fatty acids will be relatively modest and other crosslinking reactions will have to be considered. Examples are etherification of the OH groups with resol (see Section 16.2) or amino resins (Section 16.3), either thermally (stoving) or by acid catalysis. Applications are in non-yellowing wood varnishes and protective coatings, for example for steel constructions. 16.5.4.1

Long Oil Alkyds Long oil alkyds can contain from 55% oil, even up to 80%. Long oil alkyds are typically used for trade (professional painting) and DIY (do-it-yourself ) applications. Application is mostly by brush and roller and the paints are supplied in white spirit. Their Tg is low and resins will feel sticky several hours after the solvent has evaporated. Chemical drying is intensive and as a result of this the molecular weight builds up gradually in time. For a typical long oil alkyd resin, full chemical drying is obtained in a period of weeks, and can be accompanied by some yellowing. Typical oils/fatty acids that are used are linseed and soybean. Since linseed contains triply unsaturated fatty acids, it has more crosslinking potential but will also smell and yellow more. For this reason linseed oil based alkyds are mainly used for primers. For topcoats the oil of choice is soybean but safflower and sunflower are also used, the latter being superior to soybean with regard to yellowing and smell. Other oils with very high unsaturation that are often used in primers for semi-industrial applications, applied on-site, are fish oil (marine applications) and tung oil (marine and industrial construction maintenance). 16.5.4.2

Medium Oil Alkyds Medium oil alkyds contain, as the name already suggests, 35–55% oil by weight and are used in the higher-quality trade and DIY paints. In quick-drying properties and yellowing they outperform the long oil alkyds and they are mostly dissolved in white spirit. 16.5.4.3

16.5.5

Alkyd Emulsions

Alkyd paints dominated the architectural coating market for a long period until the appearance of polymer dispersions or the so-called latex paints. Specifically for wall application waterborne paints based on poly(vinyl acetate) homo- and copolymers, styrene–acrylics and pure acrylic latexes almost completely took over the market from the alkyd resins for both interior and exterior application. However, for

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wood and metal application alkyd paints still enjoy the majority of the market share, thanks to better overall paint performance. For the same application fields, alternatives to the conventional alkyd paints were desired for environmental reasons: namely, to avoid the use of (neurotoxic) solvents and solvent emissions to the air. From about 1980, aqueous alkyd emulsions have been developed and are now breaking through in the market as environmentally friendly systems. Alkyd emulsions can best be described as a dispersion of solventless alkyd resin droplets in water (o/w). When the paint dries, the water evaporates and the emulsion changes from oil-in-water (o/w) to water-in-oil (w/o). The paint properties of the dried and autoxidatively crosslinked film are largely comparable to those derived from solvent-borne alkyd paints [5]. The Inversion Process In order to keep this emulsion stable in time during storage, the particle size of the droplets must be small enough (typically 1000

rods, tubes, profiles

up to 4 m min1 low good

>300 km y1

no

no

yes

no

yes

foam panels No. of moldings to justify the investment Production rate Labor content Quality of molding

no

no

yes

inserts

no. of crosssections no

no

yes, externally no

no

yes

continuous, generally generally die dimen6m 6 m long, sions (600 long, 6m mm 250 6m diameter mm) diameter

Molded in ribs

width of the machine

in principle, none

Size limitation

yes

generally no 500–5000

low

100–5000

generally no generally no yes

moderate to high low moderate moderate good; two good; two good; one smooth smooth smooth, surfaces one semisurfaces smooth surface truck automotive, boat hulls, wind mill parts, industrial, wings hatches, electrical covers parts, ventilator housings

moderate

yes

yes

1–1000

yes

in principle, mold none dimensions

yes

press size

radomes, panels

moderate to high moderate good; two smooth surfaces

500–5000

generally no generally no yes

mold dimensions

automotive, industrial, electrical parts

low excellent; two smooth surfaces

high

>10 000

no

yes

yes

press size

16.7 Unsaturated Polyester Resins and Composites 881

882

16 Thermosets

costs. Conversely, if only a limited number of parts are required, the simpler processes such as hand lay-up or spray-up may be the cost-effective methods, since they usually involve low-cost tooling and minimal fabrication equipment. Occasionally, the shape of the part dictates that a particular process is used. For instance, cylindrical objects can often best be made by centrifugal casting or filament winding, and components having constant cross-sections that require high strength by pultrusion. The main conversion techniques for UP resins are explained in Sections 16.7.6.1–16.7.6.10. Hand Lay-up and Spray-up Both hand lay-up and spray-up processes essentially involve placing reinforcement and liquid resin onto the surface of an open mold. As the two names suggest, hand lay-up involves applying the resin and reinforcement (for example, glass fiber chopped-strand mats) by hand, while spray-up uses spray equipment to deposit resin and reinforcement (for example, chopped glass fibers) onto the mold. These relatively cheap techniques are often used to produce large, complicated, strong composite parts such as boat hulls. 16.7.6.1

Continuous Lamination Continuous lamination is another open-mold process. In this method resin is doctored onto a film of cellophane or poly(vinyl alcohol) and glass mat or chopped rovings placed on top of the resin. A second film is then placed on top of this, and the sandwich obtained is passed through rollers, which compact and remove air. They can also shape, as with corrugated type laminates. The sandwich is then passed through a heat source for curing. 16.7.6.2

Filament Winding Filament winding is a technique used for the manufacture of pipes, tubes, cylinders, and spheres and is frequently used for the construction of large tanks and pipe work for the chemical industry. In the process glass fibers are drawn through a resin bath to impregnate them with resin. The impregnated rovings are then wound under tension round a rotating mandrel. Generally the feed head supplying the rovings to the mandrel traverses backward and forward along the mandrel. By suitable design, the structures obtained, having a glass content of up to 80%, can withstand very high pressures. The filament winding process may be combined with spray-up. This hybrid process is called ‘‘chop hoop’’. The reinforcement of the resulting laminate than consists of a blend of continuous and randomly chopped fibers. 16.7.6.3

Centrifugal Casting In centrifugal casting, resin, fillers, and reinforcement are deposited against the inside surface of a rotating mold. Centrifugal force holds the material in place until the part is cured. The interior surface of the centrifugally cast parts can be given an 16.7.6.4

16.7 Unsaturated Polyester Resins and Composites

additional layer of pure resin to improve their surface appearance and to provide additional chemical resistance. Large-diameter composite pipes are produced by this technique. Pultrusion Pultrusion is a technique used for producing continuous-fiber reinforced sections in which the orientation of the fibers is kept constant during cure. The composites are made by pulling reinforcement through a resin-impregnating bath and then through a heated die, the interior of which has the desired shape. Pultruded composites, due to their generally high reinforcement levels (60–75%), have exceptionally high mechanical properties parallel to the direction of pultrusion. Examples of pultruded products are ladder rails, electrical equipment, insulator rods, light poles, window profiles, and gratings. 16.7.6.5

Cold-press Molding Reinforcing material, which may be glass mat or woven fabric, is cut to shape and placed in the GRP or light metal mold. Then the resin is poured in and the press closed with a strong deceleration toward the end of the stroke. The resin displaces air and impregnates the reinforcement. The heat of polymerization generated accelerates the cure of the resin, giving a relatively short cycle time. 16.7.6.6

Resin Infusion In this process one GRP mold is used and a plastic bag or film mounted along the mold flange. Resin is pulled through the reinforcement that is placed between the rigid and flexible molds, using vacuum. Applications include large parts, which are normally produced using contact molding. Resin infusion is a closed process and thus limits the evaporation of styrene into the atmosphere. 16.7.6.7

Resin-transfer Molding This method employs a male and a female GRP or metal mold, which are designed to fit tightly together with a rubber gasket seal. The reinforcement is placed between the molds. After the molds have been clamped together, catalyzed resin is forced into the mold by pressure (or vacuum) until resin leaves the mold through a vent hole. An overspill area is built into the mold to ensure that after trimming a good-quality molded edge is left. Low-viscosity resins are necessary to keep the pressure requirements moderate and to facilitate wetting-out of the reinforcement. 16.7.6.8

Hot-press Molding With hot-press molding the most convenient method is to use a preformed molding compound to which all the necessary ingredients have been added. Two commonly used molding compounds are SMC (sheet molding compound) and BMC (bulk molding compound). SMC is composed basically of five principal ingredients: unsaturated polyester resin, shrink-reducing thermoplastic, reinforcement, fillers, and additives (see 16.7.6.9

883

884

16 Thermosets Tab. 16.7.

Typical compositions of SMC and BMC.

Ingredient

Function

Unsat. polyester þ styrene t-Butyl peroxybenzoate Zinc stearate Magnesium oxide Pigment dispersion Calcium carbonate Thermoplastic additive Chopped rovings

Formulation [parts by weight]

resin radical initiator mold release thickening color filler shrink reduction reinforcement

SMC

BMC

100 2–3 3–4 2–3 0–8 150–200 5–20 50–100

100 1–2 3–4 0–3 0–15 200–250 5–20 45–75

Table 16.7). SMC technology comprises two distinct manufacturing steps: compounding and molding. In the compounding operation all the ingredients, except the fibers, are mixed together to form a highly viscous paste. The paste is then applied to two carrier films (usually polyethylene) to form a sandwich layer with the glass fibers in the middle (see Figure 16.9). The compounded sheets are then stored to mature in a controlled environment; in a period of three to seven days the viscosity of the compound increases from about 25 Pa s to 25 000 Pa s or higher. This thousand-fold increase in viscosity is obtained through a so-called thickening reaction. In chemical terms the added magnesium oxide reacts with the carboxylic end groups of the unsaturated polyester to obtain longer polyester chains through complex salt formation. Moreover, the obtained magnesium carboxylate salts cluster in an ionic lattice, giving rise to a fur-

Resin / filler paste Carrier film

Continuous strand roving

Paste Chopper

Compaction rollers Carrier film

Take-up roll

Fig. 16.9.

Schematic representation of the SMC machine.


ther viscosity increase. The first part of the reaction, the salt formation, is irreversible, while the ionic forces break down within the lattice again upon heating in the molding process and allow the compound to flow. When the compound is ready for molding it is cut into pieces of predetermined dimensions. The pieces are then stacked in a specific arrangement (charge pattern) in the mold so that the flow of the material is optimal. The flow is achieved by the compression action of the mold, which is normally a matched set of steel dies heated to a temperature of about 150 C. During the molding cycle (at a pressure of about 7.5 GPa), which is about 1–3 min long, the resin cures completely. When the mold opens, the part is removed and trimmed before it is sent to secondary operations, which may include cutting holes and openings using a water jet, router, punch, or drill. Very often, different SMC parts are glued together to improve structural integrity as well as to provide space for hardware assembly. In the past, molding compound growth into many high volume applications was excluded because of high polymerization shrinkage resulting in surface defects, warpage, internal cracks, and notable depression (‘‘sink’’) on the surface opposite reinforcing ribs and bosses. The ultimate solution to the problem has been the addition of certain thermoplastics to the formulation of the composite. These thermoplastics, also called low-shrink- or low-profile additives, are usually present at only 2–5 wt% of the final molded part or 7–20 wt% of the organic portion of the formulation. In all effective formulations two phases are formed in the cured composite: a crosslinked thermosetting phase and a separated thermoplastic phase well dispersed throughout the resin matrix. Upon heating of the composite during the molding process the thermoplastic is far above its glass transition temperature and the thermal expansion counteracts the volume shrinkage of the crosslinking unsaturated polyester. Upon cooling, micro-voids are created in the composite due to a higher thermal shrinkage of the thermoplastic phase than the crosslinked thermoset. The best shrinkage control systems (low-profile and class A systems), those with the most phase separation and micro-voiding, do not provide uniform pigmentation, particularly in darker colors. Shrinkage control systems that are less effective (low-shrinkage systems) provide reasonable pigment acceptance but at the cost of reduced dimensional stability and surface smoothness. A parallel technology to SMC is bulk molding compound (BMC). BMC consists basically of the same ingredients as SMC (see Table 16.7). However, its production method, appearance, and properties differ. BMC is commonly manufactured by using high-speed plowshare mixers. The different components of the paste are fed into the mixer and mixed until the compound is fully homogeneous. The chopped-glass fibers are introduced at the end of the process to prevent too much fiber degradation. BMC may be used in compression as well as in injection molding. In the latter process the compound is conveyed from the hopper to the nozzle of the barrel by screw rotation or plunger displacement. The metered charge is fed into the heated mold by fast translation of the screw or plunger. Similarly to SMC-based composites, BMC products are characterized by stiffness, temperature resistance, and di-

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16 Thermosets

mensional stability. Due to the shorter glass fibers, however, strength and impact are lower. Casting, Encapsulation, and Coating (Non-reinforced Applications) Although the majority of UP resins find their way to reinforced composites, there are also some non-reinforced applications, such as the polyester resins widely used for button manufacture. Here the resin needs to be colorless and color-stable after cure. Furthermore, when cured the resin should exhibit good hot water resistance, hardness, and impact resistance. Such resins are also used for encapsulation purposes. In most non-reinforced applications, however, use is made of resins with high filler loadings of up to 85 wt%. Cultured marble, polyester concrete, and auto-repair putties are examples of such applications. Yet another non-reinforced application is that of coatings: gel coat and topcoat are typical coatings used for polyester-based laminates, while special UV-curable unsaturated polyester resins are used in high-gloss coatings for furniture. 16.7.6.10

16.7.7

Design Considerations: Mechanical Properties of Composites

Composite designers choose from a variety of fiber reinforcement and resin systems (both thermoset and thermoplastic) to develop a part. Basically, the reinforcement provides mechanical properties such as stiffness and tensile and impact strength. The matrix material transfers the loads to the fibers, and also protects the fibers from abrasion and chemical attack. Thermoset matrices have E-moduli between 3000 and 4000 MPa while those of E-glass reinforcements are about 70 000 MPa. The tensile strength of glass fibers is around 2400 MPa and that of an UP resin about 50–60 MPa. The glass fiber content of the composite is usually between 25 and 75%. The fibers may be oriented randomly within the resin, but it is also possible to arrange for them to be oriented preferentially in the direction expected to have the highest stress. Such a material is said to be anisotropic (having different properties in different directions), and control of the anisotropy is an important means of optimizing the material for specific applications. Consider a typical region of composite of unit dimensions constructed from parallel and continuous arrangements. This region can be idealized as shown in Figure 16.10 on the left-hand side, by gathering all the fibers together, leaving the matrix to occupy the remaining volume. The total load applied in the direction of the fibers to the composite (F// ) is carried partly by the reinforcing fibers and partly by the matrix according to the generic Eq. (1). F// ¼ Ff þ Fm

ð1Þ

The fiber and matrix phases act in parallel to support the load. In terms of stress


F//

F⊥

φm

φf

φm

φf

Fig. 16.10. Loading of the composite parallel (left) and perpendicular (right) to the direction of the fibers.

(s) equation (1) can be rewritten as Eq. (2), where s// ; sf , and sm are respectively the average stress in the tensile bar, the stress in the fiber, and the stress in the matrix. A is the cross-section of the test bar, A f the total cross-section of the fibers, and A m the remaining cross-section of the matrix. s// A ¼ sf A f þ sm A m

ð2Þ

To what extent the stress is carried by the fibers depends on the ‘‘grip’’ the matrix has on the reinforcement. In the worst case, due to a total lack of adhesion between the fiber and the matrix, there is no ‘‘grip’’ at all. The matrix solely then governs the properties, and consequently the composite shows a poor mechanical performance. The perfect ‘‘grip’’, on the other hand, results from infinitely long fibers with an excellent adhesion to the matrix. In this ideal case the strain (e) in the fibers is equal to that of the matrix (iso-strain conditions). Knowing that for a material with a linear elastic behavior s ¼ Ee, wherein E is the stiffness or Young’s modulus, we can derive Eq. (3) from Eq. (2). E// e A ¼ Ef e A f þ Em e A m

ð3Þ

For infinitely long fibers the volume fraction is directly represented by the respective cross-sections. Bearing in mind that the strain is the same in all cases, we rewrite Eq. (3) to get Eq. (4), where E// ; Ef , and Em are the moduli of the composite, fiber and matrix, and jf and jm are the volume fractions of the fiber and matrix (jf þ jm ¼ 1).

887

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16 Thermosets

E ¼ Ef jf þ Em jm

ð4Þ

This relationship is known as ‘‘the rule of mixtures’’, predicting the overall modulus in terms of the moduli of the constituent phases and their volume fractions. Using essentially the same approach, but for iso-stress conditions, it is possible to calculate the corresponding relationship for the modulus perpendicular to the fiber direction. In that case the ‘‘total grip’’ of the matrix is not sufficient to deform the reinforcement, which usually has a much higher modulus than the matrix. Here, as a first approximation, the transverse modulus, E? , is determined by considering the matrix and reinforcement acting in series (see Figure 16.12, righthand side). In this situation it is not the strains of the reinforcement and the matrix that are equal, but the stresses (an idealization), as Eq. (5) states. s? ¼ sf ¼ sm

ð5Þ

The deflection of the fibers and the matrix add to give the overall transverse deflection [Eq. (6)]. e? ¼ ef þ em

ð6Þ

Since s ¼ E e for small deformations, the Eq. (7) for the transverse modulus can be derived. 1/E? ¼ jf /Ef þ ð1 jf Þ/Em

ð7Þ

Inspection of these simple equations reveals that at the usual levels of fiber addition the longitudinal modulus is dominated by the fiber modulus but the transverse modulus is more influenced by the matrix modulus. The prediction of transverse modulus is considered less reliable, in spite of its occasional agreement with experiment. In more complicated composites, for instance those with fibers in more than one direction or those having particulate or other non-fibrous reinforcements, Eq. (4) provides an upper bound to the composite modulus, while Eq. (7) is a lower bound. The formulas above are related to continuous fibers but in practice the fibers have a finite length. There is a critical length, equal to the shortest length which will allow the stress in the fiber to reach the tensile fracture stress. This length depends upon the ratio of the moduli of the two phases, the strength of the interfacial bond, and the shear strength of the polymer. Typical values for glass and carbon fibers are about 1–2 mm; in fact it is not the fiber length by itself but, more precisely, the aspect ratio, that is, the ratio of length to diameter of the fiber, that governs the properties. More detailed calculations of the mechanical properties of composites can be found in Ref. 10.

16.8 Acrylate Resins and UV Curing

16.8

Acrylate Resins and UV Curing 16.8.1

Introduction

The term ‘‘acrylate resins’’ refers to acrylate-functional polymers. This means that the reactive group, which can be either an acrylate or a methacrylate functionality (Scheme 16.25), is still present in the polymer. (This is also the distinction from acrylics, which are polymers prepared from acrylate monomers and in which the acrylate functionality is no longer present.) These polymers are generally dissolved in a reactive diluent or in a diluent mixture. Depending upon the application these resins are cured with a peroxide initiator or with the combination of light and a photoinitiator, the so-called ‘‘UV curing’’ [11]; see Section 16.8.5. When curing occurs with a peroxide, the peroxide is mixed with the resin and the mixture is then cast into on a mold.

O

O O R

H

Acrylate Scheme 16.25.

O R CH3

Methacrylate Chemical structures of acrylate and methacrylate functionality.

The reactive diluent is required for tuning the resin formulation to the application viscosity. The nature of the diluent is partly dependent on the structure of the resin. In the case of methacrylate-functional resins, methacrylate-functional diluents are mostly used, whereas in the case of acrylate-functional resins, acrylatefunctional diluents are commonly applied. Between acrylates and methacrylates there is a great difference in the homopolymerization rates. As a rule of thumb, it can be said that acrylates are ten times more reactive than their methacrylate analogs. Mixtures of acrylates and methacrylates are not often used, as acrylates are not able to increase the rate of a methacrylate resin. In that specific case, considering the reactivity ratios of this copolymerization, the methacrylate will dominate both the rate and the composition of the polymer formed. The composite resin industry is the main exception with respect to the selective choice of reactive diluents. Within this industry, styrene is used as well as reactive diluent (known in combination with unsaturated polyesters). Methacrylates are then often referred to as – chemically erroneously – vinyl esters.

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16.8.2

Chemistry

The basic chemistry upon curing is the homopolymerization of a (meth)acrylate functionality as is depicted in Scheme 16.26. This polymerization is a radical chain polymerization. The propagating radical is carbon-centered, and therefore this polymerization is sensitive to oxygen inhibition as oxygen can quench carboncentered radicals very effectively. As a result of this oxygen inhibition the top layer is generally not as thoroughly cured as the bulk of the material.

Radical O O O O O O O O R

Scheme 16.26.

P

R

O O O O O O O O R

R

P

R

R

Chain polymerization of acrylate groups.

The three-dimensional crosslinked network which is formed upon curing (Scheme 16.27) consists of two main segments: the polyacrylate chain and the polymer, to which some acrylate groups are attached. This schematic picture clearly illustrates the methods by which the properties of the network can be adjusted: the polyacrylate chain, the side groups, the number of crosslinks, and the acrylate-functional polymer.

Polymer

Polymer

Polymer

Polymer

Polymer Polymer

Scheme 16.27.

A network formed by polymerization on an acrylate resin.


16.8.3

Production

Within the acrylate-functional resins many distinctions can be made, for instance for functionality, polymerization speed, polarity, and so on. However, these distinctions generally do not result in different methods of production. The production can be divided into three main classes, from each of which a characteristic example will be given. Epoxy Acrylates The easiest acrylates to produce industrially are the epoxy acrylates; their preparation (see Scheme 16.28) starts with an epoxide-functional resin (see Section 16.4.2). In principle any epoxide-functional material can be chosen. In this reaction (meth)acrylic resin is added to the epoxide at elevated temperatures. (around 90–130 C). The (meth)acrylic acid adds to the epoxide in a ring-opening reaction resulting in an ester alcohol group. Basically this reaction is similar to the reactions used in the preparation of epoxy resins (see Section 16.4). The reactions can be either acid- or base-catalyzed; base catalysis is the more frequently used, since it limits the number of possible side reactions (for instance, transesterifications). Although these reactions can be carried out in solvent, industrially they are most frequently performed in bulk. Generally these preparations are performed in a batch-type process. 16.8.3.1

O

O O polymer

O

Diepoxide +

O O

O

OH

HO O polymer

O

O O

Epoxy acrylate OH

Acrylic acid Scheme 16.28.

Preparation of epoxy acrylates.

Acrylated oils are a special type of epoxy acrylates. They are formed from epoxidized oils such as linseed oil, and (meth)acrylic acid. Polyester Acrylates The synthesis of polyester acrylates (Scheme 16.29) is a batch process as well. In contrast to the preparation of epoxy acrylates this reaction is performed in a solvent, and generally acid-catalyzed. The preparations differ from the synthesis of unsaturated polyesters (see Section 16.7), mainly because (meth)acrylates have very reactive double bonds, which can easily homopolymerize. Consequently high reaction temperatures cannot be used. Generally the maximum temperature that 16.8.3.2

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HO

Polyester

OH

O

O

+

O

Polyester

O

O

Polyester acrylate

OH Scheme 16.29.

Synthesis of polyester acrylates.

can be applied in these reactions is 130 C. In order to remove the water which is formed in this condensation reaction in an efficient manner at these low temperatures, a solvent is selected which forms an azeotropic mixture with water. In most cases toluene is the preferred solvent. After the preparation of the polyester acrylates, the mixture is washed with water and/or dilute base in order to remove the catalyst and excess acrylic acid. Next the toluene is evaporated and the resin is obtained. Silicone acrylates also are usually prepared via an esterification reaction between a silicone diol and (meth)acrylic acid. Urethane Acrylates Urethane acrylates are produced in a batch-type fashion as well. Two procedures can be followed: 16.8.3.3

HO

Inside-out (Scheme 16.30), or

Polymer

OH

Diol

O

+

OCN R N

OCN R NCO

O O

Polymer

O

N R NCO

Isocyanate functional polymer

Diisocyanate O

OH

O

Hydroxyethyl acrylate (HEA)

O O

O

O N R N

O O

Polymer

O

O N R N

O

O O

O

urethane acrylate Scheme 16.30.

Inside-out synthesis of a urethane acrylate.


893

Outside-in (Scheme 16.31).

O

OH

O O

Hydroxyethyl acrylate (HEA) O

O

N R NCO

O

+

Isocyanate functional acrylate OCN R NCO

Diisocyanate HO

Polymer

OH

Diol

O O

O

O N R N

O O

Polymer

O

O N R N

O

O

O

O

urethane acrylate Scheme 16.31.

Outside-in synthesis of a urethane acrylate.

Each procedure has its own advantages and disadvantages, which will be briefly discussed after a description of both methodologies. For both procedures the reaction temperatures normally never exceed 90 C. Many diisocyanates can be used in these reactions. In those cases in which more than just statistical control is needed, isophorone diisocyanate (IPDI) is generally employed which has two different isocyanate groups (primary/secondary). The alcohol–isocyanate reactions are generally catalyzed with Lewis acids such as dibutyltin dilaurate (DBTDL). Tertiary amines can be used as well. With respect to IPDI, it should be noted that with DBTDL as catalyst the secondary isocyanate is approximately 10–20 times more reactive than the primary isocyanate. With a tertiary amine as catalyst, however, the primary isocyanate is about five times more reactive than the secondary isocyanate. Inside-out The preparation of a urethane acrylate according to the inside-out technology starts with the polymeric diol. The reactor is charged with the diisocyanate, catalyst, and stabilizer. Now the diol is added slowly and a strong exothermic reaction starts. The rate of isocyanate addition is generally used to control the reaction exotherm in such a way that the temperature does not exceed 45 C. The control of this exotherm is especially important when IPDI is used as diisocyanate. For a higher se16.8.3.4

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lectivity, then sometimes even a maximum temperature of 35 C is chosen. After complete addition of the isocyanate, the intermediate isocyanate-functional polymer can be isolated, although normally the subsequent reaction step is performed in the same reactor. Next 2-hydroxyethyl acrylate (HEA) is added at such a rate that the temperature rises to 60–70 C. After addition of the HEA the reaction is normally stirred at 80 C until all the isocyanate has been converted. Outside-in The basic reactions are similar to the Inside-out methodology. However, the reaction sequence starts with reaction of HEA (the outside of the polymer) with the diisocyanate instead of the polyol (the inside). The reaction vessel is charged with diisocyanate, catalyst, and stabilizer to which the HEA is added, resulting in an acrylate-functional isocyanate. Next the polyol is added, and a urethane acrylate can be obtained. 16.8.3.5

Comparing Inside-out with Outside-in The main advantages and disadvantages are summarized below. The Inside-out advantages are: higher reactor filling; and the reactive group is introduced in the last reaction step, thereby reducing the risk of gelation. The disadvantages are: high viscosities in the reactor from the start so that temperature control becomes difficult; and oligomerization of diols via reaction of two diols with a diisocyanate can occur, resulting in higher viscosities and a broader Mw distribution. The Outside-in advantages are: reduced viscosity, and therefore improved cooling and stirring; and consequently higher addition rates (shorter batch times) and a higher selectivity. The main disadvantage is the higher risk of gelation as the reactive acrylate is in the reactor from the start. For many urethane acrylates both methods can be used without any problems, and in practice they are both used. Some disadvantages of the Inside out technology related to the viscosity can be overcome by performing the reaction in a reactive diluent. However, in that case, there is a higher risk of gelation. In special cases only one method is available, namely in those cases in which the higher selectivity of the Outside-in method is required. For instance, in the case of a triol the Inside-out method will result in a gel, whereas such a three-functional urethane acrylate can be made by the Outside-in method. An extreme example is that the Outside-in technology makes it possible to use hydroxy-functional acrylics (polymers with at least ten OH groups per chain) as the polymers in the preparation of urethane acrylates. 16.8.3.6

Stabilization All the acrylate resins suffer a severe risk of homopolymerization. One can even safely state that without correct stabilization these resins cannot be made. The most often applied stabilization package consists of a phenolic stabilizer such as 2,6-di-t-butyl-4-methylphenol in combination with air; it is especially the oxygen in air that is required. Oxygen is a very efficient acrylate polymerization inhibitor 16.8.3.7


and this is one aspect in which the preparation of acrylate resins differs from most other resin synthesis. Many resin syntheses are performed under nitrogen (no discoloration, and so on) while here, without oxygen, the synthesis would almost be impossible. A direct consequence is that the evaporation of the toluene in the production of polyester acrylates is the critical step due to the reduced amount of oxygen and not the synthesis. For those cases an anaerobic inhibitor like phenothiazine can be used. Dilution The viscosity of the acrylate resins is generally not so high that dilution is required for the synthesis or for discharging the reactor at ambient temperatures. Consequently most acrylate resins can be obtained without reactive diluent, leading to a larger formulating freedom for the end formulator. The selection of the reactive diluent is generally based on factors such as dilution power, reactivity, tensile properties, surface tension (wettability), shrinkage, volatility, odor, color, and stability. 16.8.3.8

Safety Almost all acrylates and many methacrylates are sensitizers. This means that an allergic reaction can occur due to improper handling; over the years many examples have been reported where workers have become allergic due to repeated skin contact. In order to minimize skin contact, gloves should be worn at all times and all (meth)acrylates should be treated carefully, unless the producer has proven by testing that a specific (meth)acrylate is not a sensitizer. 16.8.3.9

16.8.4

Properties

The properties of the cured resin can be varied to a large extent. However, as these resins are regarded as expensive compared to the unsaturated polyester resins, they are generally used in areas in which a higher demand or control on the properties is required. An example of such an application is the chemical anchoring in which methacrylate resins are used to attach steel for balconies, bridges and suchlike to a concrete wall, floor or even ceiling. In tuning the properties the reactive diluent plays an important role as well. For instance, when the side chain of the acrylate diluent is changed from linear aliphatic to cycloaliphatic, the hardness of the network will increase without affecting the flexibility of the network. The hardness increases as well when the overall functionality of the diluents is increased; so going from a monofunctional to a difunctional diluent results in a large increase in hardness. However, in this case the network becomes harder and more brittle. When cure speed (or polymerization time) is irrelevant, the choice between an acrylate and a methacrylate gives another parameter for adjusting the hardness. In general the hardness of a polymethacrylate is 100 C higher than that of the corresponding polyacrylate.

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The polarity of the network also can be varied by manipulation of the side chains of the reactive diluent. Both apolar rubbers and polar hydrogels can be obtained through the correct choice of side chain substituent. In the above-mentioned examples, it appears that most of the mechanical properties can be achieved by correct choice of side chain. This is, however, not correct. Many of the basic mechanical properties, such as elasticity, elongation at break, and modulus, originate from the polymer in the acrylate resin and not from the side chain functionality. Generally, epoxy acrylates yield hard, tough materials, while urethane acrylates give elastic networks and polyester acrylates possess properties which lie between urethane and epoxy acrylates. 16.8.5

Introduction to UV Curing General Introduction to UV-initiated Radical Polymerization A special area of application in which the (meth)acrylate resins described above are used predominantly is UV curing. A UV curing resin formulation comprises a photoinitiator, that is, an organic compound that generates radicals upon UV irradiation [12]. The basic reaction scheme is depicted in Scheme 16.32. 16.8.5.1

hv PI

.

R

+

Scheme 16.32.

R

.

O O

Polymer

A UV-initiated radical polymerization.

This cure on demand, combined with flexibility with respect to the properties of the network, makes this system the most efficient process for the rapid production of polymeric crosslinked materials with well-defined properties. This very efficiency is the reason why photoinitiated radical polymerization is widely employed in high-performance applications where emphasis is put on the mechanical as well as the optical properties. For instance, modern dental restorative fillers are UV-curable materials. The aspherical lenses in CD applications are prepared via a UV curing process. Contact lenses are nowadays prepared in the same way. The Internet could not exist without UV curing, as the optical fibres used for data transport are protected by two UV-curable coatings. In stereolithography, threedimensional objects are prepared via UV curing, which makes this type of curing a valuable aid in modern medicine. UV curing is not limited to apparently high-tech applications, however. Most common glossy magazines have a UV-cured coating. It should be noted that, due to the time control of curing, these resin systems are ideally suited for fundamental kinetic investigations, especially combined with analytical techniques like real-time infrared spectroscopy. These investigations have


led to the conclusion that the radical polymerization is dominated by a reactive diffusion-controlled process and that bimolecular termination is the main termination mode. A standard UV-curable acrylate resin formulation consists of at least three ingredients: a photoinitiator, an acrylate resin, and a reactive diluent. The UV curing process is strongly dependent on the photoinitiator. In fact, by the choice of photoinitiator the chemistry can be altered completely. There are fundamentally different photoinitiators that can be used for three different chemistries:

radical curing, employing a radical photoinitiator; cationic curing, employing a cationic photoinitiator; and base-mediated curing, employing a photolabile base.

Around 90–95% of all UV curing processes are radical initiated polymerizations. Consequently, radical UV curing will be discussed in more detail below, while cationic and base-mediated curing will be discussed briefly in separate sections. Photoinitiators for Radical Polymerization The photoinitiator makes the resin formulation UV-curable. Only the development of thermally stable but photolabile compounds enabled the development of UV curing. For an efficient UV curing, the absorption of the photoinitiator should match, at least partly, the emission spectrum of the light source used. Two main types of photoinitiators are used: photoinitiators based on an a-cleavage process (Norrish type I), and photoinitiators based on an electron transfer followed by a hydrogen abstraction process (Norrish type II). a-Cleavage photoinitiators are monomolecular initiators, of which a-hydroxy or a-alkoxy ketones and benzoyl phosphine oxides are well known examples shown in Scheme 16.33. Norrish type II photoinitiators are bimolecular initiators. Generally an aromatic ketone is used in combination with a tertiary amine. Both aliphatic and aromatic tertiary amines can be used. A well-known example of such an initiating system is benzophenone with dimethylaminoethanol. Irrespective of the initiating system, type I or type II, it is clear that at least two different radicals are formed. Initiation proceeds in the case of type I via both radicals although at different rates. However, in the case of a type II initiating system only the amine radical initiates and the ketyl radical can be considered inert with respect to initiation. Consequently type I initiations are preferred when speed is an issue. The advantage of a type II system is that it is less sensitive to oxygen inhibition than a type I, as the ketyl radical can react with oxygen thereby reducing the active amount of oxygen. 16.8.5.2

Resin All the acrylate resins described above in Section 16.8.3 can be employed in UV curing. Low molecular weight materials are mostly used since they possess a lower viscosity. In many UV applications thin layers are employed, which makes the resin viscosity an important processing parameter. 16.8.5.3

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O

O OH

.

hv

+

.

OH

Type I Photoinitiators

O

T

O

S

hv

+

N

+

N

OH

OH

.

OH

.

Type II Photoinitiators Scheme 16.33.

Different types of radical photoinitiators. S, singlet state; T, triplet state.

However, the urethane acrylates, which generally possess higher viscosities, also have higher polymerization rates. Unsaturated polyesters (see Section 16.7) can be used in UV curing as well, but only in combination with a special reactive diluent system, which is based on vinyl ethers. Kinetic analysis has revealed that in that specific case the fumarate forms a charge transfer complex with the vinyl ether and a homopolymerization of this charge-transfer complex takes place as depicted in Scheme 16.34. An advantage of a charge-transfer complex radical polymerization is that it is less susceptible toward oxygen inhibition than methacrylate systems. This is illustrated by the fact that the first commercial UV powder system, in which the powder is sprayed (much oxygen), melted, and then UV-cured, was based on a fumarate–vinyl ether system. Reactive Diluent In liquid UV curing systems there is even more emphasis on the judicious choice of the reactive diluents, due to the application’s viscosity requirements. The reactive diluent also has a large impact on the cure speed. An example concerns polarity: apolar reactive diluents polymerize more slowly than more polar analogs. Hydrogen bonds play a role as well, which is illustrated by the fact that monomers which possess hydrogen bonds react faster than their analogs without hydrogen bonding. However, monomers with hydrogen bonds generally possess a higher 16.8.5.4


O O

O O

+ O

O O

O

R

.

+

[

O

O O

O

]

O

O O

O O O

Scheme 16.34.

Copolymerization of fumarate with vinyl ether via a charge-transfer complex.

viscosity. The functionality is another important factor, as higher-functional materials polymerize faster than lower-functional materials. The hardness or Tg of the resulting polymer is also important, as reactive diluents which give rise to high-Tg polymers polymerize faster than monomers which result in low-Tg materials (assuming the same functionality). Obviously the basic functionality is important; methacrylates react approximately ten times more slowly than acrylates. N-Vinyl amides such as N-vinyl caprolactam are sometimes used as reactive diluents as well, especially in those cases in which a high demand is put on the polymerization speed. An example of this is the fiber optic industry, in which the cure process is performed at draw speeds around 30 m s1 (irradiation times are in the order of milliseconds). When an unsaturated polyester is used as resin, only reactive diluents containing vinyl ether can be applied. Moreover, the molar ratio of fumarate bonds to vinyl ether bonds should be around 1:1 because the crosslinking reaction is a strictly alternating copolymerization. Curing Process The cure process of a sheet covered with a UV-curable resin is depicted schematically in Figure 16.11. The basic process is very simple: a sheet is put on the conveyor belt and the resin is cured under the UV lamp. After irradiation the resin is 16.8.5.5

UV lamp

Conveyor belt Fig. 16.11.

Representation of a basic UV curing process.

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fully cured. When very thin layers or high speeds are employed, curing in air becomes more difficult due to the inhibiting effect of oxygen. In these cases nitrogen inerting, that is, flushing the area under the UV lamp with nitrogen, can be used to overcome this problem. Other gases such as carbon dioxide can be used as well. A main disadvantage of acrylate polymerization is shrinkage. However, this shrinkage process is well defined since it is directly related to the number of double bonds converted. In the lens applications the quartz molds are adjusted for this shrinkage. Cationic Curing Besides photoinitiators which generate radicals, photoinitiators also exist which generate a strong acid. These initiators enable the on-demand curing of epoxide and hydroxy/epoxide mixtures. The initiators are sulfonium or iodonium based. A cationic curing of an epoxy/alcohol system is depicted in Scheme 16.35. 16.8.5.6

BF4

-

PF5

+

S

I

-

hv

+

+

H + PF5

-

Strong acid

Iodonium salt Sulfonium salt

O H

+ H

R1

O

+

+

+ OH

R1

R OH

R O+ H

R1

OH R O

+ H

R1

+

Scheme 16.35. Cationic photoinitiators and cationically photoinitiated polymerization of epoxide resins.

The acid strength of the resulting acid is strongly dependent on the counter ion. The acid strength of the system can be directly related to the polymerization rate (stronger acid results in faster polymerization). Consequently water can reduce or even inhibit these polymerizations. However, minimal amounts of alcohols or even

16.9 Rubber

water can enhance the cure speed. The highest rates are generally obtained in those cases in which the molar ratio of alcohol to epoxide is 1:5–1:10. Besides an epoxide system, vinyl ether systems also exist. The cationic vinyl ether polymerization is among the fastest polymerizations known in the UV curing industry, especially since termination is almost absent. Base-mediated Curing Base-mediated UV curing is a recent development, since UV-labile bases have only recently become commercially available. Depending on the base liberated and its base strength, various kinds of organic chemistry can be used for the network formation. Weak UV-labile bases are commonly used in lithographic processes, in which their main function is as an acid scavenger. In these processes the photochemically liberated base scavenges a thermally formed acid and no polymerization will occur on the irradiated areas. 16.8.5.7

16.9

Rubber 16.9.1

Introduction

In this final section a class of thermoset materials will be introduced, elastomers or rubbers, which has some distinct differences from the thermoset resins presented in the preceding sections [13]. First, most elastomers are characterized by very flexible polymer chains and, as a result, the glass transition temperature of elastomers, indicating the transition between a hard, rigid, glassy phase and a soft, mobile, rubbery phase, is well below room temperature. Crosslinked elastomers are relatively soft materials in their actual application (typically with a modulus of 1–10 MPa; that is, in the Sh A (Shore A) hardness range), whereas most crosslinked thermoset resins are hard materials (1 000–10 000 MPa). Secondly, elastomers are high molecular weight polymers (typically with a weight-averaged molecular weight Mw ranging from 50 000 to 500 000 g mol1 ), resulting in a high melt viscosity. Therefore, the processing temperature is higher than for thermoset resins and the processing equipment is much ‘‘heavier’’. Thirdly, most elastomers have a large number of sites per chain available for crosslinking (typically a functionality of 10–10 000 sites per chain). Consequently, the gel point is already reached at a low crosslinking conversion and any crosslinking must be avoided during mixing, processing, and shaping. Many rubber properties are directly related to the crosslink density; for example, the modulus, the elastic recovery, and the swelling resistance all increase upon increasing the crosslink density. However, these properties are not determined only by the number of chemical crosslinks, but are also affected significantly by the number of physical entanglements which are trapped upon crosslinking.

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Despite these differences in chain flexibility, molecular weight, and functionality, elastomers have the formation of a three-dimensional network in the final processing step in common with the thermoset resins such as the phenolic, epoxy, and acrylate resins. Crosslinking of a rubber renders it insoluble and, in addition, yields a material which has strongly enhanced physical properties, such as tensile properties (modulus, tensile strength, and elongation at break) and elastic recovery (under both compression and tension). Types of Rubber The most important rubber is natural rubber (NR), with an average volume of roughly 50% of the total world rubber production (approximately 15 500 kton y1 ) over the decade 1990–2000. NR is a relatively low-cost rubber collected as a latex from Hevea brasiliensis, and yields excellent physical and dynamic properties, explaining applications ranging from tires (low heat buildup) to thin-walled, soft products like gloves and balloons. Of the synthetic rubbers polybutadiene (BR) and styrene–butadiene copolymers (SBR) have also found major applications in the tire industry, while the latter is also used in foams and footwear. Acrylonitrile– butadiene copolymers, that is, nitrile rubber (NBR), have a high swelling resistance because of their high polarity, making them suitable for applications with oil and solvent contact. These so-called polydiene rubbers have unsaturation in the polymer backbone, making them relatively sensitive to oxidation, ozonolysis, and thermal degradation. Ethylene–propene–diene terpolymers (EPDM) are much more stable in this respect. Some other rubbers worth mentioning are butyl rubber, which has a low air permeability used in inner-tire tubes, and silicone rubbers and fluor rubbers, which are high-performance rubbers (high heat and solvent resistance). In summary, the range of commercial rubbers is quite large, with each rubber type being available in a large number of different grades. Because it is beyond the scope of this overview to discuss all these rubbers in detail, we will zoom in on one rubber only. EPDM (Scheme 16.36) was chosen for that purpose for two reasons. First, EPDM has a very wide structural variation in 16.9.1.1

molecular weight (distribution), long-chain branching (LCB) (distribution), chemical composition (including ethylene and propene levels, type and level of diene, but also monomer sequence distribution),

yielding an almost countless number of combinations, many of which have specific application advantages, and making EPDM a good representative for rubbers in general. Secondly, EPDM has a very good balance between processability, properties, and price, and has evolved since the early 1990s from a high-performance rubber into a mature commodity-plus rubber. With its annual world usage of approximately 900 kton y1, EPDM is today the fourth rubber volume-wise after NR, SBR, and BR, and its usage is actually the largest for a non-tire rubber.

16.9 Rubber

CH2

CH2

CH2 CH2

CH3 CH CH2 CH2

CH2=CH2 + CH2=CH-CH3 + ethene

propene

ENB

CH2 CH2 CH3

CH2 CH2

CH CH

CH2

CH2

EPDM

2

Polymerization and structure of ethylene/ propene/diene terpolymer (EPDM), with 5-ethylidene-2norbornene (ENB) given as an example for the diene. Scheme 16.36.

16.9.2

Polymerization

EP(D)M is produced via insertion polymerization of ethylene, propene, and a diene [14] (Scheme 16.36). Traditionally, Ziegler–Natta-type vanadium-based catalysts are used in combination with metal alkyl cocatalysts and sometimes promoters in order to obtain an efficient, well-controlled polymerization. The EPM copolymer cannot be crosslinked with sulfur due to the absence of unsaturation, and the efficiency for peroxide cure is not that high (Section 16.9.3). This explains why up to 10 wt% dienes are incorporated in the EPM chain; it is to allow sulfur vulcanization and to enhance the peroxide curing efficiency. The diene should have two unsaturated bonds which have quite different affinities for polymerization, in order to prevent gelation during polymerization. 5-Ethylidene-2-norbornene (ENB) and dicyclopentadiene (DCPD) fulfill this requirement, with the endocyclic norbornene unsaturation being the most reactive for polymerization (ring tension!), and upon incorporation they yield EPDM with pendent, unsaturated bonds that are susceptible to sulfur and peroxide crosslinking. 5-Vinylidene-2-norbornene (VNB) is sometimes used because the exocyclic vinyl unsaturation participates to some extent in polymerization, yielding LCB EPDM with enhanced processability. Recently, metallocene catalysts have been developed and are applied commercially, which have a much higher activity than the traditional catalysts and also have enhanced affinity for polymerization of higher a-olefins like 1-butene, 1-hexene and 1-octene. The ethene/a-olefin copolymers thus produced are a new class of materials, namely plastomers, bridging the gap between rigid polyethylene and elastic EP(D)M. Traditionally, Ziegler–Natta EP(D)M polymerization is performed in a solution process using a volatile hydrocarbon as the solvent. The ethylene and propene monomers are cooled prior to the exothermic polymerization, to avoid heating to temperatures that are so high that the catalyst activity is significantly reduced. All the monomers and the solvent have to be purified: polar moieties especially have to be removed, since they kill the catalyst. After polymerization the unreacted monomers are recycled and the polymer is recovered, a process which includes removal

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of traces of unreacted monomers and residual solvent, deactivation and washing of the catalysts, and rubber crumb formation. The activity of the metallocene catalysts is so high that catalyst removal is not necessary. Finally, gas-phase technology has been developed, and has recently been applied commercially, by which the polymerization is performed in a fluidized-bed reactor with carbon black as fluidizing agent. 16.9.3

Crosslinking

Most EPDM applications involve some sort of crosslinking [15], about 80% via sulfur vulcanization. Sulfur vulcanizates have a relatively low thermal stability, which explains the slow transition to peroxide cure in critical applications. The tensile and dynamic properties of sulfur-vulcanized EPDM are superior, whereas the elastic recovery of peroxide-cured EPDM is superior. Sulfur Vulcanization Sulfur vulcanization is the traditional crosslinking method for unsaturated elastomers. It was developed by Charles Goodyear and Thomas Hancock in the 1840s. Since then the sulfur vulcanization recipes have been further developed and optimized and today sulfur is always used in a cocktail with accelerators (mainly sulfur- and nitrogen-containing chemicals) and activators (zinc oxide, stearic acid). EPDM is an apolar rubber with relatively little unsaturation in comparison to the polydiene rubbers, and it has low solubility for the polar accelerators. As a result, the vulcanization mixture may consist of up to ten ingredients, which allows for fine-tuning of the crosslinking rate, the crosslink density, and the type of sulfur crosslink. The last two determine the physical properties of sulfur-vulcanized EPDM. In Scheme 16.37 a generally accepted mechanism for sulfur vulcanization of EPDM is presented. First, sulfur reacts with the accelerators and the activators, yielding a so-called active sulfurating species (its precise structure has still to be elucidated). Next, the crosslink precursor is formed by substitution of one of the allylic hydrogen atoms, yielding an alkenyl sulfide. Note that the unsaturation of EPDM is essential for activating the allylic position, but is not consumed during vulcanization. Next, the crosslink precursor reacts with a second EPDM chain, yielding the initial crosslink with a relatively high number of sulfur atoms in the crosslink, which upon further heating converts into shorter crosslinks. In the case of ENB as the diene, about 50 crosslink structures can be formed, because of the variation in the length of the sulfur bridge (typically one to five sulfur atoms), the three different allylic sites (C3(exo), C3(endo) and C9) and the isomerism at C8/C9 (Entgegen versus Zusammen). For normal recipes about 25–75% of the ENB units are involved in crosslink formation. Zinc salts act as ‘‘catalysts’’ for all three reaction steps. There is still a debate on the exact nature of the sulfur vulcanization reactions (ionic versus concerted). Oxidation may occur as a side reaction. Side reactions that are frequently observed during sulfur vulcanization of polydiene rubbers, such as cis–trans isomerization, allylic rearrangement, and/or the formation 16.9.3.1

16.9 Rubber

S8 + accelerator(s) + ZnO + stearic acid DT soluble sulfurated zinc complex Sm X

EPDM

allylic substitution x2

pendent sulfur (cross-link precursor)

disproportionation

Sn

initial cross-link

oxidation

oxidation products

desulfuration

DT

devulcanization

Sp

matured cross-link (p 160 C) press or calendered into a foil, and cured in a steam autoclave. EPDM strips and profiles may be continuously extruded and subsequently crosslinked in a hot-air oven, a salt bath, or an ultrahigh-frequency radiation unit. 16.9.3.3

16.9 Rubber

16.9.4


In Section 16.9.1 it was explained that EPDM is a very versatile polymer, because of the large number of structural combinations. A broad molecular weight distribution (MWD) at a given viscosity is beneficial for processability, but detrimental for network properties (a large number of dangling ends). LCB has been introduced to enhance processability. The combination of a narrow MWD with LCB yields superior materials. At ethylene levels above 55 wt%, the average length of the ethylene sequences is sufficiently large to give rise to diffuse crystalline regions, which give higher hardness, modulus, and tensile strength to EPDM. At lower ethylene levels, EPDM is a fully amorphous rubber. The type and level of diene determines the crosslinking rate and density (Section 16.9.3). The latter in its turn determines the tensile, elastic, and dynamic properties. It should be noted that EPDM, like most rubbers, is hardly used as such, but always in compounds with fillers and extender oil, because of performance improvement and cost reduction. The addition of large amounts (up to 150 phr) of reinforcing filler, mainly carbon black, results in greatly improved physical properties and UV stability at the cost of increased compound viscosity. To maintain good processability large amounts (up to 150 phr) of extender oil are added in addition to the filler. Advantages and Disadvantages The main advantage of EPDM over the polydiene rubbers, such as NR, BR, SBR, and NBR, is the saturated character of the main chain. As a result, EPDM has a relatively high resistance against oxidation, ozonolysis, heat, and UV light. This makes EPDM especially suitable for outdoor and high-heat applications, such as roof sheeting, window profiles, automotive seals (for both doors and windows), radiator hoses, cable and wire insulation, and all sorts of technical items like seals and tubes. The main disadvantages of EPDM are also related to the structure of the main chain: actually being a hydrocarbon results in a relatively low oil resistance and poor adhesion to polar substrates (metal, glass, and polar polymers). Most EPDM applications require elasticity and thus crosslinking. However, EPM is also used without crosslinking, as an impact modifier for crystalline thermoplastics such as polypropylene (PP) and polyamides, and as an oil additive. 16.9.4.1

Thermoplastic Vulcanizates We started this chapter with a description of the fundamental differences between thermoplastic and thermoset materials; we will end it with an example of a subtle blend of their respective properties. A disadvantage of the three-dimensional network of EPDM, but actually of all thermoset materials, is the lack of recyclability. Crosslinked EPDM, both the waste from production and after use, cannot be processed in the melt again like thermoplastics. Reclaiming technologies have been developed for vulcanized rubber, degrading part of the network via hightemperature and shear treatment, but these technologies are less effective for EPDM vulcanizates, probably because the EPDM chains are so stable. A break16.9.4.2

907

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16 Thermosets

through in this respect has been the development of thermoplastic vulcanizates (TPVs), which combine the elastic properties of thermoset crosslinked rubbers with the melt processability of thermoplastics. The most important TPVs from a commercial perspective are based on blends of EPDM and PP. Dynamic vulcanization of EPDM/PP blends results in a crosslinked EPDM phase, which is finely dispersed in a PP matrix. PP being the matrix in TPVs makes them melt-processable. At high EPDM loadings the thin PP layers, surrounding the crosslinked EPDM particles, act as a sort of glue and transfer the macroscopic stress to the rubber particles. Resol resins, that is, p-alkylphenol–formaldehyde condensates produced at high formaldehyde/phenol ratios and at high pH (Section 16.2.2) are used as the main crosslinking agent activated by acids for the production of TPVs. In addition to being melt-processable, TPVs have other advantages over thermoset crosslinked EPDM, such as higher oil resistance, colorability (no carbon black reinforcement), and co-extrusion with polyolefin thermoplastic parts. The volume of TPVs is growing above the average for the rubber market, and they are partly replacing thermoset EPDM in technical goods and sealing systems.

Notation

Mn Mw Tg

(number-averaged) molecular weight weight-averaged molecular weight glass transition temperature

Acronyms and Abbreviations 1K 2K BR BMC DCPD DBTDL DGEBPA EAN ECH ENB EPDM EPM HEA HDT HMTA IPDI LCB LSC LSE

one-component two-component polybutadiene ¼ butadiene rubber bulk molding compound dicyclopentadiene dibutyltin dilaurate bisphenol-A diglycidyl ether elastically active network chain epichlorohydrin 5-ethylidene-2-norbornene Ethene–propene–diene terpolymer ethene/propene co-polymer 2-hydroxyethyl acrylate heat deflection temperature hexamethylene tetramine isophorone diisocyanate long-chain branching low-styrene content low-styrene emission

References

MF MPa MWD NBR NR PF phr PP Sh SBR SMC TGIC TPV UF UP UV VNB

melamine–formaldehyde 10 6 Pascal molecular weight distribution (¼ Mw /Mn ) acrylonitrile–butadiene copolymer ¼ nitrile rubber natural rubber phenol–formaldehyde parts per hundred polypropylene Shore (hardness degree classes) styrene–butadiene copolymer sheet molding compound triglycidyl isocyanurate thermoplastic vulcanizate urea–formaldehyde unsaturated polyester ultraviolet light 5-vinylidene-2-norbornene

References 1 a] D. Stoye, W. Freitag, Resins for

Coatings, Hanser Publishers, Munich, 1996. b] Z. W. Wicks Jr., F. N. Jones, S. P. Pappas, Organic Coatings Science and Technology, Wiley-Interscience, New York, 1992, Volumes I, II. c] T. Brock, M. Groteklaes, P. Mischke, European Coatings Handbook, Vincentz Verlag, Hannover, 2000. d] W. F. Gum, W. Riese, H. Ulrich, Reaction Polymers: Polyurethanes, Epoxies, Unsaturated Polyesters, Phenolics, Special Monomers and Additives: Chemistry, Technology and Applications, Hanser Gardner Publications, Munich, 2000. e] S. H. Goodman, Handbook of Thermoset Plastics, Noyes Publications, Norwich, 1999 (2nd edition). 2 A. Knop, L. A. Pilato, Phenolic Resins, Springer Verlag, Berlin, 1990. 3 a] M. Dunky, P. Niemz, Holzwerkstoffe und Leime, Springer Verlag, Berlin, 2002. b] H. Diem, M. Gunther, ‘‘Amino resins’’, in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim, 1999. 4 a] S. Tohmura, Journal of Wood Science, 2001, 47, 451–457. b] J. J. H.

5

6

7

8

9

Nusselder, ‘‘Co-condensation of Melamine, urea and formaldehyde’’, in Proceedings of the 1998 TAPPI Plastic Laminates Symposium, Atlanta, Georgia, Tappi Press, Atlanta. a] K. Holmberg, Progress in Organic Coatings, 1992, 20, 325. b] A. Hofland, Journal of Coating Technology, 1995, 67(848), 113. c] G. Hardeman, J. Beetsma, in 14th International Conference, Coatings Community and Care, Copenhagen, 1994, paper 17. Ullmans Encyklopa¨die der Technischen Chemie, Vol. 9, Wiley-VCH, Weinheim, 6th ed. 2004. T. Misev, Powder Coatings, Chemistry and Technology, John Wiley & Sons, New York, 1992. R. Vieweg, L. Goerden, ‘‘Polyester’’, in Kunststoff-Handbuch, Vol. VIII, Carl Hanser Verlag, Munich, 1973. a] B. T. A˚stro¨m, Manufacturing of Polymer Composites, Chapman & Hall, London, UK, 1997. b] G. Akovali, New Handbook of Composite Fabrication, Rapra Technology, Shrewsburg, UK, 2001. c] D. Hull, T. W. Clyne,

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16 Thermosets D. R. Clarke, S. Suresh, I. M. Ward, An Introduction to Composite Materials, Cambridge University Press, Cambridge, UK, 1996. 10 a] A. K. Kaw, Mechanics of Composite Materials, CRC Press, Boca Raton, FL, 1997. b] F. L. Matthews and R. D. Rawlings, Composite Materials: Engineering and Science, CRC Press, 1999. 11 a] Sita Technology Ltd., Chemistry & Technology of UV and EB Formulations for Coatings, Inks and Paints, Ed. P. K. T. Oldring, 8 volumes, Wiley, London, 1998. b] J. P. Fouassier, J. F. Rabek, Radiation Curing in Polymer Science and Technology, 4 volumes, Elsevier, London, 1993. 12 a] J. P. Fouassier, Photoinitiation, Photopolymerization and Photocuring,

Hanser, Munich, 1995. b] K. D. Belfield, J. V. Crivello, Photoinitiated Polymerization, ACS Symposium Series Vol. 847, ACS, Washington DC, 2003. 13 W. Hofmann, Rubber Technology Handbook, Hanser Publishers, Munich, 1989. 14 a] J. W. M. Noordermeer, ‘‘Ethylenepropylene polymers’’, in Kirk-Othmer Encyclopedia of Chemical Technology, Wiley InterScience, New York, 5th ed. 2004. b] F. P. Baldwin, G. Ver Strate, ‘‘Polyolefin elastomers based on ethylene and propylene’’, Rubber Chemistry and Technology 1972, 45, 709. 15 M. van Duin, ‘‘Chemistry of EPDM crosslinking’’, Kautschuk und Gummi Kunststoffe, 2002, 55, 150.

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17

Fibers1 J. A. Juijn 17.1

Introduction 17.1.1

A Fiber World

Look around and see the fiber world! Clothes in a wide variety of shapes and colors, carpets and curtains in your house, sheets and blankets on your bed. Step in your car and see the safety belts, and hopefully never see the airbags. Rely on the strong cords reinforcing your tires, timing belt, V-belts. Enjoy many fiber-reinforced composites in your leisure activities: tennis rackets, golf clubs, skis, the frame of your racing bike. Go sailing and use those novel lightweight sails and strong ropes. Or maybe you’re a military man and wear a helmet and bulletproof vest based on new advanced fibers. Fiber production is a large-volume business: around 60 million metric tons per year. About 40% of this is natural fiber: cotton and wool; the remaining part is ‘‘man-made’’, synthetic fiber: polyester, polyamide, cellulose, acrylics, and so forth. More than 90% is applied in textiles or carpets. Only about 5% is used in industrial applications. And the new, ‘‘advanced’’, ‘‘high-modulus’’ fibers? Technically and commercially very interesting, but we are talking of no more than about 0.2% of the fiber capacity! You may become curious and study some yarns with a magnifying glass and discover that single fibers are as thin as 10–30 mm. With a pair of tweezers you unravel the structure and pull out fibers a few centimeters in length or discover that in some cases the fibers have infinite length. You may test yarns by pulling them between your hands – and cut your fingers when it is an aramid yarn! Gradually, you start realizing that fibers are a completely different class of polymer materials: different in shape, molecular structure, and physical and mechanical properties. 1) The symbols used in this chapter are listed at


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17 Fibers

17.1.2

Scope of this Chapter

This chapter has been written for polymer engineers, not for fiber specialists. A comprehensive impression of fiber technology can be obtained from relevant sections in an encyclopedia [1] or more specific textbooks [2–5], but this would involve reading hundreds of pages of fairly detailed text. This chapter had to be much shorter but should nevertheless give a broad survey of the fiber field. Therefore, clear choices had to be made. Natural fibers will almost completely be neglected. Man-made fibers are still in part based on natural polymer (cellulose), but fully synthetic fibers have become more important. The choice was easy: we will pay more attention to fully synthetic fibers. Many of those are spun from solution but melt-spun fibers are growing more rapidly, especially polyester. Therefore, more attention will be given here to melt spinning than to ‘‘dry’’ and ‘‘wet’’ spinning. The advantage of this approach is that most polymer engineers are already familiar with melt processing and will more easily recognize the details of a melt spinning process. The novel high-modulus, high-strength fibers, especially aramid and gel-spun polyethylene, deserve a special status. These yarns are spun from solution again, and we will discuss the specific modifications of the old wet spinning process that were required to produce ‘‘yarns stronger than steel’’. This approach has led to the following arrangement of this chapter. It starts in Section 17.2 with an introduction to fiber terminology, because this will be new for most polymer engineers. In Section 17.3 we see which properties are required to make a polymer suitable as a fiber material, and how simple the selection of a spinning process seems to be. Melt spinning is discussed in some detail in Section 17.4, with attention to typical machine parts, rheology, and orientation during spinning and drawing. Examples of simple process calculations are given. The section includes the large melt-spun fiber materials polyester, polyamide, and polypropylene. Section 17.5, which is much shorter, and inevitably more superficial, contains accounts of cellulose (rayon), cellulose acetate, acrylics, and poly(vinyl alcohol), the large dry- or wet-spun fibers. Then in Section 17.6 we discuss the typical differences in process limitations between melt spinning and spinning from solution. The chapter concludes with Section 17.7 on high-modulus, high-strength fibers: aramid and gel-spun polyethylene, but including carbon fibers. Finally, for those who have read this chapter as an appetizer, a full menu is provided in the References. 17.2

Fiber Terminology 17.2.1

Definitions: Fibers, Filaments, Spinning

Until the end of the 19th century all yarns were based on natural fibers, such as cotton and wool; all these fibers are thin (10–100 mm) and short (2–30 cm).

17.2 Fiber Terminology

For continuity in a yarn the fibers must be twisted. This is the process we call ‘‘spinning’’ – producing yarn from fibers, for example on a spinning wheel – but industrial equipment has been developed for spinning on a larger scale. More than 50% of the world’s yarn production is based on fibers. Cotton is still fiber material number one, being used in pure cotton yarns but also in blends with synthetic fibers. One natural fiber has a much greater length: silk. About 1000 m of fiber quality can be unwound from one cocoon, and such filaments are combined into a yarn. This introduces the word ‘‘filament’’ – a fiber of (almost) infinite length. And we have met the first spinning machine, the silkworm. Please note that ‘‘spinning’’ has a second meaning here: the production of continuous filaments; all synthetic fibers are spun as continuous filaments. A yarn spun from fibers is called a ‘‘fiber yarn’’ (Figure 17.1a), whereas a yarn containing endless filaments is called a ‘‘filament yarn’’ (Figure 17.1b). It seems logical that all synthetic yarns would be filament yarns, but this is not the case. Filaments are often cut into (short) staple fibers at the end of their spinning process, and then spun again into a fiber yarn, either as a blend with cotton or wool, or in a fully synthetic product. The reason is that the fibrous character of a yarn is preferred in many textile applications, and that it is difficult to imitate these tactile properties by treatment of filament yarns. Finally, we must introduce the term ‘‘cord’’ – a construction of two or more twisted yarn ends (Figure 17.1c). For example, staple fibers are first twisted into a thin yarn and two or three yarn ends are then twisted – in the opposite direction – to form a corded yarn. This is a common construction for textile knitware or fabrics, and carpet yarns. Cords are also used in industrial applications. Filament yarns are then twisted to high levels (several hundred turns per meter) and twisted yarns are combined in a second twisting step – again in the opposite direction – to form twofold or threefold cords. A well-known example is tire cord. Strictly speaking, one should distinguish between ‘‘fiber’’ and ‘‘filament’’: a fiber has a limited length and a filament is essentially endless. To avoid misunderstanding, synthetic fibers with a limited length are in most cases called ‘‘staple fibers’’. It

a b c Fig. 17.1. (a) Fiber yarn (the construction shown is in fact a loosely twisted cord); (b) filament yarn; (c) industrial cord.

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is common practice, however, to use ‘‘fiber’’ also as a general term for fiber, filament, yarn, and cord materials. 17.2.2

Synthetic Yarns

Synthetic cellulose yarns were developed between 1880 and 1910, first from a nitrocellulose solution, and later as ‘‘copper rayon’’ and ‘‘viscose rayon’’. The cellulosics are often called half-synthetic because the raw material is a natural polymer. The most important fully synthetic yarns were developed between 1935 and 1942 – polyamides (PA66, PA6), polyester (PET), and acrylic yarns (PAN copolymers). Another half-century later, many high-performance fibers were introduced, for example aramid (PPTA), gel-spun polyethylene, and carbon fiber. All synthetic yarn processes are continuous, so endless filaments are formed. For filament yarns, the spinneret contains as many holes as the number of filaments required for that particular type of yarn. For synthetic staple fiber production, the spinneret contains thousands of holes and the filaments are cut into fibers at the end of the process. 17.2.3

Titer: Tex and Denier

It is fairly difficult and inaccurate to measure the diameter of fibers or filaments. It is much easier to weigh a certain length of filament or yarn. We thus determine a linear density, or titer. The tex unit is officially included in the SI system, and has become the standard in Europe. The older unit denier stems from the silk industry and is still common in the USA and in many Asian countries. The definitions are: tex ¼ grams per 1000 meter (filament or yarn). denier ¼ grams per 9000 meter (filament or yarn). It is fairly common practice to use decitex (dtex) instead of tex (dtex ¼ grams per 10,000 meter). Numbers in dtex and denier differ by about 10%; for accurate relationships, see Table 17.1. The diameter of a filament can be calculated from the filament titer if the density

Tab. 17.1.

Conversion factors for denier, tex, and dtex.

Multiply ; to obtain m

denier

tex

dtex

denier tex dtex

1 9 0.9

1/9 1 0.1

10/9 10 1


is known. Using the definition of dtex (g 10 000 m1 ) and expressing density in g cm3 , we can calculate the diameter D in mm from Eq. (1). sffiffiffiffiffiffiffiffiffiffiffiffiffi 4 dtex D ¼ 10 pr

ð1Þ

For example, a polyester filament (r ¼ 1:38 g cm3 ) of 10 dtex has a diameter of 30.4 mm, and a filament of 1 dtex has a diameter of 9.6 mm. For yarn titers it is useful to specify the number of filaments. For example, an aramid yarn type could be indicated as 1500 denier f 1000, which means that the yarn contains 1000 filaments, each of 1.5 denier. In dtex, this yarn type would be 1670 dtex f 1000, a filament titer of 1.67 dtex. 17.2.4

Tenacity and Modulus: g denierC1, N texC1 , or GPa

After having introduced the denier and tex units the next logical step is to express breaking strength (tenacity) and modulus in g denier1 or N tex1 instead of in Pascals (1 Pa ¼ 1 N m2 ), MPa, or GPa. Fiber engineers often deviate from using the basic SI units and convert N tex1 to cN tex1 or mN tex1 , or even cN dtex1 : 1 N tex1 ¼ 100 cN tex1 ¼ 1000 mN tex1 ¼ 10 cN dtex1 . The unit g denier1 (g den1 , g d1 , gpd) stands for gram-force per denier. In conversion factors to Newtons, the gravity constant therefore appears: 1 gf ¼ 9.806 650 103 N. In combination with the conversion factor from denier to tex, this results in: 1 g denier1 ¼ 8:826 102 N tex1 ;

or

1 N tex1 ¼ 11:33 g denier1

For conversion of g denier1 or N tex1 to Pa or GPa one needs to know the density r of the yarn. The main conversion factors are given in Table 17.2. Differences in densities can be large, especially when we compare organic fibers (1–1.5 g cm3 ) with glass fibers (2.5 g cm3 ) or steel cord (7.8 g cm3 ). Therefore, aramid producers may argue that their product is five times stronger than steel (in g denier1 or N tex1 ), while steel cord producers can rightfully respond that steel has the same strength (in GPa).

Tab. 17.2.

Conversion factors for g denier1, N tex1 and GPa (express r in g cm3 ).

Multiply ; to obtain m

g denierC1

N texC1

GPa

g denier1 N tex1 GPa

1 11.33 11.33=r

8:826 102 1 1=r

8:826 102 r r 1

915

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17 Fibers

Fig. 17.2.

Examples of cross-sections of fibers and filaments.

17.2.5

Yarn Appearance

Most natural fibers have a non-round cross-section and low luster (Figure 17.2); silk is the exception. Moreover, yarns spun from these fibers have a high volume (bulk) and fiber ends make them hairy. All these factors are appreciated for applications in textiles and carpets. Synthetic yarn is most easily spun from round spinning holes, the filament yarns have a high luster, they are ‘‘flat’’ (not textured), and hairiness is absent. This may be perfect for industrial yarn applications, but it is not desirable for most textile and all carpet yarns. The cross section can be adapted by using profiled spinning holes, for example trilobes, triangles or hexagonal stars. Delustering can be achieved by the addition of fine titanium dioxide or another dulling agent, thus producing semi-dull, dull, or deep-dull yarns, as desired. To give the yarns a voluminous appearance (bulk), a texturing process is required. For low-titer textile filament yarns this is often a false-twist operation: twisting by torsion, heat setting, and subsequently detwisting, in a continuous operation. For higher yarn titers the yarn may be stuffed into a hot chamber and then pulled out with simultaneous cooling: this is called stuffer box texturing, and is the common process for staple fiber and carpet yarns. An example of a textured carpet yarn is shown in Figure 17.3(a). Blowing with air can be used as a texturing process, but also to give the yarn coherence. The yarn is led through a narrow channel or slit and air is blown perpendicularly onto it. At low yarn tension, air blowing creates filament loops. These loops give volume to the yarn and air blowing can therefore be regarded as a texturing process. The effect of air impingement on a yarn under tension is an intermingling of the filaments. This is called ‘‘interlacing’’ for improvement of the weavability of textile yarns. For industrial yarns the air impingement may be quite

a b Fig. 17.3. Textured and tangled yarns: (a) textured filament carpet yarn; (b) a knot appears when a needle is moved through a tangled yarn.


vigorous, in a so-called ‘‘tangling jet’’. If a needle is pulled through a tangled yarn a ‘‘knot’’ is formed, typically after a pull-through length of about 10 cm. The effect is shown in Figure 17.3(b). Tangled yarn packages can be easily unwound, and the yarns are weavable; the tangling operation replaces a more expensive twisting aftertreatment. 17.2.6

Textile, Carpet, and Industrial Yarns

Further aspects of yarn appearance are the yarn titer and the number of filaments: there may be thin or thick yarn bundles. This will obviously depend on the application. Textiles and carpets are often based on fiber yarns – cotton, wool, or blends, for example cotton/polyester and wool/polyamide – but 100% synthetic fiber yarns also find wide application. Fiber yarns are hardly ever found in industrial applications. Textile filament yarns have low tex values and a low number of filaments, although there is a trend toward lower filament titers, as in microfilaments (a1 dtex per filament). Textile yarns with higher titers are found in home furnishing. Textile yarns often have non-round filaments and are often textured. Carpets can be produced from fiber yarns as well as from filament yarns. The filaments or fibers are profiled and very thick, typically 10–30 dtex. Carpet yarns are always textured. Industrial yarns, including high-performance yarns such as aramid and polyethylene, are always synthetic filament yarns. They have high dtex values and a filament thickness depending on the spinning process. Wet-spun yarns usually have thinner filaments than melt-spun yarns (this is also valid for textile yarns). Industrial yarns are flat (without texture) and have round filaments, with only a few exceptions. A survey is given in Table 17.3. The mechanical properties of textile and industrial yarns differ considerably, even if we neglect the influence of texturing; see Figure 17.4. A textile yarn has a high elongation (20–50%) and a low tenacity (200–400 mN tex1 ). Industrial yarns (always flat) have an elongation below 20% and a tenacity of 600–900 mN tex1 . High-modulus, high-strength fibers (HMHS), like aromatic polyamide yarns and

Tab. 17.3.

Yarn composition in synthetic filament yarns.

Textile yarns melt-spun wet-spun Carpet yarns Industrial yarns melt-spun wet-spun

Yarn titer [dtex]

Number of filaments

Filament titer [dtex]

20–200 50–200 ca. 1500

3–200 50–200 ca. 50

1–5 1–2 10–30

500–2000 500–2000

100–400 250–2000

3–10 1–2

917

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17 Fibers

Tenacity, mN tex –1

2500

HMHS yarn 2000

1500

Industrial yarn

1000

Textile yarn

500

Elongation, % 0 0

10

20

30

40

50

Typical stress–strain curves of textile, industrial and high-modulus, high-strength yarns. Fig. 17.4.

gel-spun polyethylene, have an even lower elongation (2–4%) and a much higher tenacity (2000–4000 mN tex1 ). The initial moduli are accordingly low for textile yarns, medium to high for industrial yarns, and very high for HMHS fibers. The stress–strain curve of a typical textile yarn shows an intermediate flat section, indicating that the orientation could have been improved further still. The curves of polyamide or polyester industrial yarns have the remnant of this intermediate section although the yarns have been drawn almost to the limit. The aramid and gel-spun polyethylene processes enable complete orientation. As a result, the stress–strain curves have become almost straight lines. 17.2.7

Physical Structure

Most fibers are semicrystalline. A few of them – aramid and gel-spun polyethylene – approach 100% crystallinity. Examples of the fiber structure of PET and PPTA are shown in Figure 17.5. All fibers have a uniaxial organization: properties in the direction of the axis are completely different from those in the cross-direction. This is an essential difference from other polymer materials. The crystallization of polymers in a spinning or drawing process differs from the crystallization in quiescent melts, for example in extrusion or injection molding processes. The conditions in fiber production are highly anisotropic, and nucleation and growth rates are orders of magnitude higher. Therefore Avrami equations from the literature and values from DSC studies cannot be used to describe fiber crystallization. A direct result of the anisotropic crystallization conditions is that the crystalline


a

b

Physical structure of fibers: (a) semicrystalline structure of poly(ethylene terephthalate), PET; the broken lines are the borders of a fibril; (b) paracrystalline structure of poly( p-phenylene terephthalamide), PPTA. Fig. 17.5.

part will become almost perfectly oriented in the axis direction. The remaining amorphous regions are more disordered, and will determine the fiber properties to a large extent. Fibers drawn to a high ratio will also have a high orientation in the amorphous regions between the crystallites. It is understandable that amorphous orientation will be moderate for most textile yarns and high for industrial yarns. The total result of crystallization and orientation is a measurable overall orientation in the fiber direction, uniaxially. Fibers are always birefringent: the index of refraction in the axis direction (n== ) is larger than the index crosswise (n? ). The difference is the birefringence: Dn ¼ n== n? , and is a measure of fiber orientation. Orientation factors can be calculated from birefringence measurements ( f ¼ 0 for random orientation, f ¼ 1 for complete orientation), and are often split into crystalline orientation ( fc A 1) and amorphous orientation (0 < fa < 1). An understanding of the physical structure is important to explain fiber properties. For example, diffusion of molecules is rapid through large amorphous regions and extremely slow in crystallites. Fibers with low crystallinity and a coarse structure will therefore dye fast and deep, but their chemical stability may be low because small molecules (oxygen, ozone, water, and suchlike) can rapidly diffuse into the filaments. For industrial fibers the relationship between mechanical properties and physical structure is important. With almost perfectly oriented crystals, the orientation of

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17 Fibers

the amorphous regions largely determines the fiber modulus and its tendency to shrink. A high crystallinity, implying a low amorphous content, will contribute to modulus and will reduce shrinkage. These factors combined give the fiber engineer the possibility of developing yarns with high modulus and yet low shrinkage. A typical aspect of a fiber structure is fibril formation. Fibrils are an arrangement of many crystallites and amorphous regions in series, and are common in both natural and synthetic fibers (see Figure 17.5). Chains travel through many crystals within a fibril, but rarely transfer to neighboring fibrils. An effect of this structure is the tendency of many synthetic fibers, especially those with a high orientation, toward fibrillation. This axial splitting of fibers can be a disadvantage, but it can be applied advantageously, for example to produce the aramid ‘‘pulp’’ used as a replacement for fine asbestos fibers. The word ‘‘paracrystalline’’ is used for aramid and gel-spun polyethylene. Amorphous regions are no longer present; rather, the discussion is about defects in the crystal regions. Fiber moduli approach theoretical crystal moduli and shrinkage is virtually absent. Indeed, this is a completely different class of materials.

17.3

Fiber Polymers: Choice of Spinning Process 17.3.1

Polymer Requirements

Fibers must have a reasonable thermal stability. This means that the melting and/ or decomposition temperature must be high, preferably above 200 C. This will allow easy ironing of textiles, curing of reinforced rubber at 190 C, coating of fabrics with PVC around 190 C, and so on. All the large-volume fibers are semicrystalline, with crystalline contents of 30% or more. One can argue about the proper value of the crystallinity, which depends on the way it is measured. Sometimes there is discussion of whether a structure is really crystalline: for example, acrylics (PAN copolymers) would rather have a kind of ‘‘organized amorphous’’ or ‘‘defective crystalline’’ character. Table 17.4 shows the well-known fiber polymers, and a few which seem unsuitable as fiber material, namely polystyrene and poly(vinyl chloride). The latter finds limited fiber application, but a special syndiotactic grade with some crystallinity is then used. Polypropylene has a fairly low melting point but is nevertheless a large fiber product, because the material is cheap and versatile. Polyethylene is even lower melting and is used only as a superstrong fiber at ambient temperature. 17.3.2

Selection of Spinning Process

On paper, the selection of a spinning process seems simple. If the polymer melt is thermally stable a melt-spinning process will be preferred, because this pro-

17.3 Fiber Polymers: Choice of Spinning Process Tab. 17.4.

Properties of fiber polymers.[a]

˚

˚

Polymer

Tg [ C]

Tm or Td [ C]

Crystallinity [ %]

Polyethylene

100

130–140

Polypropylene Poly(vinyl chloride) Polystyrene Polyacrylonitrile Poly(vinyl alcohol) Cellulose Polyamide 6 Polyamide 66 Poly(ethylene terephthalate) Poly( p-phenylene terephthalamide)

20 85 100 85 70/85 230 a50 a50 75 –

170 200 (d) 275 (d) ca. 250 (d) ca. 250 (d) 200–250 (d) 225 260 255 ca. 400 (d)

>50 >80, gel-spun >50 5000 rpm! A third problem is that threading in and transfer to a new package must be fully automated; winders therefore have ‘‘revolver’’ systems which simultaneously end the winding of one package and start the next one. For textile production the capacity of winders has been increased by winding eight or even 12 packages on one winder. For industrial yarns the standard number of packages is two to four. Winders are used at the end of the spinning process, LSS or HSS, or at the end of an integrated process (spin–draw winding; see Section 17.4.9), always at high speed. Winders for separate drawing processes (see Section 17.4.9) usually operate at a1000 m min1 . At these low speeds it is possible to wind and twist the yarn simultaneously. This is done for textile yarns. For industrial yarns twist is often applied as a separate aftertreatment. 17.4.9

Drawing

A yarn spun at low speed (< 1000 m min1 ) still requires additional drawing. The elongation is still a few hundred percent and the tenacity is too low. A yarn spun at

931

932

17 Fibers

high speed (> 4000 m min1 ) may already have sufficient strength for textile or nonwoven applications, but is never strong enough for industrial use. If additional orientation is required, this can be enforced in the spin-line, in a narrow hot tube where drawing takes place. This is called hot-tube spinning (HTS) and is applied only for textile yarns. In most cases drawing is a separate operation. This can be on a separate machine, or combined with the spinning operation, but then in a separate step. The drawing process is usually carried out in two steps: cold and hot drawing. Cold drawing is the so-called neck-drawing step. For a stable process the position of the neck should be fixed, by tension and/or temperature. Hot drawing is distributed over a longer distance, in a more homogeneous deformation process. The simplest setup of a drawing machine is a drawing pin (with one yarn wrap around it) and a hotplate (the yarn loosely touches the surface) between two rolls (see Figure 17.12a). The pin is heated to about the glass transition temperature, while the hotplate is held at a temperature safely below the melting temperature. The speed ratio of the two rolls is the adjusted draw ratio (for example, 5.4). The yarn necks on the pin and completes a ‘natural’ draw (for example 3). The remaining factor (1.8) is the draw ratio over the hotplate. One lets the as-spun yarn decide how and where it wants to draw, which is not necessarily the optimum situation. More control is possible if the two steps are separated. As an example, drawing in hot gas (steam or air, Figure 17.12b) is depicted, with a set of pins for the cold drawing step and an oven for the hot drawing step. The scheme in Figure 17.12(c) shows that drawing is also possible on rolls only, provided that these are heated to

a, pin-plate drawing cold drawing on a static pin; hot drawing on the plate

c, roll-roll drawing a steamjet fixes the neck in the cold drawing step

b, hot gas drawing three-roller sets determine the initial, intermediate and end speed cold drawing on 5 static pins; hot drawing in a oven, with hot steam or air Fig. 17.12.

Drawing processes.

17.4 Melt Spinning

approximately the previously applied pin and plate temperatures, respectively. An alternative is to insert a steam jet between the rolls, which fixes the neck position. Traditionally, spinning and drawing were two separate processes. A spinning spool was made and then unwound again to feed a drawing machine. Later, the two steps were combined. This technique is called spin–draw winding (SDW). The winding speeds on spin–draw winders are always high. For example, a minimum spinning speed of 500 m min1 should be multiplied with a draw ratio (DR) of 5–6, resulting in speeds of 2500–3000 m min1 . For better economics, speeds of b4000 m min1 are preferred, which are achieved at spinning speeds of b700 m min1 . Figure 17.12(c) shows a typical set of drawing rolls (godets) for an SDW machine. Textile yarns are drawn to moderate tenacity and still high elongation. A (super)high-speed spinning process may fulfill the demands; the term fully oriented yarn (FOY) is used for the product. If a drawing process is still necessary it is often very simple, in one step, for example combined with a texturizing process. High-tenacity industrial yarns always require extensive drawing, in most cases in two steps, even if the spun yarn is already highly preoriented. When higher draw ratios are applied the stress–strain curves become steeper: tenacity is increased and elongation is reduced (see Figure 17.13). We find the endpoints of a series of curves on an envelope, often described with empirical relationships. A popular one for polyester and nylons is TE 0:5 ¼ constant (T ¼ tenacity; E ¼ elongation). A demand to raise this factor, which implies a higher breaking energy, cannot be fulfilled by adapting the draw ratio, but only by adapting a ‘‘quality’’ factor. In the example given this is an increase of the polymer molecular

Tenacity, mN tex –1

1000 900

Increasing Draw Ratio

800 700 600

Increasing Molecular Weight

500 400 300 200 100

Elongation, %

0 0

10

20

30

Stress–strain curves of polyester yarns: influence of draw ratio and molecular weight. Fig. 17.13.

40

50

933

934

17 Fibers

weight. Figure 17.13 shows realistic values of T and E for polyester textile and industrial yarns, the latter being produced from a 50% higher molecular weight. 17.4.10

Relaxation and Stabilization

High orientation of the amorphous phase results in yarn shrinkage, for example in boiling water or in hot air. The shrinkage can be partly relieved in a stabilization step: a heat treatment at (almost) constant length (DR ¼ 1). If the heat treatment is carried out at low tension the term ‘‘relaxation’’ is used (DR < 1). Almost zeroshrinkage yarns (‘‘pre-shrunk’’) can thus be produced. The equipment is the same as for drawing: hotplates, hot ovens, or rolls, often installed as an additional step in the drawing process. 17.4.11

Process Integration

It is expensive to let operators handle intermediate products and it is therefore necessary to reduce the number of process steps or to integrate process steps, in order to remain competitive. The first trend has been to increase spinning speeds and get so much orientation that the drawing step can be omitted. This is the case for textile yarns spun at high speed, ‘‘fully oriented’’ and spun-bonded polyester or polypropylene nonwovens. A second trend is to combine spinning and drawing (SDW) and, if possible, to include stabilization and texturing. An example of the latter integration is spin–draw bulk winding (SDBW) of carpet yarns. A third possibility is to couple polymerization with yarn production, by direct spinning – for example, in a continuous polyester polymerization unit coupled with staple-fiber spinning or a series of spin–draw winders. This is the ultimate wish: for monomers going in and full yarn spools coming out. 17.4.12

Rheology

For the design of polymer lines, filter packs and spinning plates, rheological data are required to make the proper calculations. Fiber polymer melts have a fairly common rheological behavior: they are viscoelastic and shear thinning. Shear Viscosity Most rheological processes during spinning are determined by shear. It is important to understand the typical shear rates in spinning machines. We will use Eq. (2), for the apparent shear rate, to work out a few examples. 17.4.12.1

g_app ¼

32Fv pD 3

ð2Þ

17.4 Melt Spinning

935

10000

polymer lines

Melt viscosity, Pa.s

filters spinning holes 1000

100 1

10

100

Shear rate, 1 s –1

1000

Fig. 17.14. Melt viscosity of polyester as a function of shear rate. ½h ¼ 0:9, at 300 C and 100 bar. An indication of the shear rate domains in a spinning machine has been added.

For a polymer line with a diameter of 4 cm through which a mass flow of 360 kg h1 (rmelt ¼ 1 g cm3 ) is pumped, we calculate g_ ¼ 16 s1 , a low-shear condition. For a flow of 5 g min1 through a 500 mm spinning hole, however, we calculate g_ ¼ 6790 s1 , a high-shear condition. In the filter package the pores can be small, but the total flow is divided over as large a surface as possible, in order to prevent a high pressure buildup. As a result, shear rates will be in the order of 100–2000 s1 . Finally, in the metering pumps the shear rates between gearwheel and housing may be in the order of 10 5 s1 . For melts with pronounced shear thinning, narrow tolerances may thus be required for a proper metering. For appropriate calculations of the flow in spinning machines one thus needs a rheological curve over about four decades of shear rate, such as Figure 17.14. A zero-shear viscosity (h0 ) or a melt flow index (MFI) gives insufficient insight. The polymer molecular weight is the most important factor determining the viscosity of melts. The scaling rule is h @ Mw3:4 . In many cases a solution viscosity (for example, intrinsic viscosity, [h]) is used as a measure for polymer molecular weight. Since for many flexible polymers ½h @ Mw0:7 we can derive another scaling rule, of melt viscosity as a function of solution viscosity: h @ ½h 5:0 . It is evident that a higher melt temperature will reduce melt viscosity, but only small variations in molecular weight can be compensated for by temperature adjustment.

10000

936

17 Fibers

Elasticity Elastic behavior is the background of melt fracture just below the spinning plate, which in practice means the immediate interruption of a running process. Elasticity can be controlled by avoiding the use of high molecular weight polymer. For example, the common grades of polypropylene have a broad molecular weight distribution and it may be required to remove the high molecular weight tail by degradation, for example with a peroxide. The resulting ‘‘fiber grade’’ is indicated as ‘‘controlled rheology’’. 17.4.12.2

Elongational Viscosity Elongational behavior is induced in the entrance of the spinning hole and in the transition region from backhole to actual capillary. In practice hardly any permanent orientation is built up in this way, however, because molecular relaxation is rapid. Spinning hole profiles are smoothened only to prevent the formation of vortices which would lead to extrudate distortion. Promoting orientation already in the spinning holes is not common for melt spinning. It could be beneficial for the orientation of melt-spun liquid-crystalline polymers, however, for example in the production of carbon fiber from pitch. More important is the elongation in the spin-line, where uniaxial deformation of the material obviously takes place. Measurement of elongational viscosity on laboratory equipment is very complicated. Fiber engineers may include it in developing spinning models, thereby using the spinning machine itself as a ‘‘rheometer’’, but will usually keep the information they gather proprietary. In situations of low elongational rate one can simply apply Trouton’s ratio [Eq. (3), where hE is the elongational viscosity and hS is shear viscosity; the subscript S is often omitted]. 17.4.12.3

hE ¼ 3hS

ð3Þ

At higher rates, and in contrast to shear thinning, strain hardening (increasing hE ) may occur, resulting in rapid orientation. In practice this is often induced or accompanied by cooling, solidification, and crystallization of the melt, making the analysis of elongational behavior even more complicated. Catastrophic strain hardening, resulting in breaking filaments, is rare for meltspun polymers. Buildup of fairly high orientation in the spin-line has become common, however, in modern high-speed spinning processes. 17.4.13

Process Calculations

Fiber engineers are notorious users of non-SI units: feet or yards instead of meters, minutes or hours instead of seconds, grams instead of kilograms, and denier or dtex instead of tex. Nevertheless, most process calculations are quite simple, without difficult conversions being necessary.

17.4 Melt Spinning

A few examples for a polyester spin–draw winding process will show this, and hopefully give an impression of how melt-spinning machines are designed. The process concerns the production of an industrial yarn with a titer of 1670 dtex f 325. We assume that spinning holes are used with a diameter of 500 mm and an L=D ratio of 1.5; there are 325 holes per spinning plate. There are eight spinning bundles per extruder. The spinning speed is 800 m min1 and the spun yarn is immediately drawn five times on the same machine (an integrated process) and then wound with a speed of 4000 m min1 . Mass Flow By definition 1 dtex ¼ 1 gram per 10,000 m. The output per bundle is then calculated from Eq. (4). 17.4.13.1

fðyarn titer in dtexÞ 10;0001 g ðspinning speed in m min1 Þ ¼ ð1670/10;000Þ 4000 ¼ 668 g min1

ð4Þ

Note that the mass flow is constant in the process, from spinning pump to winder. No mass is lost, in contrast to dry or wet spinning. For example, at the end of the spin-line the speed is 800 m min1 , five times lower than after drawing, but the undrawn yarn still has a five times higher titer. Also note that the volume flow is not completely constant, because the density increases (by about 20%) when the melt cools down, solidifies, and crystallizes. Eight yarn bundles make 8 668 ¼ 5344 g min1 ¼ 320.6 kg h1 , which would require a 120 mm, maybe 150 mm, extruder. The unit produces about 7.6 ton day1, or about 2700 ton y1. Volume Flow The density of a polyester melt is approximately 1.18 g cm3 . The volume flow per yarn bundle and spinning pump thus is 668=1:18 ¼ 566 cm 3 min1 . This flow can be achieved with a 20 cm 3 pump at 28.3 rpm. 17.4.13.2

Extrusion Speed and Elongation in the Spin-line A volume flow of 566 cm 3 min1 is extruded through 325 holes, which is 566=325 ¼ 1:74 cm 3 min1 per hole. The holes have a diameter of 500 mm (0.05 cm) and their cross-section is ðp=4Þ 0:05 2 ¼ 0:001964 cm 2 . The extrusion speed through the holes is (1.74 cm 3 min1 )/(0.001964 cm 2 ) ¼ 887 cm min1 ¼ 8.87 m min1 . The spinning speed is 800 m min1 , and the filaments are thus accelerated in the spin-line by a factor of 800=8:87 ¼ 90. In practice this is viscous flow rather than molecular orientation. Calling this a draw ratio is misleading; ‘‘draft’’, ‘‘drawdown’’ or ‘‘spin stretch factor’’ are better notions. 17.4.13.3

937

938

17 Fibers

Pressure Drop over the Spinning Holes For the calculation of shear rate it is essential to convert accurately to meters and seconds in order to obtain shear rate in reciprocal seconds (s1 ). The formula is given by Eq. (5), with Fv ¼ 1:74 cm 3 min1 and D ¼ 500 mm. 17.4.13.4

g_app ¼

32Fv pD 3

ð5Þ

The shear rate in the spinning holes in this case is g_ ¼ 2363 s1 . In the rheology graph (Figure 17.14) we see that h ¼ 220 Pa s at this shear rate. We can now calculate the pressure drop over the spinning hole with Poiseuille’s law [Eq. (6)]. DP ¼

128 L Fv h 4 p D

ð6Þ

The result is 3119335 Pa ¼ 31 bar. The pressure drop over the much wider backhole would add only a few bars. This is a ‘‘reasonable’’ pressure drop: not too high, but also not too low, because this would lead to an uneven distribution of the flow over the holes, which would result in filament titer differences within a yarn bundle. For a lower melt viscosity (for example, 100 Pa s), narrower spinning holes (350 mm) would have been necessary. The pressure drop over the filter package would add 30–50 bar, resulting in an initial pressure drop over the spinning assembly of 60–85 bar. The pressure drop over the filter does increase during the lifetime of the spinning assembly. The assembly would be taken out of the machine at 200–250 bar. 17.4.14

Polyester (Poly(ethylene terephthalate), PET)

Polyester and PET are almost synonyms. Other polyesters, such as poly(butylene terephthalate) (PBT), poly(trimethylene terephthalate) (PTT), and poly(ethylene naphthalate) (PEN), have hardly any significance as fiber materials. PET Polymer Textile-grade polyester has an intrinsic viscosity around 0.6 (degree of polymerization 100) and is spun at about 285 C, which is 30 C above the melting point. Grades for industrial yarns have an intrinsic viscosity of 0.8–0.9 (degree of polymerization 140–160) and must be spun at 300–310 C. The main degradation problem for PET is hydrolysis: each water molecule gives one chain break in the PET. There is no hydrolysis problem when a polymerization unit directly feeds the spinning machine. Handling of polymer chips requires closed systems, ultra-dry air or nitrogen, among similar precautions. Thermal degradation is unproblematic for textile yarn polyester, but becomes severe for industrial yarn polyester grades 17.4.14.1

17.4 Melt Spinning

yarn speed, m min –1

6000

6000 5000 4000

4000

3000 2000

2000

1000

0 0

0.5

1.0

1.5

Distance from spinning plate, m Fig. 17.15.

Yarn speed curves for LSS and HSS of polyester.

spun above 300 C. Even at a short residence time in the machine (below 10 min) a total drop of 0.1 in the intrinsic viscosity must often be accepted. Spinning of PET PET is a slowly crystallizing polymer. It remains amorphous in the spin-line at speeds below 3000 m min1 . At higher speeds nucleation takes place high in the spin-line, at high temperature, and these nuclei grow rapidly on their way down. The air drag forces deform the crystalline network in a ‘‘neck-like’’ fashion in the spin-line (see Figure 17.15). This necking is more pronounced for higher speeds, finer filaments, and higher molecular weights. These as-spun yarns have a low shrinkage even when the crystallinity is still low. The remaining draw ratio of such yarns is below 2. 17.4.14.2

PET Staple Fiber The majority of polyester fiber production is for textile application, and most of this is staple fiber. Staple fiber is produced by direct spinning from continuous polymerization units based on pure terephthalic acid. The spinning plates have 2000–5000 holes, and the spinning speed is around 2000 m min1 . The bundles are not wound but are laid down loosely in a container. Drawing (more than three times) and texturing is a second, separate step: spinning bundles are combined into a tow of several hundred thousand dtex and drawn, crimped, and heat-set collectively. Cotton-type fiber is the standard: 0.5–2 dtex, drawn to 20% elongation, cut to 32–40 mm in length. Wool-type fibers have a titer of 2–7 dtex; they are drawn to 40% elongation and cut to 40–120 mm in length. Staple fiber is delivered to the customer in large bales. 17.4.14.3

939

940

17 Fibers

PET Textile Filament Yarns Most textile filament yarns are also direct-spun. The output per bundle is small, and therefore hundreds of spinning positions must be fed from one polymerization unit, making the polymer melt-line system very complicated. Winders make four, six, eight, or even 12 packages simultaneously, and spinnerets and spinning pumps are clustered accordingly. Partly oriented yarn (POY) is a large product; spun at about 3500 m min1 , with a round cross-section, and draw-textured (DR @ 1:8, at about 1000 m min1 ) in a separate step. Fully oriented yarn (FOY) can be spun at b6000 m min1 , or drawn in the spin-line (hot-tube spinning, HTS), or spin-drawn. Typical yarn titers are 30, 50, 76, 110, and 176 dtex, and the filament titers 2–3 dtex, although there is a trend toward values of 1 dtex or lower. 17.4.14.4

PET Industrial Yarns Direct spinning is not very common for industrial yarns because there are only a few yarn types that would match the large capacity of polymerization units (b25 000 ton y1 ). A further complication is that the polymer must be condensed to a high molecular weight, but this can be achieved in deep-vacuum, thin-film ‘‘finishers’’. In most cases chips with a textile viscosity are solid-state postcondensed, at a relatively low temperature (about 230 C), which takes many hours but has the advantage that thermal degradation is minimized. There is a limited field of application for low yarn titers, 200–550 dtex, in sewing yarns and fine fabrics. Most industrial yarns have titers of 1100–2200 dtex. The filament titer is usually around 5 dtex, but yarns for safety belts have coarser filaments (10–15 dtex) and modern tire yarns may have finer filaments (around 3 dtex). Large spinning holes (350–800 mm) are used to handle the high melt viscosities. The standard process for PET is spin–draw winding (SDW) at speeds of 4000– 5000 m min1 . This is the cheapest process, especially when two, three, or four bundles are combined on one set of spinning and drawing godets and one winder. The yarns are strong (700–850 mN tex1 ), but may have a fairly high shrinkage. For applications such as nets, ropes, cables, and most fabrics (safety belts, conveyor belts) this is an ideal combination. For some applications almost zero shrinkage can be required (PVC-coated fabrics) and a separate, slow, drawing and stabilization process may still be applied. Tire cord is a different case. Polyester is the most important reinforcing material for radial tires. The cords run radially, from rim to rim, and their high modulus reduces the deformation of the rolling tire, and thus fuel consumption. The properties of the reinforcement in the tire depend on how well the modulus is retained during the dipping and curing processes. Therefore, tire yarns are always spun at high speeds (> 3000 m min1 ), which gives lower shrinkage and somewhat lower tenacity in the yarn. After dipping of the cords and curing of the rubber, these yarns offer the best strength/modulus combination. To reduce costs, even these 17.4.14.5

17.4 Melt Spinning

yarns are spin-drawn, which implies winding speeds of at least 6000 m min1 (high-speed spin-draw winding, HSSDW). The most important applications of polyester industrial yarns have been mentioned above: tires, other rubber reinforcement, narrow and wide fabrics, nets, ropes and cables, and sewing yarns. It is not unusual for a company to list 20 different types of polyester yarn, each type being further divided into various yarn titers and twist levels. 17.4.15

Polyamide (PA6 and PA66)

The former difference in usage between the USA (PA66) and Europe (PA6) is still evident. PA66 seems to be the larger-volume product but PA6 is still large in South American countries, eastern Europe, and India. High-melting PA46 is gradually finding application in airbag fabrics. Other polyamides (PA11, PA12) are not important as fiber materials. PA Polymer Polyamide 6, melting point 225 C, is spun at 260–280 C (290 C); polyamide 66, melting point 265 C, is spun at 290–300 C (310 C). Thermal degradation is relatively unproblematic for polyamide 6, but is more severe for polyamide 66 because of the higher spinning temperature and the tendency of polyamide 66 to crosslink. Filtration of gel particles is an issue for PA66, not for PA6. PA66 polymer meltlines must be burned out once a year because a crosslinked, charred polymer layer is built up at the hot walls. High-shear conditions in narrower polymer lines can help to remove the gel layer from the walls and prolongs the cleaning cycle. Both polyamides evolve oligomer vapor upon extrusion from the spinning holes, and snow-like deposits are formed on cold spots in the top of the spinning machine. Measures such as steam injection, suction, and regular manual cleaning must be taken to keep these under control. Polyamides are prone to oxidative degradation, resulting in yellowing. The nitrogen gas in the chips hoppers must therefore be completely free of oxygen. The reaction of polyamide with water is an equilibrium. When the chips are too dry, the polymer will postcondense in the spinning machine. When they are too wet hydrolysis will take place. The equilibrium water content depends on the molecular weight: for a higher molecular weight, the water content must be lower. There is always some loss of molecular weight by thermal degradation. This can be compensated for by postcondensation when the water content is adjusted to slightly below the equilibrium level. 17.4.15.1

PA Spinning Polyamides crystallize faster than polyester. At low spinning speeds polyamide 6 does not crystallize in the spin-line but fairly rapidly on the spinning spool, thereby also attracting water. As a result, the packages increase in volume (‘‘grow’’) and can 17.4.15.2

941

942

17 Fibers

be unwound only with difficulty when the storage conditions are not kept very constant. At speeds of 1000–4000 m min1 no proper spinning spools can be made. Above 4000 m min1 the yarns crystallize sufficiently in the spin-line and stable packages are built. Polyamide 66 crystallizes faster than PA6. As-spun yarns are always crystalline, even at low spinning speeds. Complete melting of the crystalline chips is essential because remnant crystal nuclei can even cause spherulitic crystallization. PA Staple Fiber Staple fiber is a large product, especially for applications in carpets. The production resembles that for polyester staple. The spinning speed is about 2000 m min1 , which is not problematic because the as-spun yarn is not wound but laid down in containers. The filament titer may be low (4 dtex, for velour carpet) but is usually around 20 dtex. The cross-section is profiled: it may be trilobal or a square with holes. 17.4.15.3

PA Textile Filament Yarns The yarn counts are lower than for polyester because the main applications are as hosiery yarn (for example ‘‘20 denier’’: 22 dtex f 1 or 22 dtex f 5) or in ladies’ underwear (typical yarn titers 44 dtex f 10, 78 dtex f 28, or 78 dtex f 60). The processes are, for POY, spinning at >4000 m min1 followed by a draw-texturing treatment (at approximately 1000 m min1 ), or SDW at >5000 m min1 . The elongation of nylon textile yarns is 45–50%. 17.4.15.4

PA Industrial Yarns Polyamide industrial grades have much lower melt viscosities than polyester, and are therefore spun through smaller spinning holes (250–400 mm). Polyamide yarns can be drawn to elongations slightly below 20%, but subsequent stabilization to reduce shrinkage may add a few percent elongation. Tenacities are at least as good as for polyester, but the modulus is much lower. Polyamide is applied in fabrics, especially in airbags, where PA66 has an important advantage over other fiber materials. Airbags are blown up with hot gas and PA66 does not melt because it has a high specific heat and a high melting point. In this respect, PA46 is even better. The high tenacity and breaking energy of polyamide yarns make them suitable for application in fishing nets, ropes, and cables, but the competition with cheaper polyester is fierce. For rubber reinforcement PA6 has the disadvantage of its low melting point. Nevertheless it is still widely applied in India and South America. PA66 is used on a large scale: in conveyor belts, rubber hoses, and tires. In radial tires PA66 cannot be used in the tire walls because its modulus is too low, but it is present as a cap ply around the steel belt. In oldfashioned bias-belted tires, for bumpy roads, polyamide is the perfect reinforcement. Aircraft tires also have a ‘‘diagonal’’ construction, and usually contain many layers of nylon cords. Spin–draw winding (SDW) is the standard process for the polyamides: spinning speeds are 500–1000 m min1 , winding speeds 3000–5000 m min1 . Drawing 17.4.15.5

17.4 Melt Spinning

usually takes place in two steps, cold and hot drawing, but one-step drawing is possible for low yarn counts. For better economics it is essential to process two, three, or four bundles on one set of drawing godets, and to wind them on one winder. Yarn titers are 200–600 dtex for airbags and other fine fabrics, and 1100–2200 dtex for most other applications. Filament titers are around 5 dtex. Tenacities are between 750 and 850 mN tex1 , and elongations between 18 and 25%. 17.4.16

Polypropylene (PP) PP Polymer Polypropylene is an addition polymer with a fairly broad molecular weight distribution. The strong viscoelastic effects make spinning through small holes, or slits in profiled holes, difficult. PP fiber grades are therefore made by cracking the high molecular weight tail by oxidative degradation, for example by the addition of peroxide in a twin-screw extruder. Polypropylene is full of tertiary carbon atoms which give oxidative and light stability problems. Stabilizer packages are therefore always included in fibers, with their small diameters and high specific surface. The polymer is hydrophobic. Special treatments or additions are required to make polypropylene dyeable. Polypropylene melts at about 175 C but is spun relatively hot, b75 C above its melting point. Polypropylene is often spun from large and long spinning holes (800–1500 mm, L=D ¼ 3–5). It is evident that the low melting point is a limit for industrial applications. A further disadvantage of polypropylene is its low creep resistance. 17.4.16.1

PP Spinning Polypropylene crystallizes fast, always in the spin-line, even at low speeds. 17.4.16.2

PP Staple Fiber This is probably the largest-volume spun product, for application in carpets. The usual machines resemble those for polyester and polyamide, with the spinning speed around 1000 m min1 , collection of as-spun yarn in a container, and drawtexturing a tow of several hundred thousand dtex. ‘‘Short spinning’’ is an alternative: it involves a one-floor machine, holes at a very short distance from each other, cooling over a short distance, and the spinning speed so low (@100 m min1 ) that drawing and texturing can be included on the same machine. 17.4.16.3

PP Split Fiber Much polypropylene fiber is not spun, but produced from film. A film is blown, or cast on a chill roll, drawn 6–10 times and then fibrillated. There are numerous ways of slitting, fibrillating, and cutting. Split fiber finds applications in twines and ropes (as a replacement for sisal), cheap fabrics for bags and tarpaulins, and carpet backing (replacing jute). 17.4.16.4

943

944

17 Fibers

PP Filament Yarns The production machines can be normal spin–draw winders, as for the other meltspun fibers. But here also, ‘‘compact’’ machines have been developed, for example for bulked continuous filament yarn (BCF, for carpet) with all the process steps on one machine, at an end speed of a1000 m min1 . Polypropylene is small in textile applications (in sportswear) but it has a reasonable position in high-tenacity yarns, in low-temperature applications such as ropes, cables, and geotextiles. It should be added that polypropylene can be drawn to high ratios (see Section 17.7.2). This results in very good tenacities, but the helix configuration of the isotactic chain in the crystals severely limits the modulus. 17.4.16.5

17.5

Solution Spinning 17.5.1

Preparation of Spinning Dope

If a polymer cannot be melt-spun, a solution must be made which can be spun. Chemical ractions may be involved, for example the xanthogenation or acetylation of cellulose. In general, the spinning dope is prepared in large vessels, not in extruders. Filter presses for very fine filtration are included in the equipment setup and de-aeration is carried out in a storage tank just before pumping the solution to the spinnerets. Solution spinning implies the handling of large quantities of solvent. At a polymer concentration of 20% the total mass flow for dry spinning is five times higher than the polymer mass flow. For wet spinning there is a very large additional flow of nonsolvent from the spinning bath. 17.5.2

Dry Spinning

The principle of dry spinning is shown in Figure 17.16. Dry spinning is not widely applied; cellulose acetate fibers are dry-spun, and whereas acrylics and polyvinyl alcohol can be dry spun, wet spinning is preferred. The gas in the column is preferably hot nitrogen when flammable organic solvents are used. If the solvent is water (for poly(vinyl alcohol)), hot air can be used. The column height is limited to about 5 m and the residence time of the running filaments in the column is about 1 s, during which most of the solvent must be removed from the filaments. Complete removal is not necessary because at a high polymer concentration the filaments solidify by gelation and can then be handled. Removal of solvent can be completed in later process steps, for example during additional drawing. Control of gas flows in the column, avoiding turbulence or sticking of filaments, is important. Therefore, the hot gas flow would often be downward rather than in

17.5 Solution Spinning

Fig. 17.16. Scheme for dry spinning (reproduced from Ref. 6): 1, metering pump; 2, spinneret; 3, spinning bundle; 4, drying column; 5, 6, 7, take-up system; 8, 9, inlet and outlet of drying gas.

countercurrent, and spinning holes must be a few millimeters apart, as in melt spinning. Evaporation of solvent is a relatively slow process, slower than cooling in melt spinning. It is therefore essential that the spinning filaments are thin. The common approach is to use small spinning holes (50–100 mm) and apply a ‘‘low draft’’ in the spin-line. For example, both the extrusion speed from the holes and the exit speed from the dry-spinning column would be around 300 m min1 . Even a draft below 1 is possible, which means that the die swell effect is not completely undone. Another approach is a ‘‘high-draft’’ process, spinning from large holes (500 mm) and taking up the yarn at relatively high speeds. For example, the extrusion speed through the holes could be 20–50 m min1 and the exit speed around 500 m min1 , which implies a draft of 10–25. The draft in the spin-line is viscous flow and only limited orientation is built up, a situation completely comparable with melt spinning. Cellulose Acetate There are two types of cellulose acetate fibers. When all three hydroxyl groups per cellulose unit (see Section 17.5.3.1) are acetalized, cellulose triacetate is obtained. Cellulose triacetate is spun from dichloromethane (DCM) plus 5–15% methanol 17.5.2.1

945

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17 Fibers

or ethanol. Triacetate is a small product, mainly in the form of yarns with low titers for textile applications. Cellulose acetate contains approximately 2.5 acetate groups per cellulose unit, and is actually produced by partially hydrolyzing the triacetate. The polymer is spun from a solution in acetone/water (approximately 95:5). With a large excess of drying air it is possible to remain below explosion limits. Nevertheless, all operations are carried out in closed equipment. ‘‘Secondary’’ acetate has a remarkably poor crystallinity, in fact too low to survive in a strong interfiber competition for textile applications. The main application of cellulose acetate is as tow in cigarette filters. Acrylics Most acrylic fibers are wet-spun, but dry spinning is also applied. The most common solvent is dimethylformamide, DMF. The polymerization of acrylics can also be carried out in DMF and the polymerization solution can then be directly spun. The boiling point of DMF is 153 C, making complete removal of solvent in the spinning column almost impossible. Most dry-spun acrylic production is staple fiber, and the remaining solvent is then removed during tow processing. Further information on acrylic fibers is given in Section 17.5.3.2. 17.5.2.2

Poly(vinyl alcohol) The preferred process for polyvinyl alcohol is wet spinning, but dry spinning is also possible. The solvent is water and the polymer concentration is so high (25– 50%) that solid chips can be produced at room temperature. These can be processed in a single-screw extruder, as for melt spinning. PVA dry spinning can be a low-draft or high-draft process. The low-draft as-spun yarn has a better drawability. Very high draw ratios are possible for PVA, resulting in high tenacity and modulus. For more information see Section 17.5.3.3. 17.5.2.3

17.5.3

Wet Spinning

The principle of wet spinning is shown in Figure 17.17. Wet spinning is applied for two large-scale fiber products – viscose rayon and acrylics – and one smaller product – poly(vinyl alcohol). The critical factor for each wet spinning process is how the coagulation process in the filaments proceeds. In the ternary diagram shown in Figure 17.18 the three corners represent polymer (P), solvent (S), and non-solvent (N, from the spinning bath). SD is the spinning dope composition on the line P–S. The hatched area is where phase separation takes place. The route from SD into the separation region will determine how the coagulation will proceed. (Note that a path on the line SP represents dry spinning: solvent is removed from the fiber and the polymer concentration is increased; there is no coagulation but there is gelation of the system.) If the amount of nonsolvent enter-


Fig. 17.17. Scheme for wet spinning (reproduced from Ref. 6): 1, metering pump for spinning dope; 2, spinneret; 3, spinning bundle; 4, spinning or coagulation bath; 5, godet; 6, 7, inlet and outlet of spinning bath; 8, drawing bath; 9, 10, take-up system.

ing the fiber is relatively small in comparison with the flow of solvent out of the fiber (path 1), phase separation will occur at high polymer concentrations. A homogeneous, dense structure will be formed; the continuous phase is polymer-rich. This takes a long time, however, because diffusion of solvent will become slower at higher polymer concentrations. The opposite occurs along path 2: nonsolvent enters relatively rapidly into the fiber and coagulation occurs at low polymer concentrations. A heterogeneous, porous structure is formed; the continuous phase has a low polymer concentration and the fiber will be very weak. Path 1 is preferred, but fiber formation is relatively slow then.

S SD

1 2 1

N Ternary diagram for wet spinning: P ¼ polymer, S ¼ solvent, N ¼ nonsolvent, SD ¼ spinning dope. 1, 2, coagulation paths (see text). Fig. 17.18.

P

947

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17 Fibers

It is noteworthy that ‘‘coagulation’’ is the general term for the solidification process in wet spinning, regardless of whether the solidification process is heterogeneous (coagulation) or homogeneous (gelation). Even aramid spinners use the word coagulation (see Section 17.7.1.1). In any case, solvent diffusion and coagulation will be more rapid in the skin than in the core of filaments. Skin formation is an inherent problem for wet spinning (and also for dry spinning). The main effect is that the filaments collapse: the original round cross-section is transferred into a dog bone shape. Further information is available in Ref. 6, Chapter 4. 17.5.3.1

Viscose Rayon CH2OH O

H H OH O

O

H

n

H H

OH

(1) Cellulose It is amazing to see how complicated the technology is for this first large-scale man-made fiber. The complications are caused by the insolubility of cellulose (1). Chemical modification was necessary to make the polymer spinnable. The first attempt was nitration of the hydroxyl groups, but cellulose nitrate proved more useful as gunpowder than as a basis for fiber manufacure. Xanthogenation proved more successful, although the first attempts were aiming at producing carbon filaments for electric lamps, rather than textile fibers. In the modern process (Figure 17.19), sheets of wood pulp cellulose are swollen in concentrated alkali and alkali cellulose is formed [Eq. (7), writing CellOH for a cellulose hydroxyl group]. CellOH þ NaOH ! CellONa þ H2 O

ð7Þ

The sheets are shredded and the ‘‘white crumbs’’ are aged, which implies depolymerization to a molecular weight which is suitable for textile applications or tire cord. The material is then treated in closed equipment (batchwise or continuous) with carbon disulfide in order to xanthate the cellulose [Eq. (8)]. CellONa þ CS2 ! CellOCS2 Na

ð8Þ

Byproducts give an orange-yellow color: ‘‘yellow crumbs’’ are formed. The degree of xanthation (of cellulose) is low: fewer than 0.5 hydroxyl groups need be derivatized to accomplish dissolution in dilute alkali. The polymer concentration for a textile yarn would be around 9% and the alkali concentration 5–6%. For a tire cord the polymer/alkali ratio is lower, both concentrations being 7%, for example. The

Fig. 17.19.

Spinning

Pulp

Relaxing

Aging

Shredding

Washing

Sulfidizing

CS2

Finishing Drying

Weighing

Viscose rayon process scheme (Enka tire cord process).

Stretching

Alkalization

Lye

Preparation

Pressing

Winding

Dissolving

Coning

Filtration

De-aeration

17.5 Solution Spinning 949

950

17 Fibers

spinning solution is then ripened, which implies a redistribution and slight reduction of xanthate groups. The eventual degree of xanthation would be below 0.3 for a textile yarn and close to 0.4 for a tire cord. After fine filtration and de-aeration the viscose spinning dope is spun into an acid bath, where the cellulose is regenerated [Eq. (9)]. 2 CellOCS2 Na þ H2 SO4 ! 2 CellOH þ 2 CS2 þ Na2 SO4

ð9Þ

The spinning bath for viscose contains sulfuric acid (at about 10% concentration) for the decomposition of the xanthate and neutralization of the alkali. Sodium sulfate is formed anyway, but is also dissolved in large quantities (about 20%) in the spinning bath to control the coagulation process. A further addition is zinc sulfate (< 3%), again to control coagulation. The spinning solutions – caustic plus acid – are highly corrosive. The spinnerets are made of gold/platinum, round, with diameters of a few centimeters only. The capillaries are small (50–75 mm) and close to each other (< 1 mm). The number of holes must be large because the standard filament titer of rayon yarns is around 1.7 dtex. The freshly spun filaments are very weak and a tube is often placed around the spinning bundle. Drawing is always carried out in a combined process, in most cases at least partly before the complete decomposition of the xanthate groups. Draw ratios may be low for rayon staple fiber. Tire cord is drawn to an elongation of 12–13% and a tenacity of about 500 mN tex1 . Note that these are conditioned values; in a wet state the high-modulus character is lost; the tenacity becomes lower (400 mN tex1 ) and the elongation much higher (25%). A few alternatives for the derivatization of cellulose have been found recently: direct dissolution of cellulose has been developed. For textile filament yarns, dissolution in N-methylmorpholine oxide (NMMO) is possible. The process is applied by Courtaulds and Lenzing. A solution in formic acid/phosphoric acid was found to have lyotropic behavior and tire yarns with very interesting properties could be produced (patented by Michelin). It even proved possible to use phosphoric acid alone (patented by AKZO), but the process was never commercialized. Yarns spun from solution have a different crystalline structure (cellulose II) than natural cellulose (cellulose I). The difference is that cellulose I has two intermolecular hydrogen bonds formed parallel to the glucosidic bond, whereas cellulose II has only one parallel hydrogen bond. The main effect is a large difference in crystal modulus: 130–180 GPa for cellulose I, 60–90 GPa for cellulose II. All attempts to produce man-made fibers with a cellulose I structure, and hence an even higher modulus, have remained unsuccessful, however. Applications of viscose rayon The main application of rayon is as staple fiber. Spinning is carried out from clusters of spinnerets and bundles are processed collectively as a tow. Rayon staple fiber is applied unblended, but is also used in blends with cotton and/or polyester, in outerwear. Filament yarns have become a


minor product; one may still find them applied in lining fabrics. Rayon tire cord has survived the strong competition with polyester, at least in Europe. Rayon has an unproblematic adhesion to rubber, a high modulus, and a perfect fatigue behavior, and therefore remains the ideal reinforcement for high-speed radial tires. Acrylics The crystallinity of acrylic fibers is rather ill-defined. Polyacrylonitrile (PAN) is known to be atactic, but the very strong polar interactions of the nitrile groups give PAN a definite crystalline behavior. In water a melting endotherm slightly below 200 C is measured and an extrapolated dry melting point could be as high as 320 C, above the decomposition temperature. This finding of melting point depression by water has led to suggestions that PAN could be melt-spinnable when water, or another plasticizer, is added to the polymer. So far, the technique has been considered too complicated for technical realization, however. In practice, the crystalline character of pure PAN is a problem for dissolution, drawing, and dyeing in various stages of acrylic processing. Therefore comonomers (in acrylics) are added to reduce the crystallinity. Yarns containing b 85% acrylonitrile are called ‘‘acrylics’’. Comonomers that are often used are methyl acrylate and vinyl acetate. Yarns with less than 85% acrylonitrile are called ‘‘modacrylics’’. In this case halogen-containing comonomers are often used, such as vinyl chloride, vinylidene chloride, and vinyl bromide, obviously to improve the flame resistance of the yarn. For dissolution of acrylics highly polar solvents are required to disrupt the intermolecular bonds between the nitrile groups. Frequently used organic solvents are dimethylformamide (DMF) and dimethylacetamide (DMAc). The polymer concentration is about 20%. The spinning bath is water, in most cases mixed with the organic solvent being used, in order to slow down the coagulation and precipitation. An alternative is the use of concentrated solutions (50%) of sodium thiocyanate (NaSCN) in water. The spinning bath then is a dilute solution of NaSCN in water. Acrylic solutions have the tendency to gel with time, a thermoreversible process. For spinning it is an advantage to have gelation rather than precipitation of the filaments. As a result of the gelation, as-spun acrylic filaments maintain the crosssection of the spinning holes, but are very porous. A density of 0.4–0.5 g cm3 is usual, while the density of a drawn and dried fiber is 1.17 g cm3 . The voids are very small (0.1–1 mm), and collapse during drawing and drying. Most acrylic production is for staple fiber. Huge spinning plates with up to 60 000 holes are used, and tow drawing and crimping are included in the process. Drawing takes place in a hot-water bath, which is possible because the wet glass transition temperature is about 75 C. The above-mentioned porosity of acrylics is used to control luster. Dried yarns are very lustrous but can be made dull again by a hot wet treatment. 17.5.3.2

Applications of acrylics The largest application is in fiber yarns for clothing (especially sweaters), home furnishings, covers, and blankets. The filament titers vary

951

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17 Fibers

between 1.3 and 17 dtex, the higher titers being for carpets. Tenacities are modest (200–300 mN tex1 ), and elongations are between 30 and 60%. The tenacity may seem low, but acrylics are still clearly stronger than wool. Filament textile acrylics seem to have lost the competition with polyester and nylon. There is an interesting industrial application in outdoor fabrics, based on the perfect UV stability of acrylics, which are used for awnings, tents, automobile upholstery, and outdoor furniture. These products are often pigment-dyed, for better lightfastness. A further step is when hot drawing is applied. Yarns with low comonomer levels can be drawn to ratios of 10 or more. Such yarns are used as precursors for carbon fiber production (see Section 17.7.3), but may also be applied as such, for asbestos replacement. Poly(vinyl alcohol) PVA is produced by hydrolyzing poly(vinyl acetate). PVA fiber is an almost exclusively Japanese product; some production takes place in Korea and China, based on Japanese technology. The fiber has Vinylon as a general name and Kuraray is the largest producer (Kuralon). Poly(vinyl alcohol) is water-soluble, which makes the choice of a solvent easy. And, of course, it has proved possible to make the fibers water-insoluble. Wet spinning solutions contain about 15% polymer and are spun above 70 C, because otherwise premature gelation would take place. The spinning bath (for PVA) is an almost saturated solution of sodium sulfate in water, at 40–50 C. Note that this is not the usual solvent–nonsolvent situation. Diffusion of water out of the filament is rapid, diffusion of the salt into the filaments relatively slow. The polymer concentration increases and solidificaton is by gelation. The solidification is slow, however – much slower than for viscose rayon, for example. Therefore, the spinning speeds are low. Fairly exceptional are the vertical spinning machines, spinning upward, with the spinning bath moving in co-current flow in a tube around the yarn bundle (see Figure 17.20). The spinning holes are small (100– 150 mm). Hot drawing of PVA, at 210–240 C, takes place after drying. Draw ratios (> 10) and tenacities achieved (> 900 mN tex1 ) are high, and the resulting yarns are highly crystalline (40–50%) and no longer water-soluble. For staple fiber the draw ratios may be lower, and a heat treatment for 0.5–3 min at 210–230 C is given to further crystallize the fibers. For complete insolubility in water, the hydroxyl groups can be made to react with formaldehyde to a maximum degree of 85%. This takes 10–20 min, however, and only seems interesting in tow processing. An alternative spinning bath for PVA contains alkali (NaOH, b20%), which penetrates more rapidly into the filaments. The polymer concentration in the spinning dope is somewhat higher (18%) and higher molecular weights are used. The filaments solidify more homogeneously and remain round. This process allows even higher draw ratios than spinning in a sulfate bath, for example DR > 15 and tenacity > 1300 mN tex1 . For further improvement of water insolu17.5.3.3

17.6 Comparison of Melt and Solution Spinning

Fig. 17.20. Spinning machine for PVA (reproduced from Ref. 2c): 1, vertical spinning machine, spinnerets at the bottom; 2, 3, 4, spinning bath circulation; 5, 6, godets; 7, washing; 8, drying; 9, drawing; 10, heat treatment; 11, winder.

bility, small amounts of boric acid may be added to the spinning solution, inducing crosslinking between chains. This seems to be the preferred route to high-tenacity filament yarns. The term ‘‘gel spinning’’ is often used for spinning PVA according to this route (see Section 17.7.2). Applications of PVA The largest application of PVA fibers is in paper and nonwoven fabrics, where a fraction of water-soluble fiber is often used as a binder. Further applications are in twines, ropes, fabrics (tatami mats), and tarpaulins. ‘‘Gelspun’’, very strong, PVA fibers have become an important replacement for asbestos in cement reinforcement. PVA filament yarns are used almost exclusively in industrial applications where high strength is important, such as in fishing nets, ropes and cables, reinforcement of (high-pressure) hoses, and conveyor and transmission belts. PVA is not suitable as a tire cord, however, because of its inadequate fatigue behavior.

17.6

Comparison of Melt and Solution Spinning

Having given a rough technical outline of melt and solution spinning, we can now make a comparison between the speed spinning controlling mechanisms for the two processes: cooling for melt spinning, and diffusion of solvents for solution spinning. We first make a simple analysis of the fiber formation processes, neglecting sublayer effects outside the filaments. An example of the situation is shown for cooling (melt spinning) in Figure 17.21 (at the top). Initially, the temperature will be flat throughout the filament (at the spinning temperature, for example 300 C) and the surrounding cooling air temperature is assumed to be constant (20 C). Gradually a temperature profile will be formed in

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17 Fibers

Tcenter

Toutside 0

D/2

Cooling of a filament Fourier number Fo=at D –2 –6 2 a ~ 10 m s –1 Simple approach: no sublayer

ccenter

coutside 0

D/2

Extraction of solvent from a filament Fourier number Fo=Dt/ D –2 D ~ 10–9 m2 s–1 Realistic approach: sublayer Fig. 17.21.

Comparison of melt and solution spinning.

the filament; the surrounding air remains at 20 C. This situation is shown in the figure. For solidification of the filaments we would like to have a temperature in the center of, say, 100 C. The initial temperature difference (300 20 ¼ 280 C) must be reduced to 100 20 ¼ 80 C. In relative terms, which are used in a Fourier analysis of such a cooling process, the desired ‘‘dimensionless temperature’’ is 80=280 A 0:3. For solution spinning, the analysis proceeds similarly, but now in terms of concentrations instead of temperatures. For example, we assume an initial solvent concentration of 0.8 (polymer concentration 20%) and solidification at c ¼ 0:3

17.6 Comparison of Melt and Solution Spinning

(70% polymer concentration). We assume the spinning bath consists of pure nonsolvent; hence the solvent concentration c ¼ 0 around the filaments. The desired ‘‘dimensionless concentration’’ then is: 0:3=0:8 A 0:375, not very different from the value for the dimensionless temperature in the cooling process. We proceed to the Fourier analysis for the cooling and extraction/diffusion processes. The dimensionless Fourier number for cooling is Fo ¼ at=D 2 , with a ¼ thermal diffusivity (a ¼ l=rCp ), t ¼ time required for cooling and D ¼ diameter of a filament. The thermal diffusivity is of the order of 106 m 2 s1 . The Fourier number for diffusion processes is Fo ¼ Dt=D 2 , with D ¼ diffusion coefficient, t ¼ time and D ¼ diameter. Diffusion coefficients in dilute systems are of the order of 109 m 2 s1 . The difference by a factor of 1000 between the values of thermal diffusivity and diffusion coefficient can only be compensated for by adapting the values of t=D 2 . As a first approximation, filaments in solution spinning should, for example, be made ten times thinner and process times ten times longer than in melt spinning. For cooling we had a dimensionless temperature of 0.3. A Fourier plot then shows a value of Fo ¼ at=D 2 ¼ 0:1. If we have an average diameter in the cooling process of 250 mm (2:5 104 m) we can calculate the necessary cooling time: t A 6 ms. For wet spinning we would find t A 50 ms. Experienced spinners will say that the typical difference between melt and wet spinning is correct, but that both answers are one order of magnitude wrong. Both melt spinning and wet spinning are slower by a factor of about 10 than calculated with our model, which is too simple. The mistake that we have made is obvious: we have neglected the speed limitation in the sublayer, or, even worse, in sublayers that touch each other in a bundle of filaments. Figure 17.21 (at the bottom) is a graph showing a profile in the filament and in the sublayer around the filament. The example is for concentration as the parameter, for solution spinning, but the same graph can be used for cooling by using T instead of c. Exact calculations become much more complicated, but a few qualitative statements can be made. In sublayers we are dealing with diffusion: of molecules in an air layer (cooling process and dry spinning) or of solvent molecules in a liquid layer (wet spinning). The mobility of molecules in a gas is much greater than in a liquid. For the three types of processes we thus have:

melt spinning: fast conduction in the filament, fast diffusion in the gas sublayer; dry spinning: slow diffusion in the filament, fast diffusion in the gas sublayer; wet spinning: slow diffusion in the filament, slow diffusion in the liquid sublayer.

After all, the consequences for filament titers and diameters and process speeds do not differ very much from our first rough estimate. Melt-spun filaments may have a titer of 1–5 dtex, but 20–30 dtex is still possible. Their eventual diameters range from 10 to 50 mm, but they are spun from much larger holes. Dry- and wet-spun filaments often have a titer of 1.7 dtex (13 mm), and are usually spun from holes

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17 Fibers Tab. 17.6.

Typical process data for melt and solution spinning.

Process

Typical spinning hole size [mm]

Average speed in spin-line [m minC1 (m sC1 )]

Height of spin-line or length of bath [m]

Residence time in spin-line [s]

Melt spinning Dry spinning Wet spinning

250–500 50–100 50–75

3000 (50) 300 (5) 30 (0.5)

5 5 0.5

0.1 1 1

smaller than 100 mm. Wet spinning speeds are limited to a maximum of 300–400 m min1 , whereas melt-spun yarns can be wound at speeds up to 8000 m min1 . Dry-spinning speeds are in between, at 500–1000 m min1 . Typical data are shown in Table 17.6. Not surprisingly, there are always exceptions to the rule. For example, acrylic carpet fibers up to 17 dtex are wet-spun, evidently from much larger holes and at lower speeds than are indicated in Table 17.6!

17.7

High-modulus, High-strength Fibers

In the last three decades of the 20th century many advanced fibers were developed (see the surveys in Refs. 8 and 9). Carbon fiber may be described as an inorganic fiber, but is produced by aftertreatment of organic fibers, usually acrylic yarns. High-modulus, high-strength fibers were developed from stiff-chain polymers showing liquid-crystalline behavior in solution or melt. Aramid yarns spun from solution became an important product. How flexible-chain polymers can be superdrawn was discovered, and gel-spinning of polyethylene was developed, adding a valuable product to the spectrum of high-tenacity fibers. 17.7.1

Air-gap Spinning Aramids Aramid yarns (Kevlar of DuPont, Twaron of Teijin–Twaron) are produced from poly( p-phenylene terephthalamide), PPTA (2), which is specially developed for fiber spinning and not used in any other application. DuPont had experience with poly(m-phenylene isophthalamide) in a fiber product called Nomex for hightemperature applications. The polymer is produced in dimethylacetamide and the solution is dry-spun. This cannot be done with the stiff-chain para–para analogue PPTA. The polymer does not dissolve in organic solvents. A special polymerization route had to be developed, and the discovery of lyotropic behavior of concentrated solutions in sulfuric acid then led the way to the production of a magnificent new fiber material. 17.7.1.1

17.7 High-modulus, High-strength Fibers

H

H

N

N

n

O

O

(2) PPTA (Kevlar, Twaron) The polymerization of PPTA starts at low temperature (10 C) with a solution of p-phenylene diamine (PPD) in a mixture of N-methylpyrrolidone (NMP) and calcium chloride [10]. The second monomer, terephthaloyl dichloride, TPC, is injected as rapidly as possible. A fast condensation reaction takes place, hydrochloric acid being the condensate. Much heat is evolved and a high cooling capacity is required to keep the temperature below 50 C. An oligomer is formed that becomes insoluble in the reaction medium at a degree of polymerization of about 10. In the rubbery mass further condensation must be enforced by vigorous mixing with a powerful stirrer. After neutralization, washing, and drying a yellow polymer, with a degree of polymerization of 70–100, is obtained in an irregular powder form. Dissolution of the polymer powder in water-free sulfuric acid can be achieved by freezing the acid and blending it with the powder. This ‘‘dry blend’’ melts at 60– 70 C. An alternative is direct dissolution in a kneader. The key invention for aramid spinning was the discovery of the liquid-crystalline behavior (lyotropy) of PPTA–sulfuric acid systems at polymer concentrations above 10%. Morgan (Monsanto) was the first to describe spinning of fully aromatic polymers from concentrated solutions (in organic solvents), but did not report their liquid-crystalline behavior [11]. Kwolek (DuPont) patented ‘‘optically anisotropic’’ solutions of PPTA in sulfuric acid [12] and Blades (DuPont) further specified the use of an air gap in the spinning process [13]. An example of the liquid-crystalline behavior is shown in Figure 17.22, in which the viscosity–concentration relationship of a solution in a shear field is plotted. Domains of oriented chains are formed and these domains are easily oriented in a flow field (‘‘like tree-trunks in a river’’). An elongational deformation is much more effective for achieving a high orientation. In Figure 17.23 the elongational phenomena during spinning, and their effect on molecular orientation, are shown. The polymer solution is spun at about 85 C. The polymer concentration is about 20%. Domains flow into the spinning hole and orient. The spinning holes are shaped to promote elongational flow: they have a conical entrance and a tapered capillary. In this way almost complete orientation is achieved. The filaments are quenched in cold water, which should not touch the spinneret because the spinning solution would then immediately freeze in. This makes an air gap between the spinneret surface and spinning bath necessary. An air gap is more often applied in other wet spinning processes, but in this case there is an additional very important function. A draft of 5–15 times is applied over the air gap of about 1 cm. The solidification in the cold bath is immediate: the end speed of the process is already achieved at 1 mm below the water surface. The elongational rate in the air gap is comparable with the situation of a drawing neck in cold drawing of

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17 Fibers

Fig. 17.22.

Lyotropic behavior of PPTA in sulfuric acid.

Fig. 17.23.

Orientation of aramid in spinning hole and air gap.


flexible-chain polymers. One can imagine how beneficial this draw over the air gap is, to complete the molecular orientation in the filaments. Solidification of the aramid filaments is called ‘‘coagulation’’ but is in fact a simple freezing-in of the solution. It is not a speed-limiting step. The subsequent removal of sulfuric acid from the solid filaments in the washing process is diffusion-controlled, however, and hence relatively slow. Thus, the rest of the spinning process is careful washing, neutralization, further washing, drying, and winding. The standard titer of aramid yarns is 1670 dtex f 1000 (1.7 dtex filament titer). The elongation at break is around 3%, the tenacity 2100 mN tex1 , and the modulus b 50 N tex1 (75 GPa). Filaments of 1.7 dtex and density 1.45 g cm3 have a diameter of 12 mm. Assuming a draft over the air gap of 5–15 and a polymer concentration of 20%, and neglecting changes in density, we can make an estimate of the spinning hole diameter by means of Eq. (10); hence D hole ¼ 60–100 mm. D hole ¼ Dfilament ðDR=cÞ 0:5

ð10Þ

Fibers with an even higher modulus are produced by a post-treatment of PPTA fibers above 400 C under high tension, reducing the elongation to 1.5–2% and enhancing the modulus to above 100 GPa. Special fibers for ballistics may be spun through smaller holes, and have finer filaments, for example 500 C) to a high ratio (> 10). In fact Technora is an example of superdrawabil-

959

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17 Fibers

ity (see Section 17.7.2), which is surprising for a fully aromatic polymer, but the high temperature makes the chains drawable. H

H

N

N

H

H

N

O

N

n

y

x

O

O

(3) ‘Technora’ (x=y ¼ 0:5=0:5) Applications of aramids Large fields of application for aramid yarns are: antiballistics (bulletproof vests, armored plates); reinforcement of rubber (high-pressure hoses, conveyor and transmission belts; limited application in automobile tires because the material is expensive, but wide application in bicycle tires, on account of its puncture resistance); fiber-reinforced composites (aircraft interiors, sports goods); reinforcement of optical cables; ropes and cables; protective clothing (fireworkers, metalworkers, butchers). Maybe the largest application of aramid fiber, however, is asbestos replacement, in clutches, brakes, and gaskets. To this end filaments are cut into short fibers and fibrillated in a mill. The resulting pulp can be blended with the resins and minerals to produce the clutch faces and brake linings. For replacement of asbestos in gaskets filament yarns can be applied, in the form of thick cords or fabrics.

17.7.1.2

Other Liquid-crystalline Polymers

Higher moduli and strength than in aramid can be reached with polymers having even stiffer chains. We may call them semi-ladder polymers. Examples are the polyazoles polybenzothiazole (PBT, 4) and polybenzoxazole (PBO, 5); the latter yarn is produced on small scale by Toyobo (Zylon). An other example is ‘‘M5’’ (6), a semi-ladder polymer with higher polarity, yet having a very high modulus; the product is developed by Magellan Systems International, in the USA. PBO and M5 are polymerized in polyphosphoric acid and air-gap spun from this solution. N

N

S

S

n

(4) PBT HO

N

N

N-H

n

OH

(6) ‘M5’

N

O

O

(5) PBO

N

N-H

N

n


17.7.2

Gel Spinning Theory Some flexible polymers can be drawn to (much) higher ratios than are achieved in a standard process. In the 1970s many publications were issued on the superdrawability of, especially, polyethylene. Ward and co-workers (University of Leeds, UK) showed [14] that polyethylene can be drawn 30 times, and that the modulus increased linearly with the draw ratio, up to values of 60 GPa, but the tenacities remained modest, a1 GPa. The undrawn filaments or films (strips) were melt-spun. Too high a molecular weight gave spinnability problems, and, moreover, lower molecular weights were found to be more easily drawable. Careful drawing seemed to be the key to high drawablity, in the sense that the drawing speed had to be low and/or the drawing path long. In other words, one should have a low elongational rate, e_ ¼ dv=dx. For comparison, elongational rates in common drawing processes of melt-spun yarns are in the order of 50–500 s1 . For superdrawing, values of 0.1–1 s1 are preferred, for example a homogeneous drawing on a hotplate of 1 m at a speed of 1 m s1 (60 m min1 ). Superdrawability also proved applicable to polypropylene and slightly more polar polymers, such as polyoxymethylene (POM) [15]. At the same time Pennings (DSM, later of the University of Groningen, The Netherlands) studied the fiber formation from dilute solutions of high molecular weight polyethylene. He started with fibers formed in a Couette device: these were stirring-induced fiber crystals, with the famous ‘‘shish-kebab’’ structure [16]. The best properties were obtained when fibers were slowly withdrawn from a gel layer on a rotor. The molecular weight was above 10 6 , the polymer concentration about 1%, the fiber growth rate below 1 m min1 , the moduli around 125 GPa [17], and tenacities far above 1 GPa. Smith and Lemstra (DSM, The Netherlands) combined the ideas of superdrawability and working with dilute solutions of (ultra-)high molecular weight polyethylene, and added the essential function of a continuous spinning process [18]. Gel spinning was born. To explain the concept of gel spinning and superdrawing, it is necessary to introduce the notion of entanglement of chains. If two chains form a loop, the disentangling of one of those chains depends on the length between the entanglement point and the chain end. A short chain end will fairly rapidly diffuse through the network of neighboring polymer chains and slip through the entanglement point, without chain break. The ‘‘reptation time’’ necessary for disentanglement scales with (MW) 3 , according to De Gennes [19]. For a polyethylene chain end with a molecular weight of 100 000 (approximately 4000 monomer units), the time would already be in the order of one second, a very high value for a drawing process. The consequences for gel-spinning are: 17.7.2.1

A high molecular weight is required for an eventually high draw ratio. Draw ratio scales as DR @ MW 0:5 . In gel spinning the molecular weight is higher than in

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17 Fibers

High molecular weight, dilute solution, few entanglements ↓

Gel formation by crystallization ↓ Extraction of solvent (before or during drawing) – superdrawing ↓ Gel crystals are melted and rebuilt, entanglements give coherence Fig. 17.24.

Scheme for gel spinning.

superdrawing of melt-spun polyethylene by a factor of about 10. One may thus expect a draw ratio of 30ð10Þ 0:5 A 100, which is close to reality. The number of entanglements must be reduced, but a few should remain to give the material coherence. This is only possible by working from a dilute solution. Concentrations of 5–10% seem realistic; for ultra-high modulus products an ultra-high molecular weight (g10 6 ), and an even lower concentration (2–5%), will be selected to keep the material spinnable. Figure 17.24 depicts a scheme for the gel spinning process. Gel Spinning of Polyethylene Gel-spun polyethylene is produced by DSM in cooperation with Toyobo (Dyneema) and Honeywell, formerly Allied Signal (Spectra). From the relevant patents it is fairly obvious that different solvents are used: a volatile solvent in the Dyneema process and a nonvolatile solvent in the Spectra process. A nonvolatile solvent must be completely removed before drawing, by extraction with a volatile solvent. The dried filaments have a ‘‘xerogel’’ structure, a dry, porous structure which collapses during drawing. A volatile solvent can be partly removed in the spinning step, but may well be further removed by evaporation and squeezing out in the drawing step. Dyneema yarns have filament titers of 1–2 dtex, Spectra filaments are coarser (4–11 dtex). Filaments of 2 dtex with a density of 0.97 g cm3 have a diameter of 16 mm. How is this diameter produced? If we assume a draw ratio of 100 and a concentration of 10% we can calculate the diameter of the undrawn gel filaments from Eq. (10) to be Dgel; undrawn ¼ Ddrawn ðDR=cÞ ¼ 16ð100=0:1Þ 0:5 ¼ 506 mm. We 17.7.2.2


Fig. 17.25.

Gel spinning process (reproduced from Ref. 8).

may further take into account that the density of the spinning solution is lower than that of the drawn filaments (@0.8 g cm3 versus 1.0 g cm3 ), and that there may be some draft in the spin-line. A realistic estimate of the spinning hole diameter will thus be about 1000 mm, greater than in most melt spinning processes! The spinning solution extruded from the large holes is above 100 C, in order to keep the polymer in solution and to keep the viscosity acceptable. Fiber formation for gel spinning by quenching is relatively rapid: solidification is by crystallization, rather than by evaporation or extraction of solvent. Complete removal of solvent from the thick filaments proceeds much more slowly. The quench liquid can simply be water and the spinneret cannot be immersed in the water bath because the spinning solution would then freeze in. An air gap must be used, as in aramid spinning, but without the function of increasing the molecular orientation. The extraction step depends on the solvent used. The nonvolatile solvent (highboiling, and acting as a plasticizer) must be removed completely before drawing. One can imagine that most of a volatile solvent would be removed in the initial stages of the drawing process (at 100–130 C): at least, this is how DSM (and Toyobo) illustrate it in a simplified process scheme presented in articles and folders (see Figure 17.25). The process scheme for gel-spun PE shows a stirred vessel for making up the spinning solution, but the use of a kneader in a continuous operation should be possible. It also suggests the integration of spinning and drawing, but this is improbable. As discussed in Section 17.7.2.1. the superdrawing process must be ‘‘careful’’ and slow. Even for very long drawing ovens, an end speed of 200 m min1 seems a limit. (As an example, at this speed, 3.3 m s1 , and an oven length of 33 m one would have an elongational rate (_e ¼ dv=dx) of 0.1 s1 ). The spinning speed would then have to be about 2–3 m min1 . This is technically difficult and economically unattractive. The output per hole would be as low as 0.4 g solution min1 (0.04 g polymer per minute). A ten times higher output is technically feasible. The technical implication then is that spinning and drawing must

963

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17 Fibers

be separate steps; there should be ten times more drawing than spinning positions. This can be realized if a collective drawing process is used, such as a warp of yarns drawn in a wide oven. Applications of gel-spun PE Many applications overlap with those of aramid. Polyethylene has the advantage of its lower density and often a higher tenacity. This is important in ballistics and protective clothing, where Dyneema and Spectra find wide application. A special feature in armored plates is the Spectra Shield or Dyneema UD construction: arrangements of filaments in unidirectional layers bonded by thermoplastic layers in between; the UD layers are arranged crosswise. This construction is more effective than woven structures in ballistic protection. A second large application, especially of Dyneema, is in marine twine, ropes, and nets, where it replaces polyamide and polyester yarns. Outstanding tenacity, abrasion resistance, UV stability, and low density (floating in water) give it a competitive edge, even at about ten times the selling price of the melt-spun yarns! Gel-spun polyethylene is too low-melting (about 140 C) to be applied in rubber reinforcement. In composites the curing temperature should not exceed 130 C, and a surface treatment is required for sufficient adhesion. Finally, performance under constant load is restricted because of creep limitations. Other Gel-spun or Superdrawn Fibers The ultra-high draw ratios for polyethylene are not found for any other polymer, which may not be surprising because the interchain interactions in polyethylene are so low. This makes molecular rearrangements in a drawing process easy. This ‘‘polarity concept’’ was worked out by Smook and co-workers [20]. They took values of maximum draw ratios from the literature and their own experience. For the value of the polarity of a polymer the cohesive energy was taken, which is the energy required to make a polymer chain completely free from interactions with its neighbors. This factor was made dimensionless by taking the ratio with RTd , Td (K) being the drawing temperature, and R the gas constant. The best relationship was found to be Eq. (11), where l represents the draw ratio DR; this is plotted in Figure 17.26. 17.7.2.3

ln l max ¼ 360 expðEcoh =RTd Þ 0:5

ð11Þ

Note the maximum value for polyethylene (PE) at ln l ¼ 4:6, implying a draw ratio of 100. The polar polymers, polyester and polyamide(s), which are not superdrawable and not candidates for a gel spinning process, are at the other extreme. There is an interesting group of polymers with intermediate polarity and drawability, however: polypropylene (PP; DR ¼ 47.5), poly(vinyl alcohol) (PVA; DR ¼ 30), polyacrylonitrile (PAN; DR ¼ 28), poly-l-lactic acid (PLLA; DR ¼ 20). The very good drawability of polypropylene was mentioned when discussing that polymer; it is not gel-spun, but-melt spun fibers are also highly drawable. Poly(vinyl alcohol) is wet-spun, and special versions of this process, at Kuraray,


Fig. 17.26.

(Super)drawability of flexible-chain polymers (reproduced from Ref. 20).

with high molecular weight polymer, are indeed gel-spun; the fibers replace asbestos in cement reinforcement. Polyacrylonitrile, or a copolymer with a low comonomer level, is drawn 10–15 times in the production of precursor for carbon fiber. 17.7.3

Carbon Fiber

Carbon fibers are not directly spun but are the product of a complicated aftertreatment. Nowadays, most carbon fibers (90%) are produced from an acrylic precursor. Cellulose rayon is no longer applied as a precursor. Production from pitch has been developed, but is still a small-volume business. Carbon Fiber from PAN Most textile acrylics contain 10–15% comonomers. For carbon fiber precursors lower comonomer levels are used (about 5%); comonomers are selected that promote the reactions in the aftertreatment (methyl acrylate, itaconic acid). Wet spinning is preferred because the cross-section can be controlled better then. In dry spinning skin formation can hardly be prevented and eventually the cross-section collapses into a ‘‘dog bone’’ shape, which is not desirable in carbon fiber applications. Precursor filaments are drawn to much higher draw ratios (b10) than tex17.7.3.1

965

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17 Fibers

tile yarns. Precursor filaments must be free from solid particles; sharp filtration of spinning solutions is therefore required. Textbooks even mention spinning under ‘‘clean room’’ conditions. The aftertreatment proceeds in three steps: cyclization, oxidation, and carbonization/graphitization. All steps are carried out under tension. In the cyclization step (around 200 C) N-containing rings are formed, in a ladder structure: PAN, ‘‘Orlon’’, is transferred into ‘‘Black Orlon’’. This first step is still without loss of material. In the second step this structure is made unmeltable by careful oxidation and stabilization, at temperatures between 200 and 300 C. Some hydrogen is removed, carbonyl groups and double bonds are formed, and the aromatic character is increased. The third step is a pyrolysis reaction carried out in an inert atmosphere, at temperatures which are gradually increased from 400 to 1700 C and further to 2800 C. Below 1000 C most volatile products are formed, such as H2 O, HCN, NH3 , CO, CO2 , N2 , and so on. Note that the nitrogen content of PAN is about 28%, while the weight loss of PAN to carbon fiber is about 50%. The term ‘‘graphitization’’ is not really correct because a true graphite structure is not formed. The graphite layers, condensed ring systems, are indeed there, but the layers are not strictly coordinated. This ‘‘misfit’’ in the coordination of layers is called turbostratic. Nevertheless, carbon fibers are often referred to as graphite fibers, especially in the USA. Carbon Fiber from Pitch Pitches can be produced from petroleum and coal tar, preferably from the first feedstock. A useful pitch should have a mesophase character, which means that it must contain ordered, liquid-crystalline domains. The mesophase content can be increased by solvent extraction. A pitch would typically have a molecular weight of 1000 (20 aromatic rings) and a melting point of 300 C. Pitch can be spun at about 325 C and must then be aftertreated. The cyclization step necessary for a PAN precursor can be omitted in this case. Stabilization by air oxidation and carbonization/graphitization proceeds as for a PAN precursor, but the amount of volatiles is much lower because the pitch does not contain nitrogen. 17.7.3.2

Applications of Carbon Fibers Carbon fibers are used almost exclusively in composites. Two main types are offered: one with high strength (1.9–3.9 N tex1 tenacity, 140 N tex1 modulus) and one with a high-modulus (2.2 N tex1 tenacity, 280 N tex1 modulus). Carbon fiber moduli are much higher than those of aramid and gel-spun polyethylene. 17.7.3.3

17.7.4

Other Advanced Fibers

One class of HMHS fibers has not been mentioned: liquid-crystalline polyesters, thermotropic polymers, melt-spun. In the 1970s and 1980s many compositions were studied, in most cases fully aromatic polyesters, in one case PET enriched with large quantities of p-hydroxybenzoic acid (pHBA). Only one product ‘‘sur-

Notation Tab. 17.7. Survey of properties of HMHS fibers (PET, PA, glass and steel are included for comparison).

Fiber

Aramid (standard) Aramid (HM) Technora Vectran PBO PE (standard) PE (HM) C-fiber, HS C-fiber, HM C-fiber, pitch Polyester Polyamide E-glass S-glass Steel

Density [g cmC3 ]

Elongation [ %]

Tenacity [N texC1 ]

[GPa]

Modulus [N texC1 ]

[GPa]

1.45 1.45 1.45 1.40 1.56 0.97 0.97 1.8 1.8 2.1 1.38 1.18 2.55 2.5 7.8

3.5 2.5 (1.5) 4.4 b3.3 1.5 3.5 3–3.5 1.5–2.0 0.7 0.2–2.0 10–15 18–25 1.8–3.2 4.0 –

2.1 2.1 2.2 2.0–2.5 3.7 2.6–2.8 ca. 3.5 1.9–3.9 2.2 0.5–1.0 ca. 0.8 ca. 0.8 0.6–1.2 1.2 a0.27

3.0 3.0 3.2 2.8–3.5 5.8 2.6–2.8 ca. 3.5 3.5–7.0 3.9 1.0–2.0 1.1 0.95 1.5–3.0 3.5 a2.1

50–60 75 (115) 50 46–62 180 90 120 ca. 140 ca. 280 19–380 ca. 10 ca. 5 28 35 27

72–87 109 (167) 72 65–87 280 90 120 ca. 250 ca. 500 40–800 ca. 14 ca. 6 72 87 210

vived’’ and became a small commercial product: Vectra(n) polymer was developed by Celanese; the fiber is now produced by Kuraray. The polymer has a 75:25 composition of pHBA and HNA, the latter being the code for 2,6-hydroxynaphthoic acid. Mechanical properties are on the level of a standard-type aramid. There are many polymers with high-temperature resistant fibers, most of them with textile properties (low tenacity, high elongation) (see Table 17.7). Their application is in insulation, hot gas filtration, and suchlike. Large products are meta– meta aramid (Nomex, Teijin-Conex) and polybenzimidazole (PBI). Smaller products are poly(phenylene sulfide) (PPS), several aromatic polyketones (for example poly(ether ether ketone), PEEK) and aromatic polysulfones; see Reference 9, Chapters 8 and 9.

Notation

Symbol

Name

Unit

a c Cp g_ D D DR

thermal diffusivity concentration (of spinning solutions) specific heat shear rate diameter (of spinning holes) diffusion coefficient draw ratio

m 2 s1 – (or %) kJ C1 kg1 s1 mm m 2 s1 –

967

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17 Fibers

E Ecoh e_ f fa fc Fv Fm h [h] L l l n Dn P DP r R T T Td Td Tg Tm

elongation at break (in TE 0:5 ) cohesive energy elongational rate overall orientation factor (0 < f < 1) amorphous orientation factor (0 < fa < 1) crystalline orientation factor (0 < fc < 1) volume flow mass flow melt viscosity intrinsic viscosity length (of spinning holes) thermal conductivity draw ratio (DR) index of refraction birefringence pressure pressure drop density gas constant tenacity at break (in TE 0:5 ) temperature drawing temperature (in Figure 17.25) decomposition temperature glass transition temperature melting temperature

Acronyms BCF DCM DMAc DMF DR FOY HMHS HNA HSS HSSDW HTS LSS MFI NMMO NMP PA

bulked continuous filament dichloromethane dimethylacetamide dimethylformamide draw ratio fully oriented yarn high-modulus, high-strength fibers 2,6-hydroxynaphthoic acid high-speed spinning high-speed spin-draw winding hot-tube spinning low-speed spinning melt flow index N-methylmorpholine oxide N-methylpyrrolidone polyamide

% kJ kmol1 s1 – – – m 3 s1 or cm 3 min1 kg s1 or g min1 Pa s dL g1 mm kJ m1 C1 s1 – – – Pa or bar Pa or bar kg m3 or g cm3 8.3143 kJ K1 kmol1 mN tex1 C K C C C

References

PA66, PA6 PAN PBI PBO PBT PEN PET pHBA PLLA POM POY PP PPD PPS PEEK PPTA PTT PVA SDW SDBW TPC

polyamide 66, polyamide 6 polyacrylonitrile polybenzimidazole polybenzoxazole poly(butylene terephthalate); polybenzothiazole poly(ethylene naphthalate) poly(ethylene terephthalate) p-hydroxybenzoic acid poly-l-lactic acid polyoxymethylene partly oriented yarn polypropylene p-phenylene diamine poly(phenylene sulfide) poly(ether ether ketone) poly( p-phenylene terephthalamide) poly(trimethylene terephthalate) poly(vinyl acetate); poly(vinyl alcohol) spin–draw winding spin–draw bulk winding terephthaloyl dichloride

Acknowledgments

The information in this chapter was gathered during 25 years service at Akzo Fibers Research and recently updated, extended, commented upon, and improved by former colleagues now working for Acordis and Teijin–Twaron.

References 1 Ullmann’s Encyclopedia of Industrial

Chemistry, 5th Edition, VCH Verlag, Weinheim, 1987: fibers, A10, pp. 451– 655, A11, pp. 1–84; high-performance fibers, A13, pp. 1–23; cellulose and cellulose esters, A5, pp. 375–459; silk, A24, pp. 95–106; wool, A28, pp. 395– 421. 2 International Fiber Science and Technology Series, Marcel Dekker, New York: (a) Volume 7, M. Lewin, E. M. Pearce (eds.), Fiber Chemistry, 1985 (polyester, polyamide, acrylic, polypropylene, polyvinyl alcohol, wool, silk, cotton, rayon, cellulose acetate);

(b) Volume 4, H. A. Kra¨ssig, J. Lenz, H. F. Mark (eds.), Fiber Technology, 1984 (from film to fiber); (c) Volume 6, I. Sakurada (ed.), Polyvinyl Alcohol Fibers, 1985; (d) Volume 10, J.-P. Donnet, R. C. Bansal (eds.), Carbon Fibers, 2nd Edition, 1990. 3 B. von Falkai (ed.), Synthesefasern, Verlag Chemie, Weinheim, 1981. 4 F. Fourne´, Synthetische Fasern, Carl Hanser Verlag, Mu¨nich, 1995. 5 M. Mayer, Chapter 17: Fiber Extrusion, in F. Hensen (ed.), Plastics Extrusion Technology, Hanser Publishers, Mu¨nich, 1981.

969

970

17 Fibers 6 A. Ziabicki, Fundamentals of Fibre 7 8

9

10 11 12 13

Formation, John Wiley, New York, 1976. A. Ziabicki, H. Kawai, High-Speed Spinning, John Wiley, New York, 1985. T. Nakajima (ed.), Advanced Fiber Spinning Technology, Woodhead Publishing, Cambridge, 1994. J. W. S. Hearle (ed.), HighPerformance Fibres, Woodhead Publishing, Cambridge, 2001. T. J. Veerman, L. Vollbracht, US patent 4 308 374, 1981 (AKZO). H. S. Morgan, US patent 3 414 645, 1966 (Monsanto). S. L. Kwolek, Netherlands patent 6 908 984, 1969 (DuPont). H. L. Blades, Netherlands patent 7 205 836, 1971 (DuPont).

14 G. Capaccio, T. J. Chapman, I. M.

15 16

17 18 19

20

Ward, Polymer 1975, 16, 469 (and numerous other publications). E. S. Clark, L. S. Scott, Polym. Eng. Sci. 1974, 14(10), 682. A. J. Pennings, J. M. A. A. van der Mark, H. C. Booij, Kolloid-Z. 1970, 237, 336. A. Zwijnenburg, Thesis, University of Groningen, 1978. P. Smith, P. J. Lemstra, UK patent 2 051 661, 1979 (Stamicarbon). P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, 1979. J. Smook, G. J. H. Vos, H. L. Doppert, J. Applied Polym. Sci. 1990, 41, 105– 116.

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18

Removal of Monomers and VOCs from Polymers1 Marıá J. Barandiaran and Jose´ M. Asua 18.1

Introduction

The polymerization of monomers rarely proceeds to completion and there is inevitably a level of unreacted monomer remaining in the polymer. The presence of residual monomer is undesirable, particularly when the monomers are toxic, which is the case for vinyl chloride and acrylonitrile. Monomers such as acrylates and methacrylates can also have strong (and offensive) odors, and the residual volatile components may form an explosive mixture during transportation and storage. Contamination with unreacted monomers is a problem particularly when the polymer is used for food packaging, as well as in biomedical applications and in interior paints. Moreover, polymers may contain small amounts of nonpolymerizable compounds that result from impurities in the raw materials or by-products formed from side reactions during the polymerization process. The presence of such nonpolymerizable compounds results in an increase in the level of the residual volatile organic compounds (VOCs) in the polymer and can adversely affect its ultimate properties. In addition, many polymers are produced in the presence of organic solvents, which are considered VOCs and must be eliminated before use. The increasingly strict environmental regulations [1, 2] and the higher market sensitivity to environmental and health issues are pushing the polymer manufacturing industry to reduce the amount of monomers and VOCs in polymers. Reduction of VOC emission can also lead to better workplace conditions, reduce risks of fire, reduce nuisance, and lead to economic savings. The industrial importance of VOC removal is reflected in the large number of patents that can be found in the literature. The open literature, however, is scarce. The review in 2002 by Araujo et al. [3], which summarized the principal methods employed for reducing residual monomer content, and the books edited by J. A. Biesenberger [4] and R. J. Albalak [5] reviewing the various aspects of polymer devolatilization, must be highlighted. Several definitions of VOC have been offered. The European Union (EU) in Di1) The symbols used in this chapter are listed at


972

18 Removal of Monomers and VOCs from Polymers

rective 1999/13/EC [6] defines a VOC as any organic compound with a vapor pressure equal or greater than 10 Pa at 293.15 K. The U.S. Environmental Protection Agency (EPA) [7] defines the term VOC as any organic compound that participates in atmospheric photochemical reactions. Those two definitions can lead to contradictory situations. This is the case for acetone, an organic solvent common in polymer production, that is considered to be a VOC following the EU definition, while it is exempted by the EPA’s criterion. The understanding of the origin of residual monomers and VOCs will help to design strategies to polymerize up to very high monomer conversions (the higher the monomer conversion, the less residual monomer) while simultaneously avoiding the formation of new VOCs by side reactions. However, economic constraints that limit the polymerization time as well as intrinsic characteristics of the polymerization technique (such as solution polymerization) always yield systems containing VOCs. Therefore, post-treatments are necessary in order to fulfill the legislation and market requirements. Both the polymerization strategy and the type of post-treatment strongly depend on the polymerization process used for the manufacture of the polymer. The problems associated with monomer and VOC removal in highly viscous polymers produced by mass and solution polymerizations are different from those encountered in low-viscosity, heterogeneous aqueous dispersion systems. Therefore, this chapter will be organized according to the different polymerization processes. It is focused on polymers produced by chain polymerization. In polycondensation, the problem concerns only very specific cases, such as the removal of ethylene glycol in some poly(ethylene terephthalate) routes, or the removal of phenol in phenolic resins; the problems associated with these removals are similar to those found in the treatment of polymer melts. Section 18.2 is devoted to monomer and VOC removal from polymer melts and solutions. Monomer removal in polyolefins is described in Section 18.3, and finally the treatments for VOC removal in waterborne dispersions is discussed (see Section 18.4).

18.2

Polymer Melts and Solutions

Polymers produced by mass and solution polymerization, including the polycondensation processes, represent about 45% of the total synthetic polymer production [8]. In a bulk process, polymerization at high conversions is complicated by the effect of viscosity on initiator efficiency, monomer diffusion, and heat transfer. The efficiency of the initiator radicals can be dramatically lowered due to the cage effect resulting from the low mobility of the radicals, leading to a decrease in the polymerization rate. On the other hand, in polymerizations taking place at temperatures below the glass transition temperature of the polymer, the mobility of the monomer molecules is restricted, causing a severe decrease in the polymerization rate (glass effect). The use of higher temperatures would reduce the viscosity of the system, thus avoiding some of the drawbacks associated with high viscosities. However, high

18.2 Polymer Melts and Solutions

temperatures have deleterious effects on the properties of the polymer, such as a decrease in the molecular weight and degradation of the polymer. Furthermore, some free-radical polymerizations suffer a substantial depropagation at temperatures above a certain level (ceiling point). This phenomenon is quite well known for methyl methacrylate. Biesenberger et al. [9] showed that sustained exposures in a vented extruder at temperatures higher than 250 C caused poly(methyl methacrylate) to depolymerize to methyl methacrylate. On the other hand, in spite of the much lower viscosities of solution polymerization, the economically achievable conversion is limited because, due to the low monomer concentration, the polymerization rate is extremely low at high conversions, and consequently unaffordable process times are required. 18.2.1

Devolatilization

Polymer devolatilization is a separation process in which residual VOCs are removed from the polymer matrix by application of a reduced pressure (lower than the equilibrium partial pressure of the volatile component), and/or heat. Stripping agents are also commonly used. Fundamentals Devolatilization of molten and solution polymers generally involves first the transport by diffusion of volatiles to a polymer/vapor interface and then the transport of volatiles to a gas stream through the boundary layer. The rate of removal of the volatiles through the interface can be expressed by Eq. (1), where A is the interfacial area and NVOC is the flux of the VOC through the interface given by Eq. (2). 18.2.1.1

dVOC ¼ NVOC A dt

ð1Þ

NVOC ¼ D‘cjinterface

ð2Þ

Equation (2) requires the concentration profile of the VOC in the polymer melt that can be calculated from the material balance in the polymer melt. Assuming that the convective flux is negligible, Eq. (3) expresses this balance, where D is the diffusion coefficient of the volatile in the polymer matrix. qc ¼ ‘ D‘c qt

ð3Þ

The diffusion coefficient depends on both the temperature of the system and the concentration of volatiles, and can be estimated through the Vrentas–Duda freevolume theory [10]. The boundary conditions representing the diffusion process are:

973

974


t¼0

c ¼ c0

t>0

c interface ¼ ce

c 0 being the initial interface volatile concentration. In the devolatilization of polymer solutions and polymer melts, the diffusion of the volatiles through the polymer is usually the rate-controlling part of the process [4]. Therefore, the volatile concentration at the interface, ce , is in equilibrium with the concentration of the volatile in the gas phase. The equilibrium concentration is related to the partial pressure of the volatile in the gas phase, P1 , by means of Henry’s law, Eq. (4), where H, the Henry’s law constant, depends on the temperature, pressure, and nature of the volatile. ce ¼

P1 H

ð4Þ

The interfacial area between the polymer and the gas phase is one of the key parameters in the devolatilization processes, because the rate of VOC removal is proportional to it. The devolatilizers are designed to maximize this interface area. For free-bubble devolatilization, the area of the polymer film that is exposed to devolatilization can be calculated, taking into account the geometry of the equipment. Based on those principles, several models to describe the devolatilization in singlescrew extruders [11–14] and twin-screw extruders [15, 16] have been presented. However, bubbles are produced in most of the devolatilization processes. Bubbles can be formed when the polymer melt is exposed to a pressure lower than the partial pressure of the monomer or the solvent in equilibrium with the polymeric solution. Those bubbles are composed of monomer and solvent. After formation, the bubbles grow by diffusion of monomer or solvent from the polymeric solution, and ultimately they separate from the polymer melt, releasing the monomer and solvent. Bubbles can also be formed when stripping agents are introduced into the polymer melt/solution under pressure. If the vapor pressure of the stripping agent is higher than the equilibrium partial pressure of the volatile, gas bubbles composed mainly of the stripping agent will be produced. In this case, devolatilization involves diffusion of the volatiles to the surface of these bubbles. The area of the bubbles is determined by the interplay between bubble nucleation, growth, coalescence, and rupture. Under these circumstances the determination of the interface area is not straightforward. Direct visual observation of foaming in different types of devolatilizers has been reported by numerous authors [17–21]. Several models for foam-based devolatilization are available. The fundamentals are reviewed by Lee [22]. Bubble nucleation during devolatilization of polymer melts has been explained by homogeneous nucleation [23, 24], heterogeneous nucleation [25–27], and a mixed-mode nucleation [28]. Yarin et al. [29] considered a secondary nucleation. Models that account for the growth of the bubbles, assuming either a single


spherical bubble growing in an infinite sea of liquid [30, 31] or many spherical bubbles growing together with a thin shell of liquid surrounding them [32], have been developed. More complete models consider bubble motion, coalescence, and rupture [26, 33, 34]. Implementation of Devolatilization The foregoing equations show that devolatilization may be accelerated by acting on the diffusion coefficient, the thermodynamic equilibrium, or the interfacial area. The diffusion coefficient depends on the temperature of the system and on the concentration of volatiles. An increase in the temperature results in an increase in diffusivity and a decrease in viscosity, both beneficial for devolatilization. However, many polymers are thermally sensitive, so there may be a practical upper limit on the temperature to which the polymer may be exposed, as higher temperatures may degrade the polymer. Heat stabilizers have been used to enhance the thermal stabilization [35]. On the other hand, the diffusion coefficient through the polymer strongly decreases as the concentration of the volatile approaches to zero. This drawback can be overcome by introducing an inert substance (usually water), which increases the free volume of the polymer, enhancing the molecular diffusion of the volatile to be removed [36–41]. Furthermore, the inert reduces the partial pressure of the volatile in the gas phase, increasing the driving force for devolatilization. Adding an immiscible liquid to the melt presents the additional advantage that the total vapor pressure of the system increases, and therefore the temperature and volatile concentration needed to produce bubbles in the polymer solution is lower. Recently the use of supercritical fluids in processing polymers has received much attention, because supercritical devolatilization has the potential for producing high-purity products with lower energy [42–44]. The high solubility of the volatiles in the supercritical fluid enhances the driving force for devolatilization. When the polymer is exposed to the supercritical fluid, it is swollen, and the free volume in the polymer is increased so that diffusion of the volatiles out of the polymer is enhanced. 18.2.1.2

Equipment There exists a wide variety of devolatilization equipment. According to Biesenberger [4], the processes can be classified into two main categories: non-rotating or still equipment, and rotating equipment, in which devolatilization is enhanced by mechanical agitation. Nonrotating equipment, such as the falling-strand [5, 29, 45, 46] (Figure 18.1) and falling-film [5, 47] devolatilizers, include mainly flash evaporators and other equipment with specific configurations of the flash chamber, aimed at increasing the exposed interfacial area between the gas and the polymeric phase. In this equipment, the polymer melt/solution is preheated before entering the flash chamber, where the conditions of pressure and temperature are such that the volatiles boil. Vaporization of volatiles is prompted by continuous removal of their vapor through the vacuum outlet port of the flash tank. 18.2.1.3

975

976


Feed

Vacuum

Strands Heating i jjacket

Devolatilized l melt l Fig. 18.1.

The falling-strand devolatilizer.

In those processes three different regimes can be distinguished [48]. Each of them has a different rate-limiting mechanism, which depends mainly on the difference between the equilibrium partial pressure of the volatile in equilibrium with the melt (P1 ) and the total pressure within the vacuum chamber (P). This difference (P1 P) is often called the ‘‘degree of superheat’’, SH. The first regime, termed ‘‘free boiling’’, occurs when SH is large and the liquid viscosity is low. This usually happens under volatile-rich conditions. Vapor bubbles initiate and grow very fast, causing convective mixing that enhances mass transfer to the vapor phase. Under this regime, the temperature of the melt can rapidly drop if there is not an external energy input, since the latent heat of evaporation is provided by the sensible heat of the devolatilization mass. For example, under adiabatic conditions an MMA–PMMA melt at 250 C will drop approximately 15 C for every 10 wt.% of the melt that vaporizes. As the melt temperature drops, P1 will drop and the degree of superheat will diminish. On the other hand, melt viscosity will rise, due to both the removal of volatiles and the temperature drop. These effects cause bubble growth and movement to slow down and the free-boiling regime gradually fades out. The second regime is termed ‘‘bubble growth’’ and the rate-controlling mechanism is the bubble initiation and growth. The volatile removal rate is slower than in the previous case, and consequently cool-down is reduced. Melt viscosity increases due to depletion of the volatile. In the third regime, where the viscosity of the melt is very high and the degree of superheat has been reduced due to depletion of volatiles and temperature drop, bubbles are hardly formed and existing bubbles grow very slowly. Under these circumstances, the rate of volatile loss is very slow, as it is controlled by the molecular diffusion at the melt/vapor interface. Therefore, for a very low level of volatiles, flash devolatilizers are not efficient.


Nonrotating equipment is efficient for low-viscosity polymer solutions containing large amounts of volatiles, and particularly useful when dealing with shearsensitive polymers. However, care must be taken to ensure that they do not have stagnant areas where the polymer could degrade. Excessive foam expansion and polymer entrainment in the volatile vapor stream could be serious problem. Those systems are not operable with solutions that upon flashing retain enough solvent to form a sticky mass, such as in the case of rubbery polymer solutions, because the equipment has little self-cleaning and can become fouled up [49]. On the other hand, low volatile concentrations can hardly be achieved with those equipment, since very long residence times would be required to overcome their poor masstransport efficiency, which would increase thermal degradation of the polymer. Multi-slit devolatilizers [50–52] were designed to overcome the problem of long residence times of the polymer at high temperatures. In this process, the polymeric solution is fed through an assembly of heated slits which discharge into a flash chamber where vacuum is maintained. Rotating equipment is used for devolatilization of highly viscous polymer melts/ solutions. The rotating parts serve not only to spread and move the polymer forward along the devolatilizer but also to generate and renew the polymer/gas phase interface. Typical devices are wiped-film evaporators and screw extruders. The wiped-film evaporators (Figure 18.2) employ a rotor with an array of blades attached to the rotor shaft which transport the polymer melt through the evaporator, forming films. As the product goes down along the wall, bow waves developed by the rotor blades generate a highly turbulent flow, resulting in optimum heat and mass transfer. The melt and the vapor flow either in the cocurrent fashion

Feed

VOC removal

Heating i jacket

Devolatilized t melt Fig. 18.2.

The wiped-film evaporator.

977

978


Feed hopper

Vacuum

Feed Fig. 18.3.

Metering

Devolatilization

Metering

A single-screw extruder.

(mainly for removing highly volatile contents) or in countercurrent (for removing low-volatility contents). The combination of the short residence time, narrow residence time distribution, high turbulence, and rapid surface renewal permits the wiped-film evaporator to successfully handle heat-sensitive, viscous (up to 10 4 Pa s) and fouling-type fluids. This equipment is also used as a final reactor in polycondensation reactions, where the low molecular weight species produced in the reaction must be removed to achieve high molecular weights. The screw extruders used for devolatilization (Figure 18.3) present a vent zone with a deep channel, where the polymer is exposed to vacuum. Due to the clearance between the barrel and the screw flight, a thin layer of polymer is deposited in the barrel as the screw flight rotates. Evaporation may occur either at the surface of the rotating melt pool or at the surface of the deposit films. Bubbles can be generated inside the melt pool. Due to high viscosity, they hardly diffuse through the pool. However, as the pool rotates, some of them reach the exposed pool surface, increasing the devolatilization efficiency. Both single- and twin-screw extruders have been used for devolatilization. Counter-rotating, non-intermeshing twin-screw extruders have been widely used in the isolation of polymers from solutions, emulsions, and suspensions, and in the devolatilization-driven reactive extrusion processes [53]. Co-rotating, intermeshing twin-screw extruders have been also used as bulk and solution finishing devolatilizers [54]. One of the main advantages of the twin screw over the single screws is its ability to offer additional surface removal for film exposure and the excellent interchange of pool and film polymer. On the other hand, the area under the vent of a single extruder is vulnerable to stagnation, while the twin-screw geometry reduces this possibility. However, design of twin-screw extruders is more complicated and they are more expensive than the single-screw extruders. Although selection of the devolatilization system is not an easy issue, Figure 18.4 presents a guide based on the heat sensitivity, viscosity, concentration, and economics.

18.4 Waterborne Dispersions Preconcentration (Dilute product)

Viscosity (Pa.s)

Concentration (Viscous heat sensitive)

1

Flash evaporator

50

Devolatilization (Highly viscous)

103

Trace devolatilization (Very high viscosity)

2 104

5 104

Wiped-film evaporator

Falling strand evaporator

Fig. 18.4.

979

Vented extruder

Criteria for choosing the devolatilizer equipment.

18.3

Polyolefins

Polyolefins are nowadays one of the most important groups of commercial polymers, with a consumption of about 100 million tons per year, which represents more than 40% of the total word production of polymers [55]. Because the main monomers involved in the production, ethylene and propylene, have very high vapor pressure, these polymers are produced under pressure. Therefore, the residual monomers are easily removed by depressurization and recirculated to the polymerization zone. However, in the slurry and solution processes the polymerization medium is a solvent. In those cases, the polyolefin solution or slurry leaving the reactor is discharged into a vessel at lower pressure to flash off the solvent and unreacted monomers. The slurry from the polymerization reactor can also pass directly to a centrifuge where most of the volatiles are removed and recycled to the reactor. Further removal of solvent residues is achieved by passing a heated stream of inert gas (mainly nitrogen) through the bed of pellets. In the gas-phase processes, the polymer is discharged into a cyclone separator, from which residual monomers are recovered and recompressed.

18.4

Waterborne Dispersions

Probably the main step to eliminate VOC in the polymer industry has been the substitution of solvent-based systems by waterborne products. Nowadays, about 5% of the polymers are produced by aqueous dispersion processes, mainly by suspension and emulsion polymerization [8]. The water-based polymers, however, are not totally free of monomer and VOCs, since the reaction does not reach completion and VOCs arising from impurities in the raw materials or from by-side reactions can be present. There are two main ways to reduce the residual monomer content in waterborne polymers: post-polymerization or/and devolatilization.

980


In order to develop efficient post-polymerization and devolatilization operations, it is critical to know where the residual monomer and VOCs are located, that is, if they are mainly in the polymer particles or in the aqueous phase. Table 18.1 (in Section 18.4.2.1) summarizes the partition coefficients (defined as the ratio between the concentrations of monomer/VOC in the polymer particles and in the aqueous phase) measured for several monomers and VOCs in latexes. It can be seen that the monomers are mainly located in the polymer particles. On the other hand, water-soluble VOCs such as acetaldehyde and tert-butanol are mainly in the aqueous phase. It is worth pointing out that the partition coefficient decreases (that is, the monomer partitioning shifts toward the aqueous phase) as the average monomer concentration in the system decreases [56, 57]. 18.4.1

Post-polymerization

Post-polymerization consists of adding, after the end of the main polymerization process, fresh radical-generating systems to polymerize the residual monomer. Even though residual monomers may be post-polymerized using radiation [58, 59], post-polymerization is mainly carried out using initiators. In suspension polymerization, the initiator must be oil-soluble. Those initiators, in order to be fed into the reactor, are commonly dissolved in an organic solvent, which increases the VOC content. On the other hand, the most important monomers (styrene, methyl methacrylate, and vinyl chloride) polymerized in suspension yield polymers with a high glass transition temperature (Tg ). The polymerization temperature is usually below the Tg , and hence the polymerization rate decreases sharply at high conversions due to the glass effect. Increasing the temperature of the reactor above the Tg could be a way to improve the monomer conversion. However, this would imply temperatures higher than the boiling point of the water, and in addition, long exposures to high temperatures could degrade the polymer. Therefore, post-polymerization does not seem a promising way to reduce the residual monomer in suspension polymers. In emulsion polymerization, any kind of free-radical initiation system may, in principle, be used for post-polymerization. However, some alternatives seem to be preferable. Thus, although both thermal and redox initiation systems may be used, and attempts to enhance the activity of thermal initiators by using ultrasound have been reported [60], redox systems are preferred because these systems generate a higher flux of radicals, in particular under mild conditions, leading to shorter postpolymerization times. A wide variety of oxidants and reductants can be combined in different redox systems [61, 62]. Ideally, radicals should have easy access to the place where the monomer is located. Because the residual monomer is mainly located in the polymer particles [57], oil-soluble initiators seem promising. However, as explained above, their addition would increase the VOC content. Consequently, water-soluble redox systems are more convenient. Ilundain et al. [63] demonstrated that independently of the water solubility of the monomer, water-soluble redox initiators yielding hydrophobic radicals (such as organic hydroperoxides) were ad-

18.4 Waterborne Dispersions

vantageous for monomer removal by post-polymerization. The reason is that the hydrophobic radicals can enter directly into the polymer particles whereas the hydrophilic radicals must undergo a number of propagation steps before becoming hydrophobic enough to be able to enter into the polymer particle. These growth processes take rather a long time, because under post-polymerization conditions the monomer concentration in the aqueous phase is very low, and the oligoradicals suffer bimolecular termination leading to low initiator efficiency. Emulsion polymerization is commonly used to produce film-forming polymers, and hence it is generally carried out at temperatures above the Tg of the polymers. Therefore, propagation does not become diffusion-controlled [64], and the kinetics of post-polymerization is not influenced by the reaction temperature differently than the emulsion polymerization stage [65]. The main advantages of post-polymerization treatments are that they can be carried out either in the polymerization reactor or in the storage tank, and no additional equipment is needed. However, only the polymerizable residual volatiles can be eliminated, and in some cases new VOCs are produced from secondary reactions. Thus, formaldehyde is formed when sodium sulfoxylate formaldehyde is used as the reductant [66] and acetone and tert-butanol are formed when tert-butyl hydroperoxide is used as the oxidant [67]. In addition, inorganic water-soluble initiators may be deleterious to both stability and water sensitivity of the film formed with the latexes. It is worth mentioning that some initiator systems may modify the polymer microstructure [68, 69], which can be either a problem or an opportunity to extend the range of properties achievable with a given aqueous dispersion of polymers. Modeling Post-polymerization The processes involved in post-polymerization can be summarized as follows. The radicals that are generated in the aqueous phase must add some monomeric units in order to become hydrophobic enough to enter into the particles. This critical value is referred as z, and is a function of both the monomer hydrophobicity [70] and the type of radical formed from the initiator. The radicals may suffer bimolecular termination in the aqueous phase. On the other hand, the radicals inside the particles can propagate, terminate, or desorb. Under post-polymerization conditions, however, the radical desorption rate is negligible because, due to the low monomer content in the particles, the probability of generating small free-radical species by chain transfer to monomer is very low. A mathematical model based on the mechanisms summarized above is available [71]. Optimal strategies aiming at achieving maximum monomer removal with minimum formation of VOCs have been reported [71]. 18.4.1.1

18.4.2

Devolatilization

Devolatilization of aqueous polymer dispersions is usually carried out using a stripping agent (steam and nitrogen are the most commonly used; air can also be

981

982


Vapor +VOCs

Bubble Aqueous phase

Polymer particle Stripping agent (steam, nitrogen,.)

Fig. 18.5.

The stripping of latexes.

used, but explosive vapor mixtures can be produced). Figure 18.5 summarizes the devolatilization process in a stirred tank using a stripping agent, which is usually fed in continuously until the volatile content is reduced to the desired level. Modeling From a microscopic point of view, devolatilization of aqueous-phase dispersions is a mass-transfer process, which involves the following steps: (1) diffusion of the VOCs to the particle surface; (2) transfer from the polymer surface to the aqueous phase; (3) diffusion through the aqueous phase; and (4) transfer from the aqueous phase to the gas phase (Figure 18.6) [72]. 18.4.2.1

Fig. 18.6.

Processes involved in the devolatilization of waterborne dispersions.


Assuming that the contact between the particles and the bubbles is negligible, and that the flow of the gas phase may be described by a well-mixed continuous system, the overall mass balances of the VOC in the different phases are given by Eqs. (5)–(7), where Vi is the volume of the phase i (w ¼ water phase, p ¼ polymer particles, g ¼ gas phase); Ci is the concentration of the VOC in the phase i; Ci; ej is the concentration of the VOC in the phase i that would be in equilibrium with the actual concentration of the VOC in phase j; K pw represents the overall masstransfer coefficient between the polymeric phase and the water phase, K wg is the mass-transfer coefficient between the aqueous and the gas phase; A pw and A wv are the interfacial areas between the polymer particles and the aqueous phase and between the aqueous phase and the gas phase, respectively; y is the molar fraction of the VOC in the effluent; and G is the molar flow rate of the stripping gas. Polymer particles: d ðVp Cp Þ ¼ K pw A pw ðCp Cp; ew Þ dt

ð5Þ

Aqueous phase: d ðVw Cw Þ ¼ K pw A pw ðCp Cp; ew Þ K wg A wg ðCw Cw; eg Þ dt

ð6Þ

Gas phase: d ðVg Cg Þ ¼ K wg A wg ðCw Cw; eg Þ Gy dt

ð7Þ

In the devolatilization of polymers in aqueous dispersion, the equilibria are expressed in terms of the partition coefficient of the different VOCs between the p polymer particles and the aqueous phase, k w , and the Henry’s law constant (H) for the partitioning of VOCs between the aqueous phase and the gas phase. Therefore, the equilibrium concentrations can be calculated from Eqs. (8) and (9). Cp; ew ¼ k wp Cw Cw; eg ¼

PT y H

ð8Þ ð9Þ

Tables 18.1 and 18.2 present the values of those coefficients under devolatilization conditions (residual VOC content lower than 5000 ppm) for some monomers commonly used in emulsion polymerization and for the main VOCs found in those latexes after polymerization. The polymer particle/aqueous phase interfacial area can be obtained from the particle size measurements, but the total transfer area between the aqueous and

983

984

18 Removal of Monomers and VOCs from Polymers Tab. 18.1. Polymer particle/aqueous phase coefficients under devolatilization conditions calculated by R. Salazar [100].

˚

p

Monomer

Polymer system

T [ C]

kw

Monomer content of particles (50 wt.% solids) [ %]

VAc VAc BA n-Butanol Acetaldehyde tert-Butanol

poly(vinyl acetate) VAc/BA/AA BA/S/AA VAc/BA/AA VAc/BA/AA VAc/BA/AA

65 65 65 65 65 65

10.5 10.9 240 2.5 0.5 0.5

91.3 88.2 99.6 71.4 33.3 33.3

gas phases is difficult to measure. The degree of dispersion of the gas in a liquid is a function of several factors such as the geometry of the contact equipment, the agitation device and speed, the viscosity of the medium, and the gas flow rate. In addition, the physical-chemical properties of both phases, derived by the presence of surfactants or electrolytes in the medium, affect the interfacial area. The determination of the mass-transfer coefficients is not straightforward either. For the water/gas phase, the uncertainty of the estimation of both the interfacial area and the mass-transfer coefficient is overcome through the evaluation of the volumetric mass-transfer coefficient, K wg A wg . Estimated values from different authors are presented in Table 18.3. The polymer particles/aqueous phase mass-transfer coefficient can be determined through the Fro¨ssling equation [73], where Sh is the Sherwood number, Re the Reynolds number, and Sc the Schmidt number. Sh ¼ 2:0 þ aRe b Sc c

ð10Þ

It is worth pointing out that few studies took into account the pseudoplastic behavior of the polymeric dispersions [74, 75]. There is little and sometimes contradictory information about the presence of the additional gas phase on the solid–liquid mass-transfer coefficients. Sanger et al. [76] observed that in a bubble column, K pw increased upon increasing the gas flow rate. Grisafi et al. [77], observed just the opposite in a stirred tank; this was attributed to reduction of the effective power

Tab. 18.2. Henry’s law constants at 65 C calculated by R. Salazar [100].

Compound

H [k Pa]

VAc Butyl acrylate Acetone n-Butanol

26 29 3 1.5

18.4 Waterborne Dispersions Tab. 18.3.

Estimated mass transfer volumetric coefficients.

Equation

Estimated K vw A vw [m 3 sC1 ]

Reference

K wg A wg ¼ 0:02218ðPg =VL Þ 0:5 u0:6 s 2 1:11 0:5 K wg A wg d i d i Nr me d i us 0:447 mg 0:694 ¼ 21:2 me D rD s me 2 0:67 0:5 2 3 1:29 K wg A wg di2 rN di d Nr m ¼ 1:41 103 i m rD D s 2 1:65 0:5 2 0:19 0:6 K wg A vg di2 d Nr m N di mus Nd i 0:33 ¼ 0:06 i F me rD D g s us

4:4 106

118

0:14 106

119

0:91 106

120

0:83 106

121

5:3 106

122

F ¼ ð1 þ 2:0ðlNÞ 0:5 Þ0:67 K wg A wg ¼ 8:48 108 uG0:56G0:08 N 2:7G0:2

N > 30 rpm

per unit volume used to mix the dispersion. Kim et al. [78, 79] measured experimentally the K pw of vinyl acetate diffusion into poly(vinyl acetate) and polystyrene latex particles using a vapor-phase addition method. Values in the range of 5 107 –7 107 cm min1 were obtained for diffusion of vinyl acetate into 160 nm PS latex particles. Rate-limiting Steps For suspension polymers, which are commonly high-Tg polymers and present small polymer particle/aqueous phase interfacial areas, the diffusion through the particle and the transfer from the particle surface to the aqueous phase are often the rate-limiting steps [80]. The ease of removal depends mainly on the polymer particle size, particle size distribution (the presence of large particles increases the diffusion time through the particle largely), and the presence of ‘‘glassy’’ particles. High temperatures speed up devolatilization. For emulsion polymers, the monomer devolatilization rate is often limited by the mass transfer through the interface between the aqueous phase and the gas phase [57]. For these systems, diffusion in the polymer particles is fast because the Tg of the polymer is low and the particle size is small. In addition, the mass transfer from the particles to the aqueous phase is fast because of the huge interfacial area. For water-soluble VOCs such as acetaldehyde and tert-butanol, the ratelimiting step is also the mass transfer through the liquid/gas interface. Therefore, all the process variables that increase the interfacial area between the aqueous phase and the gas phase, such as agitation, geometry of the sparger, or gas flow rate, would improve devolatilization [81]. However, the stripping of emulsion polymers may cause a substantial fraction of the surfactant added to the system to desorb from the polymer particles to form foam, thus leaving the system susceptible to coagulation. Furthermore, the me18.4.2.2

985

986


chanical agitation and the creation of large gas/liquid interfacial areas promote coagulation. In addition, the solids content of the dispersion can vary due to either the evaporation of some water or the condensation of steam. Devolatilization under Equilibrium Conditions Under some circumstances, such as when the bubble size is very small and low gas flow rates are used, the concentrations of the VOCs in all phases are at thermodynamic equilibrium. In this case, the removal kinetics of the VOCs is given by Eq. (11) [82–84], where t is the removal time and k is the apparent removal rate constant, given by Eq. (12) [84]. Mw is the water molecular weight, E is the mass of latex in the devolatilizer, and X is the mass fraction of polymer in the latex. 18.4.2.3

C ¼ ekt C0 GMw ðH=PT Þ ! # p rw E X kw 1 þ 1 rp

k¼ "

ð11Þ ð12Þ

This prediction was found to be valid for different processes, such as the stripping of styrene–butadiene latexes when a constant steam sparge rate was used [83, 85], and acrylic copolymers at different solids contents and steam sparge rates [84]. Equations (11) and (12) show that devolatilization is strongly affected by the thermodynamic equilibria of the VOCs between different phases. High values of the polymer particles/aqueous-phase partition coefficient imply that the concentration in the aqueous phase will be low and hence it will be difficult to remove the VOC from the particles. Similarly, a low value of the Henry’s law constant means that the concentration of VOCs in the gas phase is low and hence, devolatilization will be difficult. Figure 18.7 shows the kinetics of devolatilization of vinyl acetate, acetaldehyde, and n-butanol in a VAc/BA/AA latex, and that of BA in a BA/S/AA latex by stripping in laboratory-scale equipment, under equilibrium conditions. It can be p seen that the devolatilization of BA was slow due to the high affinity (high k w ) of BA to the polymer particles. The removal of n-butanol was also very slow because of its high solubility in the aqueous phase and low vapor pressure (a low value of the Henry’s law constant). Equipment Although flash devolatilization has been applied for removal of high-volatility monomers, such as vinyl chloride from poly(vinyl chloride) [86–88] or butadiene from polybutadiene [89], most of the processes described in the literature include the use of a stripping inert gas (usually steam) to improve the devolatilization efficiency. Several types of equipment have been proposed to allow fast devolatilization without affecting the stability of the dispersion and avoiding foaming. Englund [83] reviewed the different latex strippers used in commercial practice. Batch, 18.4.2.4


1

Conversion

0.8 0.6 0.4 0.2 0 0

30

60 90 120 Time (min)

Fig. 18.7. Kinetics of devolatilization of vinyl acetate, acetaldehyde, and n-butanol in a VAc/BA/AA latex, and that of BA in a BA/S/AA latex: g, acetaldehyde; k, VAc; c, n-butanol

150

180

removal rate by devolatilization from a VAc/BA/AA latex; , BA removal rate from BA/S/AA latex, under the same devolatilization conditions [100].

semibatch, and continuous tanks, using countercurrent or cross-flow gas [90–93], packed or perforated columns in countercurrent flow [94–98], and tubular devolatilizers [99] have been used. Continuous steam stripping using countercurrent flow of an aqueous dispersion and steam requires less steam than batch and cross-flow systems, although the equipment needs considerable space. On the other hand, the column continuous countercurrent stripper requires the least steam of all the processes, and little floor space. This type of stripper is attractive because the short residence time allows short exposure to high temperatures, decreasing the risk of degradation. They are used extensively in suspension processes, such as the production of poly(vinyl chloride). Those stripping systems, however, are difficult to apply to latexes, because of severe foam formation, and the risk of coagulation due to the shear and the relatively high viscosity of the latex. The low Tg of most of the latexes is an additional drawback for using column stripping systems. In these cases, tank reactors are much more convenient. The undesirable foaming can be suppressed by lowering the temperature, but this reduces the recovery of the low boiling point substances. Chemical defoaming agents may be used, but they may accelerate thermal degradation of the polymer when it is processed at an elevated temperature (as occurs with the polyvinyl chloride) and their use adds contaminants to the latex. One way to control foam formation is by a sudden increase of the pressure. This method has been proven to be very efficient in the stripping of acrylic latexes in tank reactors [100]. Many efforts have been devoted to the design of the devolatilizer to overcome this problem. The injection of the steam at a point in the autoclave such that it passes through the

987


dispersion uncondensed has been found very convenient [101]. In other cases, the gas to be drawn off is passed through porous filter elements [102] or through a condenser adapted to the top of the packed tower [94]. Other authors [90] claimed to feed the aqueous latex as a spray to avoid the contact of the droplets of spray with the side wall of the chamber, and therefore to avoid foaming, which otherwise could cause flooding of the descending latex. 18.4.3

Combined Processes

Devolatilization is an adequate technique to remove both residual monomer and nonpolymerizable volatiles. However, this method is of limited efficiency for the removal of very hydrophobic monomers, for which high flow rates of the stripping gas should be used to appreciably reduce the monomer content, with the corresponding increase in cost, and risk of foaming and coagulation. Under these circumstances, combination of post-polymerization and devolatilization is an attractive alternative. Some authors have proposed treating the dispersion initially by postpolymerization followed by stripping devolatilization [103–105]. Taylor [106] claimed a treatment in which devolatilization and post-polymerization proceeded concurrently. Salazar et al. [57] have compared the performance of concurrent combined strategies for removal of monomer and VOCs with those of other strategies including post-polymerization, devolatilization, and post-polymerization followed by devolatilization. The results obtained for the removal of BA and the VOCs present in BA/S/AA commercial latex are shown in Figure 18.8. It can be

1 0.8 Conversion

988

0.6 0.4 0.2 0 0

30

60 90 120 Time (min)

150

Fig. 18.8. Influence of the strategy on the efficiency of removal of BA from a BA/S/AA commercial latex: k, devolatilization; g, post-polymerization; c, simultaneous post-polymerization and devolatilization.

180

18.5 Summary

observed that the simultaneous process resulted in the highest efficiency of removal for monomer. For the nonpolymerizable VOCs, almost the same efficiency was obtained in all the strategies using devolatilization. 18.4.4

Alternative Processes

In addition to those conventional techniques widely applied in the industry, other alternatives that try to overcome the problems associated with the use of chemicals in post-polymerization or the high energy consumption associated with devolatilization have been implemented. The modification of the unreacted monomer into a harmless compound, or into a product easier to devolatilize, has been applied to reduce the VOC content. Ozone was found advantageous for vinylic monomer, because it can easily break the double bond and is innocuous [107, 108]. On the other hand, it could also attack the double bond of the polymer chain, causing the degradation of the polymer. Biological degradation through the use of peroxidegenerating enzymes was also shown to be able to reduce residual monomer in latex [109, 110]. This process, however, is very slow. Adsorbents, such as active carbon particles [111, 112], zeolites [111, 112], polymeric adsorbents [111], or ion exchangers [111], have been proven to be useful for VOC reduction in latexes. This treatment, however, requires additional equipment and the regeneration of the adsorbent. Extraction of the VOCs with suitable solvents has been also extensively investigated. Treatment with organic solvents was found valid for particular cases, such as the removal of high-boiling organic compounds [113] or preparation of polymeric hydrogels [114]. Application of supercritical fluids to cleaning processes was presented in 1998 as an alternative to monomer removal in the near future [115]. Kemmere et al. [116, 117] studied the potential of a post-polymerization process based on supercritical carbon dioxide to remove methyl methacrylate from poly(methyl methacrylate). This process is attractive since CO2 is very abundant, relatively inexpensive, and environmentally benign on this scale of use.

18.5

Summary

For polymer melts and solutions, the diffusion of the VOCs from the polymer to the polymer/vapor interface is the rate-limiting step. This limitation is more pronounced for highly viscous melts. Therefore, the devolatilizers must be designed to increase both the diffusion coefficient of the volatiles and the polymer/vapor interfacial area. This is achieved by using stripping agents, and equipment with rotating parts that generate and renew the interfacial area. Vented screw extruders are recommended for devolatilizing highly viscous systems. Wiped-film evaporators can handle polymer melts with viscosities up to 10 4 Pa s, and flash

989

990


and falling-strand devolatilizers are recommended for low-viscosity polymers (viscosities up to 1 Pa s). Residual monomer present in polyolefins is easily removed by depressurization. The solvent used in the slurry and solution processes is commonly eliminated by flash evaporation and subsequent contact with a heated inert gas stream. There are two main ways to reduce the VOC content in waterborne dispersed polymers: post-polymerization and devolatilization. Post-polymerization can only be applied to emulsion polymers. Water-soluble redox initiators yielding hydrophobic radicals have been found to be advantageous for monomer removal by postpolymerization. In suspension polymers the devolatilization is limited by both the diffusion through the particle (because of the usually high Tg of the polymer) and the mass transfer from the particle surface to the aqueous phase (because of the large particle size that results in a relatively small interfacial area). Therefore, the temperature and the polymer particle size play an important role in the devolatilization efficiency. In the devolatilization of emulsion polymers, the mass transfer between the aqueous phase and the gas phase is frequently the rate-determining step. Consequently, the agitation, the gas flow rate, and other process variables or design characteristics that increase the interfacial area between the aqueous phase and the gas phase will improve the devolatilization. However, care must be taken to avoid foaming and coagulation. Column continuous countercurrent strippers are often used in devolatilization of suspension polymers, while stripping in tank reactors is more convenient for low-Tg film-forming latexes.

Notation

A c ce D di E G g H K p kw Mw N NVOC P1 Pg Re Sc

interfacial area [m 2 ] concentration of the VOC [mol m3 ] volatile concentration at the interface [mol m3 ] diffusion coefficient [m 2 s1 ] impeller diameter [m] mass of latex in the devolatilizer [kg] molar flow rate of the stripping gas [mol s1 ] gravitational constant [m s2 ] Henry’s law constant [kPa] overall mass transfer coefficient [m s1 ] partition coefficient between the polymer particles and the aqueous phase water molecular weight [kg mol1 ] agitation speed [s1 ] flux of the VOC through the interface [mol m2 s1 ] partial pressure of the volatile in the gas phase [kPa] power applied to the system [W] Reynolds number Schmidt number

References

SH Sh Tg ug us V X y

degree of superheat Sherwood number glass transition temperature [ C] linear gas velocity [m s1 ] superficial gas velocity [m s1 ] volume [m 3 ] mass fraction of polymer in the latex molar fraction of the VOC in the effluent

Greek density [kg m3 ] viscosity [Pa s] effective viscosity [Pa s] interfacial tension [Nm s1 ]

r m me s

Subscripts g p w

gas phase polymer particle water phase

Acronyms AA BA S VAc VOC

acrylic acid butyl acrylate styrene vinyl acetate volatile organic compound

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Nano- and Microstructuring of Polymers1 Christiane de Witz, Carlos Sańchez, Cees Bastiaansen, and Dirk J. Broer 19.1

Introduction

Nano- and microstructures are essential for the growth of well-established technologies, such as microelectronics, but also for the development of emerging new ones in the fields of biology, nanotechnology, and information and communication technology. Since the early 1970s the production of microelectronic silicon chips has been a driving force of paramount importance for the development of micropatterning techniques. Traditionally, this field has focused on optical lithography in which a polymer film is structured using light. The patterned polymer is then used to implement and integrate the functional components of the microchip in the silicon wafer. The semiconductor industry focuses on reducing the feature size of transistors and on reaching higher integration densities for faster computation, higher performance, and cheaper production [1]. Besides microelectronics, miniaturization and high-throughput production of complex structures are also required for electro-optical and electronic devices (LCD, LED, FET), nano- and micro-electromechanical systems (MEMS), denser memories, and biosensors or biological arrays [2]. New patterning techniques and materials need to be explored in order to sustain the historical trend in size reduction in the semiconductor industry and in order to make real advances. For this, polymers are an extremely attractive platform due to their ease of processing and their tailorability, which allows fine-tuning of the properties for each application. As an example, the possibility of patterning polymers with special functionalities such as electrical, (semi-)conductivity, electroluminescence, piezoelectric, and/or dielectric properties is important for the development of plastic electronics. Well-defined structured polymer surfaces also find their application in optical and electro-optical devices [3–8]. For instance, special reflectors, made of a polymer film with a well-designed relief structure and coated with a thin-film metallic mirror, redistribute incoming 1) The symbols used in this chapter are listed at


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19 Nano- and Microstructuring of Polymers

light in a very controlled way and are widely used in reflective and transflective liquid crystal displays. Periodic relief structures are also widely employed in transmissive gratings to control light in transmission or reflection [7, 8]. First, we will briefly review different strategies for the patterning of polymer structures and, subsequently, we will focus on new patterning techniques such as photoembossing. 19.1.1

Patterning Techniques

Photolithographic techniques make use of electromagnetic (EM) radiation (UV, DUV, EUV or X-ray photons) to generate a latent image in a photopolymer [9]. The image of a mask can be transferred, either by putting it in direct contact with the photoresist and subsequently irradiating with collimated light or by projecting the image with the help of appropriate optics. Methods involving multibeam holography have also been explored [10]. Independently of the irradiation method, the EM radiation exposure induces a change in solubility in the exposed areas. A subsequent development step in a solvent removes material locally and a relief structure is obtained. This technique suffers from certain intrinsic limitations such as diffraction or, in the case of projection lithography, the depth of focus. The resolution of optical lithography increases if the wavelength of the EM radiation is reduced. The use of KrF lasers (248 nm) allows the routine fabrication of features in the order of 200 nm. In order to reduce the size of these features, shorter wavelengths need to be used. ArF excimer lasers working at 193 nm can lead to resolutions of 150 nm. Shorter wavelengths such as 157 nm (F2 excimer lasers) can be employed. New grades of modified fused silica have been developed showing weak absorption in this wavelength region, making possible the production of lithographic masks. Calcium fluoride meets also the requirements for implementing the optics for the illumination and projection systems [11]. More problematic is the use of EUV and X-ray lithography, due to the lack of transparent materials to generate the masks and the necessary optics for the lithographic process. One possibility is to use masks with openings in a stencil instead of the conventional lithographic masks consisting of metallic (for example, chromium) patterns supported on a transparent substrate. Reflection-instead of transmission-based optics also needs to be implemented. Besides the typical problems of UV lithography, damage to the mask is also induced during exposure. Although the use of these high-resolution lithographies (EUV and X-ray) has been demonstrated in a laboratory scale, their use in mass production is still restricted because of their high cost. In all the cases described the development of appropriate polymeric photoresists is of crucial importance. For example, the resolution can be enhanced by using resists with a nonlinear response that make possible the fabrication of features with a size below the diffraction limitation of the photolithographic technique [12, 13]. Electron beams (EBs) can also be employed for the fine-scale (tens of nanometers) patterning of polymer structures, because of their short (de Broglie) wavelength [14]. Here the charged character of the writing particles limits the resolution because of interparticle electrostatic interactions that produce scattering and

19.1 Introduction

therefore lack of resolution. Another important drawback of this technique is that the patterning is performed with a writing procedure and, consequently, the production of structures is laborious and expensive; this restricts their use to very specific applications. Some projection systems are being investigated in attempts to obtain a high-throughput production of patterned structures by EB lithography [15]. Other related techniques involve ion or uncharged particle beams [16]. Other non-photolithographic techniques, such as imprinting (or embossing) or cast molding, have had a large technological impact in different industrial sectors. In the case of imprinting, a rigid master is pressed against an unstructured polymer film [17]. Afterward the master is released and its inverse replica is obtained. Some popular applications of this imprinting technique are the holograms such as those used in safety features in the security industry (credit cards, passports, and money) and compact disks (CDs) [18]. It is also possible to obtain these inverse replicas by casting a polymer precursor onto the master; after curing this is released to obtain the desired inverse structure. These techniques can reach resolutions of tens of nanometers. The aspect-ratio of the replicated structures with high throughput and reproducibility is limited to values of around 1, although higher values have been reached. Among the physical origins of this limitation are the adhesion forces between the mold and the inverse replica, and their mechanical and thermal properties. Nanomachining and nanomanipulation techniques using scanning probe microscopes (SPMs) have been shown to be feasible for the patterning of polymers on a nanometer scale [19]. Polymer films have been modified by using high-force AFM. In this case the tip indents into the material, generating a hole in it. Surface profiles consisting of not only lines and dots but also more complex three-dimensional structures in a gray-scale fashion have been demonstrated. Attempts to overcome the serial nature of this AFM-based lithography have also been made (‘‘Millipede’’ project from IBM) using arrays of independent cantilevers (up to 1024 tips). In this case the hole formation rate (bit rate) is thermally assisted by heating the tip. Densities of storage up to 200 Gbytes inch 2 [31 Gbytes cm 2 ] have been achieved using this system. The movement of individual polymeric chains has also been controlled, showing the power of this new nanolithography. Other nanomanipulation techniques under development involve highly focused beams (optical tweezers) able to control the position of polymeric microbeads [20]. The techniques described above are known as top-down approaches that make use of available technologies to reduce the size of the features of the patterned structures. On the other hand, bottom-up techniques make use of the selfassembly concept to build up structures over several length scales. For example, periodic templates are generated using polymer microbeads of monodispersed size [21]. The steric repulsion between beads promotes the formation of controlled three-dimensional packed crystalline structures. These templates, or equivalent ones made of inorganic materials, can be used subsequently to generate the inverse polymeric opal also [22]. Another self-organization approach involves phase separation of block copolymers [23]. In this case the different natures of the blocks can induce their segregation, giving way to well-defined, long-range structures

997

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(lamellas, spheres, cylinders) with sizes that range from several to hundreds of nanometers, well controlled by the length of the different blocks. Although they are intrinsically attractive, these techniques have important limitations for the advanced patterning of complex structures because of the difficulty of controlling the size, position, shape, and directionality of the growth structure. Soft lithography is being investigated extensively for the large-scale generation of relief structures in metallic substrates. In microcontact printing, which is a subtechnique of soft lithography, an elastomeric stamp with a relief structure is used. The stamp is impregnated with a monolayer of a reactive ink containing, for example, thiols or silanes. The relief structure is then brought into conformal contact with a noble metal substrate (silicon or silica may also have been used). In the contact areas, a reaction occurs between the ink and the substrate, and an ink monolayer is obtained which acts as an etch-resist [24, 25]. Although this procedure is intended to be used to pattern metal layers, it has also been employed in the structuring of polymers. For example, the selective growth of polymer in the reacted areas has been demonstrated [26]. The reacted monolayer on the substrate can also act as a surface tension pattern to deposit polymers selectively. We have briefly reviewed some of the most significant available technologies for generating patterns in polymers. Each of them has specific advantages and deficiencies with respect the feature size, processability, high-throughput capability, cost effectiveness, and functionality of the final patterned structure. These and other new techniques need to be developed further in order to overcome existing problems. 19.1.2

Photoembossing

As an example of a patterning technique in polymers, we will discuss in more detail photoembossing, a photolithographic technique for the generation of polymeric relief structures. In this particular case, the relief structure develops as a result of a material flux induced by a local polymerization. The material flux deforms the surface of the film, which is permanently fixed by the photoinitiated crosslinking process without the interference of a solvent development step [9, 27, 28]. Therefore this principle is attractive in an economic sense for applications that require the structuring of large surfaces. The maskwise or holographic exposure of a thin film consisting of a blend of monomers in the presence of a photoinitiator induces a flux of material to the exposed (high UV intensity) areas; this may be accompanied by a counterflux of other material to the dark (low UV intensity) regions [29–32]. When the flux and counterflux are equally balanced in volume, a modulation in composition is obtained, whereas the surface of the film remains flat. This process is used successfully, for instance, to make volume holograms based on refractive index modulation [33, 34]. If there is an unequal volume flux, the surface of the film will become modulated because of a volume expansion of the exposed area and a corresponding volume reduction of the dark areas, or vice versa. This principle is ap-

19.2 Materials and their Photoresponsive Behavior

plied to make surface holographic gratings. The process in itself is understood and can be described quantitatively in terms of the change in the spectrum of chemical potentials of the various components during the UV exposure and polymerization [35]. Enhancement of this effect can be achieved with a blend that consists of a polymer and a polyfunctional monomer. In that case the polymer forms a stationary phase whereas the monomer diffuses to the exposed areas, resulting in pronounced surface profiles. A general drawback is that the structures form during exposure. This causes changes of the light path during exposure because the deforming surfaces refract the light in a continuously changing way during the formation of the surface relief. It is possible to overcome this problem by creating a latent image in the film during which the formation of a surface relief is minimized. The structure is then formed in a subsequent processing step, for example heating. This is established by using a blend of a polymer and a monomer which has low monomer mobility during the formation of the latent image because of its high viscosity or because of its glassy state. Diffusion and polymerization are then largely inhibited. The exposed areas contain a high concentration of reactive particles, such as free radicals, that are confined to a very localized space. Only when being heated the mobility is greatly increased and diffusion and polymerization are enabled. This then very rapidly deforms the surface to its desired shape, controlled by diffusion parameters and free surface energy. The process, which we will refer to further as photoembossing, is shown schematically in Figure 19.1. The blends may also contain a thermal initiator that initiates polymerization in the dark area simultaneously or later when heated at higher temperatures, thus fixing the surface structure. An additional, but very important, advantage is that contact masking can be applied because the film is solid and does not stick and thus cause deterioration in the mask and the film. An interesting feature of this process is that by the use of multiple lithographic exposures before the thermal development of the structures, complex surface profiles can be generated. For this multiple exposure technique the fact that the surface profile is not formed before the thermal treatment is optimally beneficial, since the second masking step can also be performed on a solid, flat surface, enabling easy mask alignment. In the rest of the chapter we will discuss structure formation during single and multiple lithographic or holographic steps, and show some examples of photoembossed structures. A newly developed method to enhance the aspect ratio of the relief structures will also be shown.

19.2

Materials and their Photoresponsive Behavior

A typical formulation for the production of photoembossed structures contains a polymer base material, a polyfunctional monomer, a photoinitiator, and, optionally, a thermal initiator. For the coating process, solvents are added. The blend is composed so that after evaporation of the solvent a solid, tack-free film is formed, al-

999

1000


solid reactive blend spun from solution substrate, e.g. glass or plastic

solvents evaporated at T < 80oC (in dark)

UV radiation at RT

photomask

free radicals immobilized by polymer two-step heating at T = 80oC and 130oC solid polymer film with fixed surface

Fig. 19.1.

Schematic representation of the photoembossing process.

lowing contact mask exposure. An example of a blend consists of poly(benzyl methacrylate) as the base polymer and di-pentaerythritol tetraacrylate as the monomer. The polymer/monomer ratio is typically 1:1. Poly(benzyl methacrylate) is a solid polymer and has a glass transition of around 54 C. It blends easily and homogeneously with the monomer. Polyfunctional acrylates like di-pentaerythritol tetraacrylate are viscous liquids and basically act as plasticizers for the polymer. If the monomer concentration is too high, the blend becomes tacky. Typically, thin films are formed by spin coating or by a doctor blade and the solvents are removed by heating for 10 min at a temperature well below the decomposition temperature of the initiators: in the case described, this is 80 C. Figure 19.2 shows the differential scanning calorimetry (DSC) traces of a sample after application as a thin film in the sample holder and drying at 80 C to remove the solvent. From DSC the sample appears stable up to 120 C, above which the thermal initiator activates the thermal polymerization of the acrylate monomer,

19.3 Single-exposure Photoembossing

Exothermic heat flux (mW)

2.5 2 1.5

1 0.5

60 min 10 min 0 min

0 -50

0

50

100

150

200

250

o

Temperature ( C) Fig. 19.2. DSC traces of the reactive photoembossing mixture (pre-heated in the dark at 80 C to remove solvents) measured after 0, 10 and 60 min of pre-exposure with a UV lamp (365 nm, 5 mW cm 2 ).

which reveals itself as the large exothermic peak in the DSC thermogram. When the sample is exposed to UV light before heating, polymerization already starts above 30 C. An isothermal DSC scan for 60 min during the UV exposure at 20 C showed no significant conversion of the acrylate monomer. This indeed shows that monomer mobility at this temperature is very limited, inhibiting further polymerization. Only when heated above 30 C, monomers are gaining diffusional freedom in order to allow propagation of the polymerization. Upon further heating, monomer conversion proceeds up to a temperature of around 80–90 C above which a small decrease is seen, most probably due to a diminishing concentration of free radicals due to termination. At temperatures above 120 C the polymerization propagation rate increases again because of free radical generation from the thermal initiator.

19.3

Single-exposure Photoembossing

The process of latent activation as demonstrated by photo-DSC is also exactly what happens when a thin film of the sample is exposed to UV through a mask. In that case a latent image is formed at the exposed sites consisting of free radicals of the photoinitiator with only minor conversion of the acrylate groups. Only after heating does further polymerization take place. The typical processing sequence has been shown in Figure 19.1. First a thin film is spun on a glass substrate. The film thickness is typically 3 mm after evaporation of the solvent by moderate heating at 80 C without premature polymerization. Cooled down to room temperature, the

1001

1002


film is exposed to 365 nm light from a mercury lamp (365 nm bandpass filter, 2.5 mW cm 2 ) through a mask. After this exposure the surface deformation is present, but is very small in relation to the desired structure. The exposed area contains a high concentration of free radicals, immobilized by the viscous matrix. By heating the film to 80 C, well above the glass transition temperature, monomer mobility is enhanced and the polymerization reaction sets in. This polymerization induces diffusion of the monomer to the exposed areas without noticeable counterdiffusion of polymeric material. This process swells the exposed areas, whereas the unexposed

Before heat treatment

After heat treatment (80 oC) Fig. 19.3. Interferometric microscope picture of a sample surface exposed with a UV dose of 0.2 J cm 2 through a checkerboard mask (feature size 10 mm) before and after heat treatment (5 min, 80 C).


areas are depleted. Figure 19.3 shows interferometric microscope picture of a surface before and after the heating step. Before heating, but after UV exposure, the surface remains relatively unstructured and flat. Directly upon heating, the surface develops itself into the relief that was imposed by the checkerboard mask. Note that in these figures the aspect ratio is heavily exaggerated: in a film of 3 mm high, differences of 1 mm are easily feasible. The structure obtained is fixed permanently either by a flood UV exposure or by heating to 130 C, at which temperature the thermal initiator finishes off the polymerization reaction. The surface profile that is created is determined by a set of parameters. First of all, of course, the mask dimensions are important, but exposure conditions (dose, temperature) and heat treatment conditions have also been proven to matter. Some examples of surface structures obtained by various mask configurations are shown in Figures 19.4 and 19.5. The structure size can be varied between approximately 1 and 20 mm. Below these dimensions the structures become small and the process is hindered by the normal limitations of lithographic processes, such diffraction. At larger dimensions the diffusion lengths of the monomer become too great, resulting in high edges and deeper central parts of the relief (compare, for instance, the two top AFM pictures in Figure 19.4), a phenomenon that is perfectly understood from the models available for this process [35]. The shape can also be affected to some extent by the exposure dose, as can be seen by comparing the two lowest AFM structures in Figure 19.4, in which an overdose of UV light has reshaped the structures that are obtained. The influence of the exposure energy on the height of the structures is demonstrated in Figure 19.6. This figure again demonstrates that surface deformation of the samples takes place only to a minor extent when they are kept at room temperature, even for a prolonged period of 24 h. The minor relief formation under these conditions is practically independent of the UV dose, but during heating, in this case in one step directly to 130 C, the structures develop and the UV dose appears to be of great importance. Too low energy gives only a small effect, for the obvious reason that photoinitiator conversion is low. A maximum in structure height is obtained at UV doses between 100 and 200 mJ cm 2 . At higher doses the structure height decreases again. This is explained by the fact that at doses that are too high (overexposure) some cross-exposure of the intended dark area cannot be avoided. At the same time the exposed area becomes too crosslinked, thus hindering further diffusion in a later stage of the process. Next to the UV dose, the periodicity also seems to have an influence on the feature’s height, as is demonstrated in Figure 19.7. In general, smaller dimensions lead to a lower relief height. This can be understood in terms of monomer deficit at smaller pitches. At greater periodicities, for instance above 20 mm, the structures have some problems filling the whole exposed area, and swelling occurs predominantly near the edges. The diffusion in the reacting medium apparently occurs typically on a length scale of some micrometers. Further optimization outside this preferred region is still possible, and an optimum balance between diffusion and polymerization reaction can be found at other diffusion and curing temperatures. However, comparing the results of the single-step (130 C) baking procedure with

1003

1004


Mask design

AFM of structure after heating step

590 nm 100 µm

100 µm

p = q = 8 µm

50 µm 50 µm 0 µm 0 µm

480 µm 100 µm

100 µm

50 µm 50 µm

p = q = 20 µm

0 µm 0 µm

1260 µm 0 µm 100 µm

100 µm

p = q = 10 µm

50 µm 50 µm 0 µm 0 µm

1020 µm 510 µm 0 µm 100 µm

p = q = 10 µm overexposure

100 µm

50 µm 50 µm

Fig. 19.4. Examples of mask designs and the surface relief that they generate after UV exposure and thermal treatment (10 min, 80 C; 10 min, 130 C).

0 µm 0 µm


Mask design

AFM of structure after heating step

724 µm 0 µm 100 µm

100 µm

50 µm 50 µm

p = q = 10 µm

0 µm 0 µm

102 nm 0 nm 100 µm

100 µm

p = q = 3 µm

50 µm 50 µm 0 µm 0 µm

280 nm 0 nm 100 µm

p = q = 10 µm

100 µm

50 µm 50 µm 0 µm 0 µm

1200 nm 0 nm 100 µm

p = 20 µm q= 5 m

100 µm

50 µm 50 µm 0 µm 0 µm

Fig. 19.5. Examples of mask designs and the surface relief that they generate after UV exposure and thermal treatment (10 min, 80 C; 10 min, 130 C).

1005


1200

Lattice height (nm)

1000 800 600 400 200 0 0.00

0.10

0.20

0.30

0.40

Exposure energy (J/cm2) Fig. 19.6. Surface relief heights after exposure through a line mask with a periodicity of 20 mm and an aperture of 50% after storage in air at RT for 25 h (c) and after a subsequent heating step at 130 C in air for 5 min (k).

1400 1200

Lattice height (nm)

1006

1000 800 600 400 200 0 0.00

0.10

0.20

0.30 –2

Exposure energy (J.cm ) Surface relief heights after exposure through a line mask with periodicities of 5 mm (c), 10 mm (D), and 20 mm (k). The films were exposed at RT and heated to 80 C for 5 min, and subsequently to 130 C for another 5 min. Fig. 19.7.

0.40

19.5 Complex Surface Structures from Interfering UV Laser Beams

the dual-step (80 and 130 C, respectively) reveals that the one-step process is about as effective as the dual-step procedure. This leads to the conclusion that the balance in diffusion and polymerization kinetics is in favor of diffusion.

19.4

Dual-exposure Photoembossing

Figure 19.3 has already shown that the surface of the reactive polymer film remains almost flat before the heating step. This is confirmed in Figure 19.6, which shows that the height of the structures before the heating step, even after prolonged storage at room temperature, remains below 100 nm. The fact that the pristine structures are small means that the optical behavior of the initially flat film is hardly affected by the exposure. This is very relevant for controlling structure shape during the exposure step. A surface that deforms redirects the rays of the masked image, or in other words the surface profiles start acting as a lens. But the fact the surface profile remains almost unaffected is of even greater importance if one wants to employ a second exposure superimposed on the first one. The procedure to perform this double exposure can be as follows. A first exposure though mask 1 is given at a dose somewhat smaller than the dose corresponding to the maximum profile as shown in Figures 19.6 and 19.7. Then mask 1 is removed and mask 2 is brought into place while the film is maintained at room temperature. A second exposure is performed with a dose equal to the dose of the first exposure step. The film now contains the latent images of two masks which will develop during the heat-diffusion step at 80 C or higher. Figure 19.8 shows some examples of the masks that are used and the resulting surface profiles obtained. The structures that were presented individually in Figures 19.3 and 19.4 are now superimposed, one on top of the other. The AFM pictures show that the formation of complex structures is very readily possible and that the second structure is not hindered in its resolution by the presence of the first structure. In the experiments shown, equal doses for the two mask exposure steps are used. Of course, one has the freedom to choose them differently and play with the dominance of each mask step. In the case where UV doses are selected at the maximum sensitivity for a single exposure or higher dose, it must be realized that local overexposure of doubly exposed areas then lead to smaller features as well.

19.5

Complex Surface Structures from Interfering UV Laser Beams

Patterned irradiation can also be obtained by means of holographic exposure. In this case two beams (or possibly three or more) coming from a laser are made to interfere in the area of the sample. In order to do this, a conventional holographic setup is employed. The s-polarized UV beam (351 nm) from an Ar laser is split in two equal-intensity beams. These are made to overlap again, obtaining as a result a

1007

1008


Mask used for 1st exposure

Mask used for 2nd exposure

AFM of resulted structure

212 nm 0 nm 100 µm 100 µm

50 µm 50 µm

p = q = 10 µm

0 µm 0 µm

p = q = 10 µm

148 nm 0 nm 100 µm 100 µm

50 µm 50 µm 0 µm 0 µm

p = q = 10 µm

p = q = 3 µm 204 nm 0 nm 100 µm 100 µm

50 µm 50 µm 0 µm 0 µm

p = q = 3 µm

p = q = 32 µm

150 nm 100 µm 100 µm

50 µm 50 µm

p = q = 3 µm

p = q = 8 µm

Fig. 19.8. Surface relief structures as measured by AFM after two mask exposures of equal UV doses.

0 µm 0 µm

19.5 Complex Surface Structures from Interfering UV Laser Beams

Fig. 19.9.

Schematic representation of the holographic setup.

sinusoidal spatial modulation of the light intensity in the interference region (Figure 19.9). The period of the modulation can be adjusted by changing the angle between the two beams. In Figure 19.10 a sinusoidal surface relief grating is shown with a periodicity of around 1 mm, such as is obtained by exposing the sample to a holographic interference and subsequently heating to 80 C as described above. Multiple holographic exposures can also be performed. As an example, Figure 19.11(a) shows a beat structure obtained by sequential exposures with two holographic gratings with close periods. First a holographic grating with a period of 1.5 mm is recorded. Afterward a second one with a period of 1.62 mm is registered. The subsequent heating step reveals a beat structure with a period of about 20 mm, the same as the beat that is obtained by addition of the two recorded sinusoidal

Fig. 19.10. Surface relief grating after holographic exposure and thermal treatment (10 min, 80 C; 10 min, 130 C).

1009


Relief height (µm)

1010

0.10 0.08 0.06 0.04 0.02 0.00

(a)

0

10

20

Position (µm)

(b) (a) Surface relief beat structure due to sequential double holographic exposure with two grating interference patterns of different periods and equal doses and thermal Fig. 19.11.

treatment (10 min, 80 C; 10 min, 130 C). (b) Schematic representation of the result of the addition of two waves of equal amplitude but different periods, L1 and L2 .

intensity profiles, as shown in Figure 19.11(b). In Figure 19.12 we see the result of exposing a film with two gratings perpendicular to each other, giving a square lattice.

19.6

Surface Structure Development under Fluids

The formation of surface relief structures starting with a flat film involves an increase in surface, which is counteracted by the free surface energy. Consequently

19.6 Surface Structure Development under Fluids

1011

Fig. 19.12. Surface relief structure due to sequential double holographic exposure with two orthogonal grating interference patterns of equal UV doses and thermal treatment (10 min, 80 C; 10 min, 130 C).

2.0

Relief height (µm)

Relief height (µm)

the surface energy is a limiting factor for the generation of these relief structures [28]. A new process has been developed to reduce interfacial interactions, favoring the formation of new surface and achieving in this way enhanced aspect ratios in the photoembossed structures. The use of a contact fluid during the development of the relief structure enhances the aspect ratio of the photoembossed structures by a factor almost 2. The effect of this process is shown in Figure 19.13. Irradiation of the film is performed through a lithographic mask (transparent and dark lines

1.5 1.0 0.5 0.0

2.0 1.5 1.0 0.5 0.0

0

10

20

30

40

Position (µm) Fig. 19.13. Surface relief heights after exposure through a line mask with a periodicity of 10 mm and an aperture of 50% after development: (a) without silicon oil; (b) with silicon oil on top.

0

10

20 Position (µm)

30

40

1012


with a period of 10 mm). After the exposure and before the baking step, half of the sample is covered with a drop of silicon oil, which wets the photosensitive film but does not diffuse into it; the other half of the sample remains uncovered. The baking is then performed at 80 C for 10 min, after which the sample is cooled to room temperature. The silicon oil is removed by rinsing the whole film with heptane. It has been checked by infrared spectroscopy of the film and of the liquids employed that neither the silicon oil nor the heptane swells or dissolves the photosensitive film. The photoembossed structure is fixed by a second heating step at 130 C for another 10 min. Figure 19.13 shows the surface profile of (left) the part cured in air and (right) the part cured with silicon oil on top and subsequently rinsed with heptane. The relief height is noticeably higher (almost double) in the part cured in contact with the fluid. The differences in shape of the developed surfaces are also remarkable. Whereas the surface baked in air shows a rounded top part, as driven by the minimization of the surface area under the action of the surface tension, the one baked in contact with the fluid shows a flatter top part and sharper edges. In addition, the top flat part corresponds quite well to the irradiated region of 5 mm. By using this special process, relief structures that reproduce the features of the mask with improved fidelity have been obtained. The research presented in this paragraph is patent pending (inventors: Dick Broer, Carlos Sańchez, and Cees Bastiaansen; applicant and owner: Dutch Polymer Institute (DPI)).

19.7

Conclusion

The generation of latent images by adjusting the monomer mobility during the UV exposure step and the subsequent development of surface structures by polymerization at higher temperatures provide accurate control over polymer surfaces. The fact that the surface deforms only after the exposure step prevents structure blur by refraction and diffraction defects, and enables multiple exposures, creating complex surfaces by multiple mask steps before heat development, or by combined holographic exposures. Reduction of interfacial interactions makes possible to obtain higher aspect ratios and more sharply defined features in the photoembossed structures.

Acknowledgments

We acknowledge Mrs. G. N. Mol, Mr. H. Nulens, Sasha Alexeev (NT-MDT) and Dr. B. van de Zande for their help, with the DSC measurements, the AFM measurements, and the interferometric microscope images. The research of Carlos Sańchez, Cees Bastiaansen, and Dirk Broer forms part of the research program of the Dutch Polymer Institute (DPI), project DPI#298.

References

Notation

Acronyms AFM CD DPI DSC DUV EB EUV FET LCD LED MEMS SPM

atomic force microscopy compact disk Dutch Polymer Institute differential scanning calorimetry deep UV electron beam extreme UV field-effect transistor liquid crystal display light-emitting diode micro-electromechanical system scanning probe microscope

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Chemical Analysis for Polymer Engineers1 Peter Schoenmakers and Petra Aarnoutse 20.1

Introduction

Polymer reaction engineers have basically two different needs for polymer analysis [1]: they want to follow the polymerization process, and to characterize the resulting polymer. These two types of analysis can be classified as process analysis and polymer analysis, respectively; different objectives are associated with each of them (Table 20.1). Process analysis enables the chemical engineer to monitor the polymerization process. For example, (s)he may want to follow the monomer conversion or the polymer molecular weight during the reaction. Process control can be realized if the results of the process monitoring are translated into specific actions. Polymer analysis concerns the product that results from the polymerization process. For example, the physical and chemical (molecular) properties may be compared with given specifications. The results of such analyses may also be used to tune the polymerization process, either for future (batch) reactions or during a continuous polymerization process. Arguably, the ultimate objective of the polymer reaction engineer is to optimize the polymerization process, using either type of measurement. The different sets of objectives and requirements for the two types of analysis imply that they are performed in quite different manners. Process analyses are preferably carried out on-line (in the reactor) or at-line (in the immediate vicinity of the reactor). Polymer analyses are typically performed in the laboratory. It is also typical for different analytical techniques to be used (Figure 20.1). These will be discussed in somewhat greater detail in subsequent sections of this chapter. We will first discuss the process analysis techniques (bottom left in the figure) in Section 20.2. The polymer analysis techniques will be discussed in Section 20.3. It should be noted that Figure 20.1 deals with chemical analysis – as does the remainder of this chapter. Physical measurements of, for example, the melt viscosity of a polymeric product are very common (at-line) process monitoring tools, but they are beyond the scope of the present chapter. 1) The abbreviations 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

20 Chemical Analysis for Polymer Engineers Tab. 20.1.

Objectives of the two types of chemical analysis for polymer reaction engineering.

Engineering objectives

R&D objectives

Process analysis

Polymer analysis

process monitoring process control process optimization polymerization mechanism reaction kinetics fast response high precision (repeatability) representative data

product characterization product control process optimization reaction mechanisms structure–property relationships quality and amount (detail) of information (often) high accuracy (correct information)

Distributions

Analytical objectives

Multidimensional and Hyphenated systems Polymer separations

Detailed molecular information Basic information (e.g. conversion)

1016

NMR spectroscopy Mass spectrometry FT-IR spectroscopy Raman spectroscopy Near-IR spectroscopy

On-line

Monomer concentration (GC or LC) Gravimetry, Titration

At-line

The main analytical techniques used in polymer reaction engineering. A distinction can be made between on-line (process) and off-line (product) analysis (left to right in the Fig. 20.1.

Laboratory

figure). On-line techniques typically provide limited information, whereas more detailed characterizations can be performed in the laboratory (bottom to top in the figure).

20.2 Process Analysis

Moreover, this is not meant to be an exhaustive treatment of all the possible methods of chemical analysis. For example, a very exciting possibility for the direct monitoring of free-radical polymerization reactions in a research environment is electron spin resonance (ESR) spectroscopy [2]. This allows a direct measurement of the concentration of radicals in the reaction mixture. In many cases catalysts are used in polymerization reactions, and determination of the fate of the catalyst during the reaction and in the product polymer may require specific analytical methods. For example, a variety of techniques exist for determining residual traces of metal catalysts in polymers. These include X-ray fluorescence characterization for solid polymers, and atomic absorption (AAS) and emission (AES) spectrometry for polymer solutions. The treatment in this chapter stops at the primary product of the polymerization reaction. Detailed studies on the structure and morphology of the ultimate polymeric product are assumed to be the domain of polymer technologists, rather than polymer reaction engineers. Methods for analyzing additives are also beyond the scope of the present chapter.

20.2

Process Analysis

Analyses during a polymerization reaction can be carried out with the aim of understanding the process (for example, the polymerization mechanism and the kinetics) or with the aim of process control. The former concerns research measurements, for which the amount of information obtained and the accuracy of the data are the prevailing criteria. The latter concerns true process analysis and the response time is a prevailing criterion. In principle, all kinds of spectroscopic techniques lend themselves to on-line measurements. Only a very few are practical. Although low-field NMR has been used to measure various material properties by applying empirical relationships, NMR is still not a realistic proposition for on-line measurements. Ironically, FTIR spectroscopy suffers from too much sensitivity. Typically, good spectra can be obtained only from very thin polymeric films (or solutions). Attenuated total reflection (ATR) probes, in which only a fraction of the IR light penetrates a very short distance into the sample, reduce the problem of excessive sensitivity. However, they aggravate the problems of variations in the baseline and nonlinear response. The latter problem also obstructs the use of UV spectrometry for monitoring polymerization reactions. Of the remaining options, near-infrared (NIR) and Raman spectroscopy are the most attractive. 20.2.1

Near-infrared Spectroscopy

Near-infrared (NIR) spectra are not very informative. In comparison with conventional (mid-)IR spectra, they are much less useful for identifying unknown com-

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1018

20 Chemical Analysis for Polymer Engineers

pounds and for diagnosing minor variations in the sample. NIR spectra contain fewer bands and much less resolution. Also, the absorption coefficients (sensitivity) are much lower in the NIR wavelength range (typically 800–1200 nm). In contemporary applications of NIR spectroscopy, the lower sensitivity is turned into an advantage: it makes it much easier to deal with the presence of strong IR absorbers, such as water, and it helps to keep the response strictly linear. Also, the spectra of individual compounds of a mixture turn out to be perfectly additive in NIR spectroscopy. Finally, unlike mid-IR spectra, NIR spectra typically show plenty of baseline, which is essential for quantitative analysis. Thus NIR spectroscopy is perfect for on-line analysis in almost all respects. The lack of resolution and of clearly defined bands for specific components or functionalities remains a distinct disadvantage. The combination of (near-)perfect linearity and spectral additivity and limited specificity and resolution makes NIR a perfect match for multivariate calibration methods or ‘‘chemometrics’’. In this case calibration is not based on a single wavelength (band) per component or functional group (one variable, thus univariate), but on many wavelengths or entire spectra (many variables, thus multivariate). The resulting spectrum is thought to be a linear combination of the individual spectra. In most cases, the spectrum corresponding to the desired property is not known. For example, the number of hydroxyl groups in a polymer can be characterized using NIR and multivariate calibration [3]. This implies that the NIR analysis must be calibrated using a training set of samples with known concentrations. The latter could be a set of (commercially) available reference materials or, more commonly, a set of samples that are analyzed in parallel using an absolute (‘‘primary’’) method. The latter must be reliable and validated. Building good calibration models is a critical step in the process. If the sample exceeds the boundaries of the training set, the NIR results are highly unreliable and may be grossly incorrect. In case of systematic deviations from the training set (for example, when a different catalyst is used in the polymerization process), recalibration is necessary. The advantages of NIR in combination with multivariate calibration (rapid on-line measurements) must be balanced against the disadvantages of indirect calibration. NIR has been used to monitor the polymerization of acrylic acid [4], for the solution polymerization of methyl methacrylate (MMA) in toluene [5], for following the emulsion polymerization of MMA and butyl acrylate [6], and for monitoring the copolymerization of MMA and N,N-dimethylacrylamide (Figure 20.2) [33]. The density of linear low-density polyethylene was monitored using NIR and a partial least-squares (PLS) calibration model [7]. 20.2.2

In-situ Raman Spectroscopy

Raman spectroscopy is based not on absorption of light, but on inelastic scattering. The wavelength of the scattered light is slightly changed because of its interaction


Absorbance [A.U.]

1.6

1.2

0.8

0.4 Offset Correction 0

6000

5000

7000

Wave number [cm–1] Fig. 20.2. Use of NIR spectroscopy in-line reaction monitoring of copolymerization of methyl methacrylate and N,Ndimethylacrylamide [33]. Spectrometer:

BOMEM MB 155; the probe was connected to the spectrometer using optical fibers and a BOMEM optical interface. The figure shows calibration samples.

with molecules. The information obtained from Raman spectroscopy is in some ways similar to that from IR spectroscopy (wavelength shifts in the micrometer range) and in some ways very different (different selection rules). For example, water is an infamous interference in mid-IR spectroscopy, but it is not detectable by Raman spectroscopy. Raman spectroscopy shares some of the advantages of NIR spectroscopy. It combines a relatively low sensitivity with a good linearity, rendering it compatible with multivariate calibration. An advantage of Raman in comparison with NIR is the much greater spectral resolution (narrower spectral bands). The greatest disadvantage of Raman spectroscopy in the analysis of polymers and the monitoring of polymer reactions is the occurrence of fluorescence, which manifests itself as disturbances in the spectral background. Raman spectroscopy is a relatively insensitive technique and because in light-scattering measurements (other than absorption measurements) the sensitivity increases with the intensity of the light source, lasers are commonly used in Raman spectroscopy. This also implies that the incident wavelength is very well defined. In general, fluorescence effects are strongest at short wavelengths. Traditionally, Raman spectroscopy is performed with lasers in the visible range as primary light sources. In FT-Raman nearIR lasers are employed, which give rise to much less fluorescence (Figure 20.3). An attractive compromise is the use of a 785 nm laser, which reduces fluorescence while offering greater sensitivity than NIR lasers.

1019

20 Chemical Analysis for Polymer Engineers -3

4.5

x 10

4

Arbitrary Raman Intensity

1020

styrene 1,3-butadiene

3.5 3 2.5 2

poly(butadiene) 1.5 1 0.5 0 1610

1620

1630

1640

1650

1660

1670

Wavenumber

(cm-1)

1680

1690

1700

FT-Raman spectra recorded during copolymerization of 1,3-butadiene and styrene [34]. Spectrometer: Bruker IFS66 FTIR/FT-Raman spectrometer with Nd:YAG laser (1.5 W, 1064 nm), N2 -cooled Ge detector. Early stages of the reaction (only polybutadiene homopolymer formed). Fig. 20.3.

20.2.3

At-line Conversion Measurements

Although chromatography is an incredibly versatile (set of ) technique(s) for performing analytical separations and quantitative analyses, the typical nature of a polymerization mixture, containing both high molecular weight polymers and low molecular weight monomers, causes serious complications. Polymer-containing samples cannot be injected easily into simple gas chromatographs, because the injector or, worse, the column will be seriously contaminated. Although several elegant solutions exist (see Section 20.3.1.1), these cannot easily be implemented and applied in at-line situations. Liquid chromatography systems can deal more easily with the combination of large and small molecules. Size exclusion chromatography (SEC; also known as gel permeation chromatography, GPC) is especially suited for the analysis of polymers, which are separated on the basis of their size in solution. Therefore, SEC is


Fig. 20.4. Use of SEC for conversion measurements. Chromatograms of two polystyrene (Acros Organics, Mw ¼ 1:4 10 5 g mol1 , PDI ¼ 2:6) calibration samples (conversion x ¼ 0:027 and x ¼ 0:401). Reprinted from Ref. [8].

often used for measuring molecular weight distributions (see Section 20.3.3.1). The mechanism underlying SEC separations concerns the partial exclusion of larger molecules from pores of the packing material. By inserting a column with very small pores (an ‘‘oligomer column’’) in the separation system, it is possible to create sufficient selectivity between the monomers and the smallest oligomers present in the reaction mixture. If suitable detectors (sufficiently sensitive and, ideally, linear) can be found for both the polymer and the monomer, then the conversion can be elegantly measured by on-line SEC. The polymerization of styrene is a good example (Figure 20.4) [8]. In this case, UV detection can be used conveniently. There has been a recent trend toward performing fast SEC separations [9], inspired by the developments in combinatorial (polymer) chemistry and the associated high-throughput experimentation. Fast SEC allows molecular weight distributions to be measured in one or two minutes. From both a practical and a theoretical point of view, the possibilities for fast SEC are much greater than was commonly believed until very recently. Fast SEC offers less resolution than conventional SEC. It only offers a rough picture of the MWD and it can only be applied to broadly distributed polymers. Yet the developments in fast SEC make it a feasible technique for determining molecular weight distributions at-line.

1021

1022


20.3

Polymer Analysis 20.3.1

Basic Laboratory Measurements Conversion In the laboratory, residual monomer concentrations are typically measured by some kind of chromatographic analysis; whether this is gas chromatography (GC) or liquid chromatography (LC) depends largely on the type(s) of monomers used in the polymerization process. Monomers that are gases or highly volatile liquids (for example, alkenes, vinyl chloride, acrylonitrile) or UV-inactive (for example, acrylates) are not readily compatible with LC. On the other hand, nonvolatile monomers (for example, diacids, diamines, bisphenol A) are not amenable to GC. When GC is being used, possible contamination of the system (injector and column) with nonvolatile polymeric material is the main issue. One option is to allow direct introduction of a polymer-containing sample (solution), but to confine the polymer to a removable part (‘‘liner’’) of the injection system. On some injection systems (specifically programmable-temperature vaporizers, or PTVs), a series of injections may be performed before the quantitative analysis of monomer is jeopardized and the liner needs to be replaced. Such an approach may be acceptable if 20.3.1.1

Fig. 20.5. Individual conversion profiles of methyl methacrylate (s) and vinylidene chloride (c) in the batch solution copolymerization of methyl methacrylate and

vinylidene chloride in dimethylformamide, estimated by GC measurements of the vapor composition of the reactor head space [35].

20.3 Polymer Analysis

the total number of analyses that need to be performed is fairly small. Another, much more sophisticated, option is the on-line coupling of an LC separation of polymer and monomer (for example, using size exclusion chromatography; see Section 20.2.3) with a GC analysis [10]. A fully automated system may be built that allows large numbers of samples to be analyzed. However, LC-GC systems are complicated to develop and to use in practice. High-level expertise is required if one is to opt for such a solution. An easier solution, that also allows complete automation, is head-space GC. An example is shown in Figure 20.5. This technique is based on measuring volatile components from the vapor phase above a nonvolatile matrix. In this case a polymer-containing sample in a vial is placed in the system. The sample may be a solid or a (viscous) liquid. In this case, the presence of a large amount of volatile solvent may be a disadvantage. Water, especially, may cause insurmountable problems. In head-space GC the vapor phase may be analyzed after equilibrium with the sample has been established (static head-space), or the sample may be purged continuously and all volatile materials may be trapped at the column inlet, to be released from there at the beginning of the analysis (‘‘purge and trap’’ or dynamic head-space). Head-space analysis (especially in the static mode) requires careful calibration and validation, but once it is in operation fully automatic analyses can be performed with relative ease. 20.3.2

Detailed Molecular Analysis FTIR Spectroscopy Infrared spectroscopy has long been one of the stalwarts of polymer analysis. One reason is that polymers and IR spectroscopy are highly compatible. As a case in point, thin films of polystyrene have been used for many years as a reference sample for calibrating the wavelength scale of conventional IR spectrometers. In most laboratories, conventional (‘‘dispersive’’) IR spectrometers have been replaced by contemporary Fourier transform instruments. The latter typically yield a higher resolution, a higher sensitivity, and a greater wavelength accuracy. The advent of FTIR has spurred on the development of highly sensitive measurements on small samples (for example, FTIR microscopy) and of very fast measurements. The range of information that can be obtained from FTIR measurements is summarized in Table 20.2. FTIR is still often used in polymer analysis, because the measurements are easy and fast and because many different types of polymeric samples can be measured with a minimum of sample preparation. However, FTIR spectroscopy also has its weak points. The resolution in the spectra is usually not limited by the quality of the spectrometer, but by the properties of the sample. In samples of polymers many bands usually overlap and, while band assignments are usually possible, the quantification of overlapping bands is difficult and prone to errors. One significant source of error is variation in the baseline of FTIR spectra. Such variations can be attributed to the fact that in FTIR spectrometers the baseline and the sample are not measured simultaneously. Also, spectral artefacts, such 20.3.2.1

1023

1024

20 Chemical Analysis for Polymer Engineers Tab. 20.2.

Information that may be obtained from FTIR spectroscopy.

Type of information

Comments

Chemical composition

clear evidence on presence or absence of specific functional groups rapid confirmation of structure chemical composition of copolymers, but quantification can be difficult useful only for low MW polymers direct measurements on polymer films thicker objects though ATR or specular reflection (surface characterization) easy characterization of powders (DRIFT) IR microscopy for small spots imaging techniques are emerging

End groups Characterization of polymers in many different forms

as scattering by the sample, may result in changes in the baseline. Other problems are possible nonlinearity in absorbance (or reflectance) versus concentration curves and the large differences in sensitivity for different absorption bands. In fact, nearIR spectroscopy is often preferred for routine quantitative measurements, because fewer problems are encountered with the linearity and with establishing the baseline. NMR Spectroscopy Nuclear magnetic resonance spectroscopy probes the spins of specific atomic nuclei in a strong magnetic field. The spin can be changed at very specific (resonance) frequencies, which depend on the chemical environment of the nucleus. Thus, NMR spectra provide very detailed chemical information on the structure of molecules (Table 20.3). The differences between the spin energy levels (with and against the magnetic field) are very small. This implies that the spins are almost evenly distributed across the different levels (Boltzmann’s law). Therefore, NMR is an insensitive technique. The difference between the energy levels, and thus the difference between the populations of the different levels, and hence the sensitivity of the NMR measurements, can be increased by increasing the field strength. Therefore there is a continuous trend toward ever-higher field strengths (and evercostlier instruments) in NMR spectroscopy. This is especially the case since (in solution NMR) the resolution also increases with increasing field strength. The two most common nuclei that are studied in NMR spectroscopy are protons ( 1 H) and carbon ( 13 C). The former has a natural abundance of almost 100%, but 13 C represents only about 1% of the carbon population. If only because 12 C cannot be observed by NMR spectroscopy, the sensitivity of ‘‘carbon-NMR’’ is significantly lower than that of ‘‘proton-NMR’’. NMR spectroscopy yields quantitative data on the number of protons or (with some precautions) carbon atoms that have a certain chemical environment. This can be directly translated into chemical composition data (see Figure 20.6). For ex20.3.2.2

20.3 Polymer Analysis Tab. 20.3.

Information that may be obtained from NMR spectroscopy.

Type of information

Comments

Chemical composition

highly detailed information on chemical structure and composition accurate quantitative data (% of specific protons; % of specific carbon atoms) at levels above about 1% good, quantitative information on average chain regularity (tacticity) information on branching (at high branching levels) easy distinction between random and block copolymers; some information on average block length information on network mobility some information on polymer morphology solution NMR gives the best resolution solid-state NMR (powders, particles, or small objects) routinely possible polymer melts studied occasionally

Molecular architecture

Physical structure Polymers in various forms

ample, the numbers of aromatic and aliphatic protons in a copolymer of styrene and isobutene unambiguously define the chemical composition. Much more detailed information is revealed by NMR spectroscopy about the specific structure of the polymer. Many different measurement techniques have been developed that reveal intricate details about the molecular structure [11], but these are beyond the scope of the current chapter. NMR spectroscopy can be used to study polymers in solution (solution NMR) or as such (solid-state NMR). In the latter case, the resolution is typically much lower, mainly because of sample anisotropy. To improve the resolution in solid-state NMR the sample can be spun at a high frequency (in the kilohertz range) and at a specific (‘‘magic’’) angle (54 ) relative to the magnetic field. The sensitivity can be increased by transferring magnetization from 1 H to 13 C nuclei. The resulting technique is referred to as cross-polarization magic-angle spinning NMR (CP-MASNMR). Mass Spectrometry Whereas (FT)IR and NMR spectroscopy have been essential tools for polymer analysis for decades, mass spectrometry (MS) only became truly relevant in the later 1990s, and the number of mass spectrometers in polymer analysis laboratories is now increasing rapidly. Our current ability to characterize (very) large molecules successfully can be seen as a small miracle. In many cases, we are now able to ionize polymer molecules and to transport large molecular ions through magnetic or electric fields with great integrity. The possibilities of contemporary MS are fantastic. However, one must be aware that there are still severe limitations. The possibilities and limitations of polymer MS will be briefly reviewed in this section. MS has been used in polymer analysis for many years, but these techniques 20.3.2.3

1025

1026


Fig. 20.6. 188 MHz 13 C NMR spectra of poly(n-butyl acrylateco-carbon monoxide-co-ethylene) samples with different monomer compositions (indicated in the figure) [36].

have relied on destruction of the polymer into smaller pieces or fragments. Pyrolysis-GC-MS is a conventional method for polymer analysis and characterization. The pyrolysis stage yields small, uncharged fragments of the polymer, such as residual monomers, which can be separated by gas chromatography (GC) and identified by MS. Pyrolysis-GC-MS yields useful information on the composition of (co)polymers, the presence of additives, and suchlike. However, the results are difficult to quantify. Similarly, in secondary-ion mass spectrometry (SIMS), the polymer is first fragmented using a beam of primary ions. The resulting fragments are charged and these fragments are subsequently analyzed by MS. SIMS can yield very useful (organic chemical) information on polymer surfaces (static SIMS), or it can yield depth profiles of materials (dynamic SIMS). In these conventional applications of MS in polymer engineering, only information on molecular fragments is obtained, but none on the intact polymer molecules.


The main breakthrough in polymer MS has been the development of soft ionization methods. This implies that molecular ions are obtained and characterized. During the formation of molecular ions, small ions (such as Hþ ) may be abstracted from the molecule, or adducts may be formed with small ions (for example, Naþ ), but no fragmentation occurs and the resulting ion is highly representative of the original polymeric molecule. Two important soft ionization techniques have emerged in recent years, namely electrospray ionization (ESI) and matrix-assisted laser desorption ionization (MALDI). Electrospray ionization (ESI) ESI requires a polymer solution to be pumped into the mass spectrometer, either as such or, conveniently, after a liquidchromatographic separation. In the latter case, a narrow, pre-separated fraction of the polymer is introduced into the mass specrometer, which greatly enhances the chances of obtaining useful mass spectra. The liquid is sprayed into the ionization chamber under the simultaneous action of an electric field of several kilovolts. The solvent should have a significant polarity and some ionic additives (salt or buffer) are typically present. It is not strictly necessary for the polymer itself to be dissolved in this polar solvent. Good results have been obtained by adding a separate (immiscible) stream of solvent to the polymer solution through a T-piece just before the ESI interface [12]. The exact mechanism by which charged molecular ions are ultimately obtained from the polymer solution has not yet been elucidated unequivocally. The spray is formed at atmospheric pressure, but then passed into the vacuum part of the ionization chamber, where the volatile components evaporate. This may result in a concentration of (nonvolatile) ions in very small droplets, which then become unstable and explode to yield a series of individual ions, including the charged polymers. An alternative mechanism, which seems to be supported by the fact that the polar, salt-containing solvent does not need to be miscible with the polymer solution, entails the complete evaporation of all solvents, followed by ionization of the polymer molecules in the gas phase. In any case, molecular ions are formed that contain one (‘‘singly charged ions’’) or more charges. The fact that multiply charged ions are being formed makes it easier to analyze molecular ions of high molecular weight, because the behavior of ions in mass spectrometers is determined by the mass-to-charge ratio (m=z) rather than by the absolute mass. However, the presence of multiply charged ions may significantly complicate the resulting mass spectra and their interpretation. Figure 20.7 shows an example of the application of ESI-MS for polymer analysis. Table 20.4 summarizes the information that can be obtained using this technique. Matrix-assisted laser desorption ionization (MALDI) In MALDI, the polymer is mixed intensively (usually from a solution) with a matrix, and possibly with other additives such as a salt. It is then deposited as a small spot on a target (MALDI plate). The deposited mixture is irradiated by a laser pulse. The matrix is selected such that it strongly adsorbs the laser light, heats up very rapidly, and induces the transfer of a single charge (usually an adduct ion) to the polymer molecule. The

1027


Extracted masses for every scan

3000

2500

2000 mass value

1028

1500

1000

500

0 0

20

40

60

Fig. 20.7. Characterization of perfluoropolyethers by on-line LC-ESI-MS [12]. LC conditions: Column: Nucleosil silica (5 mm), 150 mm 2.1 mm i.d.; mobile phase: 0.1–20% methyl-t-butyl ether in 1,1,2trichlorotrifluoroethane in 15 min; mass spectrometry: Micromass time-of-flight MS

Tab. 20.4.

80 scan #

100

120

140

operated in negative-ion mode; capillary voltage: 3 kV; desolvation temperature: 350 C; source temperature: 120 C; cone-gas flow rate: 30 L h1 ; desolvation gas flow rate: 350 L h1 . ESI coniditions: 20 mL min1 50:50 isopropanol/water added post-column; cone voltage: 50 V.

Information that may be obtained from ESI-MS.

Type of information

Comments

Absolute mass of molecular ions

limited to polar polymers with molecular weight below ca. 25 kDa structural formulas may be obtained in the case of highresolution MS only for (homo)polymers with a very narrow distribution in terms of molecular weight, chemical composition, and type of functionality prior separations (e.g., SEC, LC) allow analysis of more broadly distributed polymers by ESI-MS can be obtained from series of MS peaks (one of more different end groups) may be complicated by adduct ions may be facilitated by high-resolution MS

Molecular-weight distribution

Combined mass of end groups (homopolymers)


resulting spectra are quite variable in terms of the observed ions and their intensities, but representative spectra can be obtained by summing or averaging the spectra resulting from a large number of pulses. Only singly charged ions are observed and there is usually little or no fragmentation, although MALDI is often found to be slightly ‘‘less soft’’ (that is, more likely to induce fragmentation) than ESI [13]. Polymers that are amenable to MALDI must have some degree of polarity – polyethene and polypropene are still essentially incompatible with the technique. Another essential requirement is that the polymers be narrowly distributed in terms of molecular weight, chemical composition, and functionality. If the molecular weight distribution is not very narrow, then the smallest molecules will be over-represented in the resulting mass spectrum (‘‘discrimination’’). If one type of molecule is abundant, it may dominate the entire mass spectrum, completely suppressing the ionization of polymeric components present in lower concentrations (‘‘ion suppression’’). Selective ionization may also be observed in the case of variations in functional groups or end groups. MALDI cannot easily be coupled on-line to separation techniques such as liquid chromatography (LC), but it can be elegantly coupled off-line. Because of the high sensitivity of MALDI, one drop of LC effluent may be sufficient to prepare excellent MALDI spots. Table 20.5 summarizes the information that can be obtained using this technique. Types of mass analyzers There are quite a few different types of mass analyzers. Conventional high-resolution MS exploits magnetic-sector instruments. Quadrupole instruments have become the affordable standard for the analysis of fairly small components (up to about 1000 Da), although ion-trap systems have some specific advantages, such as the possibility of performing high-sensitivity tan-

Tab. 20.5.

Information that may be obtained from MALDI-MS.

Type of information

Comments

Absolute mass of molecular ions

limited to polymers with a certain minimum polarity, up to quite high molecular weights, provided that the sample is very narrowly distributed only for (homo)polymers with a very narrow distribution in terms of molecular weight, chemical composition, and type of functionality prior separations (e.g., SEC, LC; usually carried out off-line) allow analysis of more broadly distributed polymers by MALDI-MS can be obtained from series of MS peaks (one of more different end groups) may be complicated by adduct ions or fragmentation may be facilitated by high-resolution MS

Molecular weight distribution

Combined mass of end groups (homopolymers)

1029

1030


dem MS (MS-MS) or multiple MS (MS n ) experiments with relative ease. The big brother of the ion-trap instrument, the Fourier transform ion–cyclotron resonance (FT-ICR-MS, or simply FT-MS) spectrometer, offers extremely high resolution. It has been gaining popularity, mainly thanks to the great advances made in protein MS since the mid-1990s. However, use of FT-MS instruments is still the prerogative of the elite few. In contrast, another type of mass analyzer that was quite uncommon until recently is now proliferating rapidly in the field of polymer MS. Time-of-flight (ToF) analyzers are based on the principle of measuring the time a specific ion requires to travel a given length (in vacuum). The flight time is directly related to the m=z value. The mass range of ToF analyzers is not fundamentally limited. Thus, polymers of very high molecular weight can be analyzed after soft ionization. ToF analyzers are now the preferred type of instrument for the characterization of (natural and synthetic) macromolecules, in combination both with ESI and with MALDI. ToF combines a high sensitivity (due to a favorable duty cycle) with a broad mass range and a high spectral resolution and accuracy. High-resolution versions of ToF analyzers exploit the so-called reflectron mode, in which the length of the flight path is doubled. In addition, ToF-MS instruments have become much more accessible and much more affordable in recent years. An example of the MALDI-ToFMS technique is shown in Figure 20.8. 20.3.3

Polymer Distributions

There are many ways to measure the average properties of a polymeric sample. These will not be discussed extensively in this chapter. Among the techniques used to measure average molecular weights are those that yield number averages ðMn Þ, such as end-group titrations and end-group analysis by NMR. Such techniques work best at relatively low molecular weights. Static light scattering yields a direct estimate of the weight-average molecular weight ðMw Þ. It works best for relatively high molecular weights (above ca. 20 kDa). Measurement of the intrinsic viscosity may be used to obtain the viscosity-average molecular weight ðMv Þ. If the entire molecular weight distribution is measured, then all the desired averages and the sample polydispersity (PDI ¼ Mw =Mn ) can be readily calculated. Molecular Weight Distributions The most important molecular distribution is the molecular weight distribution (MWD) or, equivalently, the molar mass distribution (MMD). Size exclusion chromatography (SEC; also known as gel permeation chromatography, GPC) is the outstanding technique for measuring the MWD. In SEC, the retention time is related to the molecular weight by constructing a suitable calibration curve (Figure 20.9) based on the retention times (or volumes) of a set of narrowly distributed standards of known molecular weight. SEC separates on the basis of the size of molecules in solution (the hydrodynamic volume), rather than on the molecular weight. Thus, to measure accurate (in SEC terminology, ‘‘absolute’’) molecular weights, it 20.3.3.1

20.3 Polymer Analysis 100

Intensity (%)

104.1

104.1

788.0

1216.8

1645.6

2074.4

2503.2

2932.0

2402

2945

3488

Mass (amu)

Intensity (%)

100

104.1 104.1 104.1

773

1316

1859

Mass (amu) MALDI-TOF-MS spectra of polystyrenes, reflective positive-ion mode, DCTB as matrix and Agþ as a cationization agent. Top: PS with two different end groups or Fig. 20.8.

adduct ions. The two different series of peaks differ by 104.1 amu, the mass of one repeat unit styrene. Bottom: PS with four different end groups [37].

is necessary that the calibration standards be chemically identical to the sample polymer. Polystyrene standards should be used to construct a calibration curve for polystyrene samples, PMMA standards for PMMA samples, and so on. Unfortunately, suitable standards are not available for all polymers. For a number of homopolymers suitable standards are commercially available. However, for copolymers it is notoriously difficult to determine absolute molecular weights by SEC. Even if copolymer standards were available, there are many different molecules (different combinations of molecular weight and chemical composition) that exhibit the same hydrodynamic volume. Also, the size of polymer molecules in solution is affected significantly by other properties, for example the degree of branching. For these reasons, SEC is often used to measure molecular weight distributions relative to different standards, such as polystyrene. Such relative measurements may be perfectly adequate for many purposes. For example, for monitoring polymerization reactions and product specifications the precision (repeatability) of the data tends to be much more important than their accuracy.

1031


10M

Polystyrene Molecular Weight

1032

1M 106Å 105Å

100K 104Å 10K

103Å 500Å 100Å

1K

50Å

100 4

Elution volume (ml)

10

SEC calibration curves for a series of columns with different pore sizes (Polymer Laboratories, Church, Stretton, Shropshire, UK; www.polymerlabs.com/gpc). Different columns show good selectivity (shallow curves) across different molecular weight ranges. Fig. 20.9.

Detectors for size exclusion chromatography The most common detectors used in SEC are the UV absorbance detector (for polymer molecules that possess chromophores) and the (differential) refractive index detector (DRI or RI) for polymers that do not (see Table 20.6). The latter is less convenient to use in practice, and less sensitive. Therefore, the evaporative light-scattering detector (ELSD) is used increasingly for characterizing non-UV-active polymers at low concentrations (for example, polymeric contaminants or polymers separated by comprehensive twodimensional liquid chromatography). A serious disadvantage of the latter detector is that its response increases exponentially with increasing polymer concentration and is, therefore, highly nonlinear. The response of the ELSD also tends to vary more strongly with the molecular weight of the sample polymer than those of the UV and DRI detectors. A very important option is the combination of SEC with techniques that allow direct measurement of molecular weight, such as viscometry or light scattering (Table 20.7). This – in principle – eliminates the need for narrow standards of exactly the same polymer, and it alleviates the requirements on the SEC system, because slight variations in retention times due to variations in the flow rate or interactions with the column can be negotiated. This is true for viscometric detection, provided that accurate Mark–Houwink constants (K and a) are available to trans-

20.3 Polymer Analysis Tab. 20.6.

General (concentration-sensitive) detectors for size exclusion chromatography.

Detector

Abbrev.

Strong points

Weak points

UV absorbance

UV

(Differential) refractive index

(D)RI

sensitive highly linear universal (all polymers)

Evaporative light scattering

ELSD

only applicable for UV-absorbing polymers not very sensitive strong solvent signal sensitive to temperature variations needs reference cell nonlinear response response depends on polymer structure and molecular weight

Tab. 20.7.

universal (all polymers) highly sensitive

Specific (molar mass sensitive) detectors for size exclusion chromatography.

Detector

Abbrev.

Strong points

Weak points

Viscometry

Vis

Light scattering low angle right angle triple angle multi-angle

LALS[a] RALS TrALS MALS

applicable only for M > (ca.) 5000 accuracy relies on accuracy of Mark–Houwink constants or on quality of SEC measurements sensitive to calibration parameters (especially the refractive index increment dn/dc) tedious calibration procedure

Mass spectrometry electrospray ionization[b]

ESI

direct measurement of intrinsic viscosity absolute MW using Mark– Houwink constants or the universal calibration principle direct measurement of Mw (for M > (ca.) 10 000) multi-angle LS provides direct measurement of the root-mean-square radius (for M > (ca.) 50 000) absolute molecular-weight measurement information on end groups

Matrix-assisted laser desorption/ ionization[c]

MALDI

absolute molecular weight measurement information on end groups applicable to narrowly distributed polymers up to very high M

only applicable to polar polymers multiple ionization complicates spectra best for M < (ca.) 20 000 only applicable to moderately or highly polar polymers (very) narrowly distributed samples are required (in terms of MWD, FTD, etc.)

[a] Sometimes called low-angle laser light scattering (LALLS). Analogously, RALLS, TrALLS and MALLS may be found. [b] Electrospray ionization mass spectrometry can be coupled on-line with SEC. However, this does require a miniaturized SEC system (flow rate < 100 mL min1 ) or a flow splitting device after the SEC column. [c] MALDI is usually combined off-line with SEC. Samples from the effluent can be collected manually or by using a fraction collector. Devices to facilitate deposition directly from the SEC system onto a MALDI target plate are commercially available.

1033

1034


late the measured intrinsic viscosity ([h]) for the specific polymer (in the specific solvent used and at the specific temperature of operation) into the molecular weight through M ¼ K½ha . When measuring the light scattered by the polymer solution to obtain the (weight-average) molar mass, the refractive index increment (dn/dc) of the specific polymer in the specific solvent (at the temperature of operation) must be accurately known. Both an on-line viscometer and a light-scattering instrument must be calibrated using known standards. Both viscometry and light scattering require an independent measurement of the concentration of the polymer. For this purpose, one of the concentration-sensitive detectors (most often refractive index detection) is used in combination with the molecular weight selective detector. Using a combination of several detectors is common practice in SEC. A particularly powerful combination is to use refractive index detection (to measure concentration), viscometry (to measure the intrinsic viscosity), and light-scattering detection (to measure molecular weight), all together. A RALS detector is typically used instead of a MALS detector. Therefore, the rootmean-square radius of the polymer in solution cannot be measured directly and the molecular weight of large polymer molecules (M > 100 000) requires an iterative correction procedure. The on-line combination of SEC with electrospray ionization MS (SEC-ESIMS) and the off-line combination of SEC with matrix-assisted laser desorption/ ionization (SEC//MALDI-MS) are relatively recent possibilities. For the time being, SEC-ESI-MS is limited to polar polymers of relatively low molecular weight. In contrast, SEC//MALDI-MS is applicable to almost all polymers (polyethylene and polypropylene being the most notable exceptions), up to very high molecular weights. The main requirement is that the polymer samples subjected to MALDI should be narrowly distributed in terms of molecular weight, chemical composition, and functionality. The major problems of MALDI are discrimination and ion suppression. Discrimination implies that the sensitivity (response) of the MALDIToF-MS system varies with properties (for example, molecular weight, functionality) of the polymer molecules. Ion suppression occurs when a particular type of ion is present in a very high concentration and the sensitivity of less prevalent ions is (dramatically) reduced. Because MALDI requires narrowly distributed samples and because it is a very sensitive analytical technique, its (off-line) combination with chromatographic separations is highly attractive. Functionality-type Distributions Size exclusion chromatography can be used to separate polymer molecules according to their size in solution, and size can be converted to molecular weight by calibration. SEC cannot be used to separate polymers according to chemical composition or functionality. For this purpose, interactive liquid chromatography (i-LC) may be used. In i-LC the molecules of the analyte polymers interact with the mobile phase and with the stationary phase in the column. Usually, thermodynamic equilibrium is reached at any point in the column. The distribution of the polymer molecules across the two phases can then be characterized by a distribution coefficient (K ¼ c s =c m , where c s is the concentration of the polymer in the stationary 20.3.3.2


phase and c m is its concentration in the mobile phase). The retention factor (k) is proportional to the distribution coefficient and to the phase ratio [Eq. (1), where VR is the retention volume, tR the retention time, V0 is the column hold-up volume, t 0 the column hold-up time, Vs the volume of stationary phase in the column, and Vm the volume of mobile phase.] k¼

VR V0 tR t 0 Vs ¼ ¼K V0 t0 Vm

ð1Þ

Retention in i-LC is thus directly related to the thermodynamic distribution coefficient. K can be expressed in terms of fundamental thermodynamic properties, that is, the partial molar free energy associated with the transfer of one mole of analyte from the mobile phase to the stationary phase (Dg), the corresponding partial molar enthalpy, and the corresponding entropy effect (Ds), according to Eq. (2), where R is the gas constant and T the absolute temperature. RT ln K ¼ Dg ¼ Dh þ TDs

ð2Þ

Enthalpic (heat) effects arising from molecular interactions are reflected in Dh. SEC is a strictly entropic process; that is, Dh ¼ 0 and temperature has no significant effect on the elution volume. Separation between chemically different polymers can be achieved if different parts of the molecules (different co-monomers, end groups, functional groups) exhibit different interactions with the mobile phase and the stationary phase in the column. We can rewrite Eq. (2) for a homopolymer as Eq. (3), where we assume that the partial molar free energy is built up from p contributions of monomeric units ( p being the degree of polymerization) and the sum of all contributions from end groups (or functional groups). RT ln K ¼ Dg ¼ pDgmonomer

P

Dgend groups

ð3Þ

Because p is a large number for high Mr polymers, reasonable distribution coefficients (and, therefore, reasonable retention factors) can usually be obtained only if Dgmonomer A 0. A special case is the situation in which Dgmonomer ¼ 0. In this case the distribution coefficients and chromatographic retention factors are determined only by the functional groups and they are independent of the chain length ( p). This situation is known as liquid chromatography at the critical conditions or, simply, as critical chromatography. It is eminently suitable for separating polymers based on functionality. Figure 20.10 shows examples of the separation of functional poly(methyl methacrylate)s [14, 18]. All PMMA molecules without OH groups are eluted with a retention time of around 4 min in Fig. 20.10(a), irrespective of the molecular weight and (possible) other end groups present (provided that the Dg values for all the end groups other than OH are either negligibly small or very similar). The polymers with one OH end group are all eluted at around 5 min and the bifunc-

1035


HO-PMMA-OH (MD-1000X)

ELSD response

PMMA-OH 20,740 PMMA-OH 3,310 PMMA 28,300

PMMA 3,800

3

4

5

6 7 Time (min)

8

9

10 (a)

V37A1

V37A0

ELSD response

1036

V37A V37B1 V37B0

0

1

2

V37B2

3

4

Time (min) Fig. 20.10. (a) Separation of end-functional poly(methyl methacrylate)s based on the number of end groups. Column: 150 mm long 4.6 mm i.d., home-packed with Hypersil silica (3 mm particles, 100 A˚ pore size); mobile phase: 43% acetonitrile in dichloromethane; temperature: 25 C; flow rate: 0.5 mL min1 ;

V37B 5

6

(b) injection volume: 10 mL; sample concentration: 1 mg mL1 in dichloromethane; detector: evaporative light scattering [14]. (b) Separation of end-functional PMMAs prepared by RAFT polymerization [18]. Mobile phase: 40% acetonitrile in dichloromethane; other conditions as in (a).


tional (‘‘telechelic’’) PMMAs are eluted at around 8 min in Fig. 20.10(a). The elution profiles of real samples can be translated into a functionality-type distribution (FTD), provided the detector is suitably calibrated [15–17]. The separation shown in Fig.10(a) has proven to be quite robust. However, this is not usually the case for critical separations. Indeed, for carboxyl-functionalized poly(n-butyl acrylates) it proved much more difficult to achieve genuine critical chromatography [19]. Critical liquid chromatographic separations are not always easy, but they can be highly rewarding, especially for determining FTDs. Mass spectrometry (ESI-MS and MALDI-ToF-MS) is very useful for obtaining information on the total masses of various end group combinations present. In this case Eq. (4) holds, where Mion is the observed mass for the polymeric ion, p is the degree of polymerization, Mmon is the mass of the monomeric unit, Madduct is the mass of the adduct ion (typically added as a salt to the matrix/polymer mixture) P and Mendgroup is the sum of the masses of the end groups. Mion ¼ pMmon þ Madduct þ

P

ð4Þ

Mendgroup

Very accurate and precise values of Mmon and Mendgroup can be obtained by MALDI. Complicating factors are: P uncertainty in the true end group mass, which may equal Mendgroup , but also P Mendgroup þ nMmon (where n is an integer); the (broad) isotope pattern of large polymers, which, if not properly accounted P for, may cause errors in the values obtained for Mendgroup; the possibility that several different end group combinations yield exactly or P nearly the same value for Mendgroup.

It is very difficult to obtain quantitative FTDs by MS. Because the response factors are usually different for polymers with different end groups, highly specific reference materials (of accurately known composition) would be required. In some cases capillary (zone) electrophoresis can be elegantly used to separate polymers according to functionality, and to obtain accurate FTDs. This is especially the case if the end groups are charged (in a suitable buffer solution) or if they can be derivatized to yield ionizable groups. Under suitable conditions, the effect of the molecular charge on the electrophoretic mobility can be much greater than the effect of the molecular weight [20]. Chemical Composition Distributions (CCDs) For a copolymer, Eq. (3) becomes Eq. (5), where the subscript i depicts the different monomeric units. 20.3.3.3

RT ln K ¼ Dg ¼

X i

pi Dgmonomer; i

P

Dg end groups

ð5Þ

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1038


It is difficult to achieve Dgmonomer; i ¼ 0 for one particular monomer (critical chromatography) by carefully adjusting the ratio of two components of the mobile phase (a strong eluent and a weak eluent or ‘‘nonsolvent’’). It is impossible to find conditions that are critical for several different monomers simultaneously. Therefore, critical chromatography is much more useful for determining the FTDs of functional (homo)polymers than for determining the CCDs of copolymers. In the latter case, two options are open. One is to find conditions at which the separation is critical toward one type of monomer, while the second monomeric unit does not show any interaction, so that it is eluted under SEC conditions. Such conditions have been applied to block copolymers. The block for which critical conditions are maintained is made ‘‘invisible’’ and the separation reflects the block length distribution of the second block [22]. There is some discussion in the literature on whether any effect of the ‘‘invisible’’ block on the retention of the polymer molecule can be avoided [21–23]. The alternative (and more common) way to analyze copolymers is to resort to gradient elution. In this case the composition of the mobile phase is changed over time. At the initial composition, both monomers are highly retained (negative Dg). When the eluent becomes stronger, the critical conditions for one of the monomeric units will be approached. At a later point in time, this will be the case for the second monomer. In this way, a blend of polymers can be separated into its constituents. Copolymers will be eluted according to their composition (Figure 20.11). In principle, gradient elution liquid chromatography can be used to obtain CCDs of copolymers. Again, proper calibration of the detector is a significant issue if quantitative data are to be obtained. i-LC separations can also be coupled to spectrometers and other highly informative detection devices. The situation is similar, but not identical, to that described for SEC. In many cases, gradient elution is used: that is, the solvent changes as a function of time. In that case hyphenation between i-LC and viscometry or light scattering is horribly difficult. Gradient elution is not used with such devices. Thus, it is difficult to determine the (average) molar mass as a function of the chemical composition hMr iðjpol Þ. i-LC//MALDI-ToF-MS is a feasible technique; iLC-Vis and i-LC-LS are not realistic options. SEC-FTIR and SEC-NMR can – in principle – be used to obtain the average composition as a function of molar mass hjpol iðMr Þ. In principle it is possible to combine viscometry or light scattering with critical chromatography. However, since the latter technique is more practical for relatively low molar masses and the detection devices are most suitable for high masses, this is not a perfect match. Both LC-FTIR and LC-NMR can be applied in combination with solvent gradients. In both cases there are some complicating factors. In the case of FTIR, only solvent elimination interfaces can realistically be used. This implies that the effluent from the LC is sprayed into a (heated) evaporation chamber and that the nonvolatile analyte polymers are deposited on a suitable substrate (for example, a germanium disc). By moving the spray or the substrate, the entire chromatogram can be recorded. Some authors have programmed the deposition conditions to obtain optimal results for gradient elution LC-FTIR. However, as was mentioned ear-

20.3 Polymer Analysis 18 16 14

Detector signal

12 10 8 6 4 2 0 -2 0

10

20

30

40

50

60

70

80

Time (min) Gradient elution liquid chromatography of a mixture (‘‘blend’’) of a number of copolymers. Column: Supelcosil Discovery C18 , 150 mm long 2.1 mm i.d.; particle size: 5 mm; pore diameter: 180 A˚; temperature: 25 C; flow rate: 0.2 mL min1 ; injection volume: 5 mL; sample concentration: Fig. 20.11.

1.5 mg mL1 ; gradient from 5 to 95% THF in acetonitrile. Peaks from left to right: poly(methyl methacrylate) homopolymer, 20% polystyrene-co-MMA, 40% PS-co-MMA, 60% PS-co-MMA, 80% PS-co-MMA, PS homopolymer.

lier (Section 20.3.2.1), it is not easy to obtain accurate quantitative results for the copolymer composition using FTIR. In LC-NMR a solvent gradient causes severe complications associated with the suppression of the solvent signal. While suppression techniques for gradient elution LC have been developed and applied successfully, the interferences in the spectrum are more serious than they are in isocratic separations. In either case, LC-FTIR or LC-NMR, the amount of additional information obtained is limited. The LC retention axis contains information on the polymer composition. The information present in the spectra is related to this. Although additional information on structural aspects may be obtained from both FTIR and NMR spectra, the two information dimensions are far from orthogonal. This is fundamentally different for the combination of i-LC with MS, either online (most commonly using LC-ESI-MS) or off-line (most commonly using LC// MALDI-ToF-MS). In that case, the LC axis contains mainly structural information, whereas the MS axis provides information on the molar mass. A disadvantage of this combination is that fractions resulting from the i-LC separation are expected to be narrow in terms of their chemical composition distribution, but may be quite broad in terms of their molar mass distribution. In critical or pseudo-critical i-LC the very purpose of the separation is to elute all the different molar masses as one

1039

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peak. Such broad fractions are not really compatible with MS. As mentioned in section 20.3.2.3, biased results are anticipated from the MS analysis of broadly distributed samples. Better results are anticipated if fractions that are narrow both in chemical composition and in molar mass are subjected to mass spectrometry. Such fractions can be obtained from two-dimensional separations (see Section 20.3.3.5). Degree of Branching Distributions When polymer molecules are branched, their size in solution decreases. This is progressively the case with an increasing degree of branching. This also implies that molecules of equal size in solution (and thus equal SEC retention times) can have different molecular weights, depending on the degree of branching. SEC alone is not capable of revealing both the MWD and the degree of branching distribution (DBD), because the two distributions are confounded. Other information is required to yield information on the two parameters simultaneously. This extra information may be obtained from light scattering or from viscosity data, and both of these techniques can be readily combined on-line with SEC, yielding so-called hyphenated systems – that is, SEC-Vis (size-exclusion chromatography with viscometric detection) and SEC-LS (light-scattering detection). Static light scattering is most commonly coupled with SEC, but in recent years the on-line coupling of SEC and dynamic light scattering has become more readily available. The instrumentation for static light scattering is typically distinguished by the type and/or number of measurement angles. Thus, we have low-angle light scattering (LALS) and right-angle light scattering (RALS), as well as multi-angle light scattering (MALS) and triple-angle light scattering (TrALS). Often an additional ‘‘L’’ is added to the abbreviations, signifying the use of a laser source (for example, multi-angle laser light scattering, MALLS). In any case (viscometry or light scattering) a concentration-sensitive detector, typically a refractive index (RI) detector, is also installed on-line. Also quite common is the use of both viscometry and light scattering in combination with refractive index (RI) detection and SEC. Such a system with three detectors is known as a TripleSEC system. 0 are obtained from the ratio of the intrinsic viscosity of Branching parameters gvis the branched polymer ½hbr (measured by viscometry) to that of the corresponding linear reference polymer ½hlin (eluting at the same time) [Eq. (6)], or from the molecular weight of a linear reference polymer (as measured by light scattering) divided by the molecular weight of the branched polymer [Eq. (7), where a is the Mark–Houwink coefficient]. 20.3.3.4

½hbr ½hlin

ð6Þ

Mlin a ¼ Mbr

ð7Þ

0 ¼ gvis

0 gLS

The average number of branches in polymer molecules as a function of the molec-


ular weight can be estimated from Eqs. (6) or (7) and the Zimm–Stockmayer equation [24]. Complex Polymers (Multiple Distributions) We have already encountered several examples of confounded distributions. SEC may not suffice to obtain the MWD of a copolymer, because the molecules eluted at any one time may have different molecular weights and different compositions resulting in equal hydrodynamic volumes. Likewise, in the SEC separation of a branched polymer the molecules eluted at any one time may have different molecular weights and different degrees of branching, again resulting in equal hydrodynamic volumes. Just as the characterization of polymer distributions necessitates polymer separations, the characterization of two-dimensional polymer distributions necessitates two-dimensional polymer separations. Only if two distributions are fully independent do two separate one-dimensional separations suffice. This is the case if every chemical composition fraction exhibits the same MMD and every molar mass fraction exhibits the same CCD. Because this is not usually the case, one two-dimensional separation usually reveals (much) more information than two one-dimensional separations. 20.3.3.5

Comprehensive two-dimensional liquid chromatography Two-dimensional liquid chromatographic separations can be performed in the linear (‘‘heart-cut’’) format or in the comprehensive mode. In the former case, one fraction (or a few fractions) is (are) isolated from the sample and these are subsequently subjected to a second separation. An advantage of this approach is that the specific fraction(s) can be subjected to two (lengthy) high-resolution separations. A great disadvantage is that only one or a few small fractions of the sample are extensively characterized. In comprehensive two-dimensional liquid chromatography the entire sample is subjected to two different separations. The word ‘‘comprehensive’’ is justified if the final (two-dimensional) chromatogram is representative of the entire sample [25]. The recommended notation for linear (‘‘heart-cut’’) two-dimensional liquid chromatography is LC-LC, whereas comprehensive two-dimensional liquid chromatography is commonly denoted by LC LC [25]. In the case of polymer separations, the MMD is usually one of the distributions of interest. The second most important distribution is usually either the CCD or the functionality-type distribution (FTD). This implies that SEC and i-LC are attractive candidates for the two dimensions in comprehensive two-dimensional liquid chromatography. These two techniques can in principle be coupled in two different orders (either LC SEC or SEC LC, with the first dimension listed first). LC SEC has a number of prevailing advantages [26]. These include (i) the possibility of performing high-resolution (gradient) LC in the first dimension, (ii) the finite time of analysis in the second dimension, (iii) the greater choice of detectors (because the separation in the second dimension is isocratic), (iv) the possibility of changing the first-dimension LC conditions without the need to re-optimize the second-dimension conditions, (v) the relatively high concentration in the firstdimension i-LC system (not easily overloaded) and the relatively low concentration

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in the second-dimension SEC system (more easily overloaded), and (vi) the greatly reduced chance of ‘‘breakthrough’’. If the first dimension were SEC, the sample (fraction) transferred to the second-dimension LC would be dissolved in a very strong solvent, creating a great danger of detrimental ‘‘breakthrough’’ peaks [27]. A disadvantage is that the resolution in the second (fast-SEC) dimension is limited, but the series of advantages prevails. Therefore LC SEC is now the commonly employed technique. If we are to maintain the separation (resolution) that has been achieved in the first dimension in the eventual LC SEC chromatogram, we need to collect a large number of fractions. To maintain a reasonable overall analysis time, this implies that the second-dimension separation should be fast and that the resolution that can be obtained in this second dimension is limited. There have been significant developments toward fast SEC in recent years [9, 28] Moderate-resolution SEC can be performed within one or two minutes. If we want to collect 100 fractions from the first dimension, this implies that typical LC SEC analysis times are of the order of two to three hours. Indeed, these are the analysis times commonly encountered in practice. If we wish to transfer the entire first-dimension fraction to the second dimension, then the first-dimension column should have a much smaller internal diameter than the second-dimension column. Either a ‘‘miniaturized’’ first-dimension column can be used in combination with a conventional second-dimension column, or a conventional first-dimension column can be used in combination with a wide-bore (‘‘maxiaturized’’) second-dimension column. Both approaches have been demonstrated successfully. The first approach requires (very) much less solvent, produces correspondingly less waste, and is compatible with most existing LC detectors, including the molar-mass selective detectors (viscometry, light scattering) that are of great interest in polymer separations. The second approach puts fewer demands on the chromatographic (first-dimension) system in terms of extra-column band broadening and it yields larger separated fractions for subsequent off-line analysis by other methods (for example, NMR). Figure 20.12(a) shows an outline of a typical LC SEC system and Figure 20.12(b) shows an enlarged representation of the switching valve. A comprehensive two-dimensional liquid chromatography system typically consists of two liquid chromatographs that are interfaced by means of a switching valve. In the case of LC SEC the first dimension often features a gradient elution system; that is, the composition of the eluent can be programmed during the run. The valve is configured so that while one fraction is being analyzed, the next fraction is being collected (Figure 20.12(b)). Figure 20.13 features a contemporary example of an LC SEC separation [26]: in this two-dimensional separation of a series of copolymer ‘‘standards’’ of known molar mass and chemical composition, the first (gradient elution LC) dimension shows a high resolution, whereas the resolution in the second (SEC) dimension is adequate. The two separations are seen to be nearly orthogonal (that is, separation in the first dimension is (nearly) completely based on the chemical composition, whereas that in the second dimension is based on molecular size).

20.3 Polymer Analysis 2nd dimension

waste

PC

1st dimension

Detector (UV) 2nd dimension column

sample Loop 1 6-way injection valve

Loop 2

1st dimension column

(a)

LC

P Load

Inject

L1 L2

SEC

W

(b) Instrumentation for comprehensive twodimensional liquid chromatography: (a) schematic of the complete instrument; (b) preferred configuration of a 10-port switching valve. Reprinted from Ref. [26] with permission. Fig. 20.12.

Comprehensive two-dimensional liquid chromatography has seen a strong increase in popularity and in the number of applications in recent years. LC SEC has been applied to a large number of problems in polymer science. For example, the technique has been used to provide a detailed analysis of polystyrene– poly(methyl methacrylate) diblock copolymers [29], to analyze well-defined star polylactides [30], and to study to the grafting reaction of methyl methacrylate onto EPDM [31] or onto polybutadiene [32]. The main bottlenecks for the proliferation of LC SEC (and other two-

1043


1.3

1 10 2

Time SEC [min]

1044

11

1.1 3

12

1 4 6

0.9

7

8

13

9

14

5

0.8

0.5

1

1.5

2 2.5 Time LC [hours]

Fig. 20.13. LC SEC-ELSD contour plot of a mixture of homo- and copolymeric reference materials: PMMA 2900 (1), 6950 (2), 28 300 (3), 127 000 (4), 840 000 (5); S-co-MMA 20% S (6), 40% S (7), 60% S (8), 80% S (9); PS 2450 (10), 7000 (11), 30 000 (12), 200 000 (13), 900

3

3.5

4

000 (14). LC (first dimension): C18 column; flow: 4 mL min1 ; gradient: 5–70% THF in acetonitrile 0–300 min (40 C). SEC (second dimension): mixed C column; flow: 0.6 mL min1 THF. Figure reprinted from Ref. [26] with permission.

dimensional polymer separations) are now in the development of suitable calibration procedures and the associated software. Because suitable hardware for LC SEC is already commercially available, significant progress is expected in this direction.

Notation

Acronyms ATR CC CCD CP-MAS-NMR DRI ELSD ESI

attenuated total reflection critical (liquid) chromatography (or liquid chromatography at the critical conditions of adsorption) chemical composition distribution cross-polarization magic-angle spinning NMR differential refractive index evaporative light-scattering detector electrospray ionization

References

FTD FT-ICR-MS FTIR GC GPC i-LC IR LA(L)LS LC LC LC LC SEC LS MA(L)LS MALDI MS MMD MWD NIR NMR RA(L)LS RI SEC ToF TrA(L)LS Triple SEC Vis

functionality-type distribution Fourier transform ion–cyclotron resonance mass spectrometry Fourier transform infrared gas chromatography gel permeation chromatography (same as SEC) interactive liquid chromatography infrared low-angle (laser) light scattering liquid chromatography comprehensive two-dimensional liquid chromatography comprehensive two-dimensional (liquid chromatography) (size exclusion chromatography) light scattering multi-angle (laser) light scattering matrix-assisted laser desorption ionization mass spectrometry molar mass distribution (equivalent to MWD) molecular weight distribution (equivalent to MMD) near infrared nuclear magnetic resonance right-angle (laser) light scattering refractive index size exclusion chromatography time-of-flight triple-angle (laser) light scattering SEC with light-scattering, viscometry, and refractive index detection viscometry

References 1 J. A. Biesenberger, D. H. Sebastian,

2

3 4

5 6

Principles of Polymerization Engineering, Kruger, Malabar, 1993. M. Buback, M. Egorov, T. Junkers, E. Panchenko, Macromol. Rapid Commun. 25 (2004) 1004. S. C. Rutan, O. E. de Noord, R. R. Andrea, Anal. Chem. 70 (1998) 3198. N. D. Othman, G. Fevotte, D. Peycelon, J.-B. Egraz, J.-M. Suau, AIChE J. 50 (2004) 654. A. Cherfi, G. Fevotte, C. Novat, J. Appl. Polym. Sci. 85 (2002) 2510. R. A. M. Vieira, C. Sayer, E. L. Lima, J. C. Pinto, J. Appl. Polym. Sci. 84 (2002) 2670.

7 M. Watari, H. Higashiyama, N.

8

9 10

11

Mitsui, M. Tomo, Y. Ozaki, Appl. Spectrosc. 58 (2004) 248. H. A. Lousberg, H. C. J. Hoefsloot, H. F. M. Boelens, P. J. Schoenmakers, A. K. Smilde, Int. J. Polym. Anal. Char. 7 (2002) 76–92. H. Pasch, P. Kiltz, Macromol. Rapid Commun. 24 (2003) 104–108. J. Blomberg, P. J. Schoenmakers, N. van den Hoed, J. High Resolut. Chromatogr. 17 (1994) 411. P. A. Mirau, Nuclear magnetic resonance spectroscopy, in: R. F. Brady (ed.), Comprehensive Desk Reference of Polymer Characterization

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16

17

18

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23

and Analysis, Oxford University Press, New York, NY, USA, 2003, pp. 181– 219. F. P. Fitzpatrick, H.-J. Ramaker, P. J. Schoenmakers, R. Beerends, M. Verheggen, H. J. A. Phillipsen, J. Chromatogr. A 1043 (2004) 239. X.-L. Jiang, P. J. Schoenmakers, X.-W. Lou, V. Lima, J. L. M. van Dongen, J. Brokken-Zijp, Anal. Chem. 75 (2003) 5517. X.-L. Jiang, V. Lima, P. J. Schoenmakers, J. Chromatogr. A 1018 (2003) 19. R. Peters, Y. Mengerink, S. Langereis, M. Frederix, H. Linssen, J. van Hest, Sj. van der Wal, J. Chromatogr. A 994 (2002) 327. Y. Mengerink, R. Peters, C. G. de Koster, Sj. van der Wal, H. A. Claessens, C. A. Cramers, J. Chromatogr. A 914 (2001) 131. X.-L. Jiang, A. van der Horst, V. Lima, P. J. Schoenmakers, J. Chromatogr. A, submitted for publication. X.-L. Jiang, P. J. Schoenmakers, X.-W. Lou, V. Lima, J. L. M. van Dongen and J. Brokken-Zijp, Anal. Chem. 75 (2003) 5517. X.-L. Jiang, P. J. Schoenmakers, J. L. M. van Dongen, X.-W. Lou, V. Lima, J. Brokken-Zijp, J. Chromatogr. A, 1055 (2004) 123. K. Oudhoff, Capillary Electrophoresis for the Characterization of Synthetic Polymers, Ph.D. thesis, University of Amsterdam, 2004. J. Falkenhagen, H. Much, W. Stauf, A. H. E. Muller, Macromolecules 33 (2000) 3687. W. Lee, D. Cho, T. Chang, K. J. Hanley, T. P. Lodge, Macromolecules 34 (2001) 2353. H. J. A. Philipsen, J. Chromatogr. A 1037 (2004) 329.

24 S. Grcev, P. J. Schoenmakers, P. D.

Iedema, Polymer 45 (2004) 39. 25 P. J. Schoenmakers, P. Marriott, J.

Beens, LC-GC Europe 16 (2003) 335. 26 A. van der Horst, P. J.

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37

Schoenmakers, J. Chromatogr. A 1000 (2003) 693. X. Jiang, A. van der Horst, P. J. Schoenmakers, J. Chromatogr. A 982 (2002) 55–68. S. T. Popovici, W. Th. Kok, P. J. Schoenmakers, J. Chromatogr. A (2004) DOI information: 10.1016/ j.chroma.2004.05.099. H. Pasch, K. Mequanint, J. Adrian, e-Polymers (2002) Paper No. 5, CODEN: EPOLCI CAN 137:125589 AN 2002:181118 CAPLUS. T. Biela, A. Duda, S. Penczek, K. Rode, H. Pasch, J. Polym. Sci. Part A: Polym. Chem. 40 (2002) 2884. A. Siewing, J. Schierholz, D. Braun, G. Hellmann, H. Pasch, Macromol. Chem. Phys. 202 (2001) 2890. A. Siewing, B. Lahn, D. Braun, H. Pasch, J. Polym. Sci. Part A: Polym. Chem. 41 (2003) 3143. C. P. Beyers, H. F. M. Boelens, L. Klumperman, J. A. Westerhuis, Appl. Spectrosc. 58 (2004) 863. F. Bandermann, I. Tausendfreund, S. Sasic, Y. Ozaki, M. Kleimann, J. A. Westerhuis, H. W. Siesler, Macromol. Rapid Commun. 22 (2001) 690. O. Kammona, E. G. Chatzi, C. Kiparissides, J. Macromol. Sci. Rev. Macromol. Chem. Phys. C39(1), 57–134 (1999). F. J. Wyzgoski, P. L. Rinaldi, E. F. McCord, M. A. Stewart, D. R. Marshall, Macromolecules 37 (2004) 846. B. B. P. Staal, Ph.D. thesis, TU Eindhoven, 2005.

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Recent Developments in Polymer Processes1 Maartje Kemmere 21.1

Introduction

Currently, many industrial polymers are being produced in organic solvents. A typical production scheme for conventional polymer processes is shown in Figure 21.1, for which three sub-processes can be distinguished: the polymerization of the monomers, the shaping of the raw polymer, and the post-processing of the polymer. A major drawback of polymer processes in organic solvents is the inefficient removal and recovery of the solvents and monomers in each of the three subprocesses. Often the solvent recovery requires more process steps and energy than the actual production process, as indicated in Figure 21.1. For example, in the production of butadiene rubber a typical production process applies a recycle of approximately four tons of solvent per ton of polybutadiene produced [1]. Another example is the production of elastomers such as EPDM (ethylene–propylene–diene copolymer), for which typically 20 wt.% of polymer is dissolved in an excess of hexane. Annually, these conventional polymer processes add substantially to the total amount of discharged volatile organic compounds (VOCs) such as hexane and tolsolvent

solvent

P

solventt PP

S

final product

monomers SR polymerization

SR shaping

Conventional polymerization (P), shaping (S) and post-processing (PP) steps based on organic solvents, including significant solvent recovery steps (SR). Fig. 21.1.



SR post-processing

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21 Recent Developments in Polymer Processes

uene. Approximately 20 million tons of VOCs are emitted into the atmosphere each year as a result of industrial activities [2]. For these reasons, increasing environmental awareness demands more sustainable polymer processes. Currently, several technologies are being investigated to develop cleaner processes. These include the reduction and/or recycling of solvents by closed-loop operation as well as switching to solvent-free processes, for instance melt-phase polymerization [3]. However, these methods have their limitations, because many industries require process liquids for operations such as reactions, separations, and processing steps. Therefore, the possibilities of alternative green solvents to replace organic solvents are being explored, for which important examples are aqueous solvents, ionic liquids, fluorous phases, and supercritical or dense-phase fluids [3, 4]. Obviously, each of these approaches exhibits specific advantages and potential drawbacks. Ionic liquids (room-temperature molten salts), for example, have a vapor pressure that is negligible. Because ionic liquids are non-volatile, their commercial application would significantly reduce the present VOC emission. In general, ionic liquids can be used in existing equipment at reasonable capital costs [5]. Nevertheless, the cost price of the ionic liquid itself is substantial. In addition, the separation of ionic liquids from a process stream is another important point of concern. With respect to dense-phase fluids, supercritical water has been shown to be a very effective reaction medium for oxidation reactions [6, 7]. Despite extensive research efforts, however, corrosion and investment costs still form major challenges in these processes due to the extreme operation conditions required (above 647 K and 221 bar). Particularly, supercritical carbon dioxide has made its appearance as an alternative to organic solvents, because of its tunable nature, its excellent wetting characteristics, and its low viscosity [3]. Besides alternative reaction media, process and product research has resulted in a number of new classes of polymers, such as dendritic and conductive polymers. At the same time the application field of polymers has been extended, ranging nowadays from bulk materials such as polyethylene and polypropylene all the way to specialty products, for example for biomedical applications. These developments ask for well controlled polymerization processes. In this respect ultrasound has proven to be a clean and safe technology for generating radicals in situ during the polymerization reaction. In this chapter supercritical carbon dioxide will be discussed in more detail as one of the most promising green solvents for polymerizations. Subsequently, the possibilities and challenges of applying ultrasound in polymer processes are described. Finally, a short overview of the application potential of polymer processes based on supercritical carbon dioxide or ultrasound technology will be given.

21.2

Polymer Processes in Supercritical Carbon Dioxide

A supercritical fluid is defined as a substance for which the temperature and pressure are above their critical values (see Figure 21.2A) [8, 9]. Above the critical

21.2 Polymer Processes in Supercritical Carbon Dioxide

1049

1000

10000

280 K

solid

300 K

density (kg/m3)

supercritical fluid

1000 pressure (bar)

800

liquid 100

310 K

330 K

600

400 400 K

10

gas

200

A 1 200

B

0 250

300

350

400

temperature (K)

30

50

70

90

110

130

pressure (bar)

Fig. 21.2. Projections of the phase diagram of carbon dioxide A) in the pressure–temperature plane and B) in the density– pressure plane [8, 9].

temperature, the line representing vapor and liquid coexistence has disappeared. Therefore, supercritical fluids can be regarded as ‘‘hybrid solvents’’ because the properties can be tuned from liquid-like to gas-like without crossing a phase boundary by simply changing the pressure or the temperature. The unique properties of supercritical fluids provide opportunities for a variety of industrial processes. In Table 21.1 [10–13], the critical properties are shown for some components, of which carbon dioxide and water are the most frequently used as supercritical fluids. In polymerization processes, supercritical ethylene and propylene are also applied, where they act both as a solvent and as the reacting monomer [14]. With respect to the development of sustainable polymer processes, supercritical carbon dioxide (scCO2 ) has significant potential as a substitute for organic solvents. In addition to the environmental benefits, scCO2 has interesting physical and chemical properties from an engineering point of view. These include its relative chemical inertness, readily accessible critical point, and highly tunable solvent behavior. Moreover, carbon dioxide is nontoxic and nonflammable. In Figure 21.2B, the density of CO2 at different temperatures is plotted as a function of pressure. The graph clearly shows that the modification of the density requires only small changes in temperature or pressure just above the critical point. In general terms, scCO2 is a solvent with a low viscosity, high diffusion rates, and no surface tension [15]. The viscosity is in the range of 0.02–0.1 mPa s, where liquids have viscosities of approximately 0.5–1.0 mPa s and gases approximately 0.01 mPa s, respectively. Diffusivities of solutes in supercritical carbon dioxide are higher than in liquid solvents by a factor of up to 10. Binary diffusion coefficients of various substances in scCO2 have been determined experimentally as a function of CO2 density [16]. Because of the high diffusivity compared with ordinary solvents, scCO2 is often associated with high mass and heat transfer. Additionally, the

150

170

1050

21 Recent Developments in Polymer Processes Tab. 21.1.

Physical properties of various solvents [10–13].

Solvent

Tc [K ]

Pc [bar]

Polarizability a [10C19 m 3 ]

Dipole moment m [10C30 C m]

Methane Ethane Propane Ethylene Propene Ethyne Dimethyl ether Sulfur hexafluoride Difluoromethane Trifluoromethane Tetrafluoromethane Difluororethane Hexafluoroethane Carbon dioxide n-Hexane Cyclohexane Diethyl ether Methanol Acetone

190.4 305.3 369.8 282.4 364.9 308.3 400.0 318.7 351.6 299.3 227.6 386.7 293.0 304.1 507.5 553.5 466.7 512.6 508.1

46.0 48.7 42.5 50.4 46.0 61.4 52.4 37.6 58.3 48.6 37.4 45.0 30.6 73.8 30.1 40.7 36.4 80.9 47.0

26 45.0 62.9 42.3 62.6 33.3 51.6 54.6 24.8 26.5 28.6 41.5 47.6 27.6 118.3 109 87.3 32.3 63.3

0.0 0.0 0.1 0.0 0.4 0.0 1.3 0.0 2.0 1.7 0.0 2.3 0.0 0.0 0.0 0.3 1.3 1.7 2.9

Quadrupole moment Q [3:16 D 10C17 J1/2 m 5/2 ] 0.7

þ3.0

0.7 4.3

above-mentioned properties are strongly pressure-dependent in the vicinity of the critical point, making scCO2 a highly tunable solvent. For application of scCO2 as a medium in polymer processes, it is important to have insight into the specific system involved. In Section 21.2.1 a brief overview of the interactions of carbon dioxide with polymers and monomers will be given, including solubility in CO2 (Section 21.2.1.1), sorption and swelling of polymers (Section 21.2.1.2) as well as phase behavior (Section 21.2.1.3) of polymer–monomer–CO2 systems. Subsequently, an overview of processes involving polymerization (Section 21.2.2) as well polymer processing (Section 21.2.3) in scCO2 will be discussed. 21.2.1

Interactions of Carbon Dioxide with Polymers and Monomers

The thermodynamic properties of pure substances and mixtures of molecules are determined by intermolecular forces acting between the molecules or polymer segments. By examining these potentials between molecules in a mixture, insight into the solution behavior of the mixture can be obtained. The most commonly occurring interactions are dispersion, dipole–dipole, dipole–quadrupole, quadrupole– quadrupole, and specific interactions. For small molecules, the contribution of each interaction to the intermolecular potential energy Gij ðr; TÞ, is given by Eq. (1) [12], where a is the polarizability, m is the dipole moment, Q is the quadruple


moment and SI represents specific interactions such as complex formation or hydrogen bonding. "

# mi2 mj2 mi2 Q 2j Q 2i Q 2j ai aj þ C4 10 þ SI Gij ðr; TÞ A C1 6 þ C2 6 þ C3 8 r r kT r kT r kT

ð1Þ

The interactions work over different distances, with the longest range for dispersion and dipole interactions. Note that the dispersion interaction depends on the polarizability alone, and not on the temperature. Consequently, an increased polarizability of the supercritical solvent is expected to decrease the pressures needed to dissolve a nonpolar solute or polymer. Furthermore, at elevated temperatures, the configurational alignment of directional interactions such as dipoles or quadrupoles is disrupted by the thermal energy, leading to a nonpolar behavior. Hence, it may be possible to dissolve a nonpolar solute or a polymer in a polar supercritical fluid. However, to obtain sufficient density for dissolving the solutes at those elevated temperatures, substantially higher pressures need to be applied. Additionally, specific interactions such as complex formation and hydrogen bonding can increase the solvent strength of the supercritical fluid. These interactions are also highly temperature-sensitive. Solubility in Carbon Dioxide The solvent strength of carbon dioxide for solutes is dominated by a low polarizability and a strong quadrupole moment. Consequently, carbon dioxide is difficult to compare with conventional solvents, due to this ambivalent character. With its low polarizability and nonpolarity, carbon dioxide is similar to perfluoromethane, perfluoroethane, and also methane. These fluids are very weak solvents for many components. Additionally, the acidity of carbon dioxide increases the solvent strength for weakly basic solutes. In general, carbon dioxide is a reasonable solvent for small molecules, both polar and nonpolar. For many components, with the exception of water, complete miscibility can be obtained at elevated pressures. However, the critical point of the mixture, which is the lowest pressure at a given temperature where CO2 is still completely miscible, rises sharply with increasing molecule size. Consequently, most of the larger components and polymers exhibit a very limited solubility in carbon dioxide. Polymers that do exhibit a high solubility in carbon dioxide are typically characterized by a flexible backbone and high free volume (hence, a low Tg ), weak interactions between the polymer segments, and a weakly basic interaction site such as a carbonyl group [17–20]. Examples of polymers that are soluble in carbon dioxide and incorporate these characteristics include poly(alkene oxides), perfluorinated poly(propylene oxide), poly(methyl acrylate), poly(vinyl acetate), poly(alkylsiloxanes), and poly(ether carbonate). To illustrate the influence of the carbonyl group on the solubility of the polymer in carbon dioxide, Figure 21.3 shows the difference in cloud-point pressures for polymethyl acrylate (PMA) and polyvinyl acetate (PVA) [17]. Structurally, these polymers are very similar, but the carbonyl group in PVA is more easily accessible to CO2 , making it more susceptible to complex formation. As a result, the cloud21.2.1.1

1051

1052


2500 Fluid

pressure (bar)

2000

PMA 31000

1500

O 1000

PVA 125000

O

O CH3

O CH3

500 Liquid + Liquid

poly(vinyl acetate) poly(methyl acrylate)

0 280

320

360

400

440

480

temperature (K) Cloud-point curves of carbon dioxide–poly(methyl acrylate) and carbon dioxide–poly(vinyl acetate) with polymer concentrations of @5 wt.% [17]. Fig. 21.3.

point pressure of PVA is up to 1500 bar lower than that of PMA, even though the molecular weight of PVA is four times higher than that of PMA, and its glass transition temperature is 21 K higher. Sorption and Swelling of Polymers Although the solubility of polymers in CO2 is typically very low, the solubility of carbon dioxide in many polymers is substantial. When the polymer comes into contact with scCO2 , the sorption of carbon dioxide by the polymers and the resultant swelling of the polymer influence the mechanical and physical properties of the polymer. The most important effect is plasticization, that is, the reduction of the glass transition temperature (Tg ) of glassy polymers. The plasticization effect, characterized by an increased segmental and chain mobility as well as by an increase in interchain distance, is largely determined by polymer–solvent interactions and solvent size [21]. The molecular weight of the polymer is of little influence on the swelling once the entanglement molecular weight has been exceeded. The influence of specific interactions of carbon dioxide with various groups in the polymer on the swelling isotherms is shown in Figure 21.4 [22]. These polymers all have a Tg of approximately 378 K. Carbon dioxide shows little interaction with the phenyl groups in polystyrene (PS), resulting in limited swelling. Substituting one of the CH groups in styrene with a nitrogen atom renders the polymer somewhat basic, thereby increasing the swelling level as well as the swelling rate. Building carbonyl groups into the polymer has an even stronger effect. In the case of poly(methyl methacrylate) (PMMA), the swelling level is doubled as compared to PS. Besides the enhancing interactions with carbon dioxide, the sorption and swell21.2.1.2


1053

20 PMMA

CH3

swelling (%)

16 O PVP

12

O CH3

poly(methyl methacrylate)

polystyrene

8 PS

4 N

0 0

20

40

60

80

100

120

poly(vinyl pyridine)

pressure (bar) Swelling isotherms of poly(methyl methacrylate), polystyrene and poly(vinyl pyridine) in carbon dioxide at 308 K [22]. Fig. 21.4.

ing can also be increased by lowering the interchain interactions of the polymer. This is illustrated by comparison of commercial semi-crystalline low-density polyethylene (ldPE) [23] (40% crystallinity) with highly branched amorphous polyethylene [24] (< 2% crystallinity). Although both polymers lack specific interactions with carbon dioxide, the highly branched PE shows a strongly enhanced swelling (up to 25 vol.%) in carbon dioxide, whereas the ldPE exhibits a low maximum swelling (4 vol.%) at 323 K [25]. This is caused by the absence of crystallinity and the increased free volume introduced by branching. To study the CO2 -induced plasticization, different techniques can be applied, for example, gas sorption and polymer swelling [26, 27]; in-situ FTIR spectroscopy [28, 29], NMR spectroscopy [30], and dynamic light scattering [31]. Two new techniques for simultaneous sorption and swelling measurement have been proposed, based on detection of the volume change of the polymer sample measured by mercury displacement [32], and on gravimetric measurement [33], respectively. The sorption and swelling of a wide variety of polymers in scCO2 have been reported in the literature. For example, Liau and McHugh [34] have investigated the swelling of PMMA and sorption of CO2 in PMMA at temperatures ranging from 313 to 343 K and at pressures up to 272 bar by using a camera to record the change in length of the polymer sample. Wissinger and Paulaitis [26] have reported swelling and sorption in glassy polymer–gas systems for CO2 with polycarbonate (PC), PMMA, and PS at temperatures between 306 to 338 K and pressures up to 100 bar. Sada et al. [35] have studied the sorption and diffusion of CO2 in glassy polymer films of PS and PC over a temperature range of 293 to 313 K and pressures up to 30 bar by a pressure decay method. Berens et al. [36] have investigated the sorption

1054


behavior of the glassy polymers poly(vinyl chloride) (PVC), PC, PMMA, and PVA at 298 K up to 700 bar in the presence of liquid CO2 through a simple gravimetric procedure. Additionally, sorption and swelling data at 308 K have been reported by Zhang et al. [37] and up to 100 bar for PMMA, PS, polyvinylpyridine and polyisoprene as well as for block copolymers of styrene–methyl methacrylate, styrene– vinylpyridine and styrene–isoprene, respectively. Otake et al. [31] have measured the swelling of a 50 nm monodisperse PS latex in water by CO2 by dynamic light scattering at 298 K at pressures up to 350 bar. In general, the sorption and swelling of polymers by CO2 are crucial effects in designing polymer processes based a high-pressure technology, because important properties such as diffusivity, viscosity, glass transition, melting point, compressibility, and expansion will change. The plasticization effect of CO2 facilitates mass transfer properties of solutes in and out of the polymer phase, which leads to many applications: increased monomer diffusion for polymer synthesis, enhanced diffusion of small components in polymers for impregnation and extraction purposes, polymer fractionation, and polymer extrusion. Phase Behavior of Monomer–Polymer–Carbon Dioxide Systems One of the requirements for the development of new polymer processes based on scCO2 is knowledge about the phase behavior of the mixture involved, which enables one to tune the process variables properly to achieve maximum process efficiency. Important parameters in the phase behavior of the system are the solvent quality, the molecular weight, chain branching, and chemical architecture of the polymer, as well as the effect of end groups, and addition of a co-solvent or an anti-solvent. The literature available on the phase behavior of polymers in supercritical fluids has been reviewed extensively by Kirby and McHugh [38]. The usual phase equilibrium issue in polymer systems consists of determining whether phase separation occurs, and if it does, then what the phase compositions are. Although measuring is still the most accurate way at present to obtain information about the phase behavior [39–42], it is rather time-consuming. Most of the experimental work described in the literature has focused on polymer–solvent systems rather than on non-solvents. For example, binary systems of linear and branched polyethylenes in ethylene have been measured [40, 43]. In addition, the effect of carbon dioxide as a non-solvent on polyolefin–solvent systems has been studied [44–47]. When thermodynamic models are used to correlate and predict phase equilibria, the experimental effort can be reduced substantially. Modeling of polymer– supercritical fluid mixtures is particularly challenging, as the model has to be able to describe the behavior of a mixture containing molecules covering a wide range of molecular weights, which is often referred to as the ‘‘asymmetry’’ of the system. Additionally, the proximity to the critical point is difficult to model. There exist several thermodynamic models suitable for the description of phase behavior of polymer systems in supercritical media; see Chapter 3 for a detailed description of the existing models. However, the Sanchez–Lacombe equation of state (eos) and statistical associating fluid theory (SAFT) eos are the most commonly used. 21.2.1.3


The Sanchez–Lacombe model [48–50] is a lattice fluid model in which each component is divided into parts that are placed in a lattice. The different parts are allowed to interact with a mean-field intermolecular potential. By introducing an appropriate number of vacant sites (holes) in the lattice, the correct solution density can be obtained. SAFT [51–53] is based on the perturbation theory. The principle of perturbation theory is that first a model is derived for some idealized fluid with accurately known properties, called the ‘‘reference fluid’’. Subsequently, the properties of this model are related to those of a real dense fluid. By expanding this reference fluid into power series over a specified parameter, the power terms can be regarded as ‘‘corrections’’ or ‘‘perturbations’’ for the reference fluid as compared to reality. Obviously, the more the reference model approaches reality, the smaller the corrections are. Therefore, the key issue for applying perturbation theory is deriving the most suitable reference fluid. In general, the choice of a certain model to describe the phase behavior of a polymer–monomer–CO2 system is far from arbitrary, and depends on, among other factors, the particular components involved and the required detail of the description. The Sanchez–Lacombe eos has the advantage that it is very tractable and it is suitable for interpolation of data. On the other hand, the Sanchez– Lacombe model gives a poor description of systems containing specific interactions, because it is a mean-field theory with mixing rules assuming a random mixture. For that reason specific interactions can only be introduced through a temperature-dependent interaction parameter. Modeling with SAFT provides a more rigorous approach, although it is more computationally intensive. Many non-idealities can be incorporated in calculations with this eos. However, similarly to Sanchez–Lacombe, description of the binary polymer–CO2 system can be troublesome using SAFT. As an example the SAFT eos and the Sanchez–Lacombe eos have been compared for the poly(ethylene-co-propoylene)–ethylene–CO2 system [54]. Both models give a qualitative description of the increase in ethylene–PEP cloud-points upon addition of carbon dioxide as anti-solvent. For a quantitative correlation with the Sanchez–Lacombe model, a temperature-dependent interaction parameter is required. Additionally, the parameter is determined for each individual cloud-point composition. For the SAFT model, one temperature-dependent interaction parameter suffices for all compositions. For this particular system, the description of the phase behavior with the SAFT model is in better agreement with the experimental data than the description with the Sanchez–Lacombe model (see Figure 21.5).

21.2.2

Polymerization Processes in Supercritical Carbon Dioxide

With respect to the possibilities of carbon dioxide as a reaction medium for polymerization reactions, extensive reviews have been published by Scholsky [55], Shaffer et al. [56], Kendall et al. [57], and Cooper [58]. Typically, a division is made between homogeneous and heterogeneous polymerization in scCO2 .

1055

21 Recent Developments in Polymer Processes 4000 SL

pressure (bar)

3500

3000

2500

2000

A 1500 310

315

320

325

330

335

340

345

temperature (K)

4000 SAFT

3500 pressure (bar)

1056

3000

2500

2000

1500 310

B 315

320

325

330

335

340

345

temperature (K) Sanchez–Lacombe (A) and SAFT (B) correlation of the cloud-point curves for the ternary system ethylene–carbon dioxide–PEP containing 10 wt.% PEP, as a function of temperature and mass fraction of carbon dioxide: y, 0.05; D, 0.10; j, 0.15; b, 0.20 [54]. Fig. 21.5.

Both chlorofluorocarbons (CFCs) and carbon dioxide appear to be very good solvents for amorphous, low-melting fluoropolymers. Since environmental restrictions have limited the use of CFCs drastically, carbon dioxide has become a highly viable alternative solvent for the production of amorphous fluoropolymers [59]. Examples of polymerization of fluorinated monomers in a homogeneous reaction medium of scCO2 are the polymerization of fluorinated acrylates (see, for example Refs. 59–61, fluoroalkyl-derivatized styrene [62], fluorinated vinyl and cyclic ethers [63], and the telomerization of 1,1-difluoroethylene [64]. Other options to run a


polymerization under homogeneous conditions are to increase the temperature and pressure until a one-phase system is obtained [65] or to add a co-solvent. Except for certain fluoropolymers and a few other examples of polymers that are soluble in it, as given in Section 21.2.1.1, carbon dioxide exhibits a strong antisolvent effect for most high molecular weight polymers. This implies that the majority of the polymerization reactions, such as dispersion, emulsion, suspension, and precipitation polymerization, involve heterogeneous polymerization. Examples of dispersion polymerization in scCO2 are the polymerization of MMA [66] and styrene [67]. Beckman and co-workers have reported the inverse emulsion polymerization of acrylamide in scCO2 [68]. Moreover, it is possible to synthesize well-defined porous polymers by templating in scCO2 –water emulsions [69]. Macroporous beads of trimethylolpropane trimethacrylate (TRIM) can be synthesized by means of suspension polymerization [70]. Much work has been performed on precipitation polymerization, such as the polymerization of vinyl ethers and oxetanes [63], the continuous polymerization of vinylidene fluoride as well as of acrylic acid [71, 72], MMA polymerization initiated by ultrasound [73], the catalytic polymerization of olefins [74, 75] and the enzymatic polymerization to polycaprolactone [76]. An interesting extension in the application of scCO2 for polymerization purposes is the use of CO2 both as a reactant and as a solvent in a copolymerization with cyclohexene oxide [77]. From an engineering point of view several issues have to be addressed in order to design polymerization processes based on scCO2 . The previous sections have emphasized the importance of the phase behavior of the system as well as the sorption and swelling of the polymer. Additionally, in catalytic polymerization processes the solubility and recycle of the catalyst are important. Moreover, when the polymer-shaping step is to be included in the same equipment or the purification of the polymer product, this will set additional requirements on the design. Obviously, mass and heat transfer have to be investigated. In general, supercritical fluids have gas-like viscosities and diffusion rates, which allow for fast mass transfer. In addition, the thermal diffusivity of supercritical CO2 is significantly higher than for gaseous CO2 . The need for forced convection in order to facilitate sufficient mass and heat transfer during the polymerization process has to be determined for each particular type of polymerization. If the process involves a precipitation polymerization, for instance, the effect of polymer deposition on heat transfer surfaces forms a potential point of concern. Given the scale of most commercial polymerization processes, a translation from a batchwise operation as it is usually performed on a laboratory scale toward a continuous process forms another important consideration in the development of polymerization processes based on scCO2 . As an example, it has been suggested that the process concept of the catalytic polymerization of olefins in scCO2 should include a loop type of apparatus [78], as described by Ahvenainen et al. [79] (see Figure 21.6). Next to the requirements mention above, this reactor design allows adjustment of the residence time to relatively long values in order to obtain a reasonable conversion. Although many new polymerization reactions have been discovered in scCO2 since the early 1990s, the development of the engineering aspects of these pro-

1057


external recycle CO2

catalyst

1058

polymer monomer Proposed process concept of a loop reactor used for the catalytic polymerization of olefins in scCO2 [77]. Fig. 21.6.

cesses and the step toward pilot scale and industrial application had only just been initiated in the last few years. The commercialization by DuPont of the production of Teflon2 (polytetrafluoroethylene, PTFE) in scCO2 is a promising example [80]. 21.2.3

Polymer Processing in Supercritical Carbon Dioxide

Applications of supercritical fluids in polymer processing can be divided into three areas: applications where carbon dioxide does not interact with the polymer; and processing of swollen, or dissolved, polymers. Schematically, this is shown in Figure 21.7 [81, 82]. In the first of these areas, when no interaction with the polymer occurs, which is typically the case for crystalline and rigid polymers, supercritical carbon dioxide can be used as a cleaning solvent. Applications in this category include precision cleaning of surfaces, degreasing, particle removal, and dry cleaning. In particular, dry cleaning is being investigated for widespread industrial application, because traditional processes based on perchloroethylene are under pressure due to health and environmental issues. Current research in this area is focused on the development of low-cost nonfluorous CO2 -philic surfactants [18, 19] and on the development of membrane systems for the economical reuse of carbon dioxide [83]. In the case of moderate interactions of carbon dioxide with the polymer, substantial levels of sorption and swelling can occur. The swelling of the polymer significantly enhances diffusivities of solutes in the polymer matrix, allowing for rapid extraction, impregnation, shaping, and blending processes. Extraction applications include removal of residual monomers, solvents, and catalysts [84, 85]. An impor-


1059

polymer and supercritical fluid no interaction

swelling of polymer sorption and swelling

applications:: • precision i i cleaning • dry cleaning

i applications: i • extraction • impregnation • shaping • blending

dissolution of polymer

effects: • plasticization (decrease r Tg) • crystallization i (increase Tm) • changes in mechanical and surface properties u

applications: • removal of low Mw material to i improve properties • fractionation • coatings / paint

Fig. 21.7. Interactions of supercritical fluids with polymers, and their applications in polymer processes [81].

tant application of impregnation of polymers is the dyeing of textiles and fibers [86, 87]. Other impregnation applications include impregnation of polymer matrices with drugs for the production of controlled drug delivery devices [88]. Advantages in this process are the low processing temperatures and the lack of organic solvents, which need to be removed in post-processing steps. Applying scCO2 for shaping purposes is advantageous because the material properties can be tuned by varying the supercritical solvent density. This enables the removal of the solvent without passing a liquid–gas interface, which would cause the structures formed to collapse [89]. With respect to blending, an interesting application is in-situ polymerization in an existing polymer matrix after impregnation with a monomer and an initiator to form polymer blends [90]. This enables the formation of polymer blends that cannot easily be obtained by conventional methods, for example due to large differences in the melting temperatures of the pure polymers. Applications in the third category, where the polymer is dissolved in the supercritical fluid, include coating processes as well as fractionation. For example, various groups have worked on the RESS (rapid expansion from supercritical solution)-based process for spray coating using supercritical carbon dioxide as a solvent [91–94]. The main advantage of these processes is the reduction of VOC emissions. Methods of applying coatings from CO2 involve dip or spray coatings, solutions of coatings in a CO2 atmosphere and dispersion coatings. An example that has already been commercially developed is the Unicarb TM spray coating process [95], which can use highly viscous coatings and results in a narrow droplet size distribution. A potential drawback of spray coating processes using supercritical carbon dioxide is the fact that most coating materials have a low solubility in supercritical carbon dioxide. Consequently, co-solvents such as methanol are needed to enhance solubilities. An alternative to the addition of co-solvents is the use of stabilizers [96]. Typical stabilizer structures involve graft copolymers, perfluoropolyether acid and polyether carbonates.

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An extensive review on polymer processing using supercritical fluids by Kazarian [21] includes the applications mentioned above. To illustrate the possibilities of polymer processing with scCO2 from an engineering point of view, two important applications will be discussed in more detail, namely extraction (see Section 21.2.3.1) and impregnation (Section 21.2.3.2). Extraction Commercial-grade polymers contain a range of additives and contaminants such as residual monomer, solvent, and catalyst residue. Before application of the polymer in the end-product, manufactures need to remove these impurities from it, for instance to meet environmental regulatory requirements or to improve the physical polymer properties [58]. Supercritical fluid extraction (SFE) has many advantages over conventional cleaning techniques, such as a shorter extraction time, adjustable solvent strength by tuning the pressure, a wide range over which the extraction temperature can be varied, and less energy consumption [15]. A number of studies have been reported in supercritical extraction of polymer additives, extraction of monomers and oligomers, and removal of residual solvent from polymer foams [97], of which some examples will be given here. SFE can be achieved in two ways: statically or dynamically. In a static extraction, the extraction vessel is pressurized to the desired pressure with the extracting fluid and subsequently left for a certain length of time, whereas during dynamic extraction the supercritical fluid passes continuously over the sample, extracting soluble compounds and depositing them in a suitable solvent or on a solid trap. Cotton et al. [98] used dynamic extraction for the extraction of Irgafos 168 and Irganox 1010 from a commercial-grade polypropylene matrix. The rate of extraction can be hampered by two factors: the diffusion of the additive through the matrix, or the solubility of the extracted material in the supercritical fluid, can be limited. Similar results have been obtained by Lou et al. [99] for the supercritical extraction of polyethylene by applying the two-film theory. In general, if diffusion through the polymer matrix is the rate-limiting step in the extraction process, an enhanced extraction temperature will increase the extraction rate, because the diffusion coefficient increases with temperature. Moreover, the plasticizing effect will significantly enhance the diffusion rates. If the rate-limiting step is the solubility of the solute (extracted material) in the supercritical fluid, the extraction rates can be enhanced by either increasing the pressure (solvent strength) or by increasing the fluid flow rate [58]. Another example of supercritical fluid extraction is the extraction of liquid crystalline 4,4 0 -dibutylazobenzene from a polystyrene matrices [100]. Furthermore, Sekinger et al. [101] have investigated the feasibility of using SFE for extracting additives from styrene–butadiene rubber (SBR) by determining the effects of density, temperature, flow rate, and contact time on the extraction efficiency. Kemmere et al. [85] have developed a post-polymerization process which reduces the amount of residual monomer in latexes using scCO2 . Typically, the method comprises a counterflow process, in which part of the residual monomer is converted by the increased diffusion inside the polymer particles due to the swelling by scCO2 . In 21.2.3.1


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polymerization reactor

latex : / F:1.11 kg/s T:60 ºC

1

CO 2 from supply tank T: 15 ºC P: 55 bar F:: 0.796 kg/s

x 4 extractor

F:: 0.789 kg/s

P: 30 bar

T: 60°C P: 80 bar

MMA T: 11ºC

latex

2

CO2 recycle T: 11ºC P: 30 bar

5 separator

P: 30 bar F:: 0.0167 kg/s

3 HEX1 Start-up pump

7

6 compressor

HEX2

1. Latex pump flow 4.0 m3/hr head of feet 2592.0 . power . kW 12.5

2. Centrifugal pump flow 3.45 m3/hr head of feet 995.7 . power . kW 3.44

3. Heat exchanger 1 heating duty 80.5kW . U 850.0 W/m2K Tm . K 69.12 required area 1.4

4. Extractor

5. Separator t

height 4.25 m diameter 0.60 m wall thickness 0.024 m

height 3.38 m diameter 1.12 m wall thickness 0.045 m

6. Compressor s (screw)) total power 53.66 kW U

Fig. 21.8. Process flow sheet for the removal of residual monomer from latex-products using scCO2 [84].

addition, the amount of residual monomer is further reduced by the extraction capacity of scCO2 [102]. Figure 21.8 shows the flow diagram of the extraction process. A viability study, including equipment sizing and economic evaluation, has shown that the removal of residual monomer from latex-products using scCO2 in principle yields a process which is both technically and economically feasible, and capable of meeting future requirements [85]. Other examples of extrapolation of SFE results obtained on a laboratory scale toward production equipment are described by Brunner [103] and Perrut and co-workers [104]. Impregnation and Dyeing The same properties that make scCO2 attractive for extraction purposes can be exploited in the impregnation and dyeing of polymeric materials [105]. By applying supercritical conditions, it is possible to impregnate polymers with pharmaceuticals, flavors, and fragrances, or with additives such as pigments, stabilizers, and 21.2.3.2

7. Heat exchangerr 2 heating duty (-)51.57 . kW 850.0 W/m2K Tm 57.00 K required area 4.52

1062


plasticizers. A wide variety of polymer systems have been impregnated using scCO2 , for example PMMA [106], PS, PE, PC, PVC, and polypropylene (PP) [107], and polydimethylsiloxane (PDMS) [108, 109], as well as poly(ethylene terephthalate) (PET) [110, 111]. The key parameter that determines the feasibility of the impregnation process is the equilibrium distribution or partition coefficient of the solute that is being impregnated into the polymer. There are two different methods of supercritical fluid impregnation of additives into polymer products. The first involves diffusion of a compound that is soluble in the supercritical fluid. The impregnation takes place when a polymer matrix is contacted with the supercritical fluid containing the solute. On depressurization, the CO2 flows out of the polymer matrix, leaving the solute trapped in it [112]. For example, Ma and Tomasko [113] have investigated the impregnation of high-density polyethylene (hdPE) with the nonionic surfactant N,N-dimethyldodecylamine-N-oxide using scCO2 . They have observed that the penetration and absorption into the polymer produces no structural changes or losses of mechanical strength. However, as a result of the swelling by scCO2 , the wetting properties of hdPE are changed permanently, in contrast to conventional techniques using an aqueous solution. In the second route to impregnation, the compound typically has a low solubility in the supercritical phase; dyeing is a typical example (see, for example, Refs. 106, 107, 111, 114). As described earlier in Section 21.2.3, impregnation applications based on scCO2 technology include polymer blending and the production of polymeric drug delivery devices. However, the largest application of impregnation with scCO2 at present is the dyeing of textiles and fibers. The main motivation is the replacement of large amounts of water in the dyeing process with benign scCO2 , which can reduce the wastewater in the textile industry significantly [85, 86]. The ease of recovering the carbon dioxide by reducing the pressure and recycling it without the necessity to clean the carbon dioxide provides a strong environmental advantage. For these reasons, there has been a significant effort to commercialize supercritical fluid dyeing of PET [112], for which the first pilot plant has been built [115].

21.3

Ultrasound-induced Radical Polymerization

For the development of sustainable polymer processes, ultrasound is an interesting technology, as it allows for polymerizations without the use of initiator. The radicals are generated in situ by cavitation events [116, 117], which make possible a clean and intrinsically safe polymerization reaction. As a result of the high strain rates outside the bubble, cavitation can also induce chain scission [118, 119], which provides an additional means to control the molecular weight of the polymer produced. In Sections 21.3.1 and 21.3.2 the physical background of ultrasoundinduced cavitation and radical formation will be described. Subsequently (see Section 21.3.3), an overview of the several types of ultrasound-induced polymerizations will be given, namely bulk, precipitation, and emulsion polymerization.

21.3 Ultrasound-induced Radical Polymerization

Finally, in Section 21.3.4, the breakage of polymer chains by cavitation will be discussed, including the possibility of synthesizing block copolymers. 21.3.1

Ultrasound and Cavitation in Liquids

Ultrasound passes through an elastic medium as a longitudinal wave, which is a series of alternating compressions and rarefactions. This means that liquid is displaced parallel to the direction of motion of the wave. Ultrasound comprises sound waves typically in the range of 20 kHz to approximately 500 MHz. The frequency ( f ) and the acoustic amplitude (PA; max ) are the most important properties to characterize the pressure wave. The variation of the acoustic pressure (PA ) of an ultrasound wave as a function of time (t) at a fixed frequency is described by Eq. (2) [120]. PA ¼ PA; max sinð2 p f tÞ

ð2Þ

The use of ultrasound can be divided into two areas: low-intensity, high-frequency ultrasound (2–500 MHz, 0.1–0.5 W cm2 ) and power ultrasound with a high intensity and a low frequency (20–900 kHz, >10 W cm2 ); see Table 21.2. The first does not alter the state of the medium through which it travels and is commonly used for nondestructive evaluation and medical diagnosis [121]. This type of ultrasound cannot be used for reactions. Contrarily, power ultrasound uses the energy to create cavitations, which involve the formation, growth, and implosive collapse of microscopic bubbles in a liquid. These bubbles are generated when the ‘‘negative’’ pressure during the rarefaction phase of the sound wave is sufficiently great to disrupt the liquid. The implosive collapse of the bubbles can produce extreme temperatures and pressures locally for very short times, due to compression of the gas phase inside the cavity [122, 123]. These hot-spots can lead to irreversible changes such as the formation of excited states, bond breakage, and the generation of radicals. Power ultrasound is applied for cleaning purposes, treatment of kidney stones, plastic welding, and chemical reactions [124].

Tab. 21.2.

Overview of different types of ultrasound, including the various applications.

Power ultrasound, 20–100 kHz

Sonophoresis, 20 kHz, low power

Therapeutic ultrasound, 1 MHz, high power

Nondestructive ultrasound, I2 MHz, low power

Sonochemistry Welding Cleaning Cell disruption Sterilization Kidney stones

transdermal drug delivery

therapeutic massage controlled release

flaw detection medical diagnoses

1063


400 350 300

h

250

Grow t

-6

pse Colla

Radius (10 m)

1064

200 150 100 50 0

Formation 0

10

20

Hot-spot 30

40

50

60

70

80

-6

Time (10 s)

Fig. 21.9. Schematic representation of bubble growth and collapse in water at ambient pressure irradiated with ultrasound and the resulting hot-spot due to adiabatic

compression. The radius of the cavitation bubble as a function of time has been calculated using the dynamic bubble model based on the Rayleigh–Plesset equation [136].

Sonochemistry comprises the chemical effects that are induced by power ultrasound. The ultrasound waves cause the molecules to oscillate around their mean position in a liquid. During the positive-pressure cycle the distance between the molecules decreases, while during the negative-pressure period the distance increases. At a sufficiently high intensity a critical distance between the molecules is exceeded during the negative-pressure period, and a cavity is formed [125]. Due to the presence of nuclei such as dissolved gases and solid impurities, cavities are formed at far lower sound pressures than theoretically predicted [126]. After the formation of a cavity, a critical acoustic pressure has to be overcome to initiate the explosive growth of this bubble. This explosive growth is followed by an implosive collapse (see Figure 21.9). During this collapse, the contents of the bubble are heated almost adiabatically, which leads to local short-lived hot-spots in the liquid. Depending on the specific conditions, bubble wall velocities, and pressures and temperatures in the bubble, can increase up to 1500 m s1 , 200 bar and 5000 K, respectively [127]. The acoustic pressure amplitude determines the growth of a cavitation bubble and consequently the chemical effects upon collapse. The amplitude of the pressure wave can be measured directly with a hydrophone or calculated using a calorimetric method [128, 129], by which it is possible to determine the ultrasound power (Q US ) that is transferred to the liquid. With the ultrasound power, the density of the liquid (r), the speed of sound in the medium (v), and the surface area of the ultrasound source (AUS ), the acoustic amplitude can be calculated according to Eq. (3). The ultrasound intensity is the power input divided by the surface area of the source [130].


PA; max

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q US pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2rv ¼ 2 r v IUS AUS

ð3Þ

The Blake threshold pressure (PB ) describes the critical pressure to initiate the explosive growth of a cavitation bubble [Eq. (4)] [131]. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 4 s u PB ¼ P0 Pv þ s u s t3 3 Pv R 30 P0 þ 2 R0

ð4Þ

Equation (4) assumes that the external pressure (P0 ), the vapor pressure (Pv ), the surface tension (s), and the equilibrium radius of the bubble (R 0 ) determine the negative pressure in the liquid required to start an explosive growth of a cavity [132]. The Blake threshold pressure is based on a static approach and is only valid when the surface tension dominates all dynamic effects, such as mass transfer and viscosity. The influence of the medium characteristics and the physical conditions on the implosion can be calculated with a dynamic bubble model [120, 133]. The dynamic movement of a bubble in a sound pressure field can be described with the Rayleigh–Plesset equation. By combining the Rayleigh–Plesset equation with a mass and energy balance over the bubble, the temperature and pressure in the bubble can be calculated [134–136]. The model also describes the dynamic movement of the bubble wall, which results in a calculated radius of the cavitation bubble as a function of time (see Figure 21.9). The explosive growth phase and the collapse phase of the bubble can be distinguished clearly. Moreover, when dynamic effects are more important than the surface tension, the cavitation threshold can be calculated with the dynamic model, while the Blake threshold pressure cannot be used in these conditions. 21.3.2

Radical Formation by Cavitation

Because of the extreme conditions during a cavitation event, radicals can be formed. Several parameters affect cavitation and thereby the polymerization reaction, since the radical formation rate is directly influenced by the cavitational collapse. The number of radicals formed due to sonification is a function of the number of cavities created and the number of radicals that are formed per cavitation bubble. The bubble wall velocity during collapse and the hot-spot temperature determine the rate at which radicals are formed, both inside and outside a single bubble. These two parameters depend on the physical properties of the liquid as well as on the physical and chemical processes occurring around the cavity. The most important properties and processes occurring in a cavitation bubble are depicted schematically in Figure 21.10. The number of cavities is determined, for instance, by the impurities in the liquid, the static pressure, the ultrasound intensity, and the vapor pressure. This emphasizes the complexity of the influences on the overall

1065

1066


Fig. 21.10. Schematic representation of a cavitation bubble, including the processes and physical properties that determine the cavitation process.

radical formation rate [137, 138]. In the following sections, the effect of the most important parameters will be discussed. Reaction temperature When the reaction temperature is altered, the liquid properties will change. Although these properties (viscosity, surface tension, sound velocity, vapor pressure, and so on) all have an influence on the chemical effect of cavitation, the change in vapor pressure dominates the other liquid properties. As the temperature is raised, the vapor pressure in the bubble is increased, which cushions the implosion of the cavity. This results in a lower local temperature inside the cavity at higher overall temperatures. Consequently, fewer radicals are formed per cavitation bubble. On the other hand, a higher vapor pressure can lead to easier bubble formation due to the decrease in the cavitation threshold. In most cases, however, an increase in reaction temperature will result in an overall decrease in the radical formation rate. Therefore, ultrasound-induced reactions exhibit the opposite behavior to common radical reactions [139]. Static pressure A high static pressure can prevent the formation of cavitation bubbles, as the Blake threshold pressure increases. This implies that fewer or no cavitations are formed at higher static pressures. To counteract this effect a higher acoustic pressure is required, which will result in a more violent collapse of a cavitation bubble. Viscosity At increasing viscosities the growth and collapse of a cavitation bubble is retarded, due to the higher drag of the liquid. At a certain viscosity the drag force becomes too high and the bubble has insufficient time to grow to a critical radius.


If this radius is not reached, no collapse will occur. Additionally, the higher viscosity slows down the collapse, which makes it possible for heat to be transferred to the liquid. Due to this heat transfer, lower hot-spot temperatures are reached and subsequently fewer radicals are being generated. Ultrasound intensity At first the radical formation rate will increase to a maximum with increasing ultrasound intensity [137]. This is caused by the higher cavitation intensity per bubble and the larger number of cavitation bubbles. At ultrasound intensities that are too high, however, a cloud of cavitation bubbles is formed near the ultrasound source, due to which the pressure wave is no longer transmitted efficiently to the liquid. As a result, the cavitation intensity and consequently the radical formation rate decrease with a continued increase in intensity [137]. An optimum radical formation rate with ultrasound intensity can thus be found. Ultrasound frequency The frequency of ultrasound has a significant effect on the cavitation process. At very high frequencies (> 1 MHz), the cavitation effect is reduced as the inertia of a cavitation bubble becomes too high to react to fastchanging pressures. Most ultrasound-induced reactions are therefore carried out at frequencies between 20 and 300 kHz. Typically a frequency of 20 kHz is used for ultrasound-induced polymerizations and polymer scission reactions [140]. 21.3.3

Cavitation-induced Polymerization

Generally, free-radical polymerization consists of four elementary steps: initiation, propagation, chain transfer, and termination. When ultrasound is used to initiate polymerization, radicals can be formed both from monomer and from polymer molecules. This implies that due to radical formation by polymer scission an additional elementary step is involved in ultrasound-induced polymerization, as indicated in Figure 21.11. The radicals from polymer and monomer are generated by two different mechanisms. The monomer molecules are dissociated by the high temperatures inside the hot-spot, whereas the polymer chains are fractured by the high strain rates outside the bubble [141]. The majority of the radicals in an ultrasound-induced polymerization reaction originate from the polymer chains [142]; see Figure 21.12. It has to be noted that the radicals are only formed in the immediate vicinity of the ultrasound source where cavitation occurs. Subsequently, these radicals are dispersed throughout the reactor. In the literature several types of ultrasound-induced polymerizations have been reported, namely bulk, precipitation, and emulsion polymerization. Bulk Polymerization Most ultrasound-induced bulk polymerizations are performed at room temperature [143]. This low temperature is chosen because radical formation induced by 21.3.3.1

1067

1068


Initiation M + Cavitation

2 R•

Ri = 2 k dmon [ M ]

Propagation

M n• + M Chain Transfer to monomer M n• + M

to polymer M n • + Mm Termination by combination M n• + Mm•

by disproportionation M n• + Mm• Polymer Scission M n + Cavitation

Mn+1

R p = k p [ M ] [ M n .]

Mn + M•

Rctm = k ctm [ M ] [ M n. ]

Mn + Mm•

Rctp = k ctp [ M n . ] [ M m ]

Mn+m

Rtc = k tc [ M n .] [ M m .]

Mn + Mm

Rtd = k td [ M n. ] [ M m .]

Mm• + Mn-m• Rd = 2 k dpol [ M n ]

Fig. 21.11. Reaction mechanism of ultrasound-induced radical polymerization, assuming intrinsic polymerization and avoiding thermal initiation.

ultrasound appears to be more efficient at lower temperatures [128]; see also Section 21.3.2. In contrast to conventional thermal initiators such as potassium persulfate, ultrasound can initiate a polymerization reaction at ambient temperatures. It should be noted, however, that the propagation rate increases with an increasing temperature. Therefore, a suitable well-tuned system has to be chosen to optimize ultrasound-induced polymerization. In the literature, several bulk polymerizations initiated by ultrasound have been reported; examples are the polymerization of methyl methacrylate [116, 144, 145], ethyl methacrylate [146], n-butyl methacrylate [117, 141], and styrene [147]. Since the basic reaction kinetics are well known [144, 148] the ultrasound-induced polymerization of MMA is by far the most studied system. Typically, the resulting molecular weight of the PMMA obtained is in the range 1 10 5 –6 10 5 g mol1 . An important parameter during ultrasound-induced bulk polymerizations is the viscosity. As the reaction proceeds, the polymer chains formed cause a dramatic increase in the viscosity, resulting in slower growth and collapse of the cavity. As

3.0x10

-9

2.5x10

-9

2.0x10

-9

1.5x10 1.0x10

5.0x10

MMA PMMA

3.5x10

-4

3.0x10

-4

2.5x10

-4

2.0x10

-4

1.5x10

-4

1.0x10

-4

5.0x10

-5

-9

-9

-10

0

1

2

3

4

Kd PMMA (1/s)

Kd MMA (1/s)


5

Weight % PMMA Radical formation rate constants from MMA and PMMA as a function of the weight percentage of polymer dissolved at a temperature of 293 K and an ultrasound intensity of 62 W cm2 [142]. Fig. 21.12.

the cavitations become less effective, the radical formation rate both from monomer and from polymer will decrease. Subsequently, the reaction rate decreases as shown in Figure 21.13. At a conversion of approximately 20% the collapse is no longer sufficiently strong to induce hot-spot temperatures that are able to generate monomeric radicals. Moreover, the strain rates outside the collapsing bubble are not large enough to produce polymeric radicals by polymer scission [116]. As a consequence, the polymerization ultimately stops at this conversion, which represents a serious drawback for the development of ultrasound-induced bulk polymerization toward industrial application. Precipitation Polymerization As described in Section 21.3.3.1, ultrasound-induced bulk polymerizations are limited to relatively low conversions, because a strong increase in viscosity upon reaction hinders cavitation. In order to obtain higher conversions, precipitation polymerization forms a potential solution. Because the polymer produced is precipitated from the reaction medium, the viscosity and consequently the radical formation rate are expected to remain virtually constant. From this perspective, liquid carbon dioxide is a suitable reaction medium, because most monomers have a high solubility in CO2 , whereas it exhibits an anti-solvent effect for most polymers. Moreover, CO2 is regarded as an environmentally friendly compound, which is nontoxic, nonflammable, and naturally abundant. Since higher pressures are required for CO2 to act as an anti-solvent [56, 149], the possibility of ultrasoundinduced cavitation in pressurized carbon dioxide systems has been studied [73, 150]. 21.3.3.2

1069


10.0

Conversion (%)

1070

298 K 313 K

7.5

5.0

2.5

0.0

0

1

2

3

4

Time (h) Fig. 21.13. Conversion development of an ultrasound-induced bulk polymerization of MMA at 298 and 313 K [150]. The broken lines indicate the initial reaction rate.

During pressurization of a liquid, the Blake threshold pressure [Eq. (4)] increases, which implies that higher acoustic pressures are needed to produce cavitations. Obviously, no cavitation occurs when the Blake threshold pressure exceeds the maximum acoustic pressure that can be applied with the currently available equipment. The vapor pressure of the liquid, however, can counteract the static pressure (see Figure 21.14). As a result of this, the Blake threshold is reduced in liquids with a high vapor pressure, such as CO2 , ethylene, and ammonia. This enables cavitation at increased static pressures [73, 150]. Additionally, the dynamic movement of the bubble in pressurized CO2 has been calculated using the dynamic bubble model based on the Rayleigh–Plesset equation. According to the calculations, the bubble exhibits a similar movement to that of water at ambient pressure. To validate these simulations, cavitation experiments have been performed in pressurized CO2 –MMA systems using the radical scavenger 1,1-diphenyl-2picrylhydrazyl [151]. The radical formation rate appears to be in the order of 1:5 104 s1 , from which no noticable difference occurs upon varying the MMA–CO2 ratios. Moreover, ultrasound-induced polymerizations of MMA in CO2 -expanded MMA have resulted in high molecular weight polymers [151]. Emulsion Polymerization The generation of radicals by ultrasound can also be applied in emulsion polymerization, which comprises a free-radical polymerization in a heterogeneous reaction system, yielding submicron polymer particles in a continuous aqueous phase. Ultrasound can be applied for emulsification purposes as well as at higher conversions in the emulsion polymerization process. 21.3.3.3


70 60

Pressure (bar)

50 40

Blake threshold H2O Blake threshold CO2 Vapour pressure H2O Vapour pressure CO2

30 20 10 0 275

280

285

290

295

300

305

Temperature (K) Fig. 21.14. Calculated Blake threshold and vapor pressure for water and carbon dioxide at 58.2 bar [73].

At the beginning of the polymerization, emulsification and nucleation govern the course of the process. The monomer droplets have to be small enough to provide a negligible resistance to monomer transport from the droplets through the aqueous phase to the growing polymer particles. Only in the case of sufficient emulsification, can intrinsic polymerization kinetics be assumed. This implies that in conventional emulsion polymerization vigorous stirring is required to disperse the monomer droplets. When ultrasound is applied as an energy source for emulsification, a very uniform emulsion of small monomer droplets is obtained without additional stirring. The high implosion velocity of the cavitation bubbles ensures that the monomer droplets are well dispersed. A special case involves mini- or microemulsion polymerization. Depending on the surfactant system used and the emulsification induced by ultrasound, the monomer droplets become so small that they can serve as nucleation loci. In ordinary emulsion polymerization, the main locus of nucleation is within the monomer-swollen micelles. Typically, polymer particles produced by ultrasound-induced emulsion polymerization have a diameter of approximately 50 nm [152, 153]. During emulsion polymerization induced by ultrasound, the redundancy of initiator is advantageous in terms of process control and safety. Moreover, initiator residues do not contaminate the product. In contrast to ultrasound-induced bulk polymerizations, high conversions can be obtained in ultrasound-induced emulsion polymerization [154, 155]. Since a heterogeneous reaction system is involved, in which the polymer is insoluble in the continuous aqueous phase, the viscosity of the water phase does not increase upon reaction. The cavitation events occur in the continuous phase, producing radicals mainly from water and surfactant molecules

1071


[156]. Subsequently, the oligomeric radicals enter the monomer-swollen polymer particles and continue to polymerize. Examples of ultrasound-induced emulsion polymerization are described for styrene [153, 155, 157], methyl methacrylate [152, 156, 158–161], and n-butyl acrylate [162] and the copolymerization of vinyl acetate and butyl acrylate is also reported [154]. 21.3.4

Cavitation-induced Polymer Scission

In terms of product properties, the molecular weight distribution is an important characteristic of polymers. In the polymer industry a post-processing step is often applied to alter the molecular weight of the polymers, for example for the peroxideinduced degradation of polypropylene [163]. In this process, fracture of the polymer chain occurs at a random site. An alternative method is ultrasound-induced polymer scission, which involves a much better-controlled, nonrandom process [119]. This enables relatively straightforward production of the desired molecular weight as well as the formation of block copolymers, as described in Section 21.3.5. It has been shown that ultrasound-induced polymer breakage is a direct consequence of cavitation because, under conditions that suppress cavitation, no degradation is observed. In this nonrandom scission process the polymer is fractured at the center of the chain [164, 165]. This is clearly shown in Figure 21.15, in which a polymer with a molecular weight of 90 kg mol1 is produced from a polymer with an initial molecular weight of 180 kg mol1 . It is often thought that the extreme temperatures inside the bubble upon implosion contribute to the degradation.

6 5

dwt/d(logM)

1072

4

0 min 5 min 10 min 20 min 40 min

3 2 1 0 4 10

5

10

Molecular weight (g/mol) Fig. 21.15. Molecular weight distributions of an ultrasoundinduced polymer scission of PMMA into MMA with an initial Mn of 18:0 10 4 g mol1 [151].


However, temperature degradation implies a random process and thus does not explain the scission in the middle of the chain. Ultrasound-induced chain fracture arises from the high strain rates on the polymer chain upon implosion. The nonrandom fracture in the middle of the chain by cavitation can only occur when the polymer chain is in a nonrandom conformation and thus completely stretched [166, 167]. In the absence of an external force a polymer chain in solution is randomly coiled. Upon bubble collapse, the entire molecule will move along with the fluid. However, due to the velocity profile near the cavities, friction between the polymer chain and the liquid will occur. Under sufficiently strong flow conditions, the solvent drag force causes extension of the polymer molecule and finally full stretching. If the polymer chain is stretched, the maximum stress due to the drag force will be in the center of the polymer chain, which is in analogy with flow-induced polymer scission [168]. If the drag force on the stretched polymer molecule exceeds the bond strength, scission will occur in the middle of the chain; otherwise the chain is not fractured and a limiting molecular weight (Mlim ) is reached [169]. This limiting molecular weight is no longer fractured as the forces acting on the chain are smaller than the force required to break a bond. The kinetics of ultrasonic scission can be directly ascribed to the implosion velocity of the cavitation bubbles and the polymer that is fractured. The scission rate and the limiting molecular weight are thus influenced by the ultrasonic wave, the solvent properties, and the polymer structure. The influence of ultrasound intensity, liquid viscosity, and liquid temperature on the degradation rate and the limiting molecular weight have been studied by Price et al. [170]. A higher intensity results in a faster scission and a lower limiting molecular weight. Lower temperatures and lower viscosities have similar influences on the scission rate and Mlim . These three effects can be ascribed to the higher implosion velocity, which induces higher strain rates around the bubble and therefore lead to a lower limiting molecular weight and a higher scission rate. The initial molecular weight of the polymer influences neither the implosion velocity of a cavity nor the limiting molecular weight that is reached. However, the molecular weight has a large influence on the fracture rate of the polymer chains. A higher scission rate is observed for polymers with a higher initial molecular weight [170]. This is a consequence of the higher solvent drag force on the longer chains. As a result, narrow molecular weight distributions can be produced by ultrasound-induced polymer scission, because a high molecular weight polymer is fractured faster. Independently of the initial molecular weight, a similar limiting molecular weight is obtained for different experiments, as the final conditions are equal after several breakage events of the higher molecular weight polymers [167]. 21.3.5

Synthesis of Block Copolymers

An interesting application of ultrasound-induced scission and polymerization is the synthesis of block copolymers, which are used in many applications where dif-

1073

1074


Ph

Ph

Ph

ultrasound

Ph

)n

)n ultrasound

Ph

Ph

)n•

Ph

Ph

)n (

)n

•

)n

Fig. 21.16. The production of a block copolymer from two homopolymers with ultrasound [117].

ferent polymers are connected to yield a material with hybrid properties [171], for instance as a compatibilizing agent between immiscible polymers [172]. Anionic and living radical polymerization reactions are the most important techniques for synthesis of block copolymers. Alternatively, ultrasound can be used to produce these block copolymers, for which two different methods are applicable. The first route to synthesis of block copolymers by ultrasound starts with the dissolution of a homopolymer in a different monomer [173]. Subsequently, ultrasonic scission of the polymer chains generates polymeric radicals, which initiate the polymerization reaction with the monomer present. In this way ultrasound provides the controlled formation of block copolymers, examples of which are polyethylene with acrylamide [173], and PMMA with styrene [174]. Secondly, dissolving two different homopolymers in a nonreactive solvent can lead to block copolymers as well (see Figure 21.16). In this case, the polymeric radicals generated have to undergo termination by cross-combination [170, 175]. If no cross-combination occurs, the original homopolymers are reproduced again. The advantage of the second method is that block copolymers can be produced from homopolymers of which the polymer–monomer systems are immiscible. Using this approach the synthesis of the block copolymer of polystyrene and poly(methyl phenyl silane) [170] as well as the block copolymer of poly(vinyl chloride) and poly(acrylonitrileco-butadiene) [171] has been described.

21.4

Concluding Remarks and Outlook for the Future

In this chapter, supercritical carbon dioxide and ultrasound have been discussed as two examples of emerging technologies in polymer processes. Replacing the traditional organic solvents with supercritical carbon dioxide provides possibilities for

21.4 Concluding Remarks and Outlook for the Future

the development of sustainable polymer processes. Additionally, the tunability of the solvent and the relatively easy separation of the polymer from the reaction medium are important advantages. However, applying scCO2 as a clean solvent in polymer processes is not the simplest route, because it implies, among other things, high-pressure equipment, complex phase behavior, new measurement techniques, and the development of novel process concepts rather than extending the conventional technologies. Currently, there is a lack of integrated tools between the research on a laboratory scale and the industrial-scale application, mainly caused by the absence of pilot-scale facilities. Recent innovative tools, such as supercritical reaction calorimetry [176] on the liter scale, could fill this gap by allowing the determination of engineering, process, scaleup, and safety data [177]. Additionally, several process design calculations have shown that polymer processes based on scCO2 technology can be economically feasible, depending on the value of the product and the process conditions. Moreover, further developments will reduce costs of supercritical application substantially. It is expected that the major application of supercritical carbon dioxide will first be in the food and pharmaceuticals industry because of additional marketing advantages, such as the GRAS (generally regarded as safe) status. Nevertheless, the fact that DuPont is commercializing the production of fluoropolymers in scCO2 illustrates the possibility of applying supercritical fluid technology in polymer processes as well. In addition, the long-existing ldPE tubular process (approximately 2500 bar, 600 K) proves that a high-pressure polymerization process performed on a large scale can survive in a highly competitive field. With respect to controllability of polymerization processes, ultrasound has significant potential as a clean and safe technology. After the production of most types of polymers, catalyst and initiator residues contaminate the product. Since ultrasound generates the radicals in situ, no initiator or catalyst is required to start a polymerization reaction. In addition, high-temperature free-radical polymerization is emerging in order to avoid the use of initiators that will remain present in the polymer chain at the end of the process [178]. The mechanism of thermal initiation at high temperatures is still not well described. However, in analogy with initiation by ultrasound at low temperatures, high-temperature processes open the route for the sustainable production of polymers composed only of monomer units. An additional advantage of ultrasound is the intrinsically safe operation, because turning off the electrical power supply will immediately stop the radical formation and consequently the polymerization reaction. Besides polymerization, polymer scission can also occur through irradiation with ultrasound, due to the rapid flow around a cavitation bubble. At a sufficiently small chain length, further fracture of the polymer is prevented. In this way ultrasound-induced polymer scission provides an additional means to control the molecular weight of the product. At the moment the major challenge of ultrasound-induced polymer processes is the scaleup, mainly in terms of energy conversion of the ultrasound generator to the probe. Although no large-scale industrial polymerization process based on ultrasound exists yet, commercial applications in other fields such as ultrasound cleaning and sterilization prove that ultrasound is a readily available technique

1075

1076


and relatively simple to implement in existing industrial equipment. Still, the application of ultrasound for polymerization purposes requires a thorough, multidisciplinary understanding of both ultrasound parameters and liquid properties, including physics, chemistry, and engineering. In general, the various subjects in this chapter have been addressed from an engineering point of view, for which industrial application is one of the important issues. Although significant effort is being put into new developments by a large number of research groups, the development trajectory from the concept idea, via the laboratory, bench, and pilot scale, toward industrial implementation is often long and not easy. To reduce the boundaries between the academic approach and industrial practice, collaboration between industrial R&D, research institutes, and universities is essential to reduce costs and to exploit existing know-how and experimental facilities, as well as to reduce the development time. Given the economics of an emerging technology as compared to long-existing processes, it is a challenge to implement new process concepts at reasonable costs. For these reasons, in the short term the number of large-scale industrial polymer processes based on supercritical fluid or ultrasound technology will be limited, for which stimulation from government and research consortia can contribute substantially to facilitate the development trajectory. Nevertheless, the progress made in research today will enable the development of sustainable and well-controlled polymer processes for the future.

Acknowledgments

The author thanks Marc Jacobs and Martijn Kuijpers for their contribution to this chapter.

Notation

AUS f IUS P0 PA PA; max PB Pc Pv Q Q US R0 Tc Tg

surface area of ultrasound source [m 2 ] frequency [Hz] ultrasound intensity [W/cm2 ] external pressure [bar] acoustic pressure [bar] maximum acoustic amplitude [bar] Blake threshold pressure [bar] critical pressure [bar] vapor pressure [bar] quadrupole moment [J 1=2 m 5=2 ] ultrasound power transferred to liquid [W] equilibrium radius of cavitation bubble [m] critical temperature [K] glass transition temperature [K]

References

speed of sound in medium [m s1 ] polarizability [m 3 ] potential energy [J] dipole moment [C m] speed of sound in medium [m s1 ] density [kg m3 ] surface tension [N m1 ]

v a Gij m v r s

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1081

1083

Index a AAS, see Atomic absorbtion spectroscopy AES, see Atomic emmision spectroscopy Ab-initio calculation 781 Abstraction reaction 784 A–C, see Air-to-close valve Accelerated weathering test 824 Accumulation 556, 580 Acetaldehyde 89 Acid-catalyzed esterification and alcoholysis 86 Acidolysis 86, 100 Acid strength 900 Acid value 860, 865 Acoustic amplitude 1064 Acoustic attenuation spectroscopy 624 Acrylate resins 889 Acrylated oils 891 Acrylic fibers 946 Acrylic solutions 951 Acrylics 951 Action spectrum 800 Activated anionic mechanism 348 Activation energy 565, 744, 747 Activation spectrum 799 Activation volume 744 Activity 18 Activity coefficient 19, 29, 34 Actuator 629, 652 Adaptive control 664, 668 A/D converter 626 Additives 836 Adiabatic polymerizer 662 Adiabatic temperature rise 308, 546, 565 Adsorbents 989 Advanced control 657 Advancement process 109, 854 AES, see Atomic emission spectroscopy After-oils 930

Aging prevention for plastic material 818 Agitation requirement 289 Agitator 289 Agitator power 534 Air-gap spinning 921, 956 Air-to-close (A–C) valve 652 Air-to-open (A–O) valve 652 Alarm 589, 626 Alcohol-isocyanate reaction 893 Alcoholysis, acid-catalyzed 86 Alcoholysis process 857 Alkyd constant 858 Alkyd emulsions 861 Alkyd resins 855 Aminolysis reactions 100 Amino resins 843 Amorphous orientation 919 Amorphous phase 682 Analog filter 627 Analytical pyrolysis 790 Analytical techniques, overview 1016 Analytical tools 12 Anhydrides 86 Anionic polymerization 325 Anisotropic crystallization 918 Anisotropy 886 Antagonistic effect 826 Anti-misting agent 714 Antioxidant 818 Antiozonant 824 Antirad 810 Anti-reset windup 643 A–O, see Air-to-open valve Apolar rubbers 896 Apparent shear rate 934 Apperant equilibrium constant 99 Applications of acrylics 951 Applications of aramids 960 Applications of carbon fibers 966


1084

Index Applications of gel-spun PE 964 Applications of PVA 953 Aramid 921, 956, 959 Aramid2/Kevlar2 5 Aspect ratio 888 Association behavior 326–327, 330, 715 Association number 328–329 Atactic polystyrene 721 At-line analysis 1015 At-line conversion measurement 1020 Atomic absorbtion spectroscopy (AAS) 1017 Atomic emission spectroscopy (AES) 1017 Attraction term 37 Autoacceleration 762 Automatic control 650 Auto-oxidation mechanism 782 Autoxidative drying 858 Auxiliary devices 596 Average branching density 491 Average copolymer composition 478 Average molecular weight 123, 269 Average number of radicals per particle 258, 260 Average sequence length 474 Average value 597 Axial-flow 289 Azeotrope 859 Azeotropic distillation 865 Azo initiator 154 Azomethine structure 841

b Back-biting 67, 189 Backflow 648 Backhole 927 Balances 604 Base-mediated curing 901 Basicity of HAS 821 Batch control 669 Batch copolymerization 478 Batch emulsion polymerization 251 Batch polymerization 204 Batch process 195, 579 Batch reactor 335, 669 Bead 216, 233, 238 Beer-Lambert law 607, 618 Bellows pressure gauge 602 Belt balance 605 Belt process 353 Bernoulli equation 607, 611 Bias 640 Bidirectional fiber 879 Bifunctional initiator 334, 1035 Bimetallic thermometer 601

Bimodal distribution 343 Bimolecular deactivation 386 Binary format 626 Binomial distribution 456 Birefringence 919 Birth conversion 495 Bisphenol-A 850 Bisphenol-A glycidyl resins 841 Black body 601 Black orlon 966 Blake threshold pressure 1065 Blending process 1058 Block copolymer 335, 338, 342, 344, 351, 690, 714, 1073 Block copolymer application 691 Block diagram 629 Block length distribution 1038 Boltzmann superposition principle 730–731 Boltzmann’s law 1024 Bonds 114 Bottom-up technique 997 Boublik-Mansoori hardsphere equation 45 Branch point 504, 711 Branched architecture 472, 502 Branching 272, 445 Branching behavior 700 Branching distribution 672 Branching moment distribution 454 Branching parameter 1040 Branching process theory 13 Breakage 218, 229 Breaking strength of a covalent bond 814 Breakthrough in SEC systems 1042 Brittleness 855 Bubble coalescence 974 Bubble growth 974, 976 Bubble nucleation 78, 974 Bubble pressure method 622 Bulk (compression) modulus 725 Bulk molding compound (BMC) 883, 885 Bulk polymerization 1067 Buoyancy method 606 Burgers vector 747 Butterfly valve 654

c Cactus 127 Cage effect 765 Calorimetry 297 Capacitance measurement 608 Capillary electrophoresis 1037 Capillary hydrodynamic fractionation (CHDF) 624 Capillary number 221

Index Capillary viscometer 619 Capsule 603 Carbenium ion 352 Carbon fiber 791, 956, 965, 966 Carbon nanotubes 692 Carbon NMR 1024 Carboxilic anhydrides 853 Carboxylic acids 853 Carpet yarn 916–917, 926 Cascade control 650, 660 Cascade theory 480–482 Casting 886 Casting system 549 Catalan number 484 Catalysis by metallic compounds 87 Catalyst 451, 842, 1075 Catalyst activity 377 Catalyst deactivation mechanism 375 Cationic curing 900 Cationic photoinitiator 897 Cavitation 654, 1063 Cavitation-induced polymerization 1067 Ceiling temperature 172, 565 Cellulose acetate 944–945 Cellulose rayon 965 Cellulose triacetate 921 Central filter 926 Centrifugal casting 882 CFD, see Computational fluid dynamics CHDF, see Capillary hydrodynamic fractionation Chain branching 761 Chain conformation 692 Chain growth 564 Chain-growth polymerization 11 Chain mobility 765 Chain orientation 766 Chain retraction 741 Chain scission 190, 714, 771, 803, 809 Chain transfer 166 Chain-transfer constant 167 Chain-transfer mechanism 374 Chain-transfer reaction 775 Chain transfer to hydrogen 386 Chain transfer to polymer 188 Chains of beads and springs 697 Chain walking mechanism 382 Characteristic ratio 723 Charlesby-Pinner plot 809 Check valve 648 Chemical analysis 1015 Chemical composition distribution (CCD) 390, 394, 1037 Chemical defect 795

Chemical defoaming 987 Chemical degradation 758 Chemical derivatization 774 Chemical distribution 341 Chemical engineering 14 Chemical heterogeneity 339 Chemical reaction engineering 5 Chemical resistance 874 Chemicrystallization 766 Chemiluminescence 778, 789 Chemometric analysis 301 Chromatographic retention 1035 Chromel/alumel electrode 601 Chromophore impurities 795 Chromophores 793 Classes approach 444 Closed-loop control 305 Closure problem 438 Cloud-point 1051 CMC, see Critical micelle concentration Coagulation 294, 946, 959 Coagulation risk 987 Coagulative nucleation 253, 267 Coating process 1059 Coaxial device 620 Cobalt irradiation 806 Co-condensation 845 Cold drawing 932 Cold-press molding 883 Column hold-up volume 1035 Combined scission/branching 501 Comonomer distribution 344 Compartmentalization of radicals 254 Compatible blends 811 Complex flowsheets 205 Complex modulus 731 Composite formation 692 Composition measurement 620 Composition of copolymer 292 Composition of monomer 292 Compositional drift 670 Comprehensive two-dimensional LC instrumentation 1043 Compressibility factor 43 Compressible flow 613, 625 Compton scattering 807 Computational fluid dynamics (CFD) 205, 290 Computer simulation 780 Concentrated solutions and melts 698 Concentration of monomer in the polymer particles 258 Concentration of radicals in the aqueous phase 262

1085

1086

Index Concentration-sensitive detector 1033 Concerted multidisciplinary approach 15 Condensation 11 Condensed mode 421 Conditional Monte Carlo sampling 502 Conductivity measurement 621 Cone-and-plate viscometer 620 Connectivity 472, 482, 502 Contact ion pairs 324 Contact thermometer 599 Continuos stirred tank reactor (CSTR) 584 Continuous lamination 882 Continuous operation 536 Continuous polymerization 204 Continuous steam stripping 987 Control loop 629 Control of feed 587 Control system 629 Controlled degradation and crosslinking 812 Controlled depressurization 588 Controlled reaction 546 Controlled thermal degradation 791 Controlled variable 628, 639 Controller 628, 639 Controller action 640 Controller gain 640 Controller output 640 Controller tuning 644 Conversion of monomer 291 Converter 626 Cooling 570, 689 Cooling belt 842 Cooling coil 570 Cooling experiment 573 Cooling rate 686 Cooling speed 929 Coordination catalyst 373 Coordination polymerization 12 Coordinator 7 Copolymer 225, 241, 243 Copolymer averaged rate, coefficient for propagation 183 Copolymer composition 338 Copolymerization 179, 338, 413, 451, 473 Copolymer sequence distribution 181 Copolymer-solvent systems phase behavior of 48 Copper/constantan electrode 601 Corded yarn 913 Coriolis flowmeter 617 Couette device 961 Couette-Taylor reactor 288 Coulter-counter particle size analyzer 623

Counter 609 Covalent polarized bond 324 Covalent species 352 Crack advance 750 Crack formation 769 Crack tip 750 Cracked specimen 750 Craze-bulk interface 748 Craze fibril 749 Craze length 748 Craze nucleation 749 Craze stress 749 Crazing 748 Creep behavior 709 Creep response 708 Critical-angle refractometer 621 Critical length for entry of radicals 267 Critical liquid chromatographic separation 1037 Critical micelle concentration (CMC) 264, 622 Criticality classes 558 Cross-aggregation 329 Crosslink definition 177 Crosslink density 834 Crosslinked polymer 748–749 Crosslinkers 467 Crosslinking 272, 771, 809, 868, 871 Crosslinks 834 Cross model 703, 740 Cross-tie fibril 750 Crystaf, see Crystallization analysis fractionation Crystal thickness 689 Crystalline domains 688 Crystalline morphology 688, 690 Crystalline orientation 919 Crystalline phase 682, 766 Crystalline polymer 681, 688 Crystallinity 1053 Crystallizability 682 Crystallization 680, 686, 918 Crystallization analysis fractionation (Crystaf ) 369 Crystallization enhancement 687 Crystallization rate 687 Crystallization reduction 687 Crystallization temperature 686 CSTR, see Continuos stirred tank reactor Cumulative distribution 407 Cure 835 Curing 564, 871 Cyclic monomers 346 Cyclization of side chains 763

Index

d D/A converter 626 Damping function 741 Data highway 651 DCS, see Distributed control system Deadband 647 De-aeration of the melt 924 Dead chain 467 Dead time 636, 639 Deborah number 708 Decomposition 337, 556, 564 Defoaming agent 987 Degradation detection method 767 Degradation reaction of nylon-6 100 Degree of aggregation 329 Degree of branching distribution (DBD) 1040 Degree of crystallinity 766 Degree of polymerization 126, 159 Degree of superheat 976 Delustering 916 Dendrimers 712 Denier 914 Denisov cycle 820 Dense grafting 712 Densimetry 297–298 Density balance 617 Depolymerization 356, 566, 973 Depropagation 172, 973 Depropagation in copolymerization 188 Derivative controller (D controller) 641 Derivative kick 642 Derivative mode filter 642 Derivative time 641 Detection time 589 Detector for SEC 1032 Development process 2 Development trajectory 1076 Deviation 597 Deviation variable 632 Devolatilization 971 Diacids 865 Dialkyltin catalyst 89 Diaphragm pressure gauge 603 Diaphragm valve 652 Dibutyltin dilaurate (DBTDL) 893 Dicyclopentadiene (DCPD) 873, 903 Diethylene glycol 89 Diethylenetriamine 853 Differential refractometer 621 Diffusion constant 764 Diffusion control 230 Diffusion-controlled chain termination 165, 229

Diffusion-controlled reaction 190 Diffusivity 1049 Digital control system (DCS) 642 Diglycidyl ether of bisphenol-A (DGEBPA) 850, 873 Digraph 114 Dilatant 619 Dilatometer 619 Dilute solution 696 Dimensional temperature dependance 1003 Dimensionality 431 Dimethylol urea rate of formation 104 Diols 864 Dioxane 90 Dipole moment 1050 Direct esterification process 91 Dislocations 747 Dispersion polymerization 256, 1057 Dispersion term 45–46 Disproportionation 166, 440 Distributed control system (DCS) 627, 651, 668, 672 Distribution coefficient 1034 Distribution of active polymer chains 269 Distribution of feed 535 Distribution of inactive chains 269 Distributive properties 431 Disturbance variable 628, 640 Doi and Edwards model 738, 740 Doolittle’s equation 734 Dormant end group 345 Double-screw reactor 356 Drag-reducing agent 714 Draw ratio 748, 932–933 Drawing 931 Drop coalescence 222, 235 Drop-in technology 373 Drop mixing 224, 240 Drop size 216 Drop size distribution 222–223, 227 Drop stabilizer 214, 223 Dry spinning 921, 944, 955 Dual-exposure photoembossing 1007 Dulling agent 916 Dumbbell model 697, 703 Dumping 588 Dyads 64 Dyeing 1061 Dynamic control 553 Dynamic head-space 1023 Dynamic light scattering 624 Dynamic modulus 731 Dynamic oscillatory flow 709 Dynamic SIMS 1026

1087

1088

Index Dynamic stability 584 Dyneema 962, 964 Dyneema process 962

e EKF, see Extended Kalman filter Elastic behavior 936 Elastic polymer 706 Elastically active network chain 121 Elastically active network junction 121 Elastomer 725, 727–729, 739, 901 Elastomeric green strength 715 Electro-chemical potential method 621 Electron beam accelerator 806 Electron beam lithography 996 Electron spin resonance (ESR) spectroscopy 776, 1017 Electronic balance 605 Electro-pneumatic valve positioner 650 Electrospray ionization (ESI) 1027, 1033 Electrostatic spraying 867 Eleostearic acid 856 Elimination reaction 324, 337 Elongation rate 936 Elongational viscosity 936 Embrittlement 749 Emergency cooling 575, 587 Emerging technologies 7 Emulsification 1071 Emulsion polymer 985 Emulsion polymerization 249, 980, 1057, 1070 Emulsion polymerization reactor 286, 305 End conversion 494 End-to-end distance 723, 738 End-to-end distance of polymer coil 693 End-to-end vector 723, 727, 738 Energy dissipation by the stirrer 578 Energy dose 1003 Energy requirement 8 Energy transfer 808, 811 Enforced cooling 928 Enolization 345 Entanglement 729, 738–740, 749 Enthalpy balance 292 Entropic free volume model 36 Environmental and safety issues 15 Environmental aspects 13 Environmentally friendly 255 Enzymatic polymerization 1057 EPDM, see Ethylene-propylene-diene rubbers Epichlorohydrin (ECH) 108, 850 Epidemic model 790 Epitaxial crystallization 687

Epoxy acrylates 891 Epoxy resins 108, 849 Epoxy-novolacs 841 Equal percentage 652 Equation of state model 39 Equilibrium equations 259 Equilibrium monomer concentration 347, 354 Equilibrium morphology 274 Error 628, 640 Ester enolate anion 345 Esterification 86, 863 Esterification acid-catalyzed 86 Ether bridges 845, 839 Ethoxylation 351 Ethylene 451 Ethylene oxide 349 Ethylene-propene-diene (EPDM) 368, 902 5-Ethylidene-2-norbornene (ENB) 903 Evaporative light-scattering detector (ELSD) 1032 Exchange of bonds 115 Exchange reaction 66 Excited state quencher 808 Exothermic reaction 541, 565 Expansion factor 613 Expansion thermometer 601 Exponential smoothing filter 627 Exposure time 74 Extended Kalman filter (EKF) 667 Extensional thickening 705 External impurity 795 Externally applied stress 767 Extraction of VOC 989 Extruder 75, 76, 549 Extrusion 816, 866, 923, 937

f Fail safe 587 Failure scenario 554 Falling-film devolatilizer 975 Falling-strand devolatilizer 975 False twist operation 916 Fatty acid 855 Fatty acid process 857 Feedback controller 628 Feedforward control 659, 666 FENE, see Finitely extensible nonlinear elastic Fiber modulus 915 Fiber polymers 920 Fiber reinforcement 886 Fiber tenacity 915 Fiber terminology 912 Fiber yarn 913, 917

Index Fibers, definition 912 Fibril 748 Fibril extension 748 Fibril formation 920 Fick’s second law 798 Filament 913 Filament number 915 Filament textile acrylics 952 Filament titer 914 Filament winding 882 Filament yarn 913 Fillers 692, 878 Film-formation temperature 273 Film-forming equipment 71 Filter constant 642 Filter package 926, 935, 938 Filtration 926 Finish 929 Finish roll 930 Finite element method 432 Finitely extensible nonlinear elastic (FENE) dumbbell model 703 First dimension separation 1042 First-order system 632 First-order system plus dead time (FOPDT) 636, 645 First-shell substitution effect (FSSE) 64 Flame retarding agents 849 Flash evaporator 975 Flexibility 866 Flexibilization of amino-resins 849 Flexible foams 111 Float method 606 Flooding 575 Flory distribution 442 Flory-Huggins, multicomponent 70 Flory-Huggins theory 21, 30 Flory’s chain length distribution 387 Flow-induced degradation 817 Flow measurement 608 Fluidized-bed reactor 82, 417, 420 Fluorescence effect 1019 Fluoropolymer 1057 Foam 111 Foam formation 987 Foam-based devolatilization 974 Food preservation 812 FOPDT, see First-order system plus dead time Force balance 614 Force field calculation 782 Formaldehyde 108, 346, 838 Formalin solution 842 Formulation 250 Fouling 570, 925

Fourier analysis 954 Fourier number 955 Fourier transform IR spectroscopy (FTIR) 621, 1023, 1024 Fourier transform ion-cyclotron resonance spectrometer 1030 Fractionation 1059 Fracture 741–742, 748 Fracture behavior 751 Fracture resistance 740, 750–751 Fragment 472 Fragmentation 178, 190 Fragmentation in polymer analysis 1027, 1029 Free boiling 976 Free-bubble devolatilization 974 Free ions 324 Free-radical polymerization 1067 Free-radical polymerization, heterogeneous 12 Free-radical polymerization, homogeneous 12 Free-radical polymerization mechanism 157 Free volume 734–735, 764 FSSE, see First-shell substitution effect FTIR, see Fourier transform IR spectroscopy Fugacity 18 Fugacity coefficient 19, 42 Fully oriented yarn (FOY) 933, 940 Fumaric acid 871 Functional monomer 250 Functionality 835 Functionality-type distribution (FTD) 1034, 1041

g Galerkin h–p method 434 Gas chromatograph feedback controller 666 Gas chromatography (GC) 297–298, 621, 1022 Gas-phase process 416 Gas-phase reactor 420 Gas-phase stirred-bed reactor 417 Gate valve 656 Gauss normal differential distribution function 598 Gaussian chain 707 GC, see Gas chromatography Gel 272 Gelation 714, 837 Gel effect 193, 568–569 Gel permeation chromatography (GPC) 622, 664, 771, 1020 Gel point 128, 771, 835

1089

1090

Index Gel spinning 921, 961, 962 Gel-spun polyethylene 918 964 General-purpose finishes 930 Generation 487 Geometric similarity 295, 534, 536 G-factor 808 Glass filament 875 Glass transition 721, 732–733, 735, 745, 683, 764, 901 Glassy polymer 742–743, 750 Glycerol 856 Glycidyl group 841, 850 Glyptals 857 Gold distribution 333–334 GPC, see Gel permeation chromatography, see also SEC Grade transition 669 Gradient elution liquid chromatography 1038–1039, 1042 Graft copolymers 232 Graph theory 502 Graphite nanosheets 692 Graphitization 966 GRAS (generally regarded as safe) 1075 Gravimetric method 619 Gray 808 Green solvent 1048 Grotthus-Draper law 793 Group contribution Flory model 36 Group contribution methods 35

h Hagen-Poiseuille law 619 Half-bonds 114 Half-life 162 Half-sandwich metallocene 381 Hand lay-up process 882 Hansen solubility parameters 32 Hard sphere term 44 Hardening 835 Hardness 895 HAS, see Hindered amine stabilizer HCSTR, see Homogeneous continuous stirred tank reactor HDC, see Hydrodynamic chromatography HDPE, see High-density polyethylene Head-space gas chromatography 1023 Heart-cut 1041 Heat accumulation 561, 581 Heat balance 292, 554, 559, 562 Heat-balance calorimetry 303 Heat-flow calorimetry 303 Heat loss 562 Heat release 559

Heat removal 194, 560 Heat resistance 779 Heat-resistant polymer 780 Heat-sensitive substrate 869 Heat-shrinkable product 813 Heat stabilizer 975 Heat transfer 214, 227, 235–236, 242, 290, 539, 972 Heat-transfer coefficient 293, 560 Heat-transfer resistance 400 Heaviside step function 632 Helmholtz energy 44 Helmholtz free energy 726 Henry’s constant 974 Henry’s law 974 Heterogeneous kinetics 789 Heterogeneous nucleation 252, 264 Heterogeneous oxidation 789 Heterogeneous polymerization 1055 Heterogeneous reaction model 827 Heterolytic bond scission 760 HEUR, see Hydrophobically associting polymer Hexamethylenetetramine (HMTA) 838 Hexamethylolmelamine 841 Hierarchical approach 658 High monomer conversion 972 High-density polyethylene (HDPE) 366 High-energy radiation 805 High-impact polypropylene 417 High-impact propylene/ethylene copolymers 368 Highly branched popylethylene 1053 High-modulus high-strength fibers (HMHS) 917 High-pressure laminate 843 High-pressure liquid-liquid equilibria 25 High-pressure vapor-liquid equilibria 25 High-speed spin-draw winding (HSSDW) 941 High-speed spinning (HSS) 931 High-temperature gel permeation chromatography 369 High-tenacity fiber 956 Hildebrand theory of 32 Hindered amine stabilizer (HAS) 819 Hindered phenols 818 HMTA, see Hexamethylenetetramine Holdup scaling factor 537 Hollow particle 273 Holographic photoembossing 1007 Homocondensation 845 Homogeneous continuous stirred tank reactor (HCSTR) 335

Index Homogeneous continuous stirred tank reactor, oscillating feeds 343 Homogeneous deformation 749 Homogeneous nucleation 252, 266 Homogeneous polymerization 1055 Homolytic degradation 760 Hookean elasticity 708 Hooke’s law 724–725 Hosiery yarn 926 Hot cooling 574 Hot drawing 932 Hot-press molding 883 Hot-tube spinning (HTS) 932, 940 Hybrid polymer particles 256 Hybrid polymer-polymer particle 273 Hydride elimination 385 Hydrodynamic chromatography (HDC) 624 Hydrogen bonding 767 Hydrolysis 866 Hydrometer 617 Hydroperester 783 Hydroperoxide 761, 783, 858 Hydrophobically associating polymer (HEUR) 715 Hydrostatic method 607 Hydroxyl value 865 Hydroxymethylation 102 Hyperbranched polymer 712

i I-controller 641 Ideal elastic response 706 Ideal measured value 597 Ideal PID controller 643 IMC, see Internal model control Implicit penultimate unit effect 186 Impregnation 1058, 1061 Increase in viscosity 195 Indicator 597, 626 Induction period 777 Industrial-scale SSP 82 Industrial yarn 916–917, 924, 926, 928 Inferential control 668 Infinite dilution 70 Infrared spectroscopy (IR) 620, 660 Inherent viscosity 664 Inhibition 170, 588 Initiator efficiency 156, 972 Initiator residue 1075 Initiation 162, 581, 583, 760, 802 Inorganic powder 243 Inorganic solid 217 Input/output (I/O) 651 Insertion polymerization 903

Inside-out technology 893 In-situ Raman spectroscopy 1018 Instantaneous average molecular weight 269 Instantaneous property 387 Instrument gauge 597 Integral conrtroller 641 Integral of the timeweighted absolute error (ITAE) 644 Integral of the timeweighted absolute error tuning rules 645 Integrating process 638 Integrator plus dead time process 639 Interactive liquid chromatography (i-LC) 1034 Interfacial process 94 Interlacing 916 Interlocks 589 Intermolecular chain transfer 356 Intermolecular transfer to polymer 174 Internal impurity 795 Internal model control (IMC) 668 Internal viscosity 765 Interparticle heat transfer 400 Interparticle mass transfer 400 Intramolecular chain transfer 356 Intramolecular H abstraction 784 Intramolecular ring-forming reaction 67 Intramolecular transfer 189 Intraparticle heat transfer 400 Intraparticle mass transfer 400 Intrinsically safe operation 1075 Inventory scaleup factor 543 Inverse emulsion polymerization 256 Inverse polymerization 239 Inversion process 862 Ion pair 324, 352 Ion pair association 715 Ion supression 1029 Ionic liquid 1048 Ionic polymerization 323 Ionization constant 352 Ionization method 621 Ionomer 715 Ion-trap system 1029 iPP, see isotactic polypropylene Iron/constantan electrode 601 Irreversible polycondensation 132 Irreversible termination transfer 331 Isolation valve 648 Isophorone diisocyanate (IPDI) 893 Isotactic polypropylene (iPP) 746, 798 Isothermal cloud-point curves 43 Isothermal operation 579

1091

1092

Index ITAE, see Integral of timeweighted absolute error

j Jacobsen-Stockmayer theory 348

k K-BKZ (Kaye-Bernstein, Kearsley, and Zapas) formalism 741 Kevlar 956 Kinetic chain length 159 Kinetic energy correction factor 612 Kneaders 356 Kolmogorov length 221 Kubelka-Munk equation 797

l Labile structure 775 Laboratory measurement 1022 Lactamate anion 348 Lambert-Beer law 797 Laminar flow scale 539 Laminar plug flow reactor 335 Laminate flooring 849 Laplace-Kelvin equation 78 Laser for Raman spectroscopy 1019 Late transition metal catalyst 373, 381 Latent heat of evaporation 574 Latent radical image 1001 Latex 987 LCB, see Long-chain branch LDPE, see Low-densisty polyethylene Length truncation 434 Level measurement technique 606 Life cycle analysis 9 Lifetime prediction 824 Limiting molecular weight 1073 Limiting swelling 259 Limiting temperature 347 Linear AB step polymerization 480 Linear chain 498 Linear energy transfer 807 Linear low-density polyethylene (LLDPE) 17, 367 Linear polycondensation, kinetically controlled 129 Linear polymer 268 Linear polymerization 435 Linear scission 473 Linear viscoelasticity 709, 729, 740–741 Linearly elastic dumbbell model 703 Linoleic acid 856 Liquid chromatography (LC) 1022 Liquid-crystalline behavior 957

Liquid-crystalline polymer 960 Liquid-filled thermometer 601 Liquid level measurement 605 Lithium-based polymerization 327 Living anionic polymerization 690 Living chain 467 Living polymer 331 Living polymerization 325–326 LLDPE, see Linear low-density polyethylene Logarithmic relationship 800 Long oil alkyds 861 Long-chain approximation 410 Long-chain branch (LCB) 174, 371, 395, 902 Long-chain hypothesis 159 Longitudinal modulus 888 Loop circulation 570 Loop reactor 288 Lorentz-Lorenz law 621 Loss modulus 710, 731 Loss of control 563 Low angle light scattering (LALS) 1033, 1040 Low-density polyethylene (LDPE) 17, 153, 366 Low molecular weight polyamine 853 Low-pass filter 627 Low-pressure liquid-liquid equilibria 20 Low-pressure vapor-liquid equilibria 20–21 Low styrene content resin 872 Lower critical solution temperature 24 Lytropic behavior 956

m Macrocycle 348, 355 Macroheterogeneity 789 Macromolecular brush 712 Macromolecules empirical description 4 Macromonomer 175 Macroparticle 401 Macroscopic property 750 Magnetic-inductive flowmeter 617 MALDI, see Matrix-assisted laser desorption ionization Maleic anhydride 871 Manipulated variable 311, 628, 640 Manual control 650 Mark-Houwink relation 696 Mass analyzer, type 1029 Mass balance 117, 290 Mass flow 613, 616, 937 Mass polymerization 972 Mass spectrometer 622 Mass spectrometry (MS) 1025, 1037 Mass-to-charge ratio 1027 Mass-transfer coefficient 983

Index Mass-transfer process 982 Master 662 Master curve 734 Mastication of rubber 817 Matrix-assisted laser desorption ionization (MALDI) 1027, 1029, 1033 Maxiaturized second-dimension column 1042 Maximum drop diameter 219 Maximum technical temperature (MTT) 557 Maximum temperature of the synthesis reaction (MTSR) 556 Maxwell model 707, 709, 711 Mean square radius of gyration 693 Measurement and control 13 Measurement chain 596 Measuring error 597 Mechanical energy 561 Mechanical energy balance 607 Mechanical modulus 683 Mechanical tests 768 Mechanism for radical entry 261 Mechanochemical degradation 813 Mechanochemical synthesis 818 Medium oil alkyds 861 Melamine resins 106 Melamine-formaldehyde (MF) resins 843 Melamine-urea-formaldehyde (MUF) resins 8 Melt flow index (MFI) 935 Melt index 620, 664 Melt spinning 920, 922–923, 926, 953, 955 Melt viscosity 740 Melting point 686 Metal catalysts 373, 381 Metal deactivator 824 Metal injection molding (MIM) 792 Metallocene 5, 373, 451, 483 Metallocene catalyst 371, 379 Metastable morphology 275 Method of characteristics 131 Method of moments 198, 406, 408, 410, 413, 435 Methodology 9, 11 Methylene bridge 839, 845 Methylol groups 840 Micelle 251 Microdomain 789 Microemulsion polymerization 256, 1071 Microheterogeneity 789 Microparticle 401 Micro-region 7 Microscopic mass balance 72 Microstructure 326

Microstructure/property relationship 250 Microwave radar 608 Mid-chain radical 177, 190 Mid-chain scission 772 Midrange infrared spectroscopy (MIR) 297, 300 Mie scattering 624 MIM, see Metal injection molding Miniaturized first-dimension column 1042 Miniemulsion polymerization 256, 1071 MIR see Midrange infrared spectroscopy Mixing 195, 227, 242, 289 Mixing rules 41 Mixing time 538, 540 Modacrylics 951 Model identification 637 Model predictive 671 Model predictive control (MPC) 668, 672 Model-based control 206, 664 Modeling of distributions 201 Modern process 13 Molar mass sensitive detector 1033 Molecular dynamics simulation 782 Molecular weight averages 622 Molecular weight distribution (MWD) 267, 331, 341, 622, 664, 771, 1021, 1030, 1031, 1072 Moment distribution 476 Monad 64 Monitoring techniques 298 Monodispersed micron-size particles 257 Monomer composition 292 Monomer concentration 258 Monomer droplet 251 Monomer feed 582 Monomer partitioning 259 Monomer reactivity ratio 181 Monomethylol urea rate of formation 104 Monomolecular deactivation 386 Monoradical assumption 446 Monte Carlo 202, 485, 790 Mooney viscometer 620 Morphology 217, 232–233 Morphology of block copolymers 691 Morse potential 814 Morton-Flory-Huggins equation 260 Moving-average filter 627 Moving packed-bed reactor 82 MPC, see Model predictive control MTT, see Maximum technical temperature Multi-angle light scattering (MALS) 1033 Multiaxial stress state 749 Multidirectional fibers 879 Multidisciplinary approach 7, 10

1093

1094

Index Multiexposure photoembossing 1007 Multigrain model 401, 413 Multinomial distribution 120 Multi-objective goal 6 Multiphase stirred tank 542 Multiple distributions 1041 Multiple holographic exposure 1007 Multiple-site catalyst 392 Multiple steady state 656 Multiplicity 584 Multiplicity of solution 563 Multi-product plant 657 Multiradicals 467 Multi-slit devolatilizer 977 Multitubular polymerization reactor 543 Multivariable 664 Multivariate calibration 1018 Multi-way partial least squares (PLS) 671 Multi-way principal component analysis (PCA) 671 Multizone circulating reactor 425 Multizone reactor 419

n Nanoclay 692 Nanocomposites 692 Nanomanipulation 997 Nanoparticles dispersion 692 Nanophase separation of block copolymer melts 714 Nanostructured material 692 Narrow-band 602 Nascent crystallization 687 Natural rubber (NR) 902 Near-infrared (NIR) spectroscopy 297, 300, 1017–1019 Negative feedback 640 Neopentyl glycol 865 Newtonian fluid 571, 708 Newtonian viscous behavior 708 Newton’s law of viscosity 619 Nitrile rubber (NBR) 902 Nitrogen inerting 900 Nitroxide radical 819 NMR, see Nuclear magnetic resonance Nomex 956 Non wovens 206 Nonlinear controller 311 Nonlinear polymer 272, 699 Nonlinear viscoelasticity 740–741 Nonpolymerizable compounds 971 Nonradical degradation mechanisms 763 Nonrandom factor model 34 Nonrandom two-liquid (NRTL) theory 34

Non-reinforced applications 886 Non-self-regulating 631 Normal stress differences 741 Norrish photoprocess 787, 803, 897 Novolac application 107, 840, 842 Nozzle 611 Nuclear magnetic resonance (NMR) 621, 775, 1024 Nucleating agents 687 Nucleation 747 Nucleation rate 295 Number-average chain length 412 Number fraction 489 Number of monomer units 473 Number of polymer particles 264 Numerical fractionation 202 Numerical inversion of generating functions 126 Nylon-6 98, 101

o Off-line analysis 1042 Offset 641 Olefin copolymerization 388 Oleic acid 856 Oligomer column 1021 On-aim 657 One component formulation 836 On-line analysis 1015 On-line monitoring 296 On-line sensor selection 297 On-off controller 646 Onset temperature 308 Optical clarity 688 Optical pyrometer 602 Optical scanning electron microscopy 623 Optimization 306 Orange peel effect 868 Organic solvent 1047 Orifice plate 611, 613 Orlon 966 Oscillatory baffled reactor 234, 239 Osmotic pressure 694 Output unit 596 Outside-in technology 894 Oval gear flowmeter 609 Oval wheel counter 610 Overcooling 583 Over-damped 632 Overlapping chains 694 Overshoot 646 Oxidation resistance 907 Oxidative degradation 798 Oxirane 850

Index Oxygen diffusion constant 798 Oxygen inhibitor 894 Oxygen uptake 776 Ozone absorption 796

p PA, see Polyamide Palmitic acid 856 Panel board 627, 650 Paper coating 256 Paracrystalline 920 Paraffin 872 Paraformaldehyde 102 Parallel filter 926 Parametric sensitivity 562, 580 Particle board 849 Particle fragmentation 401 Particle morphology 273 Particle nucleation 264 Particle size distribution (PSD) 223, 234– 235, 294, 623, 672 Particle size stability 584 Partition coefficient 259, 1062 Partly oriented yarn (POY) 940 PCA, see Multi-way principal component analysis PC-SAFT application 47 PD controller 642 Pendent chain 117 Pentaerythritol 856 Peracids 774, 783, 788 Performance 14 Peroxide 155, 774, 871 Peroxide curing 905 Peroxide decomposer 821 Peroxy radicals 762 Personal computer (PC) controller 651 Phantom network 122 Phase behavior 1054 Phase equilibria 18 Phenol 838 Phenolic resins 838 Phillips catalyst 373, 378 Phosgenation 94 Photodegradation 793 Photoembossing 998–1001 Photo-Fries rearrangement 803 Photoinitiated radical polymerization 896 Photoinitiation 802, 897 Photo-ionization 621 Photolabile base 897 Photolithographic techniques 996 Photo-oxidation 793, 796, 855 Photoresponse 999

Photostabilizer 822 Physical association 715 Physical degradation 757 Physical properties 571 Physicochemical degradation 758 PI controller 642 PID controller 642, 645 PID feedback controller 659 Piezoelectric pressure transducer 604 Piezoresistive pressure transducer 604 Planar zigzag (trans) conformation 681 Planck radiation law 602 Plasticization effect 1052 Platinum thermometer 600 PLS, see Multi-way partial least squares Plug 652 Plug flow reactor 335 Pneumatic position controller 650 Pneumatic pressure 598 Pneumatic signal 598 Point-contact 606 Poiseuille’s law 938 Poisson distribution 333–334, 750 Polar additive 326, 343 Polar hydrogel 896 Polarizability 1050 Polarographic method 621 Polyacetal 346 Polyacrylonitrile (PAN) 951 Poly(alkyl methacrylate) 712 Polyamide (PA) 917, 923, 941, 942 Polybenzothiazole (PBT) 960 Polybenzoxazole (PBO) 960 Poly(butyleneterephthalate) (PBT) 938 Polycarbonate 733 Polycondensation 11, 95, 132, 432, 481, 547, 863, 869, 972 Polydispersity 333, 335, 622 Polyester 687, 853, 891, 923 Polyesterification 859 Polyetherimide 780 Polyethylene (PE) 682, 687, 920 Poly(ethylene terephthalate) (PET) 685, 687, 938–940 Poly(ethylenenaphthalate) (PEN) 938 Polyhedral silsesquioxane (POSS) 692 Polyimide 780 Polyisobutene 734 Polymer branching 711 Polymer coil size 693 Polymer composition 179 Polymer composition feedback controller 666 Polymer conformation 692, 784 Polymer degradation, physical factor 763

1095

1096

Index Polymer distribution 1030 Polymer line 924, 935 Polymer melt 972 Polymer microstructure 249 Polymer microstructuring 995 Polymer morphology 766 Polymer nanostructuring 995 Polymer particle 252 Polymer processing 1058 Polymer reaction engineering development 5 Polymer reaction engineering disciplines 9 Polymer reaction engineering history 4 Polymer recycling 792 Polymer scission 1072 Polymer solution thermodynamics 18 Polymer-like structures 715 Polymeric additive 714 Polymeric fluid 703 Polymeric material 3 Polymeric network 834 Polymerization-induced diffusion 999 Polymerization kinetics 377, 383 Poly(methyl methacrylate) (PMMA) 154, 682, 685, 733, 1036 Poly(m-phenylene isophthalamide) 956 Polyolefin 365, 979 Polyoxyalkylene 349, 353 Polyoxymethylene (POM) 346, 356, 961 Poly(p-phenylene terephthalamide), (PPTA) 956 Polypropylene (PP) 367, 682, 923, 943, 964 Polypropylene degradation 782 Polystyrene 153, 920 Polystyrene films 1023 Polystyrene-poly(methyl methacrylate) analyzer 1043 Poly(tetrafluoroethylene) (PTFE) 682, 812, 1058 Polytetrahydrofurane (THF) 356 Poly(trimethylene terephthalate) (PTT) 938 Polytropic reaction 579 Polyurea 109 Polyurethane 109 Poly(vinyl acetate) (PVA) 154, 217, 224, 232, 444, 458, 683 Poly(vinyl alcohol) 683, 946, 952 Poly(vinyl chloride) (PVC) 216, 230, 686, 744, 823, 920 Population balance 262, 408–409, 414, 432 Porosity 231–232, 237 Positional form 643 Post-polymerization 980 Post-processing 13 Post-treatment of VOCs 972

Pot life 871 Potassium alkoxide 350 Powder coating 866 Power consumption 289 Power number 236 Power ultrasound 1063 PRE, see Polymer reaction engineering Precipitation 714 Precipitation polymerization 1057, 1069 Precursor particle 267 Predici 202 Prediction of the gel point 121 Pre-polymerized catalyst 401 Pressure drop 536, 938 Pressure limitation 546 Pressure measurement 602 Pressure-reducing valve 648 Pressure relief 588 Pressure temperature level and flow (PTLF) 658 Preventive measure 586 Primary initiation 802 Primary particle 401 Primary polymer 485, 503 Primary reactor 662 Primary recycling 792 Principle of equal reactivity 62 Principle of similarity 295 Probabilities of extinction 120 Probabilities of reaction 120 Probability generating functions 480 Process development 5 Process gain 632 Process or measured variable 628 Process scheme 5 Process system integration 6 Process under development 14 Production rate 116, 297 Product quality 290, 297 Productivity 583 Products-by-process 249 Programmable logic controller (PLC) 651 Propagation 162, 760 Propagation rate coefficient 327 Proportional band 641 Proportional controller (P controller) 640 Propoxylation 351 Protection strategy 558 Protective coatings 861 Protective effect 808 Proton NMR 1024 Pseudo bulk 270 Pseudo distribution 445, 449, 451, 458, 473 Pseudo-kinetic rate constant 414

Index Pseudoplastic 619 Pulsed-flow reactor 288 Pulsed-laser-induced polymerization 163 Pultrusion 883 Pure delay 636 PVT behavior of polymer melts 41 Pyrolysis 790 Pyrolysis-GC-MS 1026

q Quadrupole instrument 1029 Quadrupole moment 1050 Quantization error 626 Quasi-elastic light scattering (QELS) 623 Quasi-steady-state assumption (QSSA) 159, 489 Quench collar 929 Quencher 823 Quenching 588, 927 Quick opening 652 p-Quinone structures 841

r Radial cooling 928 Radial flow 289 Radiation chemistry 807 Radiation-induced grafting 812 Radiation pyrometer 601 Radiation sterilization 811 Radical antioxidant 818 Radical chain polymerization 890 Radical entry 260 Radical exit 262 Radical formation 1065 Radical photoinitiator 897 Radical polymerization technique 690 Radical reactivity ratio 187 Radical-scavenging stabilizer 818 Radical site 444 Radioactive gamma-ray 607 Radiolytic degradation 805 Radiometrically 618 Radius of gyration 502 Radius tap 613 Raman spectroscopy 297, 300, 621, 660 Ramp function 638 Random polymerization 57 Random scission 498, 771–772 Range 597 Rapid expansion from supercritical solution 1059 Rapid stress-induced crystallization 687 Rate laws 158 Rate-limiting steps 985

Rate of crystallization 81 Rate of nucleation 85 Rate of polymerization 159, 258 Rate of termination 262 Rate theory 815 Ratio control 660 Ratio of specific heat 613 Rayleigh-Plesset equation 1065 Rayleigh scattering 623–624 Rayon tire cord 951 Reaction calorimetry 302, 572, 580, 667, 1075 Reaction diffusion 192 Reaction dynamics 567 Reaction engineering 14 Reaction injection molding (RIM) 111 Reaction injection molding process 349 Reaction order 329 Reaction temperature 1066 Reactive diluent 889, 898 Reactor blends 368, 417 Reactor temperature control 664 Recirculated loop reactor 343 Recirculating tubular (loop) reactor 353 Recombination 440 Recombination termination 498 Recursive method 68 Redox initiation system 980 Reduced fluid density 47 Reduction of risk 585 Redundant system 586 Ree-Eyring model 744 Reflux 575 Reflux condenser 290 Refractive index 297, 621 Refractive index detector (DRI or RI) 1032, 1040 Regulating flaps 654 Regulator 628 Reinforced composites 879 Reinforcement 875 Reinforcing material 692 Relative gain array (RGA) 665 Relative reaction rate 498 Relaxation 934 Relaxation time 702, 732–733, 735, 737 Renewable resource 8 Repeatability 597 Replication 997 Replication phenomenon 401 Reptation 738 Reptation time 739 Reset time 641 Reset windup 641

1097

1098

Index Residence time distribution (RTD) 235, 242, 287, 491 Residual monomer 971, 1060 Resin 835 Resin infusion 883 Resin stability 848 Resin-transfer molding 883 Resistance temperature detector 599 Resistance thermometer 599 Resols 107, 840 Restrictor device 611 Retardation 170 Retarded cooling 929 Retention factor 1035 Reversibility 347 Reversible alternating polycondensation with FSSE 133 Reynolds number 219, 237 Rheological behavior 696, 715, 934 Rheological control 714 Rheological effects of branching 700 Rheology 294, 502, 702 Rheology modifier 712 Ricinic acid 856 Right-angle light scattering (RALS) 1033, 1040 Rigid foams 111 Rigid rod-like polymer 701 RIM, see Reaction injection molding Ring-opening oxidation 805 Ring-opening polymerization 346 Ring-opening reaction 852 Risk parameter 307 Risk-reducing measure 586 Root mean square end-to-end distance 693 Rotameter 614, 616 Rotated helical conformation 681 Rotating disk contactor 71 Rotating filters 926 Rotating lobe flowmeter 610 Rotating vane meter 611 Rotating viscometer 619 Rouse model 697, 703, 736–737 Rouse relaxation time 738 Rovings 875 Rubber 343 Rubber elasticity 721, 725, 727, 729 Rubberlike elasticity 706 Rubbery plateau 722, 729, 739, 743 Runaway incident 307, 553 Russell mechanism 778

s Safe operation 307

Safe region 308 Safety 13, 290, 296 Safety barrier 575 Safety valve 648 SAFT, see Statistical association fluid theory Sampling of monomer units 504 Sampling period 643 Sanchez-Lacombe lattice fluid theory 40 Sanchez-Lacombe model 1055 Satellite drops 221, 224 Scalability 533 Scale 597 Scaleup 237–238, 295, 560, 569 Scaleup criterion 296 Schulz-Flory 61, 334, 336 Scission 762, 771, 785 Scission probability distribution function 772 Scission rate 1073 Seat 652 SEC, see Size exclusion chromatography, see also Gel permeation chromatography Secondary initiation 802 Secondary particle 401 Secondary reaction 786 Secondary reactor 662 Secondary recycling 792 Secondary relaxation 733, 744, 749 Secondary relaxation temperature 749 Secondary suspending agent (SSA) 231–232, 243 Secondary-ion mass spectrometry (SIMS) 1026 Second-order system 635 Second-shell substitution effect 64 Seeding 569, 585 Segmental diffusion 191 Segregated stirred tank reactor 336 Segregation 235, 241 Self-assembling behavior 690 Self-operated regulator 647 Self-polycondensation 119 Self-regulating 630 SEM, see Scanning electron microscopy Semi-batch control 669 Semi-batch operation 580 Semi-batch process 196 Semi-batch reactor 335, 670 Semicontinuous reactor 254 Semi-crystalline low-density polyethylene (ldPE) 1053 Semicrystalline polymer 682, 688, 722, 725, 733, 743, 746, 751 Semi-ladder polymers 960 Seniority/priority distribution 502

Index Sensible heat 561 Sensing element 596 Sensitivity to inhibition 87 Sensitizer 895 Sensor 596 Sensor signal processing 625 Sequence distribution 179 Sequence length 478 Sequence length distribution 473 Sequential polymerization 57 Set point 628, 640 Set-point trajectory 306 Settling time 646 Shape-forming process 687 Shear compliance 730 Shear flow 703 Shear force 578 Shear modulus 725, 730, 747 Shear rate 938 Shear thickening 619 Shear thinning 619, 702, 740 Shear viscosity 934 Sheet molding compound (SMC) 883 Shewhart (x-bar) chart 671 Shift factor 734–736 Shish-kebab structure 961 Short-chain branch 382 Short oil alkyds 861 Short-chain branching (intramolecular transfer to polymer) 177 Shoulder 467 Shrinkage 934 Shrinkage control 885 Silicone acrylates 892 SIMS, see Secondary-ion mass spectrometry Single charged ions 1027 Single-exposure photoembossing 1001 Single-loop controller 627, 649–651 Single-screw extruder 924, 974, 978 Single-site catalyst 383 Single-train process 534 Singular value decomposition analysis (SVD) 665 Size distribution 220 Size-exclusion chromatography (SEC) 622, 664, 1020, 1021, 1033 Sizing 875 Skin formation 948 Slave 662 Slurry process 416, 422 Smart transducer 604, 625 Smith-Ewart Case 263–264 Soft lithography 998 Solar spectral irradiance 796

Solar spectrum 796 Solid-state NMR 1025 Solid-state polycondensation (SSP) 80 Solubility 694, 1051 Solution NMR 1025 Solution polymerization 972 Solution process 416, 423 Solution spinning 921–922, 944, 953 Solvent-based system 979 Solvent or bulk media effect 63 Solvent-separated ion pair 324 Solvent/transfer agent 167 Sonic densitometer 618 Sonic velocity 654 Sonochemistry 1064 Sorption of polymer 1052 Sound speed 1064 Span 597 Sparged reactor vessel 542 Spatial location of atoms in models of network formation 128 Specific gravity 617 Specific heat 683 Specifically programmable-temperature vaporizer (PTV) 1022 Spectra process 962, 964 Spectra filament 962 Spectroscopic technique 299 Spectrum 1018 Spherolite 747 Spin stretch factor 937 Spin-box 925 Spin-draw bulk winding (SDBW) 934 Spin-draw winding (SDW) 933 Spinnability 922 Spinning 913, 918, 925, 931, 938 Spin-trapping 776 Spray coating 1059 Spray tower 343 Spray-up process 882 Spring 652 SSP, see Solid-state polycondensation Stabilization 894 Stabilizer 216–217 Staged models for describing rotating disk contactor 76 Standard 596–597 Standard deviation 598 Standard liquid 853 Staple fiber 913 Stark-Einstein law 793 Star-shaped block copolymer 344 Starved feed 670 State estimator 667

1099

1100

Index State variable 305 Static head-space 1023 Static light scattering 624 Static mixer 545 Static pressure 1066 Static SIMS 1026 Statistical associated fluid theory (SAFT) 44, 47, 1054 Statistical copolymer 338 Statistical process control 671 Staudinger 4 Steady-state model feedforward control 660 Steam stripping 987 Stearic acid 856 Stefan-Boltzmann law 601 Stem position 652 Step growth 565 Step-growth polymerization 11 Step response 632 Steric factor 784 Stiffening behavior 707 Stiffness 887 Stirred-bed gas-phase reactor 421 Stirred-bed reactor 82, 417, 420 Stirred-tank continuous mode 286 Stirred-tank reactor 286, 537 Stirred-tank reactor batch mode 286 Stirred-tank reactor semibatch mode 286 Stirred tanks in series 542 Stirrer power 561 Stockmayer’s distribution 61, 388, 406 Stoichiometric coefficient 115 Storage modulus 710, 731 Strain 709 Strain gauge 604–605 Strain rate 1073 Strands 875 Stress-activated chain scission 815 Stress-concentrating effect 750 Stress gradient 749 Stress intensity factor 750 Stress relaxation behavior 708 Stress-induced chemical reactivity 813 Stress-induced morphologies 767 Strip-chart recorder 627 Stripping 985 Stripping agent 973 Stuffer box texture 916 Styrene 871 Styrene polymerization 169 Styrene-butadiene copolymers (SBR) 902 Styrene-butadiene latexes 986 Sulfur vulcanization 904 Supercritical carbon dioxide 1048–1049

Supercritical fluid 975, 989 Supercritical fluid extraction (SFE) 1060 Supramolecular organization 759 Surface energy 259 Surface tension 622 Surface tension reduction 1010 Surfactant 251, 862 Surlyn 715 Suspension polymerization 213, 239, 980, 985, 1057 Sustainability 8 Sustainable integrated PRE 9 Sustainable new polymer processes 15 Swelling 575–576, 1052 Synergetic approach 7 Synergistic effect 826 Synthetic cellulose yarn 914 Synthetic polymer 1 Synthetic polymer material 5 Synthetic staple fiber production 914 Synthetic yarn 914 Synthetic yarn spinning 916

t Taffy process 109, 854 Tandem reactor technology 417 Tangled yarn 917 Taylored polymer 691 Technora 959 Temperature control 567, 579 Temperature measurement 599 Temperature rising elution fractionation (Tref ) 369 Tensile strength 705 Terminal double bond incorporation 497 Terminal double bond (TDB) 444 Terminal double bond moment distribution 459 Terminal double bond propagation 454 Terminal model 180, 388 Termination 165, 324, 337, 477, 762 Tertiary recycling 792 Tetrahydrofuran 90 Tetramethylolmethane 856 Tex 914 Textile filament yarn 917 Textile yarn 916–917, 924, 933 Thermal behavior 679 Thermal conversion 560, 580 Thermal degradation 778 Thermal expansion coeffcient 683 Thermal initiation 169, 980 Thermal oxidative degradation of polypropylene 782

Index Thermal polymerization of styrene 169 Thermal runaway 254, 546 Thermal stability 575, 779 Thermal stabilization 821, 975 Thermal time constant 574 Thermocouple 600 Thermocouple pyrometer 601 Thermodynamic model 1054 Thermodynamics 12 Thermoplastic elastomer 112 Thermoplastic vulcanizates 907 Thermoplastics 2–3 Thermosets 2, 833 Thermosetting 833 Thin layer resin 897 Throughput scaling factor 537 Time-averaged probability 258 Time constant 632 Time-independent fibril failure criterion 750 Time-of-flight (ToF) analyzer 1030 Time-temperature superposition 734, 736 Time to maximum rate under adiabatic conditions 557 Tire cord 940 Titer 914 Tobolsky and Eyring model 815 Topological scission 472 Topology 502 Total radiation pyrometer 602 Totalizer 605 Trail generating functions 127 Transducer 596–597 Transesterification 863 Transesterification of diphenylcarbonate (DPC) and BPA 96 Transfer of information 6 Transfer of proton 337 Transfer reaction 325, 338, 352 Transfer-to-monomer 168, 386 Transfer-to-polymer 449 Transformation variable 480 Transition metal 761 Transition-state theory 781, 815 Translational diffusion 192 Transmitter 596–597, 625 Transparent semicrystalline polymer 688 Transverse modulus 888 Tri-block copolymer 342 Triethylenetetramine 853 Trim 652 Trimethylolpropane 856 Trimethylolpropane trimethacrylate (TRIM) 1057 Trioxane 355

Triple angle light scattering (TrALS) 1033 Triple-detector instrument 623 TripleSEC system 1040 Tromsdorff effect 568 Trouton’s ratio 936 True thermodynamic constant 99 Tube model 738–739, 741 Tubular reactor 287, 343, 543, 546 Tuning fork densitometer 617 Tuning rules 646 Turbidimetry 624 Turbine flowmeter 611 Turbulence 218, 235, 242 Turbulent regime 539 Twaron 956 Twin-screw extruder 924, 974, 978 Two component formulation 836 Two-dimensional liquid chromatography 1041 Two-dimensional separation 1042 Two-phase flow 588 Two-point controller 647

u Ultimate gain 646 Ultimate period 646 Ultra-high molecular weights 257 Ultrasound 297, 299, 980, 1057, 1062 Ultraviolet (UV) absorbance 794, 822, 1032 Ultraviolet curing 869, 896, 899 Ultraviolet detector 624 Ultraviolet lamp 899 Ultraviolet lithography 996 Ultraviolet region 621 Unidirectional fiber 879 UNIFAC model 36 Unimolecular reaction 130 UNIQUAC theory 34 Univariate calibration 1018 Unsaturated polyester resin 869 Unzipping 773 Upper and lower control limit 671 Upper critical solution temperature 24 Urea-formaldehyde (UF) resin 843 Urethane acrylate 892, 894 Uron UF resin 105 UV, see Ultraviolet

v Valve characteristic 652–653 Valve coeffcient 653 Valve position controller 650 Valve resistance 631 Valve stem 652

1101

1102

Index van der Waals 722 van der Waals mixing rules 46 Vanadium catalyst 379 Vapor velocity 576 Velocity form 643 Velocity gradient 705 Venturi tube 611, 613 Vibrating flow U-tube 618 Vibrating-reed viscometer 620 Vinyl chloride 217, 220, 224, 227 Vinyl content 326 Vinyl ester resin 873 5-Vinylidene-2-norbornene (VNB) 903 Vinyl polymerization 541, 546 Viscoelastic response 707 Viscometric detection 1032 Viscometry 1033 Viscose rayon 948, 950 Viscosity 213, 229, 240, 571, 578, 695, 698, 702, 734, 740, 746, 860, 1049, 1066 Viscosity index improvement 714 Viscosity measurement 619 Viscous force 219 Viscous shear 218, 221, 243 Visible region 621 Vitrification 848 VK column 101 Volatile compounds 788 Volatile organic compound (VOC) 971, 989, 1047 Volume flow 937 Volume fraction 696 Volume fraction mixing rules 46 Volume scaleup factor 533 Volumetric flow 612, 616 Volumetric flow rate 294 Volumetric growth rate 294 von Mises criterion 743, 749

w Water-based finish 929 Waterborne dispersions 979 Water tolerance 848 WAXD, see Wide-angle X-ray diffraction

Weber number 218–219, 237 Weight distribution in interfacial synthesis 96 Weight measurement 604 Weight-average chain length 413 Weight-average molecular weight 125 Weight-fraction sampling 486 Wet spinning 921–922, 946, 955 Wicker-tube reactor 288 Wide-angle X-ray diffraction (WAXD) 681 Wien’s law 601 Wilson correlation 576 Wilson plot 572 Winders capacity 931 Winding 931 Wiped-film evaporator 977 Wiped-film reactor 71 WLF equation 736, 746 Work hardening 729, 742–743, 747

x Xerogel structure

962

y Yarn appearance 916 Yield 725, 741, 743–744, 746, 748 Yield drop 742 Young’s modulus 722, 887

z Z-average degree of polymerization 126 Z-average molecular weight 125 Z-domain 480 Zener element 732 Zero-one system 268 Zero-shear viscosity 715 Ziegler-Natta catalyst 370, 372, 378, 903 Ziegler-Natta EP(D)M polymerization 903 Ziegler-Nichols controller tuning parameter 646 Zimm model 698, 737 Zimm regime 699 Zimm-Stockmayer equation 1040 Zylon 960

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