Handbook of Polymer Reaction Engineering

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

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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 Joa˜o B. P. Soares and Leonardo C. Simon

8.1 8.1.1 8.1.2

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

Contents

8.1.3 8.2 8.2.1 8.2.2 8.2.3 8.3

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

8.3.1

Comparison between Coordination and Free-radical Polymerization 383

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

9

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

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

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

XIII

XIV

Contents

9.6.2.1 9.6.2.2 9.6.3 9.6.3.1 9.6.3.2 9.6.3.3

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

9.6.4

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

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

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

9.9.4.1

Graph Theoretical Connectivity Matrices

510

Contents

9.9.4.2 9.9.4.3 9.10 9.10.1 9.10.1.1 9.10.2

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

9.10.2.1 9.10.2.2 9.10.3 9.10.3.1 9.10.3.2 9.10.3.3 9.10.4 9.10.4.1 9.10.5

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

10

Scaleup of Polymerization Processes E. Bruce Nauman

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

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

522

533

Volume 2 11

11.1 11.2 11.2.1

Safety of Polymerization Processes Francis Stoessel Introduction 553

553

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

554

XV

XVI

Contents

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

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

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

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

Contents

12

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

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

595

XVII

XVIII

Contents

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

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

13

Polymer Properties through Structure Uday Shankar Agarwal

13.1 13.1.1 13.1.2 13.1.3 13.1.4 13.1.5 13.1.6 13.1.7 13.1.8

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

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

660

679

Contents

13.3.4 13.3.4.1 13.3.4.2 13.3.5 13.3.5.1 13.3.6 13.3.7 13.3.8 13.3.9 13.4

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

14

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

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

15

15.1 15.2 15.2.1 15.2.1.1 15.2.1.2 15.2.1.3 15.2.1.4 15.2.2

721

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

757

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

XIX

XX

Contents

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

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

799

Contents

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

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

16

Thermosets 833 Rolf A. T. M. van Benthem, Lars J. Evers, Jo Mattheij, Ad Hoﬂand, Leendert J. Molhoek, Ad J. de Koning, Johan F. G. A. Jansen, and Martin van Duin Introduction 833 Thermoset Materials 833 Networks 834 Advantages 835 Curing Resins 835 Functionality 835 Formulation 836 Production 837 General Areas of Application 837 Phenolic Resins 838 Introduction 838 Chemistry 838 Resols 840 Novolacs 840 Epoxy-novolacs 841 Discoloration 841 Production 842 Properties and Applications 842 Amino Resins 843

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

818

XXI

XXII

Contents

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

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

869

Contents

16.7.6 16.7.6.1 16.7.6.2 16.7.6.3 16.7.6.4 16.7.6.5 16.7.6.6 16.7.6.7 16.7.6.8 16.7.6.9 16.7.6.10

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

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

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

XXIII

XXIV

Contents

16.9.4.2

Thermoplastic Vulcanizates Notation 908 References 909

17

Fibers 911 J. A. Juijn

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

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

907

Contents

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

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

XXV

XXVI

Contents

18

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

19

19.1 19.1.1 19.1.2 19.2 19.3 19.4 19.5 19.6 19.7

20

20.1 20.2 20.2.1 20.2.2 20.2.3

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

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

971

986

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

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

1007

Contents

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

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

21

Recent Developments in Polymer Processes Maartje Kemmere Introduction 1047

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

1047

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

1083

XXVII

XXIX

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

XXX

Preface

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

XXXI

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

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

XXXII

List of Contributors

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

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

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

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

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

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

List of Contributors

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

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

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

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

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

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

XXXIII

XXXIV

List of Contributors

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

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

List of Contributors

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

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

XXXV

1

1

Polymer Reaction Engineering, an Integrated Approach Th. Meyer and J. T. F. Keurentjes 1.1

Polymer Materials

Synthetic polymers can be denoted as the materials of the 20th century. Since World War II the production volume of polymers has increased by a factor of 50 to a current value of more than 120 million tonnes annually (Figure 1.1). The consumption per capita has also increased over the years to a worldwide average of approximately 20 kg per annum in the year 2000. In terms of volumetric output, the production of polymers exceeds that of iron and steel. The enormous growth of synthetic polymers is due tot the fact that they are lightweight materials, act as insulators for electricity and heat, cover a wide range of properties from soft packaging materials to ﬁbers stronger than steel, and allow for relatively easy processing.

25 120

Annual Production 106 to/an

20

100 Consumption, kg/hab

15

80 60

10 40 5

20

World Population , 109 people

0 1940 Fig. 1.1.

1950

1960

1970 1980 Year

1990

2000

0 2010

Polymer production and the evolution of the population since 1940 [1].

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

2

1 Polymer Reaction Engineering, an Integrated Approach Tab. 1.1.

Applications and 2002 Western European markets for the major thermoplastics [1].

Thermoplastic

Market [10 3 tonnes]

Applications

LDPE

7935

PP

7803

PVC

5792

HDPE

5269

PET PS/EPS

3234 3279

PA

1399

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

ABS/SAN PMMA

788 317

Moreover, parts with complex shapes can be made at low cost and at high speed by shaping polymers or monomers in the liquid state. The polymer market can be divided into thermoplastics and thermosets. The major thermoplastics include high-density polyethylene (HDPE), low-density polyethylene (LDPE), polyethylene terephthalate (PET), polypropylene (PP), polystyrene (PS and EPS), poly(vinyl chloride) (PVC), polyamide (PA), poly(methyl methacrylate) (PMMA) and styrene copolymers (ABS, SAN). The most important applications of thermoplastics are summarized in Table 1.1. The total Western European demand for thermoplastics was 37.4 million tonnes in 2002, a growth of about 9% as compared to 2001 [1]. Thermoplastics are used not only in the manufacture of many typical plastics applications such as packaging and automotive parts, but also in non-plastic applications such as textile ﬁbers and coatings. These non-plastic applications account for about 14% of all thermoplastics consumed. The major thermosets include epoxy resins, phenolics, and polyurethanes (PU), for which the major applications are summarized in Table 1.2. It has to be noted, Tab. 1.2.

Applications and 2002 Western European markets for the major thermosets [1].

Thermoset

Market [10 3 tonnes]

Applications

PU Phenolics

3089 912

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

Epoxy resins

420

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

Building and construction 17.6%

Fig. 1.2.

Agriculture 2.5%

Packaging 38.1%

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

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

3

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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

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1 Polymer Reaction Engineering, an Integrated Approach

Energy Raw material Safety

Natural gas

Coal, Oil

Capacity

Pollution control

Water

Process Scientific methodology

Quality control Air

Flexible production

Recycling Clean processes

Optimization

Green solvents

Intensification

From empiricism to strategy

1980

Renewable feedstock Integrated and inherent safety

Plant operation

Market

Fig. 1.3.

Less energy demanding processes

Integrated heat recovery

1990

Multidisciplinarity

2000

2010

Changing priorities in industrial chemical engineering research.

the solution of problems of practical, industrial, and societal importance. Since the 1970s, a changing focus in chemical reaction engineering can be observed, which is summarized in Figure 1.3. To deal with more stringent requirements in terms of energy consumption requires a shift from heat loss minimization toward novel intensiﬁed process concepts that intrinsically require less energy. Safety should now be considered as an intrinsic plant property rather than a responsive action, and the plant needs to be ﬂexible to be able to respond quickly to changes in the market. Last but not least, new concepts will be required to provide a basis for sustainable future developments, that is, the use of renewable resources and processes based on ‘‘green’’ solvents. As a result of this changing focus, a shift toward a multidisciplinary approach can be observed. For PRE this implies the combination of several disciplines such as polymer chemistry, thermodynamics, characterization, modeling, safety, mechanics, physics, and process technology. PRE problems are often of a multi-scale and multifunctional nature to achieve a multi-objective goal. One particular feature of PRE is that the scope ranges from the micro scale on a molecular level up to the macro scale of complete industrial systems. PRE plays a crucial role in the transfer of information across the boundaries of diﬀerent scale regions and to provide a comprehensive and coherent basis for the description of these processes [19]. As depicted in Figure 1.4, there is a direct link between time and size scale, from which it is obvious that the micro and macro scales are not related to the same time scale [20]. As an example, molecular dynamics calculations are addressing a time scale in the order of femto- to nanoseconds, whereas process system integration evolves on the scale of years. Engineers have traditionally been working at the meso scale, which is represented by the middle portion of Figure 1.4, using phe-

1.4 Toward Integrated Polymer Reaction Engineering Time scale System integration Environmental, Global modeling

Years Engineering design Process models

Days

Continuum Models Heterogeneous

Minutes

Phenomenological Models Microstructure

Milliseconds

Molecular level Elementary reactions Molecular modeling Chemical equilibrium

Nanoseconds

Picoseconds

Atomic level Fundamental Quantum techniques Quantum chemistry

Femtoseconds 1Å

10Å

100Å

1µm

1mm

1m

1km

Size scale Fig. 1.4.

Activities in PRE with their corresponding time and size scales.

nomenological and continuum models. Today these limits are pushed in two directions, both toward a more fundamental understanding and at the same time toward a more global scale. In the past, the ‘‘micro-region’’ has traditionally been the domain of physicists and chemists, whereas the ‘‘macro-region’’ has been the ﬁeld, rather, of process or plant engineers. Today, it becomes obvious that only using a multidisciplinary, parallel, and synergetic approach can lead to successful developments. Polymer reaction engineering will play an essential role as the core and the coordinator of this complex process.

1.4

Toward Integrated Polymer Reaction Engineering

As will be obvious from the foregoing discussion, PRE is composed of many disciplines all linked together. These disciplines can be either mature or emergent, but they have a common gateway (see Figure 1.5). Although there is not necessarily a direct connection between them, there exists a common core in which the diﬀerent disciplines make their own speciﬁc contribution to a general objective. The frontiers in PRE are determined by what we know, understand, and are able to quantify, and these frontiers are moving with growing knowledge, competences, and experience. Eﬀorts to push these limits will induce innovative developments leading to emerging technologies and products, and will also strengthen the multidisciplinary approach. In general terms, PRE can be deﬁned as the science that

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1 Polymer Reaction Engineering, an Integrated Approach

Materials sciences Thermodynamics

Novel processes

Inherent safety

Materials Application

Environment Recycling, Disposal

Modeling and simulation

Polymer Reaction Engineering (PRE)

Process integration optimization

New products

Polymer chemistry Reaction kinetics

Post-reaction processes

Novel processes

Nano-, MicroBio-

Measurement and control

Fig. 1.5.

The expanding sphere of polymer reaction engineering.

brings molecules to an end-use product. We can either consider it like a black box (Figure 1.6) or we can try to deﬁne the interconnected disciplines that compose this black box (Figure 1.7). Provided the required product properties can be met, we expect that sustainability is the common denominator for all the disciplines involved in this process. The process of transforming raw materials into valuable end-use products is not a one-way procedure but rather an iterative process in which we try to optimize all the parameters involved. The selection of the proper chemistry and technology should include an evaluation of environmental, safety, and economic parameters. Moreover, questions regarding the possible use of renewable resources and minimizing the energy requirement will have to be answered. Deﬁning PRE in this manner appears to be very close to the procedure of life cycle analysis (LCA) [21].

Raw materials

Fig. 1.6.

Polymer Reaction Engineering

PRE as a black box process.

End use product

1.5 The Disciplines in Polymer Reaction Engineering

9

Sustainability

Materials sciences Thermodynamics

Raw materials

Novel processes

Inherent safety

Materials Application

Energies Needs

Environment Recycling, Disposal

Profit Modeling and simulation

Laws Economy

Satisfaction

Integrated PRE Knowledge

Process integration optimization

New products

Polymer chemistry Reaction kinetics

Post-reaction processes

Nano-, MicroBio-

Novel processes Measurement and control

Renewable

Fig. 1.7.

Products

The integrated approach for sustainable PRE.

Life cycle analysis is a tool assisting decision making in the engineering process. LCA includes the information on the history of the materials used, and the different process and raw material alternatives, as well as the ﬁnal product requirements. LCA is an instrument driven by environmental considerations against a background of technical and economic speciﬁcations, and involves the so-called 3P concept (people, planet, and proﬁt). The LCA-based PRE methodology (Figure 1.8) [22] leads to an optimization of all the parameters involved and a reduction of the costs. This seems to be contradictory at ﬁrst sight, but integrating all the aspects often leads to cost reductions. In our view, the use of this approach will lead to a ‘‘sustainable integrated PRE’’.

1.5

The Disciplines in Polymer Reaction Engineering

The diﬀerent disciplines involved in PRE can be represented using the academia– industry dichotomy (Figure 1.9). The interests of the two types of players are not identical: the diﬀerences are similar to the diﬀerences in their mission statements. Nevertheless, we can observe that a great overlap is present in the middle zone,

10

1 Polymer Reaction Engineering, an Integrated Approach

waste management

Specification - technical - economic - ecologic - safety

recycling

Balances

raw material

product use

- energy - material - emission - waste - sewage

synthesis

processing

Evaluation leads to closed loop assessment of costs

Fig. 1.8.

Life cycle analysis of parts, methods, products, and systems.

where interests, tools, and knowledge are similar, thus providing a strong basis for partnership. As stated above, PRE is composed of a large number of disciplines, which are described in more detail in the following chapters of this handbook. These disciplines are interconnected by a synergetic and multidisciplinary approach, and com-

Academia Fundamental kinetics

Novel technologies

Thermodynamics

Molecular modeling

Measurement and control Reactor design

Environment

Polymer physics Polymer chemistry

Safety Quality assurance

Applied modeling

Process modeling

Market economics

Industry Fig. 1.9.

Overlap of industrial and academic disciplines.

1.5 The Disciplines in Polymer Reaction Engineering

11

Polycondensation

Fig. 1.10. Product-driven PRE, based on an orthogonal relationship between science and engineering.

mercial products are the ﬁnal achievement resulting from this methodology. This could be expressed by an orthogonal representation (Figure 1.10) where polymer sciences are linked with engineering sciences. Every type of polymerization will have its own speciﬁc features, models, and engineering aspects involved. From Figure 1.10 it will be obvious that only teamwork, bringing together several ﬁelds of expertise, can lead to the ﬁnal objective. 1.5.1

Polymerization Mechanisms

Polymerization reactions can be classiﬁed depending on the reaction mechanism involved and can be either step-growth or chain-growth. These mechanisms diﬀer basically with the time scale of the process. In step-growth polymerization (like polycondensation), the polymer chain growth proceeds slowly from monomer to dimer, trimer, and so on, until the ﬁnal polymer size is formed at high monomer conversions. Both the chain lifetime and the polymerization time are often in the order of hours. In chain-growth polymerization (like ionic or free-radical polymerization), macromolecules grow to full size in a much shorter time (seconds being the order of magnitude) than required for high monomer conversion. High molecular weights are already obtained at low monomer conversion, which is in great contrast to step-growth polymerizations. Also, unlike step-growth polymerization, chain-growth polymerization requires the presence of an active center. Condensation polymers are the result of a condensation reaction between monomers, with or without the formation of a condensation by-product (Chapter 3). Examples of polymers produced by condensation are polyamide[6.6], (Nylon 6,6) the result of the intermolecular condensation of hexamethylenediamine and adipic acid, and polyamide[6], (Nylon 6) which is the product of intramolecular condensation of a-caprolactam. This type of reaction is generally sensitive to thermodynamic equilibrium and requires the removal of the by-product, which is often volatile.

12

1 Polymer Reaction Engineering, an Integrated Approach

The polymers produced by condensation reactions can be either linear or nonlinear, depending on the number of functional groups per monomer. The polymerization process can be performed in bulk (liquid or solid state) or as an interfacial polymerization. Free-radical polymerization (FRP) can be performed homogeneously (in bulk, solution, or suspension; Chapters 4 and 5) or heterogeneously (emulsion, precipitation; Chapter 6). The active site is always a radical that can be unstable (classical FRP) or stabilized as in pseudo-living FRP. Radicals can be formed by the homolytic bond rupture of initiators (molecules sensitive to homolytic cleavage, such as peroxides, photosensitive molecules, or bisazo compounds) or by complex mechanisms creating radicals from monomer units using thermal or high-energy sources, such as X-rays, g-irradiation, or UV. This type of polymerization usually comprises several steps: initiation, propagation, various transfer mechanisms, and termination. In ionic polymerizations a cation or anion is the active site (Chapter 7). A heterolytic process leads to charged parts of molecules that can induce the polymerization by nucleophilic or electrophilic processes. These reactions generally evolve at low temperatures (even as low as 120 C) due to the high reactivity of ions. Also, they are very sensitive to impurities present in the monomer or solvent. These reactions are not always terminated, so lead to living polymerization. This process is often used to build tailor-made copolymers. Coordination polymerizations require a transition metal catalyst (Chapter 8). Polyoleﬁns are often produced by this kind of reaction where the catalyst (Ziegler– Natta, for example) acts as the active site but also as the steric regulator, which makes it possible to build polymers with a deﬁned tacticity. Nowadays a great research eﬀort is devoted to the synthesis of new transition metal-based catalysts, such as metallocenes, to produce new products. 1.5.2

Fundamental and Engineering Sciences

Apart from the various polymerization mechanisms involved, a large number of other disciplines will have to be involved, according to the matrix depicted in Figure 1.10. Thermodynamics is essential to understand the physicochemical properties of the individual reactants, solvents, and products involved (Chapter 2). Also, it provides information on the interaction between the various components present in the reaction mixture, from which phenomena such as phase behavior and partitioning can be derived. This information is usually accessible by using the appropriate equation of state for a given system studied. A close collaboration between chemical physicists, chemists, and chemical engineers is required to take full advantage of this fundamental knowledge. Polymer solutions (solid, bulk, solution, complex media) have to be characterized by several speciﬁc analytical tools (Chapter 20). Techniques such as NMR, ESR,

1.5 The Disciplines in Polymer Reaction Engineering

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

13

14

1 Polymer Reaction Engineering, an Integrated Approach

the techniques involved. Devolatilization, post-process reaction, and extraction are some of the methodologies employed for this purpose. Stability and degradation of polymers (Chapter 15) become relevant especially during post processing or moulding processes. Temperature, oxidation and mechanic stresses are the main contributors to product degradation. Currently, there is a strong emphasis on the synthesis of novel functional polymers shaped on a nano scale (Chapter 19) and the development of sustainable production processes (Chapter 21). The latter includes process intensiﬁcation as a methodology, the use of ‘‘green’’ solvents, and the use of renewable resources. Many of the new processes under development are focusing on one or more of these topics, for which the use of supercritical ﬂuids is currently being implemented on an industrial scale.

1.6

The Future: Product-inspired Polymer Reaction Engineering

Innovation times in industry have shown a steady decrease since the 1970s. Classic thinking is that process development becomes increasingly important as industry matures [23]. This is due to the fact that in an early phase of the lifetime of an industry, when product concepts are still being created, the rate of product innovation exceeds the rate of process innovation. This period continues until a dominant design has emerged and opportunities for radical product innovation decrease. In this phase, the shift is toward process innovations to reduce cost price. The half-time of product innovation (time-to-market) in the early 1970s was about ten years. Currently, two years is often considered long. This acceleration of innovation time is the result of competitive pressure in the market. As a rule of thumb, the ﬁrst company to enter the market with a new product can get up to 60% of market share, so there is a high reward for being ﬁrst. As has been discussed above, chemical engineering has been the basis for polymer reaction engineering in the past. In recent discussions, however, it has been emphasized that a need exists to refocus chemical engineering toward productdriven process engineering [24, 25]. The thinking about a process should then start with the customer or consumer: which of the two depends on the structure of the supply chain. The wishes of the consumer and consumer-perceived product properties have to be translated into physical and chemical product properties. In this way, the main physical attributes of a product are determined, including an idea about the microstructure. Next, a functional analysis is performed to determine the lowest number of transformations needed to create the product; this is followed by a morphological analysis [26]. Finally, a conceptual process design exercise is performed to generate possible process routes to achieve the desired product properties. This sequence of events is the core of product-inspired polymer reaction engineering. A key characteristic of this approach is the fact that it avoids the classical ‘‘unit operation trap’’, because it does not ﬁx the mindset to consider only traditional reactor design and separation process steps to build a process.

References

1.7

Concluding Remarks

In the foregoing we have presented a general framework for sustainable polymer reaction engineering. Its most important characteristic lies in the concerted multidisciplinary approach, rather than focusing on individual competencies. Given the volume of polymer production, it will be of major importance that environmental and safety issues become an integral part of the development process. In combination with tools such as life cycle analysis and product-inspired PRE, this will allow the development of sustainable new polymer processes.

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

12 13

14

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

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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.

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2

Polymer Thermodynamics1 Theodoor W. de Loos 2.1

Introduction

The phase behavior of polymer solutions plays an important role in polymer production and processing. Many polymers are produced by solution polymerization. Solvent choice, solvent recovery and the removal of traces of solvent from the polymer product are important factors in these processes. An example is the production of linear low-density polyethylene (LLDPE), which is a copolymer of ethylene and a 1-alkene. Hydrocarbons are used as solvents in this process. The reactor conditions are limited at high temperature by the onset of a liquid–liquid phase split, characterized by a lower critical solution temperature, and at low temperature by crystallization of LLDPE. The pressure must be high enough to keep the ethylene in solution. Another well-known example is the production of low-density polyethylene (LDPE). In this process ethylene is compressed together with an initiator and the LDPE is formed by radical polymerization. The reactor pressure chosen must be high enough to dissolve the polymer in its monomer. In practice reactor pressures are higher than 200 MPa. Since the conversion of ethylene to LDPE is incomplete, LDPE has to be separated from unreacted ethylene, which is recycled to the reactor. To save energy this is done by pressure reduction in two steps, which involve a high-pressure vapor–liquid ﬂash. From a thermodynamic point of view polymer solutions are complicated solutions. A polymer is not a single component but a multicomponent mixture characterized by a molecular weight distribution or by average molecular weights, such as the number-average molecular weight Mn or the weight-average molecular weight Mw . In the case of a copolymer diﬀerent types of copolymers are possible, for example random copolymers and block copolymers, and the comonomer content may vary. Because of their asymmetric nature the entropy of mixing of polymer solutions is much lower than in the case of a mixture of two low molecular weight compounds; also, the pure solvent and the polymer have rather diﬀerent free 1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

18

2 Polymer Thermodynamics

volumes. Because of this, polymer–solvent systems often show a liquid–liquid phase split. In this chapter it is not possible to give an in-depth treatment of the thermodynamics of polymer solutions. For further reading, see Refs. 1–6.

2.2

Thermodynamics and Phase Behavior of Polymer Solutions 2.2.1

Thermodynamic Principles of Phase Equilibria

The equilibrium conditions for phase equilibria can be derived in the simplest way using the Gibbs energy G. According to the second law of thermodynamics, the total Gibbs energy of a closed system at constant temperature and pressure is a minimum at equilibrium. If this condition is combined with the condition that the total number of moles of component i is constant in a closed system [Eq. (1), where nai is number of moles of component i in phase a], the equilibrium conditions given by Eq. (2) can easily be derived for a system of p phases and N components [7]. X

nai ¼ constant

ð1Þ

a

mai ¼ mib ¼ ¼ mpi

for i ¼ 1; 2; . . . ; N

ð2Þ

The chemical potential of component i in phase a is deﬁned by Eq. (3), where g is the molar Gibbs energy. 0 X 1 q nai g a B C i @ A mai ¼ ð3Þ qnai P; T; nj0i

2.2.2

Fugacity and Activity

In the thermodynamic treatment of phase equilibria, auxiliary thermodynamic functions such as the fugacity coeﬃcient and the activity coeﬃcient are often used. These functions are C closely related to the Gibbs energy. The fugacity of component i in a mixture, fi , is deﬁned by Eq. (4a) together with (4b). C dmi 1 RT ln fi at constant T C fi lim ¼ 1 P!0 Pi

ð4aÞ ð4bÞ

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

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

ð5Þ

The fugacity coeﬃcient can be calculated from an equation of state by Eq. (6) or (7) [8]. C ð P RT dP ð6Þ Vi RT ln fi ¼ P 0 0X 1 " # n i RT ð V C @ A qP RT þ RT ln i RT ln fi ¼ ð7Þ qn V PV i V; T; nj0i y According to Eq. (4), the equilibrium relation in Eq. (2) can be replaced by Eq. (8). C C C fi a ¼ fi b ¼ ¼ fi p

for i ¼ 1; 2; . . . ; N

ð8Þ

C The activity a i is deﬁned as the ratio of fi and the fugacity of component i in the standard state fi 0 at the same P and T [Eq. (9)]. C fi ðP; T; xÞ ai 1 0 fi ðP; T; x 0 Þ

ð9Þ

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

ð10Þ

The activity coeﬃcient of component i, gi [Eq. (11)], is a measure of the deviation from ideal solution behavior, so the fugacity of a nonideal liquid solution can be written as Eq. (12). gi 1

ai a iid

ð11Þ

C fi ¼ x i gi fi 0

ð12Þ

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

19

20

2 Polymer Thermodynamics

In this approach the standard state fugacity fi 0 of a liquid component is usually the fugacity of the pure liquid component, and is closely related to the vapor pressure Pisat of that component. On the vapor-pressure curve of a pure component, we have conditions according to Eqs. (14). fi L ðPisat ; TÞ ¼ fi V ðPisat ; TÞ ¼ fVi ðPisat ; TÞPisat

ð14Þ

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

ðP

dmiL Pisat RT

! ¼ fi L ðPisat ; TÞ exp

ðP

viL dP Pisat RT

! ð15Þ

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

fi ðP; TÞ ¼

fVi ðPisat ; TÞPisat

exp

ðP

viL dP sat RT Pi

! ð16Þ

At low pressure the fugacity coeﬃcient and the exponential term are close to 1, so Eq. (17) holds. fi L A Pisat

ð17Þ

2.2.3

Equilibrium Conditions

For low-pressure vapor–liquid equilibria (VLE) the equilibrium condition of Eq. (18) is usually written as Eq. (19), in which fi 0 is calculated with Eq. (16) or (17) C and ji with Eq. (6) or (7); x i and yi are the liquid-phase and vapor-phase mole fraction of component i, respectively. C C fi L ¼ fi V

ð18Þ

C x i gi fi ¼ yi ji P

ð19Þ

0

The truncated virial equation or the ideal gas equation is often used as the equation of state. In the latter case all fugacity coeﬃcients are 1, so the simplest form of Eq. (19) is Eq. (20). x i gi Pisat ¼ yi P

ð20Þ

In the case of low-pressure liquid–liquid equilibria (LLE) the equilibrium condition is given by Eq. (21). C C fi 0 ¼ fi 00

or

ðx i gi fi 0 Þ 0 ¼ ðx i gi fi 0 Þ 00

ð21Þ

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

The two liquid phases are indicated by prime ( 0 ) and double prime ( 00 ). If for both liquid phases the standard fugacity is chosen as the pure liquid component, Eq. (21) reduces to Eq. (22). ðx i gi Þ 0 ¼ ðx i gi Þ 00

ð22Þ

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

or

C C ðx i ji PÞa ¼ ðx i ji PÞb

or

C C ðx i ji Þa ¼ ðx i ji Þb

ð23Þ

2.2.4

Low-pressure Vapor–Liquid Equilibria

Since polymers have no vapor pressure and as a consequence the vapor phase does not contain polymer, the equilibrium conditions for low-pressure vapor–liquid equilibria of polymer solutions as given by Eq. (20) are only applicable to the solvent s as in Eq. (24), or in a case where the weight fraction of polymer wp is used as a composition variable as in Eq. (25), where Ws is the weight fraction based activity coeﬃcient of the solvent. P ¼ xs gs Pssat

ð24Þ

P ¼ ð1 wp ÞWs Pssat

ð25Þ

The relation between a mole fraction based activity coeﬃcient gi and a weight fraction based activity coeﬃcient Wi of component i is given by Eq. (26), where wi and Mi are the weight fraction and molecular weight of component i, respectively. gs and Ws can be obtained from a correlation of experimental data using a suitable model or from a predictive model (see Section 2.3). N X wj gi wi ¼ ¼ Mi Wi x i M j j¼1

ð26Þ

2.2.5

Flory–Huggins Theory and Liquid–Liquid Equilibria

The Flory–Huggins theory of polymer solutions [1] is based on a rigid lattice model in which a polymer molecule is assumed to consist of r segments of the size of a solvent molecule. The Flory–Huggins expression for the Gibbs energy of mixing Dmix G for Ns moles of solvent and Np moles of polymer is given by Eq. (27). Dmix G ¼ Ns ln js þ Np ln jp þ js jp ðNs þ rNp Þw RT

ð27Þ

21

22

2 Polymer Thermodynamics

The ﬁrst two terms on the right-hand side of Eq. (27) represent the so-called combinatorial entropy of mixing, and the third term is an approximation for the enthalpy of mixing. In practice w, the Flory–Huggins interaction parameter, is used as an adjustable, temperature-dependent parameter. js and jp are the segment fractions of solvent and polymer, respectively, deﬁned by Eq. (28).

js ¼

Ns Ns þ rNp

and jp ¼

rNp Ns þ rNp

ð28Þ

According to the original Flory–Huggins theory w is given by Eq. (29), in which ess ; epp , and esp are the interaction energies between two solvent molecules, two polymer segments, and a solvent molecule and a polymer segment, respectively.

wz

ð2esp ess epp Þ T

ð29Þ

Often r is approximated by the ratio of the molar volumes of pure liquid polymer and pure solvent. The segment fractions of solvent and polymer are then equal to the volume fractions of solvent and polymer, fs and fp . Equation (27) is derived using many assumptions and approximations (for a discussion, see Ref. 8), but on the basis of this rather simple expression many features of the phase behavior of polymer solutions can be explained. The expressions for the mole fraction based activity coeﬃcients of solvent and polymer are Eqs. (30) and (31), respectively. jp 1 þ 1 jp þ jp2 w ln gs ¼ ln ð1 jp Þ þ r r

ð30Þ

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

ð31Þ

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

2.2 Thermodynamics and Phase Behavior of Polymer Solutions 0.40

0.50

r=1000 0.60

χ

r=100 0.70

0.80

0.90

r=10

1.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

φp Fig. 2.1. Liquid–liquid equilibria for polymer–solvent systems at diﬀerent polymer chain lengths r. Calculated from the Flory– Huggins theory.

the two-phase region, shifts with increasing r to lower values of w, higher temperature, and lower values of jp . At r ¼ y the critical point is found at a limiting value of w ¼ 12 , T ¼ y, and jp ¼ 0. This follows directly from the critical point conditions [Eq. (32), where G is the Gibbs energy per mole of lattice sites]. qG qjp

! ¼ P; T

! q2G ¼0 qjp2

ð32Þ

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

crit

1 1 2 1 þ pﬃﬃ ¼ 2 r

jpcrit ¼

1 pﬃﬃ 1þ r

ð33Þ ð34Þ

23

24

2 Polymer Thermodynamics

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

T

L1+L2

L

L1+L2 Solvent

φp

Polymer

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

Fig. 2.2.

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

T

L1+L2

L

L

L1+L2 Solvent Fig. 2.3.

φp

Polymer

Hourglass-shaped two-phase liquid–liquid equilibria.

2.2.6

High-pressure Liquid–Liquid and Vapor–Liquid Equilibria

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

25

26

2 Polymer Thermodynamics

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

Fig. 2.4.

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

2.2 Thermodynamics and Phase Behavior of Polymer Solutions

Fig. 2.5. Liquid–liquid equilibria in the polystyrene– methylcyclohexane system at the indicated molecular weights of polystyrene (in g mol1 ). Reproduced with permission from Ref. 14.

ethane þ LDPE. For this type of systems it is not possible to make a clear distinction between vapor–liquid equilibria and liquid–liquid equilibria. The polymer-rich phase can be considered to be a liquid phase, but the solvent-rich phase is liquidlike at low temperature, but vaporlike at high temperature. De Loos et al. [20] measured cloud-point curves for LLDPE systems plus hexane, plus heptane, and plus octane (see Figure 2.10). In this case also, the polymer is more soluble in the higher molecular weight solvent, which is shown by the shift of the LCSTtype cloud-point curve to higher temperatures. An increased degree of branching of the polymer leads to a better mutual solubility [21].

27

28

2 Polymer Thermodynamics

T

L1+L2

φp

Solvent Fig. 2.6.

L

Polymer

Closed-loop liquid–liquid equilibria.

c

P b

a

a

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

Fig. 2.7.

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

2.3 Activity Coeﬃcient Models

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

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

2.3

Activity Coeﬃcient Models

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

29

30

2 Polymer Thermodynamics

Cloud-point isopleths of LDPE in various n-alkane solvent at 5 wt.% polymer. The short dash–long dash curve is the solidiﬁcation boundary of LDPE. Reproduced with permission from Ref. 17.

Fig. 2.9.

in the free volume of polymer and solvent can be accounted for by adding a free volume contribution [4]. Further, Eq. (27) can be extended to account for the polydispersity of the polymer. The system-dependent parameters in these models can be adjusted to experimental data or predicted from a group contribution approach [4, 26]. 2.3.1

Flory–Huggins Theory

For a solution of a polydisperse polymer with m polymer components in one solvent, the Flory–Huggins expression for the Gibbs energy of mixing per mole of lat-

2.3 Activity Coeﬃcient Models

Fig. 2.10. Cloud-point curves of poly(E-co-1-octene)–n-alkane systems at P ¼ 3 MPa. Mn ¼ 33 kg mol1 , Mw ¼ 124 kg mol1 , Mz ¼ 420 kg mol1 . The systems show LCST phase behavior. Reproduced with permission from Ref. 20.

tice sites can be written as Eq. (35). The sum is only over the polymer components, m P ji is the segment fraction of polymer component i, and jp ¼ ji ¼ ð1 js Þ is the i¼1 overall polymer segment fraction. m X Dmix G ¼ js ln js þ ji ri1 ln ji þ js jp g sp RT i¼1

ð35Þ

g sp ¼ f ðT; jp Þ

ð36Þ

To avoid confusion with the polymer–solvent interaction parameter, the symbol g sp [Eq. (36)] is used instead of w, which is independent of concentration. Equations (37a)–(37c) are examples of expressions used for the temperature and composition dependence of g sp [27–29]. g sp ¼ a þ

b þ cT þ djp þ ejp2 T

c T g sp ¼ a þ 1 djp

ð37aÞ

bþ

ð37bÞ

31

32

2 Polymer Thermodynamics

aþ g sp ¼

b þ c ln T T 1 djp

ð37cÞ

We can derive Eq. (38) for the solvent, and Eq. (39) for polymer component i, where rn is number-average chain length of the polymer, which is deﬁned by Eq. (40) and which is proportional to the number-average molecular weight of the polymer. ! qg sp 2 1 ð38Þ j lnðxs gs Þ ¼ ln js þ 1 j þ g sp js rn p qjp p ri jp qg sp 2 lnðx i gi Þ ¼ ln ji þ 1 j ri js þ ri g sp jp ð39Þ rn qjs i m X

rn ¼

n i ri

i¼1 m X

ð40Þ ni

i¼1

Using Eqs. (37) and (38), vapor–liquid and liquid–liquid equilibria can be calculated as discussed above. Reference 5 gives details of liquid–liquid equilibrium calculations in polydisperse polymer–solvent systems. In Figure 2.11 the experimentally determined phase behavior of three PEG–water systems is compared with the calculated phase behavior of these systems using Eq. (37c) to represent g sp as a function of temperature and polymer segment fraction. The parameters were ﬁtted to the data [29]. The description of the inﬂuence of pressure on polymer–solvent phase behavior is also possible using a pressure-dependent g sp [27, 30–32]. However, this approach is purely empirical. 2.3.2

Hansen Solubility Parameters

According to the regular solution theory of Hildebrand the w-parameter can be approximated by Eq. (41) [8], where vs is the molar volume of the solvent and ds and dp are the solubility parameters of solvent and polymer, respectively. Since these solubility parameters are pure component parameters, Eq. (41) combined with Eq. (27) results in a predictive model. However, since many simpliﬁcations are involved, the results of this model can be considered as only a rough estimate. Following the slogan ‘‘like dissolves like’’, a good solvent for a polymer is a solvent for which ds and dp have similar values. w¼

vs ðds dp Þ 2 RT

ð41Þ

2.3 Activity Coeﬃcient Models

Fig. 2.11. Closed-loop liquid–liquid equilibria in the PEG– water system. Symbols: experimental data, M ¼ 3:35 kg mol1 (0), M ¼ 8 kg/mol (5), M ¼ 15 kg/mol (4); curves ﬁtted using Eq. (37c). Reproduced with permission from Ref. 29.

Hansen suggested reﬁning the solubility parameter theory by the introduction of contributions from dispersive interactions (d), polar interactions ( p) and hydrogen bond formation (hb), as in Eq. (42) [33]. w¼

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

ð42Þ

Recently, Lindvig et al. [34, 35] showed that Eq. (42) systematically overestimates the inﬁnite dilution activity coeﬃcient of the solvent and proposed an alternative expression, Eq. (43). w¼a

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

ð43Þ

33

34

2 Polymer Thermodynamics

These authors showed that for a number of polymer–solvent system with a ¼ 0:6 this method performs similarly to group contribution methods using volume fractions to represent the segment fractions in the Flory–Huggins model. Values of solubility parameters are tabulated by Barton [36]. 2.3.3

Correlation of Solvent Activities

The combination of Eq. (35) with one of the Eqs. (36) or similar empirical equations is not very successful for describing the phase behavior of polymer–solvent systems with strong interactive species and of systems which only diﬀer in molecular weight. Since the mid-1990s activity coeﬃcient models have been proposed based on a combination of the Flory–Huggins type of expression for the combinatorial entropy of mixing and segment-based local composition models to account for the contribution from energetic interactions (the residual contribution to the activity coeﬃcient). In 1993 Chen [37] proposed a correlative model that used a combination of the Flory–Huggins expression for the combinatorial entropy of mixing and the nonrandom two-liquid (NRTL) theory [38]. The same approach was followed by Wu et al. [39]. These authors used Freed’s expression [41, 42] for the combinatorial entropy of mixing, which is more accurate than the Flory–Huggins expression, in combination with the NRTL theory. The nonrandom factor model of Haghtalab and Vera [42] was modiﬁed by Zafarini-Moattar and Sadeghi [43] for polymer solutions. In 2004, Pedrosa et al. [44] suggested use of the UNIQUAC theory [45] instead of the NRTL theory and tested various combinations of combinatorial contributions and residual contributions using a database of 70 low-pressure VLE systems. These authors have concluded that the combination of a segment-based NRTL or UNIQUAC residual term with a good combinatorial term is able to produce the best correlations of experimental VLE data and can also be used to predict the inﬂuence of the molecular weight of the polymer on polymer–solvent VLE. As an example we will give here the expression for the activity coeﬃcient of the solvent for a model which combines the p free-volume combinatorial term with a segment-based Wu-NRTL residual term [44]. The p free-volume combinatorial term combines combinatorial contributions and free-volume contributions [45]. The activity coeﬃcient of the solvent is given by Eq. (44). ln gs ¼ ln gscombfv þ ln gsres

ð44Þ

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

fsfv f fv þ1 s xs xs

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

ð45Þ

2.3 Activity Coeﬃcient Models fv

xiv fv fi ¼ X i fv xj vi

ð46Þ

j

The free volume of a component is deﬁned by Eq. (47), where vi is the molar volume of liquid component i, vi; vdW is the hard-core volume or van der Waals volume of this component and can be calculated using the Bondi tables [46], and p is a correction factor, which is calculated from Eq. (48) in the p free-volume model [45]. fv

vi ¼ ðvi vi; vdW Þ p

ð47Þ

vs vp

ð48Þ

p¼1

The residual term is given by Eq. (49) together with Eqs. (50), where asp and a ps are adjustable interaction parameters, a is the NRTL-nonrandomness parameter, which was ﬁxed by Pedrosa et al. at 0.4. " gsres

2 tps Gps

tsp Gsp2

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

¼

qs X p2

# ð49Þ ð50Þ

X i is the eﬀective mole fraction of segments of species i and qi is the eﬀective segment number of species i, which are given by Eqs. (51) and (52); rs ¼ 1 and rp ¼ r. x i qi Xi ¼ X xj q j

ð51Þ

j

1 qi ¼ ri 1 2a 1 ri

ð52Þ

2.3.4

Group Contribution Models

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

35

36

2 Polymer Thermodynamics

of experimental data. This makes the group contribution methods purely predictive. However, since details of molecular structure are not considered, group contribution methods for activity coeﬃcients are in general less accurate than correlative models. Oishi and Prausnitz [47] proposed writing the solvent activity coeﬃcient for polymer–solvent systems as the sum of three terms [Eq. (53)]. ln gs ¼ ln gscomb þ ln gsfv þ ln gsres

ð53Þ

The combinatorial contribution gscomb accounts for diﬀerences in size and shape fv of the molecules. The free-volume contribution gs accounts for changes in free volume due to mixing, caused by the large diﬀerence between the free volumes of pure solvent and polymer. For ordinary liquid mixtures in far from critical conditions, this term is usually negligible. The residual contribution gsres accounts for energy interactions. In the approach of Oishi and Prausnitz, the combinatorial contribution is represented by the Staverman–Guggenheim expression, a modiﬁcation of the Flory–Huggins equation, also used in the UNIFAC group contribution model [48]; for the residual contribution also, the corresponding expression of the UNIFAC model is used. The free-volume contribution is calculated from the Flory equation of state [49]. This group contribution model and those of Chen et al. [50] and Danner and High [51] are discussed in Ref. 4. Here we will discuss the entropic free volume model of Elbro et al. [52] and Kontogeorgis et al. [53], and the group contribution Flory (GC–Flory) model of Bogdanic et al. [54, 55], which is a modiﬁcation of the model of Chen et al. [50]. In the entropic-free volume model, the activity coeﬃcient of the solvent is given by Eqs. (44)–(48) with p ¼ 1 [52]. The residual contribution is represented by the residual contribution of the UNIFAC model with temperature-dependent interaction parameters [53]. The liquid molar volumes needed for the calculation of the free volume of a component can be taken from experiment or calculated from the Tait equation [4] or by the group contribution method of Elbro et al. [56]. This model is relatively easy to use. In the GC–Flory model, the solvent activity coeﬃcient is calculated from Eq. (53). The combinatorial term is calculated by means of the original Flory–Huggins expression, Eq. (54). ln gicomb ¼ ln

fi f þ1 i xi xi

ð54Þ

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

RT v~1/3 þ C E attr v v~1/3 1 v

ð55Þ

2.3 Activity Coeﬃcient Models

v~ ¼

v v

ð56Þ

The molar hard-core v is calculated from the pure component hard-core molar volumes vi using a linear mixing rule [Eq. (57)]. The same type of mixing rule is used for the number of external degrees of freedom parameter C [Eq. (58)]. v ¼

X

x i vi

ð57Þ

x i Ci

ð58Þ

i

C¼

X i

The pure component hard-core volumes and C parameters are calculated from the group contribution expressions in Eqs. (59) and (60), X

vi ¼ 21:9662

uðiÞ m Rm

ð59Þ

m

Ci ¼

X

uðiÞ n

CT0 ; n þ CT; n

n

1 1 T T0

þ

X n

R X n Cn0 Rm

ð60Þ

m ðiÞ

where un is the number of groups of type n in molecule i and T0 is a reference temperature. R n is the normalized van der Waals volume of group n as used in the UNIFAC method. The attraction term E attr is related to the UNIFAC model by Eq. (61), z is the lattice coordination number, chosen to be equal to 10, and qi , the surface area of molecule i, is given by Eq. (62) 2 E attr ¼

X1 i

2

X

yj expðDeji /RTÞDeji

6 6 j X zqi x i 6 4eii þ

3

7 7 7 yk expðDeki /RTÞ 5

k

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

X

vðiÞ n Qn

ð61Þ ð62Þ

n

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

eij0 v~

and

Deij ¼ eij eii

ð63Þ

37

38

2 Polymer Thermodynamics

eji0 ¼

X

yðiÞ m

X

m

yðn jÞ enm

ð64Þ

n

enm ¼ ðenn emm Þ 1/2 þ Denm

ð65Þ

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

ð66Þ

j

x i qi yi ¼ X xj q j

ð67Þ

j ðiÞ

u Q Xn n yðiÞ n ¼ uðiÞ m Qm

ð68Þ

m

Note that indices m and n refer to groups m and n, and i and j to molecules i and j. The resulting expressions for the free-volume contribution and the residual contribution in the activity coeﬃcient are Eqs. (69) and (70). ln

fv gi

ln gires

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

ð69Þ

ð70Þ

k

To apply this method, the liquid molar volumes of the mixture and of the pure components need to be known. At a given pressure and temperature these values can be calculated from the equation of state. However, since eji according to Eq. (63) is volume-dependent, this involves an iterative procedure similar to that described by Danner and High [4] for the method of Chen et al. [50]. Figure 2.12 shows the experimental solvent activity of the system poly(propylene oxide)– benzene at 347.85 K compared to the correlation by UNIQUAC and predictions by the GC–Flory model. The result of the correlation is almost perfect. The predicted solvent activities by the GC–Flory model are also very close to the experimental values. In Figure 2.13 a comparison is shown of experimental solvent activity

2.4 Equation of State Models

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

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

2.4

Equation of State Models

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

39

2 Polymer Thermodynamics

log Ω∞cal

40

log Ω∞exp Fig. 2.13. Comparison of calculated activity coeﬃcients at inﬁnite dilution using the GC–Flory model with experimental values for many polymer–solvent systems. Reproduced with permission from Ref. 54.

and co-workers and can be used to model the phase behavior of mixtures of small and large molecules, including polymers, over a wide range of pressure and temperature [57–60]. Here, we will discuss only the two equations of state methods: the relatively simple Sanchez–Lacombe (S–L) lattice ﬂuid model [61, 62], and the statistical associating-ﬂuid theory (SAFT) [63, 64], which has now become one of the standard equations of state for polymer solutions. 2.4.1

The Sanchez–Lacombe Lattice Fluid Theory

Like the Flory–Huggins model, the Sanchez–Lacombe lattice ﬂuid theory is based on the assumption that segments of solvent molecules and polymer molecules occupy the lattice sites of a rigid lattice, but vacant lattice sites are also allowed. The number of vacant lattice sites, and as a consequence the total number of lattice sites, are pressure-dependent, and in this way compressibility is introduced.

2.4 Equation of State Models

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

ð71Þ

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

v n0 þ rn ¼ v rn

ð72aÞ

p~ ¼

P Pv ¼ P e

ð72bÞ

kT T~ ¼ T

ð72cÞ

The parameters with an asterisk ( ) are the so-called characteristic parameters. In practice r, the number of segments per molecule, e , the interaction energy parameter, and v , the molar volume of a lattice site, are used as the independent purecomponent parameters; n is the number of moles of the component and n0 is the number of moles of vacant lattice sites. If v is the volume per mole of segments, the total volume of the system is given by V ¼ nrv. For high molecular weights r is large, so it can be concluded from Eq. (71) that the density of polymer melts is not very dependent on molecular weight and that the PVT behavior of polymer melts follows the corresponding states principle (see Figure 2.14) [62]. For mixtures the same equation of state is used, but the characteristic parameters r; e , and v are composition-dependent. Neau [65] gives an overview of diﬀerent mixing rules proposed in the literature. Often-used mixing rules are given in Eqs. (73)–(75). In Eq. (75) k ij is an adjustable binary interaction parameter which equals zero for i ¼ j and which can ﬁtted to binary experimental data. r¼

X

v ¼ e ¼

ð73Þ

x i ri

X

ji vi

XX

ji jj

ð74Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃ eii ejj ð1 k ij Þ

ð75Þ

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

ð76Þ

41

42

2 Polymer Thermodynamics

Fig. 2.14. Corresponding states behavior of various liquid– polymer PVT data according to the Sanchez–Lacombe model. Symbols represent experimental data; curves are calculated from Eq. (71). Reproduced with permission from Ref. 62.

Neau [65] also showed that the expressions for the chemical potential used in earlier literature to calculate phase equilibria are thermodynamically inconsistent. According to Neau, the correct expression for the fugacity coeﬃcient for the SL model is Eq. (77).

2.4 Equation of State Models

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

z1 X xj r j

!"

nr qv v qn i

#

nj 0n i

ð77Þ

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

p~v~ Pv ¼ r RT T~

ð78Þ

In Figure 2.15 [66] experimental isothermal cloud-point curves of the linear low density polyethylene þ hexane system are compared with the results of a ﬁt of these data using the Sanchez–Lacombe equation of state. The pure component parameters of hexane were calculated from the critical point of hexane and its acentric factor [67]. The pure component parameters of the polymer were obtained from a simultaneous ﬁt of PVT data and the data presented in Figure 2.15. The equations solved were those described by Koak and Heidemann [68]. The binary interaction parameter was linearly dependent on temperature. The polymer was

10 9 8

P / MPa

7 6 5 4 3 2 1 0 0.00

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

0.10

0.15

0.20

0.25

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

0.30

0.35

43

44

2 Polymer Thermodynamics

represented by 36 pseudo-components. The SL theory is very well able to describe the experimental phase behavior. 2.4.2

Statistical Associating-ﬂuid Theory

The statistical associating-ﬂuid theory (SAFT) developed by Chapman et al. [63, 64] is based on the thermodynamic perturbation theory of Wertheim [69]. Since it ﬁrst appeared, many diﬀerent versions of SAFT have been published. The diﬀerent SAFT versions and their application have been reviewed by Muller and Gubbins [70]. For polymer solutions the SAFT version of Huang and Radosz [71, 72] is the most widely used. In 2000, a promising new version of SAFT for polymer solutions called PC-SAFT (perturbed chain-SAFT), was proposed by Gross and Sadowski [73]. Here we will restrict ourselves to these two SAFT versions. The basics of both equations of state are equal, and can be written as separate contributions to the molar Helmholtz energy a. The molar residual Helmholtz energy a res, which is the diﬀerence between the molar Helmholtz energy of the system and the molar Heltmholtz energy of the same system in the ideal gas state in the same conditions of temperature, pressure, and composition, is calculated as the sum of the contributions of a hard chain term a hc , a dispersion term a disp , and an association term a assoc [Eq. (79)]. a res ¼ a a ig ¼ a hc þ a disp þ a assoc

ð79Þ

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

qA p¼ qV T; n qA mi ¼ qn i V; T; nj0i

ð80Þ ð81Þ

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

2.4 Equation of State Models

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

X

x i mi

ð82Þ ð83Þ

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

ð84Þ

In this equation zk is given by Eq. (85), where r is the number density, dii is the temperature-dependent hard-sphere diameter obtained from Eq. (86); mi , the hard-sphere diameter si, and the energy parameter ei are pure component parameters. p X x i mi ðdii Þ k zk ¼ r 6 3ei dii ðTÞ ¼ si 1 0:12 exp kT

ð85Þ ð86Þ

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

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

ð87Þ ð88Þ

SAFT Dispersion Term In the Huang and Radosz version of SAFT [71, 72] the Chen–Kreglewski dispersion term is used. This term is obtained from a ﬁt to the physical property data of argon and is given by Eq. (89) [76], where t is a constant equal to 0.74048. The constants Dij are given by Chen and Kreglewski [76]. 2.4.2.2

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

ð89Þ

45

46

2 Polymer Thermodynamics

For mixtures, the van der Waals one-ﬂuid mixing rules or the volume fraction mixing rules can be used. The van der Waals mixing rules are given by Eq. (90), where the combining rules are Eqs. (91) and (92), in which k ij is an adjustable binary interaction parameter. XX u ¼ kT

i

j

XX

uij 3 d kT ij ð90Þ

x i xj dij3

j

i

dij ¼

x i xj

di þ dj 2

ð91Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uij ¼ ð1 k ij Þ uii ujj

ð92Þ

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

ð93Þ

x i mi d 3i fi ¼ X xj m j d 3j

ð94Þ

j

The combing rule for uij is again given by Eq. (92). The PC-SAFT Dispersion Term In SAFT the dispersion term represents the interactions between individual segments, while in PC-SAFT the dispersion term represents the interactions of chains of segments. The expression for a disp derived by Gross and Sadowski is Eq. (95), where the terms on the right-hand side are deﬁned by Eqs. (96) and (97). 2.4.2.3

a disp a1 a2 ¼ þ RT RT RT

ð95Þ

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

2

u kT

2

s3

6 X i¼0

bi ðmÞh i

ð96Þ

ð97Þ

2.4 Equation of State Models

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

pNAv rmd 3 6

ð98Þ

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

m1 ðm 1Þðm 2Þ a 1i þ a 2i m m2

ð99Þ

bi ðmÞ ¼ b0i þ

m1 ðm 1Þðm 2Þ b1i þ b2i m m2

ð100Þ

The constants a 0i –b2i are ﬁtted to the thermophysical properties of n-alkanes and are given by Gross and Sadowski [73]. SAFT and PC-SAFT Applications Both SAFT and PC-SAFT contain pure component parameters: the energy parameter e or u, the hard-sphere diameter s, or the hard-sphere volume v 00 , and the number of segments m per molecule. For small (solvent) molecules these parameters are obtained from a ﬁt of vapor pressure data and saturated liquid volume data. Since they do not have a vapor pressure, this ﬁt is not possible for polymers, and the pure component polymer parameters are obtained from a ﬁt to PVT data of the molten polymer or from a ﬁt to PVT data and binary phase equilibrium data. For the description of a mixture one needs one binary interaction parameter k ij per binary, which has to be ﬁtted to phase equilibrium data. If necessary, k ij can be made temperature-dependent. In general, phase equilibria are very sensitive to the k ij value. In Figure 2.16 [77], experimental results from a system of ethylene þ HDPE are compared with modeling results using SAFT and Sanchez–Lacombe models. In both cases k ij is taken to be linearly dependent on temperature. Due to the polydispersity of the polymer, the cloud-point curves show a dip in which the critical point is located. If in the modeling the polymer is assumed to be monodisperse, this behavior cannot be reproduced. The ﬁgure shows that both SAFT and Sanchez– Lacombe models give a reasonable description of the experimental phase behavior, although at high and low polymer concentrations the deviations become larger. The same system was modeled by Tumakaka et al. [78] using SAFT and PCSAFT. The results are presented in Figure 2.17, which clearly shows that in this case PC-SAFT gives a better result than SAFT. In the same paper these authors present PC-SAFT modeling results for LDPE þ solvent systems at constant polymer concentration. The pure LDPE parameters were ﬁtted to the experimental data of ethane þ LDPE. These parameters were subsequently used to describe the LDPE þ ethane, þ propene, þ propane, þ butane, and þ 1-butene systems, using a 2.4.2.4

47

48

2 Polymer Thermodynamics

Fig. 2.16. Isothermal cloudpoint curves of the HDPE þ ethylene system. Mn ¼ 43 kg mol1 , Mw ¼ 118 kg mol1 , Mz ¼ 231 kg mol1 . Symbols: experimental data; curves: modeling results: (a) SAFT model; (b) Sanchez– Lacombe model. Reproduced with permission from Ref. 77.

temperature-independent k ij . The results are very good (see Figure 2.18), but it should be kept in mind that the modeling is restricted to one polymer concentration. Extension to Copolymers For the modeling of the phase behavior of copolymer–solvent systems, the copolymer can be treated as a homopolymer with eﬀective pure component parameters. Examples of this approach are given by McHugh and co-workers [79, 80]. The disadvantage of this approach is that the pure component polymer parameters depend on the type and composition of the copolymer. Pure component polymer parameters are obtained from binary polymer–solvent phase equilibrium data. With these parameters it is possible to model the phase behavior of the same polymer with another solvent. A better approach is the copolymer SAFT approach of Radosz and co-workers [81–83], in which the copolymer parameters are estimated on the basis of the molecular weight and structure only. For an AB-type copolymer there are three binary interaction parameters, the interaction parameters between A segments and segments of the solvent molecule, the interaction parameter between B segments 2.4.2.5

2.4 Equation of State Models

Fig. 2.17. Isothermal cloud-point curves of the HDPE þ ethylene system. Mn ¼ 43 kg mol1 , Mw ¼ 118 kg mol1 , Mz ¼ 231 kg mol1 . Symbols: experimental data; curves: modeling results. Reproduced with permission from Ref. 78.

and segments of the solvent molecule, and the interaction parameter between A and B segments. The ﬁrst two binary interaction parameters can be obtained from the phase behavior of the two homopolymer systems, while the third has to be ﬁtted to some copolymer–solvent data. Once these parameters are known, predictions can be made for copolymer–solvent systems with the same type of copolymer but with a diﬀerent copolymer composition. The same approach was followed by Gross et al. [84] for the PC-SAFT model. The result is known as copolymer PCSAFT. The two parameters that characterize the polymer structure are the fraction of type A segments in the polymer molecule and the bonding fraction which gives the fraction of bonds between segment types A and B. The original literature gives details. Figures 2.19 and 2.20 show experimental cloud-point curves and PC-SAFT modeling of the ethylene þ poly(E-co-EA) and ethane þ poly(E-co-BA) systems, respectively [85], at a polymer concentration of 5 wt.%. The model correctly predicts the change in the location of the cloud point with changing comonomer concentration in the polymer. Especially interesting is the ethylene þ poly(E-co-EA) system, in that the curve of cloud-point pressure as a function of the EA concentration in the polymer shows a minimum. This behavior is correctly described by the model.

49

50

2 Polymer Thermodynamics

Fig. 2.18. Constant composition cloud-point curves of LDPE– solvent systems at 5 wt.% polymer. Symbols: experimental data; curves: modeling results using PC-SAFT with one temperature-independent k ij for each system. Reproduced with permission from Ref. 78.

2.5

Conclusions

Polymer–solvent systems behave in many respects in the same way as systems of low molecular weight components. Diﬀerences between polymer–solvent systems and low molecular weight systems are mainly caused by the fact that polymers have no vapor pressure, that polymers are composed of many components of diﬀerent molecular weights, and that there is a large diﬀerence between the free volumes of the solvent and of the polymer. The thermodynamic models for polymer–solvent systems are less advanced than for systems of low molecular weight compounds. In general low-pressure vapor– liquid equilibria can be described very well with a variety of models. Once the adjustable parameters in these models are ﬁtted to experimental data, reliable predictions can be made for other conditions, for example at a diﬀerent temperature or for a system with the same solvent and the same type of polymer with a diﬀerent

2.5 Conclusions

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

the range 113–157 kg mol1 . Symbols: experimental data; curves: PC-SAFT calculations. Reproduced with permission from Ref. 85.

molecular weight. Vapor–liquid-equilibria can be predicted with diﬀerent group contribution models. Most recent models give reliable predictions of comparable accuracy for diﬀerent polymer–solvent systems. A disadvantage of this type of model is that the parameters for the groups of interest should be available. In general the thermodynamic modeling of low-pressure liquid–liquid equilibria is more diﬃcult than for vapor–liquid equilibria. This also holds for polymer– solvent systems. Reliable prediction methods are not available. The correlation of liquid–liquid data using the extended Flory–Huggins type models [27–29] gives reasonably good results. Again, once the adjustable parameters in these models are known, predictions can be made for other conditions. High-pressure ﬂuid-phase equilibria can only be modeled using equations of state. However, the equation of state models contain adjustable binary interaction parameters that have to be ﬁtted to data. Small variations in these parameters in general have a large inﬂuence on the predicted phase equilibria. The most promising models for high-pressure phase equilibria of polymer solutions are the SAFT and the PC-SAFT ones.

51

52

2 Polymer Thermodynamics

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

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

Notation

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

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

Notation

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

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

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

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

Subscripts d hb i; j mix p s vdW

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

53

54

2 Polymer Thermodynamics

Superscripts 0

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

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

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

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55

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

45

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47 48

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53

54 55 56

57 58 59 60 61 62 63

64

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

65 E. Neau, Fluid Phase Equilib. 2002,

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

67 68 69 70 71 72 73 74 75

76 77

78

79 80

81

82 83 84

85

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

57

3

Polycondensation1 Ma´rio Rui P. F. N. Costa and Rolf Bachmann 3.1

Basic Concepts 3.1.1

A Brief History

The concept of producing larger molecules starting from smaller ones containing suitable groups (such as carboxyls and hydroxyls), by reaction to create a bond between inert chemical moieties by splitting oﬀ a smaller molecule (a condensation), can be traced back to the middle of the nineteenth century [1]. It was discovered when preparing oligomers of ethylene glycol, but, in spite of the good impact the work had at that time, it fell into oblivion a little later. Phenol–formaldehyde (in the form of resins and varnishes) was the ﬁrst commercial product (in 1907) consisting of an entirely synthetic polymer [2]; its formation was also an example of polycondensation; but this early development was purely empirical and contributed very little to scientiﬁc progress. Staudinger’s concept of the existence of macromolecules came two decades later [3]. It paved the way to the systematic study aimed at producing synthetic ﬁbers carried out by Carothers at DuPont in the late 1920s and early 1930s [4], from which stems the still-used concepts of addition and condensation polymerizations. In his famous book, Flory [5] justly criticized this terminology, since closely related polymerization mechanisms involving reactions of groups with active hydrogen atoms with epoxides and isocyanates do not lead to the splitting of a byproduct; he proposed a classiﬁcation into ‘‘step growth’’ and ‘‘chain growth polymerization’’. These expressions became widely used, even if they are quite vague; much better would be ‘‘random’’ and ‘‘sequential’’ polymerization [6]. A survey of important examples of polycondensation reactions can be found in Table 3.1.

1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

58

3 Polycondensation Tab. 3.1.

A survey of technically important polycondensation reactions.

Functional groups or monomers

Connecting group

By-product

aCOOH Carboxylic acid

aOH hydroxyl

aCOOa ester

H2 O

aCOOR Ester

aOH hydroxyl

aCOOa ester

RaOH

aCOOH Carboxylic acid

aNH2 primary amine

aCONHa amide

H2 O

ClCOCl/aCOCl Phosgene/chloroformate

aOH hydroxyl

O

O C

HCl O

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

aCl chloro end group (as in ClC6 H4 Cl)

aSa sulﬁde

NaCl

NaOa Sodium alkoxide/phenoxide, e.g.,

aCl chloro end group, e.g.,

aOa ether

NaCl

NaO

CH3 C CH3

aNbCbO Isocyanate

Cl ONa

O S O

aOH hydroxyl

Cl

O

O C

none NH

urethane aNbCbO Isocyanate

aNH2 primary amine

O C NH

none NH

urea CH CH2 O

1,2-epoxide aCH2 OH Methylol

aNHa secondary amine

none

OH N CH2 CH

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

CH2NH

O C

H2 O NH

N,N 0 -dialkylurea AraCH2 OH Aromatic methylol

Ar 0 aCH2 OH aromatic methylol

AraCH2 aAr 0

HCHO

In some polycondensations, new functional groups are created by reactions of the monomers. Thus, methylol end groups are formed by the initial reactions of formaldehyde with other monomers, and their condensation reactions shown in the last two rows of Table 3.1 occur in later stages of formaldehyde/urea and formaldehyde/phenol polymerizations (see Section 3.3.4).

3.1 Basic Concepts

3.1.2

Molecular Weight Growth and Carothers’ Equation

Carothers was mainly interested in polyesteriﬁcations, and later in polyamidations, of compounds with the structures AXA and BYB, or sometimes AXB, in which X and Y are inert chemical moieties whereas A and B are the active carboxyl and hydroxyl or amine groups, giving rise to a connecting ester or amide group Z and splitting oﬀ a by-product (water) W [Eq. (1)]. AþB!ZþW

ð1Þ

If reacting groups A and B do not belong to the same molecule, and thus do not create a ring, this reaction leads to a decrease of one polymer molecule (comprising the starting monomers). A count of the polymer molecules and their mass in the system can easily be carried out with the help of p, the conversion of reference groups A. Starting with 1 mol of monomer AXB, the remaining moles of polymer are 1 p. Polymer mass per mole of AXB is equal to its molecular weight MAXB , minus the mass lost in byproduct formation, p MW , yielding Eq. (2), allowing the number-average molecular weight (mass of polymer/moles of polymer) to be predicted.

Mn ¼

MAXB pMW 1 p

ð2Þ

With two monomers, on introduction of the initial stoichiometric ratio r ¼ ½B0 /½A0 between mole concentrations of groups A and B, a similar reasoning leads to Eq. (3).

Mn ¼

MAXA þ rMBYB 2pMW 1 þ r 2p

ð3Þ

High molecular weights (say, above 20 000, needed for use as ﬁbers) are possible only by reaching high conversions when monomer molecular weights are a few hundreds. With two diﬀerent monomers, stoichiometric ratios very close to unity are also needed. Well-known relationships for number-average degree of polymerization xn are obtained from Eqs. (2) and (3) by replacing the molecular weights of monomers by 1 and the molecular weight of by-product by 0. The eﬀect of ring formation on Mn of open chain molecules can be taken into account by introducing pc , the conversion of end groups yielding rings, in Eqs. (2) and (3). For instance, when there are two monomers AXA and BYB, Eq. (4) is obtained.

Mn ¼

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

ð4Þ

59

60

3 Polycondensation

Average degrees of polymerization and sol fraction in XA f polycondensations without intramolecular reaction, versus conversion p of A groups with monomer functionalities f ¼ 2 and f ¼ 3, for equilibrium or batch reaction starting from monomer.

Fig. 3.1.

Prediction of pc and the average molecular weight of rings is not a trivial task. Carothers has stated that the fraction of rings in bulk polycondensations is negligible, except when ﬁve- or six-membered rings (or higher in the case of rings containing Si, P, and similar heavier elements lower in the periodic table) can result. Recent experimental results show this concept is not valid in the case of irreversible polycondensations (see Section 3.1.7). Monomers with three or more functional groups lead to the formation of branched polymers. A network of macroscopic dimensions (a gel) appears for high enough conversions and monomer branching. In Figure 3.1 are presented classical predictions (see Section 3.4.3), of average degrees of polymerization and weight fraction of ﬁnite molecules (sol fraction) for the single monomer polycondensations of XA3 and XA2 . Notice that, in contrast with linear polycondensations, which require high conversions for obtaining high values of x n , low-value maxima (4 if f ¼ 3) at gel point are observed with these nonlinear polycondensations. Close to full conversion, the sol is nearly pure unconverted monomer. Stockmayer [7] has obtained the distribution for the fractions of the initial monomer units present in each ﬁnite polymer molecules (Eq. (5), where Px is the set of isomers with x repeating units X). x½Px /½X ¼

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

ð5Þ

3.1 Basic Concepts

Fig. 3.2. Equilibrium distributions of monomer units x ½Px =½X for XA f polycondensations without intramolecular reaction versus conversion p of A groups with monomer functionalities f ¼ 2 and 3.

If the functionality f ¼ 2, the even more notorious Schulz–Flory [8] distribution is obtained [Eq. (6)]. x½Px /½X ¼ xð1 pÞ 2 p x1

ð6Þ

This latter distribution has a relative maximum for x G x n, in contrast to the general Stockmayer distribution, which decreases monotonically (see Figure 3.2). More complex chemical systems lead to molecular weight distributions that are qualitatively similar: linear polycondensations have most often geometrical or nearly geometrical distributions with polydispersities M w /M n close to 2, whereas nonlinear polycondensations lead to extremely broad distributions near the gel point, but always dominated in number and even in weight by the lower molecular weight species. This chapter concentrates on processes for making either high molecular weight, mainly linear polymers, or else low molecular weight, branched oligomers which will be later used to form networks. In any case, consideration of initial stoichiometry, and possible change of functionality due to side reactions, is essential to predict average molecular weights, and to avoid premature gelation in the processes described here. It is also possible to prepare low molecular weight mixtures of ring molecules for further reactive processing (using, for instance, anion polymerization), starting

61

62

3 Polycondensation

with monomers [8] or even with recycled polymer [9], by dilution with inert solvents and a suitable catalyst (see also Section 3.1.7). The discussion of these otherwise interesting processes falls outside the scope of this chapter. 3.1.3

The Principle of Equal Reactivity and the Prediction of the Evolution of Functional Group Concentrations

Carothers’ successful synthesis of high molecular weight, aliphatic polyesters could only be carried out thanks to his care in reducing the eﬀect of the hydrolysis reaction through use of high vacuum. Further research carried out by his former collaborator P. J. Flory in order to assess the eﬀect of molecular weight on chemical kinetics would eventually lead to the establishment of the principle of equal reactivity. A parallel research, which also has contributed to the establishment of the same concept and to the prediction of equilibrium chain length distribution (CLD), has been led by G. V. Schulz [10] in Germany. Assuming there is no mutual interference of end groups in the same molecule (which does often happen with groups attached to aromatic rings), the speciﬁc rate of consumption, RA , of end groups A by an irreversible reaction reaction with groups B can be written as Eq. (7). RA ¼ k½A½B

ð7Þ

Equation (7) states that a single apparent second-order rate constant k should describe all intermolecular reactions of the involved groups A and B for a certain concentration of catalyst(s) (which may be A or B themselves) and solvent, regardless of the polymer molecule to which they belong. A change of composition of the reaction medium will often cause k also to change. If the reaction is reversible, a more general expression [Eq. (8)] for RA should be used. RA ¼ k½A½B k Z ½Z

ð8Þ

In the case where, as often happens, a by-product W exists, the apparent ﬁrst-order reverse rate constant k Z is related, through Eq. (9), to apparent equilibrium ratio K and by-product concentration ½W . k Z ¼ k½W /K

ð9Þ

3.1.4

Eﬀect of Reaction Media on Equilibrium and Rate Parameters

Polycondensations must be carried out in bulk or with a concentration of solvent as low as possible (viscosity being the limiting factor) in order to minimize the formation of rings. If the end groups have polarities and/or hydrogen acceptor or do-

3.1 Basic Concepts

nor aﬃnities very diﬀerent from the bonds, this is likely to lead to changes in the value of apparent constant k, which will be observed as a dependence on conversion or stoichiometric ratio. Known examples are the apparent increase in reactivity with conversion in esteriﬁcations and the strong eﬀect of water concentration on apparent equilibrium and rate constants in amidation systems (see Sections 3.3.1 and 3.3.3 respectively). Should some thermodynamic model of the chemical system be available, knowledge of activity coeﬃcients g of chemical groups A; B; Z and by-product W would allow the apparent equilibrium ratio K to be related to the thermodynamic K 0 (a true constant, a function only of temperature) through Eq. (10). K ¼ K0

gA gB ½Z½W ¼ g Z gW ½A½B

ð10Þ

This kind of computation can be done using the UNIFAC contribution group method [11], but its empirical character and lack of available parameters at high temperatures limit drastically its usefulness. Solvent or bulk media eﬀects can be modeled by taking advantage of group contribution methods [12]. Multiple kinetic experiments with diﬀerent media compositions are needed in order to compare the free energy of interaction involving the hypothetical transition state group (see Ref. 13 for an example). No such studies have ever been tried with polymerization reactions. Progress in computational quantum mechanics should ultimately make the practical use of these correction methods more widespread. In nonlinear polycondensations, it is possible to observe a limiting conversion, the topological limit [14] when unreacted groups in network are too far apart to meet. Rate constants will fall before reaching the topological limit if glass transition temperature Tg is attained (the glass eﬀect). This situation occurs with many thermosetting systems of practical interest, epoxies being the most studied, as discussed in the reviews by Mita and Horie [15] and Dusˇek [14]. A new theory for describing this phenomenon, leaving aside the long-used ‘‘free volume’’ concept, was recently put forward by Corezzi et al. [16]. It is based upon the Adam–Gibbs [17] entropy theory of glass transition. Formation of more covalent bonds by polymerization decreases the number of available statistical conﬁgurations of the chemical system, thus decreasing conﬁgurational entropy, suggesting Eq. (11) relating structural relaxation time tr to number-average degree of polymerization x n . tr ¼ t0 exp½b g ðTÞx n bg ¼

b g0 T T0

ð11Þ

Function b g ðTÞ is empirical and could be diﬀerent from that in Eq. (11) [16]. There is for the moment no published use of this theory to ﬁt apparent kinetic constants.

63

64

3 Polycondensation

A possible suggestion is the following (untested) correlation with yet another two parameters, k 0 and Cg , based on the Rabinowitch equation for diﬀusional eﬀects on chemical reactions [Eq. (12)]. 1/k ¼ 1/k 0 þ Cg tr

ð12Þ

In Dusˇek’s terminology, diﬀusion control of chemical reactions may be speciﬁc, when the bond formation rate depends on the mobilities of individual reacting molecules or substructures, or else nonspeciﬁc, when the reaction rate depends only on the average properties of the system. Experimental evidence points toward prevalence of nonspeciﬁc rate control: for instance, gel conversion in epoxide resin cure is the same regardless of glass eﬀect [18]. Lacking a better alternative, an empirical dependence of rate parameters on conversion, for a given starting composition, must be tried. Examples of this approach will be discussed along with their respective chemical systems (bulk polyester and polyamide formation). It is seen that the glass eﬀect could be taken into account using a similar procedure. 3.1.5

Polycondensation Reactions with Substitution Eﬀects

When the reaction of a functional group changes the reactivity of its neighbors, a substitution eﬀect exists. A ﬁrst-shell substitution eﬀect (FSSE) is the simplest kind of departure from ideal behavior. It is the situation where reactivity is aﬀected only by the reaction of the functional groups attached to the same monomer unit. FSSEs are encountered in many polycondensations, as reactivities of functional groups in monomers are often diﬀerent from the reactivities of end groups in polymers because of mutual steric, resonance, or electrostatic interactions. For instance, a glycol HOaXaOH shows no substitution eﬀect in an esteriﬁcation reaction if the reactivity of the hydroxyls in aCOOaXaOH is the same: a single kind of group aOH needs to be considered and the description of the reaction scheme is much simpler. A second-shell substitution eﬀect (SSSE) occurs when the reactivity of a functional group is aﬀected by the reaction of all the groups attached not only to the same monomer unit, but also to the units linked to that one. This behavior has been recognized in urea/formaldehyde formation (see Section 3.3.4.2). Linear polycondensations AXA þ BYB with FSSEs in both root units X and Y are examples found in some important processes, such as the esteriﬁcation of terephthalic acid and ethylene glycol leading to poly(ethylene terephthalate) (PET). Four diﬀerent rate constants ki are needed to describe forward reactions, and possibly an equal number of apparent rate constants (kiZ , i ¼ 1 . . . 4) are required to describe reverse reactions:

3.1 Basic Concepts k1

AXA þ BYB T AXZY B þ W k1Z k3

AXZ þ BYB T BYZX Z þ W k3Z

k2

AXA þ BYZ T AXZY Z þ W k2Z

k4

ð13Þ

AXZ þ BYZ T ZXZY Z þ W k4Z

Recall that apparent ﬁrst-order rate constants of the reverse reactions kiZ are related to the apparent second-order rate constants of the forward reactions ki through the apparent equilibrium ratios Ki and the concentrations of by-product: Ki ¼

ki kiZ ½W

ð14Þ

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

ð15Þ

RZA ¼ 2k2 ½AXAð½ZD þ ½ZB Þ þ 2k4 ½ZD 2 ½ZA ð2k3 ½BYB þ k4 ½ZB Þ 2k Z ½ZA RZB ¼ 2k3 ½BYBð½ZD þ ½ZA Þ þ 2k4 ½ZD 2 ½ZB ð2k2 ½AXA þ k4 ½ZA Þ 2k Z ½ZB

65

66

3 Polycondensation

In conclusion, descriptions of substitution eﬀects not only require knowledge of more kinetic parameters, but also may lead to added mathematical diﬃculties, even for just the prediction of basic characteristics of the chemical system, such as the concentrations of functional groups.

3.1.6

Exchange Reactions

Exchange reactions without formation of by-products, such as the acidolysis and aminolysis reactions present in polyamidations, do not increase number-average molecular weight, but can be an important cause of relaxation of chain length distributions toward the equilibrium. Kotliar has presented a general review of these reactions [19]. Direct bond exchanges in the complete absence of by-product (such as amide– amide exchange) are at best very slow (if they indeed exist) and, as these reactions can be safely neglected, only exchange reactions involving an end group and a bond should be of any importance. The exchange of bonds Z1 and Z2 (necessarily formed with a common by-product W), can be described through Eqs. (16). k1

XA þ YC T XZ1 Y þ W k1Z

E k12

k2

XB þ YC T XZ2 Y þ W k2Z

ð16Þ

XA þ X 0 Z2 Y T XZ1 Y þ X 0 B E k21

These reactions modify the counts of functional groups of similar chemical nature (for example, distinguishable kinds of amides/amines/carboxylic acids). If there is only a single kind of bond, there is no net creation or destruction of functional groups or bonds, but a reshuﬄing of pieces of the reacting molecules takes place. In order not to violate the condition of chemical equilibrium, rate constants of forward and reverse exchange reactions can be related to equilibrium constants of the condensation reactions in which the same functional groups are involved. In this example, the equilibrium ratio of the exchange reaction must be in accord with Eq. (17). E k12 K1 ¼ E k21 K2

ð17Þ

This operation will reduce the number of kinetic parameters in the kinetic scheme. Also, several of the various equilibrium constants can also be related through a reasoning due to Gordon and Scantlebury [20] for the XA f polycondensation, which consists of checking the formation of the same ﬁnal products through successive equilibria but with diﬀerent intermediate stages.

3.1 Basic Concepts

3.1.7

Ring-forming Reactions

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

L n ! Cn þ W cZ

ð18Þ

k

k aE

L m þ Cn ! L mþn cE

ð19Þ

k ðnÞ

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

ð20Þ

67

68

3 Polycondensation

kijcE ðnÞ ¼ kijE k c ðnÞ k c ðnÞ ¼

1 ð3/2pnb 2 Þ 3/2 NAL

ð21Þ ð22Þ

In Eq. (22), the small contributions of the distances of the ends of functional groups b were assimilated to the length of a bond, and NAL is the Avogadro– Lo¨schmidt number. If there is only one kind of bond, an equilibrium constant of cyclization K c ðnÞ can be deﬁned from the equilibrium in Eq. (23). K c ðnÞ ¼

½Cn ½L m ¼ k c ðnÞ/n ½L nþm

ð23Þ

3.1.8

Modeling of Polymerization Schemes

Methods for predicting molecular weight distributions at chemical equilibrium and for irreversible polycondensations are presented with some detail below; see Sections 3.4.3 and 3.4.4. Systems at chemical equilibrium are amenable to description by the domain of calculus of probabilities known as the theory of branching processes [27]. Batch reactions starting from monomers can sometimes be described by the same solutions, increasing the practical importance of such an approach. Exchange reactions also drive molecular weight distributions toward equilibrium. Many reactions of great technological interest, such as melt polyesteriﬁcations, can be tackled using this approach. Applicability of statistical–probabilistic methods far from chemical equilibrium cannot be guaranteed, and the kinetic approaches described in Sections 3.4.4 and 3.4.5 are a better choice. Use of statistical–probabilistic methods for describing polycondensations started with Flory, who used them to compute the equilibrium chain length distribution of linear systems and later was able to predict gelation conditions for multifunctional monomers. Stockmayer [7] could extend this method to the computation of chain length distributions and average molecular weights of nonlinear polymers. Gordon [20, 28] recognized that nonlinear polymers in the absence of intramolecular reaction can be described by a Galton–Watson branching process. Relatively complex chemical systems, presenting substitution eﬀects, became tractable, and even some properties of polymer networks relevant to rubber elasticity theory [29] and average radius of gyration [30] as well as other polymer properties [31–33] could be computed. Alternative approaches to Gordon’s branching theory have later been developed, mainly the so-called recursive method [34, 35], which is in a number of ways simpler to understand and use, but lacks the power of the older theory in many situations.

3.2 Mass Transfer Issues in Polycondensations

Reversible polycondensations can be tackled using the concept of molecular fragments introduced in Section 3.1.5. It is possible to establish closed sets of rate equations for those fragments in many important cases (the main diﬃculty being the presence of higher-order substitution eﬀects). For a more detailed discussion, together with a short analysis of the much simpler case of linear polycondensations with a single kind of bond (a single monomer AXB or two monomers with diﬀerent groups AXA þ BYB), see Section 3.4.5. It is noteworthy that CLD in those chemical systems (and probably in similar alternating polyesteriﬁcations and polyamidations) is nearly always fairly close to equilibrium, the main discrepancies occurring as far as the concentrations of the ﬁrst linear oligomers are concerned. So, a method which predicts those concentrations correctly is what is really needed in practice. The main factor controlling the CLD should be the presence of multifunctional impurities, which should be carefully tracked by the kinetic model. Prediction of the whole CLD should be possible only with Monte Carlo methods, with their usual drawbacks [36]. Nevertheless, polycondensations are easier to simulate by Monte Carlo than other polymerizations, since all reactions have more or less the same time scale. These simulations are invaluable for investigating the eﬀect of space correlations between reacting groups and their eﬀect on ring formation and elastic properties [37]. Prediction of molecular size distribution (which hopefully can be determined by size exclusion chromatography) is also a useful result [38] which is diﬃcult to obtain otherwise.

3.2

Mass Transfer Issues in Polycondensations

This section deals with situations where mass transfer eﬀects of by-products (devolatilization, solid-state polymerization) and monomers (interfacial polymerization) become so important as to cause a spatial change in polymer molecular weight distributions. 3.2.1

Removal of Volatile By-products

Devolatilization in irreversible polycondensations is carried out in the later stages of the process and is similar to other polymerizations. For reversible polycondensations, however, it diﬀers from the devolatilization of other polymers in a number of ways.

Reaction and removal of by-product are intimately connected. Monomers may be removed in substantial quantities. Removal of volatiles takes place during the whole course of the reaction.

69

70

3 Polycondensation

In the ﬁrst stages of the reaction, reversible polycondensates readily form boiling liquids and care has to be taken to avoid removal of monomers, foaming, or product entrainment. Equilibrium constants are of the order of unity for polyesters and polycarbonates, and of the order of hundreds for polyamides. In order to obtain end-group conversions above 0.99, this implies a concentration of by-product ½W below 104 times the concentration of bonds in the former two systems. The high temperature of the processes increases the vapor pressure of the by-products and their equilibrium solubility decreases, but even so partial pressures of some millibars for polyester and polycarbonate processes are required; they may be obtained by using a combination of vacuum and inert stripping gas. These low partial pressures and consequently nearly inﬁnite dilution often justify the application of Henry’s law to describe the vapor–liquid equilibrium according to Eq. (24), in which PW is the equilibrium vapor pressure of by-product W, HW is its Henry constant for the polyS mer melt (in the usual range of 10–1000 bar), wW is mass fraction in the melt, PW is the vapor pressure of pure W, and WW is its weight fraction activity coeﬃcient. S PW ¼ HW wW ¼ PW WW wW

ð24Þ

The second part of Eq. (24) is useful below the critical temperature of the byproduct, when some thermodynamic model is available. For instance, given the densities of melt and by-product respectively as rP ; rW , Flory–Huggins theory leads to Eq. (25). WW ¼

rP expð1 þ wÞ rW

ð25Þ

Similar expressions hold for the volatile monomers, which at high dilutions can be treated independently. At low conversions, the inﬁnite dilution is not applicable and the multicomponent Flory–Huggins equation [Eq. (26)], for example, should be used instead. Here the fYi are the volume (or segment) fractions of the various components, i ¼ 1 up to NY being the volatile components and the polymer being component NY þ 1; the yYi are their mole fractions, VYi the molar volumes, and the wij the binary interaction parameters. 0 1 W Yi ¼

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

ð26Þ

j0i

At high conversions, mass transfer resistance to by-product removal is a key factor, not because of low diﬀusivities D of the by-products (in the range 109 to 1011 m 2 s1 ), and consequent mass transfer coeﬃcients, but because of low interfacial areas. Because of the huge increase in viscosity of the melt when the reacting mixture changes from the initial mixture of monomers/oligomers to high polymer, a possi-

3.2 Mass Transfer Issues in Polycondensations

ble solution is the use of diﬀerent continuous stirred reactors (CSTRs) in series, with diﬀerent kinds of stirrers. Other processes consist of simple unstirred bubble columns. In either case, these reacting mixtures at the beginning of the processes resemble boiling liquids. Stirring helps the formation of bubbles by cavitation and breaks existing bubbles, increasing interfacial area. Inert gas injection will also help (but loss of volatile monomers becomes more diﬃcult to counterbalance). These mechanisms become ineﬃcient for high enough molecular weight with the consequent higher viscosity (above 200 Pa s), and special ﬁlm-forming equipment must be used. Equipment for devolatilization of residual monomers and solvents without chemical reaction is used in most polymerization processes, vented extruders being a common choice. They are also used in the ﬁnal stages of reversible polycondensations [39], as well as other specially developed devices that we discuss next. The basic design of these latter systems consists in a partially ﬁlled horizontal cylindrical vessel in which the reaction mixture moves axially with the help of a screw or a rotor with blades. The impeller periodically extracts part of the bulk liquid, leaves it exposed as a thin ﬁlm to the gas phase for a short time t f , and remixes it again with the main stream as it ﬂows toward the outlet. According to the mechanism of creation of ﬁlm, three classes of devices can be distinguished [40–42]:

Wiped-ﬁlm reactors (WFRs) [43], where the impeller blades throw the polymer melt against the inner surface of the vessel (Figure 3.3), subjecting it to a high shear rate (in the range 10 3 –10 4 s1 ) in the gaps between their tips and the wall. This high shear is favorable to conveying pseudoplastic, high-viscosity polymers, but can conversely bring about problems for chains which break under shear, such as polybutylene terephthalate. Vented extruders work using a similar principle in their central section (where deeper screws lead to partial ﬁlling of the channel) and their reactor models can be considered a variant of this class. Rotating disk contactors (RDCs) (Figure 3.4), in which the disks periodically dip into the pool of reacting mixture. The polymer ﬁlm they carry is exposed to the gas phase before being mixed again with the bulk liquid. Falling-strand or falling-ﬁlm evaporators.

Fig. 3.3.

Diagrammatic representation of a wiped-ﬁlm reactor.

71

72

3 Polycondensation

Fig. 3.4.

Diagrammatic representation of a rotating disk contactor.

The currently used description of homogeneous diﬀusion of volatile by-products in polymer media during reversible polycondensations is due to Secor [44]. It considers polymer molecules immobile. The ﬂux of small molecules has a negligible convective contribution; only the diﬀusional ﬂux with respect to the polymer needs be considered, and the microscopic mass balance of a generic volatile component (usually, but not always, a by-product) Yi and a group A i belonging to the polymer may be written, neglecting density variations, as in Eq. (27). q½Yi ¼ DYi ‘ 2 ½Yi þ RYi qt

ð27Þ

q½A i ¼ RA i qt As the resistance to mass transfer in the gas phase may be neglected, the diﬀusive ﬂux of evaporation N_ Yi , in moles per unit area and unit time, will be obtained with the help of a mass transfer coeﬃcient kfYi . There are two equivalent alternative definitions [Eq. (28)], one using a driving force in terms of mole concentrations, the other in terms of activities (asterisks mark a concentration or activity value at the interface, considered to lie at y ¼ 0). q½Yi a ¼ kfYi ð½Yi ½Yi Þ ¼ kfY ðaYw aYi Þ N_ Yi ¼ DYi i qy jy¼0

ð28Þ

Combining this expression with the microscopic mass balances in the polymer ﬁlm, it is possible to predict mass transfer coeﬃcients for simple geometries and to take into account possible coupling with chemical reactions. For instance, if the polymer ﬁlm is immobile, has a constant depth L, and mass transfer occurs along the y direction with a plane geometry, time-averaged mass transfer coeﬃcient of

3.2 Mass Transfer Issues in Polycondensations

volatile species kfi after an exposure time t f would be computed by solving Eq. (27) with initial and boundary conditions as given in Eqs. (29). ½Yi jt¼0 ¼ ½Yi 0 ;

½A i jt¼0 ¼ ½A i 0

½Yi jy¼0 ¼ ½Yi ;

q½Yi ¼0 qy jy¼L

DYi

ð tf

kfYi t f ¼

0

ð29Þ

q½Yi ð y; tÞ dt qy jy¼0

½Yi 0 ½Yi

In view of the kinds of gas–liquid contact described above, an obvious model is provided by the penetration theory (inﬁnite depth L), where a portion of ﬂuid with uniform concentration proﬁle is suddenly exposed to the gas phase, and is replaced by fresh ﬂuid after an exposure time t f . For an inﬁnite ﬁlm depth L and negligible chemical reaction, this yields the well-known result [Eq. (30)] for the time-averaged mass transfer coeﬃcient. sﬃﬃﬃﬃﬃﬃﬃ DYi pt f

ð30Þ

kfYi ¼ 2

Pell and Davis [45] were the ﬁrst to actually measure a diﬀusion coeﬃcient for a volatile by-product of a polycondensation, using PET formation in ﬁlms of varying depth (although obtaining values of D much higher than those nowadays accepted). An early example of discussion of coupling of diﬀusion/chemical reactions in these systems may be found in Ref. 46. The ﬁrst model associating the axial transport along the reactor (direction z) with the cross-ﬂow transfer of volatile by product (direction y) (see Figure 3.5) is due to Amon and Denson [47] (Ault and Mellichamp [46] considered that all the polymer was in the ﬁlm) and was developed for WFRs. It assumes plug ﬂow in the pool, which implies a negligible hold-up of the liquid in the ﬁlm. A time-averaged mass

Volatile by-product

x y JT

Low M polymer inlet

Polymer outlet

z=0

Fig. 3.5.

Simple model for a WFR.

z

z=L y

73

74

3 Polycondensation

transfer coeﬃcient is obtained at each axial position by solving Eqs. (27) and (29), and the mass balance of the pool is written as Eqs. (31), where u z is the axial superﬁcial velocity (volumetric ﬂow rate of polymer Q divided by cross-section area of the pool) and a v is the ﬁlm area per unit volume. uz

d½Yi ¼ RYi kfYi a v ð½Yi ½Yi Þ dz

uz

d½A i ¼ RA i dz

ð31Þ

½Yi jz¼0 ¼ ½Yi 0 Since the main mass transfer area is the ﬁlm on the barrel wall, the exposure time would be calculated [47] through Eq. (32), where d T is the inner barrel diameter, L x is the ﬁlm perimeter (L x G pd T , if the nip is small) and n_ is the screw rotational speed in rotations per unit time. tf ¼

Lx pd T n_

ð32Þ

A better model [48] takes into account the movement of the ﬁlm along the wall with velocity u x ¼ pd T n_ between coordinates x ¼ 0 and x ¼ L x and therefore adds a convection term, leading to Eqs. (33), where the ½A i f and ½Yi f are the concentrations of the species in the ﬁlm. There is no need to consider cylindrical geometry for the ﬁlm, since its thickness is low. q½Yi f qt

þ ux

q½Yi f qx

¼ DYi

q 2 ½Yi f qy 2 ðL

q½Yi q½Yi ux þ uz ¼ R Yi þ qt qz L

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

0

þ RYi

q½A i f qt

þ ux

q½A i f qx

¼ RA i

½Yi f ðt; yÞ dy u z ½Yi

ðL 0

½A i f ðt; yÞ dy u z ½A i

ð33Þ

½A i jz¼0 ¼ ½A i 0 ðtÞ

½Yi f jx ¼0 ðt; y; zÞ ¼ ½Yi ðt; zÞ ½A i f jx ¼0 ðt; y; zÞ ¼ ½A i ðt; zÞ q½Yi f qy

¼0 jy¼L

½Yi f ðt; 0Þ ¼ ½Yi

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

3.2 Mass Transfer Issues in Polycondensations

Unwrapped views: Film

Fig. 3.6.

Scheme of ﬂow and mass transfer in single-screw vented extruders.

A scheme of ﬂow and mass transfer in single-screw vented extruders, also illustrating some key geometrical parameters, is shown in Figure 3.6. As polymer ﬂows inside the channel along a trajectory in a helix, with a total length LB /sin y, in which y is the angle of the screw, the coordinate z will be taken along this helicoidal path. Dimensions of the channel will be LW (width) by H (depth), with a fraction fL ﬁlled with liquid. As the movement of screw pushes the polymer pool, a fraction passes through the space between the barrel and the screw, forming the desired evaporating ﬁlm. If the ﬂuid is Newtonian, the ﬁlm width is h/2, where h is the clearance. The average transverse velocity of the screw dragging the ﬁlm, vT , is given by Eq. (34). ð34Þ

vT ¼ pd T n_ sin y

The ﬁlm which is wiped from the channel re-enters at a distance d upstream of its departure point given by Eq. (35) [50] and the time of exposure of the ﬁlm t f is given by Eq. (36). d ¼ pd T fL cos y

ð35Þ

t f ¼ ð1 fL Þ/n_

ð36Þ

The concentrations ½A i f ðzÞ and ½Yi f ðzÞ respectively for nonvolatile and volatile components in the ﬁlm, back-mixed at axial position z, are obtained by solving Eq. (27) for t ¼ t f with initial conditions described by Eqs. (37) (a more exact model would consider convection as in Eq. (33)): ½Yi jt¼0 ¼ ½Yi ðz þ dÞ;

½A i jt¼0 ¼ ½A i ðz þ dÞ

ð37Þ

75

76

3 Polycondensation

Assuming plug ﬂow in the channel (a trivial change would be to add an axial dispersion coeﬃcient), the mass balances in the channel taking into account also the devolatilization from the pool (mass transfer coeﬃcients kfpYi ) thus becomes those given in Eqs. (38).

uz

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

uz

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

½Yi jz¼0 ¼ ½Yi 0

ð38Þ

½A i jz¼0 ¼ ½A i 0

The presence of the delay d may be circumvented by using a Taylor expansion about z [50] in order to obtain a system of second-order ODE. Also according to Refs. 49 and 50, the exposure time of the pool may be estimated through Eq. (39).

t fP ¼

H pd T n_ sin y

ð39Þ

Another integration of Eq. (27) for t ¼ t fP with a trivial modiﬁcation of Eq. (30) will provide an estimation of the mass transfer coeﬃcients of the pool kfpYi . Twin-screw extruders have the advantage of being self-cleaning and can work with extremely high viscosity, above 10 6 Pa s [53], making them a good choice for polyamide and polyurethane ﬁnal stages of reaction, thanks to the possibility of using reduced space times. Their detailed modeling is more diﬃcult than with single-screw devices, and few fundamental studies [54] have been carried out. Rates of mass transfer have been predicted for a co-rotating twin screw using penetration theory and experiments have been done with a transparent device (for observing whether bubbles were present or not in the devolatilization zone). Observed results of kf a v were proportional to the speed of rotation to the power of 0.5, as penetration theory predicts, but were three times lower than theoretical predictions, which has been attributed to the very low liquid hold-up and consequent lack of coverage. In contrast to WFRs and vented extruders, use of staged models for describing rotating disk contactors is a natural choice [55], since the J compartments with liquid hold-ups Vm j can be approximated as CSTRs. Taking into account the possibility of back-ﬂow, the overall volume ﬂow rate leaving the jth CSTR, Q j , will be divided into a fraction b j going back to CSTR j 1 and 1 b j going to tank j þ 1, except for the ﬁrst CSTR, in which b 1 ¼ 0. Also, Q Jþ1 ¼ 0, and the gas phase will be considered well mixed with uniform temperature (see Figure 3.7). Thus, mass balances at a steady state of volatile and nonvolatile components in the jth compartment ( j ¼ 1, J) may be written as in Eqs. (40).

3.2 Mass Transfer Issues in Polycondensations

Fig. 3.7.

Staged model of an RDC.

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

ð40Þ

¼ Q j ½Yi j RYi Vm j þ kfYij avj Vm j ð½Yi j ½Yi Þ Mass transfer coeﬃcients can be computed using penetration theory as described above. Murakami et al. [56] obtained correlations for hold-up and mixing time in RDCs. Fractional dead space (between 0 and 18% for their experimental data) can be predicted from the mixing time. This dead space should obviously be kept to a minimum, because polymer staying there will degrade due to secondary reactions to possibly discolored or gelled material, and product quality may be seriously harmed. Local ﬁlm thickness and hold-up have been correlated to physical properties (surface tension, viscosity) and geometrical parameters [57]. Residence time distribution has been shown to become narrower when viscosity grows [58]. Interfacial area [58] can be correlated with relative ﬁlling level H/d T and the number of disk rings per unit length ( J/LT ) [Eq. (41)]. av ¼

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

ð41Þ

An experimental study on mass transfer in disk-ring contactors of diameter dR using low-viscosity acrylamide solutions as a model ﬂuid [60] has led to the correlation of Eq. (42), but this correlation should only be used as a ﬁrst approximation [40], in view of the complexities introduced by polymer viscoelasticity. 2 0:5 kf dR d n_ ¼ 1:59 R Sh ¼ D D

ð42Þ

Falling-ﬁlm or falling-strand devolatilizators receive increasing attention as they require no heavy and expensive machinery: Polymer is simply pumped through small slits or holes. Eﬃcient surface renewal is achieved by shear thinning during the fall [59]; strands and ﬁlms become very thin and may be completely depleted of

77

78

3 Polycondensation

volatiles. Depletion or possible tearing determine the optimal height of strands/ ﬁlms for a given viscosity and initial ﬁlm diameter. Reaction considerably increases the eﬃciency of these devices (by a factor of up to ten) as it replenishes the volatile by-product. Guides to the strands/ﬁlms, such as wires, thin rods, or grids, will further enhance the role of the reaction while reducing the eﬀect of shear thinning and tearing. In all these devices, an additional component of interfacial area is provided by the gas bubbles, which can result from sparging with inert gas (widely used for devolatilization without chemical reaction) or from boiling. Observed rates of mass transfer are often several times higher than predicted [50] and this discrepancy has been linked to the presence of bubbles. The Laplace–Kelvin equation predicts that an isolated gas bubble should redissolve if its size is below a critical threshold, and conversely it should grow if its radius r b exceeds the critical value given by Eq. (43) [50], where s is the surface tension, Pme is the local pressure over the bubble (it may be simply the hydrostatic pressure, but in other circumstances it may be controlled by medium elasticity [61]), and Pb is the pressure inside the bubble, equal to the sum of the partial vapor pressures due to inert gases and volatile components if physical equilibrium and gas-phase ideality hold. So, it is possible that no bubbling occurs if the pressure is high enough or the content of volatiles is too low, but this is often not the case. rbc ¼

2s Pb Pme

ð43Þ

Bubble nucleation in polymer media is usually heterogeneous [62]. It starts when shear stress manages to detach the small bubbles which are stuck in cracks and crevices of vessel walls, internals, or suspended dust. Once the source of heterogeneous nucleation is spent, only high supersaturation will restart boiling. Of course, in many practical situations there are already small air bubbles in the polymer, and they will start foaming too. Favelukis et al. [63] have conﬁrmed this view, and have developed a theory for bubble growth which may explain the higher mass transfer rates obtained with vented extruders and similar devices for high rotation speeds. A predictive theory for bubble nucleation was developed in this same research [64]. Gestring and Mewes [65] have studied polymer degassing both with and without bubbling using a transparent drum with a rotating blade (similar to the screw of a vented extruder). Measured values of mass transfer rates without bubbling are three times lower than predicted by penetration theory, because neither pool nor ﬁlm is well stirred – which could explain the failure of predictions in Ref. 54. Rates of mass transfer in the presence of foaming were about 40 times higher than in the bubble-free regime. Trace devolatilization with the help of a stripper agent has a greatly enhanced efﬁciency. An important recent result is that, when it forms, foam grows until a limiting volume is reached, regardless of initial volatile content and presence of stripper gas [66], and thus the devolatilization section in vented extruders should

3.2 Mass Transfer Issues in Polycondensations

have enough room for that expansion, and residence time should also be suﬃcient to allow the ﬁnal density to be reached. A patent [67] for enhancing mass transfer in this class of reactors proposes the introduction of inert gas into the polymer in order to force the formation of bubbles (the forced gas sweeping process). An experimental and theoretical model has also been presented [68] and will be brieﬂy summarized here. The reactor was a rotating disk contactor for making bisphenol A (BPA) polycarbonate. The reactor model uses a staged approach, and the crux is the prediction of the mass transfer rate of by-product (phenol). The two relationships expressed in Eqs. (44) for the volume of a gas bubble Vb as a function of the gas ﬂow rate Q g [69] and of its rising velocity u b in a laminar regime [70] were the key data. 1/4 15mQ g 3/4 4p 2rg 3 Q g d b 1/2 2gd b ub ¼ 1þ Cd u b Vb Vb ¼

Cd ¼

ð44Þ

16 þ1 Re

In these equations m; r, respectively, are the viscosity and density of the liquid, g is the acceleration of gravity, d b is the bubble diameter, and Cd is the drag coeﬃcient. The mass transfer coeﬃcient was predicted using penetration theory, and the exposure time was computed [Eq. (45)] as the rising time of a bubble in the melt (at height hR above the gas injection point). tf ¼

hR ub

ð45Þ

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

Nb Vm

ð46Þ

Q g tf Nb ¼ Vb The observed bubble frequency agreed with the theoretical predictions, as also did the proﬁles of x n versus reaction time. It is interesting to ﬁnish this complex section with such a case study, suggesting that for some problems ‘‘classical’’ Chemical Engineering of the 1960s can still help in present-day industrial and scientiﬁc problems.

79

80

3 Polycondensation

The design and operation of most polycondensation reactors for devolatilization of volatile by-products (namely vented extruders) is clearly a very diﬃcult problem because of the complexity of the ﬂows (with or without foaming). Further progress is likely to require a heavy use of computational ﬂuid dynamics, as simpliﬁed models seem to have arrived at their limits. 3.2.2

Solid-state Polycondensation

Current industrial processes for the production of high molecular weight, linear, aromatic polyesters and polyamides, for use as plastics and ﬁbers, use solid-state polycondensation (SSP) for their last stages. This is the kind of process that will be treated in this section: the polycondensation of semi-crystalline, low molecular weight polymers to high molecular weight ones, occurring below the melting temperature of high polymer, but well above the glass transition temperature. We will disregard polycondensation of crystalline monomers, which is also covered in the review by Pilati [71]. Because of the need to provide enough interfacial area to allow the removal of volatile by-products, the polymer has to be in a powdery form. One of the several optimization problems of these processes is to specify an economical starting particle size. A practical diﬃculty is the possible tendency of the particles to stick, which will make the process unfeasible. It is counteracted by starting with polymers with sufﬁciently high crystallinity, and by adding glass beads. Another problem may be the sublimation of oligomers, as in nylon-6, which may clog the bed. A precrystallization step for PET, to make it attain at least 40% crystallinity before SSP starts, is present in industrial processes since early 1970’s. The main reason for using SSP is the achievement of molecular weights higher than would be possible in melt polycondensation, owing to the selectivity gain of polycondensation with respect to degradation reactions. Therefore, in a batch SSP, a maximum in molecular weight versus time is expected, and this maximum will occur at shorter times and will lead to lower molecular weights as the temperature is increased. The key assumptions made in order to interpret SSP are due to Gostoli, Pilati et al. [72, 73] (see Figure 3.8):

All chain end groups belong to the amorphous regions. No reactions occur in crystalline regions. Chemical reactions follow the same kinetics as in melt, taking into account the change in the volume of the reaction media aﬀecting functional group concentrations. Equilibrium CLD holds locally.

The overall polydispersity of polymer will be greater than the equilibrium value (2

3.2 Mass Transfer Issues in Polycondensations

Fig. 3.8. Phase separation in solid-state polycondensation; A, B are the end groups, W is volatile by-product.

for linear polycondensations) because of the radial proﬁle of Mn in the diﬀusioncontrolled regime. In the same way as in heterogeneous catalysis, eﬀective diﬀusion coeﬃcients DYie (for ﬂuxes with respect to the total geometric area) are decreased relative to the values in the melt DYi because of the obstruction due to the crystalline phase at a volume fraction fcr and because of the tortuosity factor tD (which depends on the structure of the solid, values of 1.5 to 3 being common); the relationship is given in Eq. (47). DYie ¼ DYi

1 fcr tD

ð47Þ

Notice that volume and mass fractions wcr of the crystalline phase are diﬀerent because of the slight diﬀerence in density with respect to the amorphous phase. An unavoidable complication is thus the description of the build-up of the crystalline phase, which aﬀects mass transfer and chemical reactions by increasing functional group concentrations in the amorphous phase. The rate of crystallization is often described by the Avrami equation [Eq. (48)]. wcr ¼ 1 expðkcr t ncr Þ

ð48Þ

The exponent ncr is a function of nucleation and growth type, which is not constant for the entire course of crystallization. Mallon and Ray [74] have put forward in-

81

82

3 Polycondensation

stead of Avrami equation an empirical rate law in terms of residual amorphous phase volume fraction. The microscopic mass balance of polymer functional groups and volatile components given in Eqs. (27) has to be modiﬁed in order to take into account the variable reaction volume due to polymer crystallization. Here we do not follow the notation in Ref. 74; rather, we use concentrations and rates of reaction per unit volume of amorphous phase instead of concentrations per unit volume of particle ½A i p ¼ ½A i /ð1 fcr Þ, and so on, in order to use the same kinetic rate laws [Eqs. (49)] as in the melt. q½ð1 fcr Þ½Yi 1 fcr ¼ ‘ DYi ‘½Yi þ ð1 fcr ÞRYi qt tD

ð49Þ

q½ð1 fcr Þ½A i ¼ ð1 fcr ÞRA i qt Examples of the use of this approach with PET and nylon-6,6, including a successful comparison with available experimental data, can be found in Ref. 74. Industrial-scale SSP is carried out in moving packed bed, ﬂuidized bed, and stirred bed reactors; Mallon and Ray have published a brief discussion of idealized models of these reactors [75]. Fluidized beds have a serious drawback because of the high consumption of gas needed to keep the bed in a ﬂuidized state, and residence time distribution is unfavorable to high conversions. Stirred beds in series are a possible solution, depending on economic details. A model for SSP of nylon-6,6 in a moving bed reactor, considering its complex geometry and its start-up and shutdown operation [76], can serve as a guide for dealing with more complex real-life situations. 3.2.3

Interfacial Polycondensation

Typical chemical systems are fast reactions between two difunctional monomers, AXA þ BYB. The ﬁrst monomer (diamine, bisphenolate) is dissolved in a water solution (in alkaline media in both cases), and the other monomer, with low water solubility (acid chloride, phosgene), is usually dissolved in an organic solvent. Either the neutral form of AXA is in an appreciable amount (in the case of amines), or a phase transfer catalyst is needed (as in polycarbonate synthesis), since ionized forms will not dissolve in the organic phase. A decrease in the pH is often used to quench interfacial polyamidation. The chain extension of water dispersions of isocyanates with water-dissolved amines, in order to make polyurea dispersions, shares some similarities with the former (amine þ acid chloride) systems. Another common feature among these chemical systems is the presence of a parasite reaction consuming end groups B by reaction with water.

3.2 Mass Transfer Issues in Polycondensations

Addition of a monofunctional chain stopper to the organic phase is advisable in order to control the ﬁnal M n . In most cases (namely polyamides and polyureas), the polymer is insoluble in monomer BYB or in its solvent, and precipitates as soon as it forms, often yielding a shell through which AXA has to diﬀuse. Polycarbonates are an exception, as they are completely soluble in the methylene chloride solvent chosen for their production. Also, the organic phase is continuous in this latter case, which is a rather exceptional situation for interfacial polycondensations. Also with the exception of polycarbonates, interfacial polycondensation is mostly used in the production of specialty products, such as membranes [80–82] and microcapsules [83–87]. Early important contributions on interfacial polycondensations are described by Morgan [77] as well as in Refs. 78 and 79. These studies show that the reaction occurs in a layer close to the interface, on the organic side; the adjective ‘‘interfacial’’ is thus rather misleading. Reaction is very fast and mass transfer resistance is an important factor. Models have for a long time concentrated on describing the velocity of consumption of water-soluble monomer and consequent rate of ﬁlm growth [83–87]. Although a possible framework for describing the simpler case of polycarbonate formation has been presented by Mills [88], the ﬁrst attempt at predicting molecular weight distributions for more typical interfacial polycondensation systems is due to Karoda, Kulkarni et al. [89, 90], based on experimental data by Johnson [91]. This work will be the basis of the analysis next presented in this section. Assuming an apparent second-order rate constant in the organic phase k of the order 10 2 to 10 4 m 3 /kmol s [77] and a concentration of functional groups B in the bulk organic phase ½Bb ¼ 1 kmol m3 , the characteristic reaction time for consumption of monomer AXA is of the order of 104 to 102 s. In the absence of polymer, the diﬀusion coeﬃcient of AXA in organic solvent þ BYB, assuming a molecular weight of up to a few hundreds, should be of the order of 1010 to 1011 m 2 s1 , and this yields a characteristic width of reaction zone LR ¼ 108 to 107 m. So, polymerization occurs in a thin shell beneath the ﬁlm of precipitated polymer. There will be no functional groups B at the water interface and a steep gradient of concentration of monomer BYB will be rapidly established inside the organic phase. Instead of solving microscopic mass balances for the concentration proﬁles, Karode et al. [89, 90] use average concentrations in the reaction zone and consider its thickness LR constant (see Figure 3.9). Furthermore, the rate of movement of the interface between the organic phase and the precipitated polymer ﬁlm is considered to be slow, so that a pseudo-steady approximation for the diﬀusion of AXA through the polymer ﬁlm and for diﬀusion of BYB inside the organic phase should be valid. Mass balances of monomers in the reaction zone and in bulk aqueous and organic phases resulting from these assumptions are given in Eqs. (50).

83

84

3 Polycondensation

[AXA]blk

[BYB]blk

[BYB]blk

He [AXA]blk [AXA]

[BYB]

Hi [AXA]

[BYB]

LP Swollen

aqueous

polymer

phase

film

Reaction zone

Fig. 3.9.

Bulk

LR

x Bulk organic phase

Model for interfacial polycondensation.

LR

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

LR

d½BYB ¼ LR RBYB þ kf BYB ð½BYBblk ½BYBÞ dt

ð50Þ

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

d½BYBblk ¼ av; org kf BYB ð½BYBblk ½BYBÞ dt

There are two partition coeﬃcients He and Hi for monomer AXA: He is the ratio of bulk concentration in the aqueous phase ½AXAblk to concentration in the outer interface of polymer ﬁlm (thickness L p ). Hi is the ratio of monomer concentration in the polymer ﬁlm to concentration in the organic phase.

Vaq and Vorg are the volumes of aqueous and organic phases and av; aq ; av; org are their interfacial areas per unit volume (trivially related). No mass transfer resistance is assumed to exist outside the polymer ﬁlm (although it can be easily included), but it is considered inside the organic phase, with the help of a mass transfer coeﬃcient kf BYB (which can be obtained by penetration theory). Introduc-

3.3 Polycondensation Processes in Detail

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

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

ð51Þ

Polymer precipitation is taken into account through the model of Kamide et al. [92] with a phenomenological rate of nucleation k nuc; n (nil for n ¼ 1, taken as independent of molecular weight for n > 1) [93]. The ‘‘saturation’’ concentrations of polymer species are taken as the values of their concentrations in the lower branch of the spinodal curve for the liquid–liquid equilibrium with organic solvent. The earlier paper by Karode et al. [89] considered only spinodal decomposition. A coherent ﬁlm is predicted to form when the sum of projected areas for all phase-separated polymer nuclei (assumed spherical) is equal to the interfacial area. Film thickness is predicted through the overall mass balance of precipitated polymer. Besides thermodynamic data on the liquid–liquid equilibria of polymer/solvent and hydrophilic monomer polymer/solvent, this model needs the rate of nucleation k nuc , considered as an adjustable parameter, it also tries to ﬁt the diﬀusion coeﬃcient of monomer in the polymer ﬁlm DAXA , and, as it postulates a constant width of the reaction zone, it has also to ﬁt the kinetic parameters. In spite of its limitations in predictive power (a natural consequence of the complexity of the phenomena involved), this approach has given an important new insight on these processes. Film permeation properties should depend on the mode of phase separation, nucleation giving better crystallinity. Molecular weight depends on the competition between reaction and precipitation of polymer: a more powerful solvent should lead to higher molecular weight. The existence of a sharp maximum of average molecular weight as a function of the concentration of monomer in the organic phase when the two monomer ﬂuxes toward the reaction zone are balanced, remarked upon by P. W. Morgan, could at last be explained by this model – 40 years later.

3.3

Polycondensation Processes in Detail 3.3.1

Polyesters Structure and Production Processes Two kinds of polyesters will be discussed in this section: 3.3.1.1

Linear, crystalline, high molecular weight (M n between 15 000 and 100 000), used as plastics and ﬁbers, the most important being poly(ethylene terephthalate) (PET) and poly(butylene terephthalate) (PBT);

85

86

3 Polycondensation

Unsaturated low molecular weight (Mn between 1000 and 10 000), often branched, used as macromonomers for synthesis of thermosets (polyester resins), or thermosetting materials by themselves (alkyd resins). They are prepared from several monomers, namely phthalic and maleic anhydrides, adipic acid, isophthalic acid, natural fatty acids or triglycerides, and a great variety of multifunctional alcohols. In a few special cases, they may be saturated and/or linear for use as macromonomers in the production of polyurethanes or other polymers.

Acid-catalyzed Esteriﬁcation and Alcoholysis The two main reactions in these processes are esteriﬁcation and alcoholysis, which share similar mechanisms, with a hypothetical tetrahedral intermediate. Reactions between anhydrides and alcohols are much faster than carboxyl þ alcohol reaction, and full conversion of anhydrides (unless they are in excess) will occur in less than one minute while being melted and blended with the rest of the mixture. This stage is slightly exothermal and care is needed to avoid sudden boiling of the reacting mixture. Kinetic parameters of this reaction (apart from selectivities with respect to hydroxyls) are usually not needed. Acidolysis reactions have a diﬀerent mechanism, involving mixed anhydrides [94]. Their rate is comparable to the esteriﬁcation reactions only above 250 C. Esteriﬁcation also occurs in high-temperature alcoholysis of aromatic esters, as carboxyl end groups are formed by side reactions, and should also be considered in the kinetic modeling of these processes. In spite of being the ﬁrst reaction ever studied [95], esteriﬁcation has been under investigation ever since, and much knowledge has accumulated, even if some points are still less clear. The basic kinetic model for polyesteriﬁcation was established by Flory and is summarized in his classic book [5]. Esteriﬁcation was shown to be acid-catalyzed, it is ﬁrst-order with respect to hydroxyls and, with respect to carboxyls, its order is either one in the presence of foreign strong protic acids, or two in their absence [Eq. (52)]. 3.3.1.2

R COOH ¼ kscat ½OH½COOH 2

ðno foreign acidÞ

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

ð52Þ

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

3.3 Polycondensation Processes in Detail

groups, and such a relationship (Eq. (53) where p is carboxyl conversion) has been proposed by Chen and Wu [98, 99]. k ¼ kA expðapÞ

ð53Þ

In Eq. (53) the empirical parameter a is a function of the initial stoichiometric ratio r. It was theoretically linked to the dependence of the dissociation equilibrium constant of the carboxylic acid on the dielectric constant of the medium and to the dependence of the latter on the carboxylic acid concentration through conversion p. A good ﬁt of experimental data has been obtained with the parameter a varying in the range 0.2 to 1.2 both for the self-catalyzed and the foreign acid-catalyzed esteriﬁcations of adipic acid with diﬀerent diols. In their modeling of unsaturated polyester resin formation, Beigzadeh et al. [100] and Zetterlund et al. [101] have also found the Chen–Wu relationship more useful than the rather cumbersome empirical models of Paatero et al. [102] or Lehtonen et al. [103]. Zetterlund et al. [101] have also presented interesting experimental data on the simultaneous self- and cross-catalysis by two carboxyl groups, namely those formed by the addition of maleic and phthalic anhydride to 1,2– propanediol; they show there is an anti-synergistic eﬀect: that is, the total rate reaction of the two carboxyl groups is lower than the sum of the rates of reaction of the individual acids with the same concentration and at the same temperature. Catalysis by Metallic Compounds Self-catalyzed esteriﬁcation is often too slow to be of practical use, especially because hydroxyl-terminated polymers are either sought or are a consequence of the process (aromatic polyesteriﬁcations are carried out with a large excess of hydroxyls at the beginning of the process, because of the low solubility of the diacid), and strong protic acids are not advisable, as they would catalyze polymer hydrolysis if allowed to remain with the polymer. Even volatile catalysts such as methanesulfonic acid are avoided. Therefore, metallic salts are currently used as catalysts, both for esteriﬁcation and alcoholysis. Strong bases, such as lithium hydroxide, can also be used, but for alcoholysis only (as in polycarbonate formation). Metals in metallic complexes can catalyze esteriﬁcation and alcoholysis through two distinct mechanisms [117]: 3.3.1.3

Metals of groups II–III (such as Zn, Mn, Ce, Pb), usually introduced as carboxylates, complex the oxygen in carbonyl esters preferentially. Metals of groups III–VI (namely Ti, Sb, Ge, Bi), usually introduced as alkoxides, dialkyltin oxides R2 SnO and carboxylates such as dibutyltin dilaurate, coordinate with the acylic oxygen of esters. Their activity with respect to esteriﬁcation and alcoholysis has been compared by Habib and Ma´lek [105, 106] and Chung [107], who found volcano-shaped relationships with diﬀerent optima of activity of the several metals in terms of metal electronegativity according to Tanaka [108] for the glycolysis of dimethyl terephthalate

87

88

3 Polycondensation

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

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

kHE

ð54Þ

! XCOOCH2 CH2 OH þ XCOOCH3 XCOOCH2 CH2 OOCX þ CH3 OH kHE =KHE

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

3.3 Polycondensation Processes in Detail

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

O C O

O H C CH O CH2

Scheme 3.1.

OH C O

CH O

+

CH2

Vinyl end group formation from PET.

O C

89

90

3 Polycondensation

This is a ﬁrst-order reaction, with no eﬀect of catalysts or additives. Further alcoholysis of vinyl esters yields acetaldehyde [Eq. (a)]. XCOOCHbCH2 þ HOCH2 CH2 OOCY ! XCOOCH2 CH2 OOCY þ CH3 CHO ðaÞ DEG units are observed to form mainly in the ﬁrst stages of the process. The presence of a hydroxyl ester group is indispensable, as it can be shown experimentally that DEG forms in negligible amounts if ethylene glycol is heated alone in the absence of acid catalysts [126]. Reimschuessel [127] has suggested the attack of ethylene glycol or a hydroxyethyl end group as a possible source of the DEG moieties (see Scheme 3.2). The analogous intramolecular etheriﬁcation is the source of side product dioxane [125] (see Scheme 3.3).

O C

O

O C OH

Scheme 3.2.

C

CH2 O O CH2

+

H O

+

CH2 CH2 OX

O C CH2 O CH2 CH2 O CH2 OX

Formation of DEG moieties in PET.

O C OCH2CH2 O OH

O +

C OH

CH2 CH2 O O CH2 CH2

CH2CH2 Scheme 3.3.

Dioxane formation in PET.

Poly(butylene terephthalate) is also subject to analogous side reactions [128], the formation of 1-butenyl end groups (Scheme 3.4) followed by splitting-oﬀ of 1,3-butadiene or tetrahydrofuran (Scheme 3.5). This latter reaction is not aﬀected by the metallic catalyst, but is catalyzed by the carboxyl end groups, making production of PBT by direct esteriﬁcation of terephthalic acid and 1,4-butanediol disadvantageous. Side Reactions in the Formation of Unsaturated Polyesters A recent review on the chemistry of unsaturated polyesters has been published by Malik et al. [129]. Besides cis–trans isomerization [130], the other important side reaction is Ordelt reaction [131–133] (see Scheme 3.6) which increases branching and consumes double bonds. Both reactions are reversible and acid-catalyzed. 3.3.1.5

3.3 Polycondensation Processes in Detail

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

OH + C O

H2C CH CH CH2 Scheme 3.4.

CH2 O H2C CH CH2

+

HO

O C

O C

Vinyl end group formation from PBT.

O C O Scheme 3.5.

CH2 CH2 CH2 CH2 O

THF splitting-oﬀ from PBT.

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

OH + C O

CH2 CH2 CH2 CH2 HO

RO HC CH2 R'OOC COOR''

Ordelt reaction.

Modeling of Processes of Aromatic Polyester Production Continuous processes are currently used in the manufacture of PET. Several models have been developed, with the aim of contributing to a better design and operation. A brief discussion of their main assumptions and predictive capacities follows. Since polycondensation in solid-state and in ﬁlm-forming devices has been analyzed previously, only the speciﬁc aspects of the initial process stages still needs to be covered. Earlier models for continuous processes based on DMT [134, 135] study the inﬂuence of variables, such as the initial stoichiometric ratio, reactor temperatures, and average residence times for CSTRs in series connected with distillation columns, on process performance. They show it is possible to optimize conversion and minimize formation of side products. Even if kinetics and physical equilibria are now much better known, their qualitative conclusions should still hold. PBT production, for which no published process models have been found, should be described by a similar approach. Yamada et al. [136] and more recently Kang et al. [137] have presented models of the direct esteriﬁcation process of terephthalic acid (TPA) with ethylene glycol (Figure 3.10). As TPA has a low solubility because of its high melting point, the ﬁrst reactor in the train (‘‘esteriﬁcation’’ reactor) is operated at a higher pressure and 3.3.1.6

91

92

3 Polycondensation

Fig. 3.10. Simple model for WFR used for the direct esteriﬁcation process of terephthalic acid (TPA) with ethylene glycol.

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

3.3 Polycondensation Processes in Detail

Description of the branched structure of the resins is less problematic than in the case of alkyds, because the conversion of double bonds by Ordelt reaction is in the range 10 to 20%, so the molecules are almost comb-like and the approximations used by Yang and Pascault [140] should be reasonable. Modeling of alkyd resin production is a rather formidable task because of the high number of distinguishable chemical groups, the branched structure of the polymer, a nonnegligible amount of intramolecular reaction, and side reactions of the double bonds in fatty acids. The usual problems found in previously discussed polyesteriﬁcations, namely lack of data for liquid–vapor and liquid–solid equilibria and associated mass transfers, are also present. Industrial processes [141–143] use batch stirred reactors, connected to partial condensers in order to recover volatile glycols. Azeotropic distillation with xylene (for instance) is often used. Monomers may be added in several steps to overcome solubility problems. Vacuum and inert gas sparging is also used in diﬀerent stages. Catalysts (mostly alkyltin salts) are used to convert most of the carboxyl groups, as speciﬁcations often require less than 1 mg g1 KOH acid value. Long reaction times are sometimes avoided through diﬀerent techniques for eliminating residual carboxyls. The same plant usually produces diﬀerent varieties of resins, according, for instance, to fatty acid content. New compositions are often sought in order to improve end use properties or to compensate for ﬂuctuations of raw material prices. Therefore, it is often necessary to look for initial amounts of monomers which satisfy stoichiometric constraints (such as mole ratio and fatty acid content), will not lead to gelation, and meanwhile keep an acceptably high M n (for instance). An early review of the foundations of the macromolecular chemistry of alkyds has been presented by Kienle [143] and simple methods for predicting gelation conversion have been reviewed by Jonason [144]. These predictions are not very accurate, but good data on intramolecular reactions and diﬀerences of reactivity of functional groups will be needed if better control of physico-chemical properties is sought. 3.3.2

Polycarbonates General Introduction Polycarbonates are polyesters of carbonic acid formed by reaction of diols (aromatic, aliphatic or a mixture of both) with a derivative of carbonic acid. The ﬁrst preparations of polycarbonates were reported by Einhorn in 1898 [155], by reaction of phosgene with resorcinol or hydroquinone in a pyridine solution. Bischoﬀ and van Hedenstro¨m in 1902 [156] obtained the same aromatic polycarbonates via transesteriﬁcation with diphenyl carbonate (DPC). Thus the main routes to polycarbonates were established early, but the properties of the products seemed uninteresting. Around 1930 aliphatic polycarbonates were studied by Carothers and van Natta [157]. These carbonates have low melting points and thermal resistance and are not commercially interesting as stand-alone thermoplastics. Low molecular 3.3.2.1

93

94

3 Polycondensation

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

CH3 C CH3

n NaO

Scheme 3.7.

ONa + n COCl2

-2n NaCl O

CH3 C CH3

O C O

n

Overall stoichiometry of bisphenol A polycarbonate formation.

The synthesis proceeds in two steps: phosgenation of BPA forming oligomeric carbonates with phenolic and chloroformate end groups; and polycondensation of the oligomers (see Scheme 3.8). For the phosgenation, BPA is ﬁrst dissolved in an

3.3 Polycondensation Processes in Detail

*

...

*

...

*

...

m

m

m

O C Cl

+

O C Cl

+ NaO

O C Cl

+

Scheme 3.8.

...

*

n

Na

4 NaOH

*

...

*

...

*

...

m

m

m

O C

...

n

*

O

+ NaCl

+ NaCl

Na + NaCl + Na2CO3 + H2O

Formation of polycarbonate by interfacial polycondensation.

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

95

96

3 Polycondensation

Dispersion of the two phases. Rates always depend on mixing. Eﬀective kinetic rate constants can be formulated as a function of energy dissipation or interfacial area. The type of emulsion. Both types of emulsions, oil in water (o/w) as well as water in oil (w/o), can be found. The partition of phenols between the phases and its pH dependence [177]. Silva and Kosky [178] studied the reaction of hydrolysis taking into account the diﬀerent phases and the partitioning of BPA between them. Monofunctional phenols with better solubility in the organic phase show a better eﬃciency as chain terminators. Mass transfer to and across the boundaries. Intrinsic (kinetic) reaction rates.

Due to the low reaction temperature and the use of chain terminator, the molecular weight distribution in interfacial synthesis is kinetically controlled and may be far from thermodynamic equilibrium. In two parametric studies Mills [179] and Munjal [180] have tried to model the full molecular weight distribution of polycarbonate. Varying the ratio of mass transfer/kinetic rates, they show how mass transfer limitations can lead to a higher polydispersity or a higher oligomer content. Melt Transesteriﬁcation The melt process is based on the transesteriﬁcation of diphenylcarbonate (DPC) and BPA (see Scheme 3.9). 3.3.2.3

O n HO

OH + n

OCO

O O

OC

+ 2n

OH

n Scheme 3.9.

Formation of bisphenol A polycarbonate by transesteriﬁcation.

The melt process requires no solvent and polycarbonate is produced directly without the need for cleaning, drying, and devolatilization. The only by-product, phenol, is removed and can be recycled to the production of DPC or BPA. The process, however, requires high temperatures (190–320 C) so that the polymer remains molten and the viscosity can still be handled. With improved quality of the starting materials and improved high-viscosity reactors these temperatures no longer aﬀect polycarbonate quality and most grades can now be produced via the melt process. The transesteriﬁcation is a reversible reaction with an equilibrium constant close to unity, so that the by-product phenol has to be removed for the reaction to proceed. Pressures below 0.1 hPa may have to be applied for the ﬁnal stage.

3.3 Polycondensation Processes in Detail

Small amounts of basic catalyst (< 0.01 mol% of alkyl-ammonium, or phosphonium-salts) are mixed together with the DPC and BPA and the reaction is started at the lower temperature. Conversion is driven by the removal of phenol and the pressure is successively reduced while the temperature is increased. Simple reactors such as CSTRs, tubular reactors or falling-ﬁlm evaporators can be used for the ﬁrst stages. Viscosity increases dramatically from 5 mPa s up to 1000 Pa s as conversion increases and special reactors are needed for dealing with the high viscosity at high conversions [149–151]. The melt process depends on the reaction rates, the thermodynamic phase equilibrium, and mass transfer between the phases. Detailed mechanistic studies of the Li-catalyzed melt process have been published by Choi et al. [152], and of the solubility of DPC and phenol in polycarbonate by Webb [153]. The liquid–gas equilibrium has to take into account at least two components: phenol and DPC. For a semi-batch operation for the ﬁrst stages, optimal variations of pressure and temperature can be calculated based on the above relationships plus the assumption of phase equilibrium, or on a simple relationship for the mass transfer of each volatile component Yi (Eq. (55), with the mass transfer rates per unit volume Ji of component Yi , mass transfer coeﬃcient of component i kfi , interface area per unit volume a v , and equilibrium concentration ½Yi at the interface). Ji ¼ kfi a v ð½Yi ½Yi Þ

ð55Þ

Mass transfer coeﬃcients may be obtained by ﬁtting to process data. Including DPC loss, production capacity (reaction time), and unwanted side products in a cost function, the optimization leads to a balancing of mass transfer and reaction rate. This means that an optimal process is neither entirely mass-transfer nor kinetically controlled. To avoid side reactions that impair product quality, the lowest temperature that kinetics and mass transfer allow is chosen. The results of the semi-batch optimization can be transferred to the design of a staged, continuous process [154]. The balancing of reaction versus mass transfer rate can similarly be applied for the last stages. The relative DPC loss compared to phenol removal is quite high for these stages. A high fraction of phenol end groups compared to OH end groups is desirable for product quality. Therefore, excess DPC has to be used at the beginning of the reaction. Unfortunately, a high fraction of phenol end groups reduces the concentration of one reaction partner and hence reaction rates [146]. Mass transfer and hence the speed with which phenol can be removed are reduced with increasing viscosity. Reactors are needed that provide a high rate of surface renewal. These may be wiped-ﬁlm evaporators, single- or twin-screw extruders, reactors of the rotating disk type or falling-strand evaporators [145, 149, 150]. Whereas the ﬁrst three reactor types create surface actively, in falling-strand evaporators the melt is simply pumped through small holes. High surface renewal rates are achieved by small holes, but also by shear thinning of the falling strands [59]. The eﬀect of shear thinning ﬁlms can also be achieved in rotating disk reactors with disks that contain holes. Apart from surface renewal, mass transfer is de-

97

98

3 Polycondensation

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

Polyamides Introduction Since their discovery by Carothers [181], aliphatic polyamides such as nylon-6,6 and nylon-6 are important textile ﬁbers and plastics. Similar polyamides produced by melt reaction of aliphatic diacids þ diamines, or by hydrolytic polymerization of lactams, have some interest as engineering plastics, and will also be discussed in this section. Aliphatic, low molecular weight, branched polyamides of dimer fatty acids with di- and triethylenediamine are among the most widely used curing agents for epoxide resins. The chemistry of their formation is similar to that observed for the aforementioned crystalline linear polyamides. Aromatic polyamides are specialty products [182], used for high-performance ﬁbers and composites, which are produced by solution or interfacial processes from acid chlorides and amines. 3.3.3.1

Kinetic Modeling The only published kinetic data on nylon-6,6 reaction are due to Ogata [183, 184], and there is in general a great scarcity of information on the amide þ acid reaction. Much more data are available on nylon-6 formation, such as in Refs. 185–187. Because of the obvious identity in average chemical group composition of both polymers, it should be possible to reconcile both kinetic laws [188]. However, nylon-6,6 data have been obtained at high water concentrations and relatively low temperatures, and data on nylon-6 are in the opposite situation. Degradation reactions of nylon-6,6 through adipic acid chain ends (see Section 3.3.3.3) complicate the interpretation of kinetic data at higher temperatures. Schaﬀer, McAuley et al. [189] have recently obtained experimental data on the nylon-6,12 system, which does not present these problems, and these kinetics are now much better understood. Apparent equilibrium and kinetic constants in these systems are seen to depend on water concentration. It obviously changes the activity coeﬃcients of functional groups, but no thermodynamic model has ever been used to completely describe the mixture. The activity of water can be directly measured from knowledge of its vapor pressure, and it has been claimed that a correlation based upon a Flory– Huggins model can predict it [190], but no model exists for taking into account the group interactions. Some interesting ideas can be found in Ref. 191, but no actual thermodynamic model has been developed, concentrations of hypothetical species having been used throughout that paper. The crux of the treatment by Schaﬀer et al. [189] lies in the empirical correlation 3.3.3.2

3.3 Polycondensation Processes in Detail

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

ð56Þ

As the overall composition of the system may be described by two variables, such as [H2 O] and [COOH], a dependence on [COOH] might be added in Eq. (56). However, this is not needed, as no eﬀect of the mole ratio [COOH]/[NH2 ] on the apparent equilibrium constant has ever been detected. Insertion of the above relationship into the mass action law provides a relationship between apparent equilibrium constant K a and true thermodynamic constant K 0 [Eq. (57)]. K a ¼ K 0 gH2 O ðbA þ mA ½H2 OÞ

ð57Þ

It is seen that parameter bA is absorbed by the unknown thermodynamic constant, and only the ratio g A ¼ mA /bA can actually be found from experimental data. The correlations found for the activity coeﬃcient of water are given in Eqs. (58). 3613 ðNylon 6,12Þ gH2 O ¼ exp 9:624 T gH2 O

2258 ðNylon 6,6Þ ¼ exp 6:390 T

ð58Þ

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

K a ¼ K a0

K a0

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

bA DH DS exp ¼ þ RT0 R gH2 O ðT0 Þ

ð59Þ

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

R CONH R CONH

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

99

100

3 Polycondensation Tab. 3.2.

Equilibrium and rate parameters for nylon formation.

Parameter

Units

Estimate

95% conﬁdence interval

DH K a0 gA Ec kc0 km

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

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

G1.42 G6.2 G0:45 102 G7.7 G0:71 104 G15.4

The optimum values of parameters for the third-order model are shown in Table 3.2. Their approximate correlation matrix can be found in the same reference. It shows that parameters describing equilibrium are highly correlated with g A , and the pre-exponential factor is highly correlated with k m . Further, there is little change in parameters for the second-order model, which yields ku0 ¼ 2:64 10 7 mg mol1 h1 and has a similar sum of weighed squared residuals. Thus, it is not yet possible to determine the reaction order without performing experiments with excess of diamine or diacid, which have not yet been reported at the time of writing. The authors state the parameters thus obtained for nylon-6,12 should hold for the other aliphatic polyamides, just correcting the equilibrium constant, which is lower by a factor of 0.4 to 0.5 in nylon-6,12 relative to nylon-6,6. There are several studies concluding that the apparent order of reaction changes from two to three as conversion grows. The reason might be a nonideality eﬀect similar to the one observed in esteriﬁcations. Miller [192] has carried out an experimental study on aminolysis and acidolysis reactions using carefully dried model compounds. As in esteriﬁcations, acidolysis is slower and is explained by a mechanism involving the formation of intermediate anhydrides. Its activation energy is 27 kcal mol1 , whereas aminolysis has the much smaller activation energy of 13 kcal mol1 . The rate of aminolysis was shown to be ﬁrst order in carboxylic acid. These results are valuable not only for dealing with block polymers, but also in kinetic modeling, particularly with cyclic lactams: a narrow CLD will not occur because of the reorganization brought about by aminolysis. Hydrolysis of caprolactam is autocatalytic. Mallon and Ray [191] have suggested that its initial rate is determined by the presence of impurities. The same authors have also remarked that the rate constant of addition of caprolactam to amine end groups is about what would be expected for an aminolysis reaction. It should also be possible to predict the concentrations of higher cyclic oligomers, but the only usable data concern the equilibrium concentrations, and experimental conﬁrmation has not yet been possible [191]. Nonoxidative Thermal Degradation Reactions The main degradation reaction of nylon-6 is decarboxylation through interaction of a carboxyl end group and caprolactam or an amide in the polymer chain (see 3.3.3.3

3.3 Polycondensation Processes in Detail

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

X

HO C N OH O C

CH2

CH2

N

X CH2 k2

H2C

CH2 CH2 Scheme 3.10.

N

O C

- H2 O

CH2

H2C

C

O

k1

CH2 - CO 2 CH2

C H2C H2C

CH2 CH2 CH2

Nonoxidative thermal degradation of nylon-6.

Nylon-6,6 also degrades in the absence of oxygen, to a much greater extent than nylon-6, and it eventually gels, as explained by the simpliﬁed mechanism in Scheme 3.11 [195, 196]. XNHCO(CH2)4COOH k1 O

NH Y O

O C X NH

k2

C

O

CH2

CH2

C

k3

- CO2

N

Scheme 3.11.

X N

k4 + 2 NH2

Z

Y

+ NH2

XNH

CH2 CH2 X

- H2O

Z

Z

+ 2 NH3

Nonoxidative thermal degradation of nylon-6,6.

Cyclopentanone units are created, with a decrease in molecular weight, either from carboxyl end groups or by intrachain reaction. Losses of CO2 and NH3 lead to an imbalance of end groups, a nuisance for dyeing, and also to crosslinking. Process Modeling Recent progress in kinetic and reactor modeling make it possible to assist the reactor design and process operations with unprecedented exactness. A recent analysis of the nylon-6,6 process [138] looked for improvements based on expansion of the solid-state polymerization, but the gain was minor compared to what happened with PET. The reason is the much reduced sensitivity of polyamides to by-product removal as compared to polyesters, since the equilibrium constant of the former is much greater. Since the early 1970s, many researchers have modeled nylon-6 processes [197], as reviewed by Kumar and Gupta [198]. There is an interesting optimization problem inherent to this process, which consists in adding just as much water as is needed to start polymerization, and to get rid of it in the later stages. Because of 3.3.3.4

101

102

3 Polycondensation

the autocatalytic nature of the caprolactam hydrolysis, back-mixing increases conversion. A simple and widely used reactor is the VK column (simpliﬁed continuous column), essentially a vertical tube at atmospheric pressure [193, 199], stirred by the boiling action of water leaving the reactor mixture in the top zone. Extensive pilot-plant tests carried by Jacobs and Schweigman [199], and further data from industrial plant, have lead to a simple model of the VK column, consisting in one or two CSTRs in series, followed by a plug-ﬂow reactor. Other designs [200] have improved performance by preventing water evaporation at the top of the column through the use of above-atmospheric pressure. The bottom one-third of the columns has a homogenizing function, not only physical but also chemical, through the aminolysis reaction [201]. Vacuum stripping with a low residence time is used to eliminate most of the large amount of caprolactam which remains because of the chemical equilibrium of the back-biting reaction. An alternative is a hot-water extraction step, which will also extract higher cyclic oligomers. 3.3.4

Polymerizations with Formaldehyde: Amino Resins (Urea and Melamine) and Phenolics Formaldehyde Solutions in Water Formaldehyde is a gas at room temperature. It may be sold as a low molecular weight, solid polymer (paraformaldehyde), and more conveniently as 37% or 55% water solutions, which usually contain some methanol. Under such conditions, nearly all the formaldehyde is transformed into methanediol and higher oligomers (see Scheme 3.12), usually end-capped by methanol, in order to reduce the average molecular weight and prevent precipitation of paraformaldehyde. 3.3.4.1

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

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

Formaldehyde/water/methanol equilibria.

These reactions are not very fast at room temperature: characteristic reaction times are of the order of minutes. The various equilibrium constants have been measured using NMR [202] and a model describing the vapor–liquid equilibrium in that system has been developed. Amino Resins Reaction between a water solution of formaldehyde with urea, melamine, and similar molecules (such as acrylamide) leads to hydroxymethylation of the nitrogens 3.3.4.2

3.3 Polycondensation Processes in Detail

[Eq. (b)] and further condensations produce the so-called amino resins [Eq. (c)], of which 80% are based on urea [203], the rest being nearly all produced from melamine (1). XaNH2 þ HCHO ! XaNHaCH2 OH

ðbÞ

XaNHaCH2 OH þ YaNH2 ! XaNHaCH2 Y þ H2 O

ðcÞ

NH2 N H2N

N N NH2

1 Scheme 3.13.

Melamine.

Since melamine is made from urea and ammonia, it is more expensive. Melamine resins are therefore chosen when one can get an appreciable beneﬁt from their better hydrolytic or thermal resistance. Urea–formaldehyde (UF) resins are mainly used as adhesives for wood. Laminated sheets (tables and counter tops) are a major application for melamine resins, which stay in the outer decorative surface. Molding compounds, their ﬁrst big application, is still a major market, taking advantage of their extreme hardness and heat resistance. Coatings, textile ﬁnishing, paper additives, leather tanning and foundry binders, for which methanol- or butanol-etheriﬁed resins are usually employed, are important markets discussed in Ref. 203. A major problem with the use of UF resins is their formaldehyde emission due to hydrolysis. Formation of melamine resins is much less reversible and therefore food contact with them is allowed. Besides earlier classic data on the kinetics of reactions between urea, formaldehyde, and UF oligomers by de Jong and de Jonge [204–206], only experiments by Price et al. [207] at higher temperatures in a sealed reactor are of immediate use to establish a kinetic model of the chemical system. Kumar and Sood [208] have proposed an FSSE model for the early stage of this polycondensation. A modiﬁed version of that model introduces the groups presented in Scheme 3.14, where their ﬁve urea monads U0 . . . U4 have been kept but three formaldehyde monads have been used instead of two. Formation of tetrasubstituted urea is known to be negligible. Both de Jong and de Jonge, and Price et al., have considered that hydrolysis reactions are unimolecular. Kumar and Sood [208] have considered it could be bimolecular, which seems to be reasonable. Available experimental data could not decide for any of the alternatives, since the water concentration was always the same; this matter needs to be solved, because higher initial concentrations of formalde-

103

104

3 Polycondensation

U0 =

O C

H2N

O C

U3= H2N F0 = H

O C

NH2

U1 =

U4 = N

H2N

O C

U2 =

NH-

-HN

O C NH-

O C -HN

N

F2 = -CH2-

F1 = -CH2OH H

Scheme 3.14.

Monads in the FSSE model of urea/formaldehyde polycondensation.

hyde are often used nowadays. Chemical transformations according to this new model are shown in Scheme 3.15.

U 0 + F0

k1 →

←  k

U 1 + F1 + W

k6 →

U 0 + F1

h2

k

U 1 + F0

2 → U + F1 + W ←  2 k

k

U 1 + F1

h1

7 → U + F2 + W ←   2 k h2

k

U 1 + F0

5 →

←  k

k

U 3 + F1 + W

U 1 + F1

h1

U 2 + F0

k4 →

←  k → h1

Scheme 3.15.

←   k

U 3 + F2 + W

k

U 4 + F1 + W

U 2 + F1

h1

←  k

8 →

h2

9 → U + F2 + W ←   4 k h2

k5

U 3 + F0

U 1 + F2 + W

←   k

h1

k10

U 4 + F1 + W

U 3 + F1

→ ←   k

U 4 + F2 + W

h2

Kinetic scheme of FSSE model of urea/formaldehyde polycondensation.

A possible simpliﬁcation consists in distinguishing only rate constants of forward reactions according to the number of hydrogens involved, and therefore k1 ¼ 2k2 ¼ 4k5 and k3 ¼ k4 , k6 ¼ 2k7 ¼ 4k10 and k8 ¼ k9 , reducing the number of unknown parameters. Since the only information in the experimental data of Price et al. [207] is formaldehyde concentration versus time, it is not possible to estimate rate constants k6 to k10 from them. Although only at low temperatures, rate constants k1 ; k3 , and kh1 (this one measured from the rate of formation/hydrolysis of both monomethylol urea and dimethylolurea) are nevertheless available from other sources, such as Ref. 209. A crucial check of the FSSE hypothesis is the equality of the ﬁrst-order hydrolysis constants of methylol groups in mono- and dimethylolurea: the rate constants per mole of the chemical substances should be double for dimethylolurea. The value of this ratio is 1.68, standard deviation 0.42, for 15 values reported by Landqvist with diﬀerent buﬀers (pH 6, 7, 9.2, 10) at temperatures 20, 30 and 40 C.

3.3 Polycondensation Processes in Detail

However, that ratio is 6.9 according to de Jong and de Jonge; the activation energy of the hydrolysis is the same (20 kcal mol1 ), though, according to both research studies. The equilibrium constants of the ﬁrst and second hydroxymethylations of urea are, respectively, 990 and 253 at 35 C, and there is a decrease of about 3 in the forward rate constants of the successive substitutions. Interestingly, the rate constants for the reaction of methylenediurea with formaldehyde or monomethylolurea are identical, respectively, to those observed for urea þ formaldehyde and urea þ monomethylolurea [210, 211]. So, there seems to be enough evidence to take FSSEs into account for urea, but unfortunately it seems there might be no such thing as a single ‘‘aCH2 OH’’ group, and SSSEs should be needed for fully describing this chemistry. For simplicity, we will keep using the above model in the discussion. Rate constants are known to depend on pH (although not much between pH 4 and 9); there is catalysis by OH and Hþ , so these constants should be written as in Eq. (61) [204]. ki ¼ ki0 þ kiOH bOH c þ kiH bHþ c

ð61Þ

Rate constants k6 to k10 in this scheme can be estimated from data of reactions in acid media between urea, mono- and dimethylolurea [206]. In the same work, reaction between methylols was found negligible (the temperature was at most 50 C). For these reactions, the terms ki0 ; kiOH can be neglected; the reactions are very slow at pH > 4. More recent work has concentrated on analysis supported by quantitative 13 C NMR [212–216] and size exclusion chromatography [217] has completed this view of the chemistry. At high temperatures and alkaline pH, methylol groups form methylol ether bridges and uron rings 2 (Scheme 3.16).

2

HOCH2N

N HOCH2

O C O H

Scheme 3.16.

O C

N

N CH2

N

N CH2

O C NH-CH2OCH2N O C O

N CH2

O C

N

+ H2O

+ H2O

2 Reactions at alkaline pH in urea/formaldehyde polycondensation.

The preparation and possible industrial uses of uron UF resins may be found in Ref. 218. NMR shows that these intra- or intermolecular ether bonds are destroyed with liberation of formaldehyde at acid pH.

105

106

3 Polycondensation

Another question, that should not be overlooked, is the incomplete solubility of the polymer in water. Except at low conversions, the reaction medium resembles a colloidal dispersion [219]. Gelation may be physical, before or instead of being chemical [220]. Amino resins are nearly always made in batch processes, consisting of thermostated reaction kettles connected to a condenser. The current method of synthesis has three stages [221]:

synthesis of methylolated oligomers at pH 8 to 8.5; acid condensation at pH 4 to 5; addition of urea to decrease the ﬁnal stoichiometric ratio to 1 to 1.3.

The goal is to reduce formaldehyde emissions [222], while keeping the properties of products (such as wood panels) at an acceptable level. Since the formaldehyde/ urea molar ratio had to be decreased, the process has also become more diﬃcult to control. Modeling of the process has increased its potential importance in this context, but the diﬃculties of putting it into practice are considerable, because of the daunting complexity of the chemistry. Notice that it should be integrated with the modeling of the cure stage (a complex combination of heat, mass, and mechanical modeling, in the case of wood panel manufacture). Cure is performed with ammonium chloride as catalyst, which also acts as a formaldehyde scavenger. Its chemistry is not fully understood, because of the much higher temperatures than in resin synthesis, which lead possibly to ladder structures. Melamine resins have a similar chemistry, the main diﬀerence being the reduced importance of hydrolysis reactions. They are prepared using two stages, alkaline addition of formaldehyde, and acid polycondensation. The initial stage of the melamine/formaldehyde reaction has been studied by Nastke et al. [223], who succeeded in providing evidence not only of hydroxymethylation, but also of the formation of methylene and methylene ether bridges using polarography. The functionality of melamine is six, with a negative substitution effect of about 40% [224, 225]. As with UF, formaldehyde addition is acid- and base-catalyzed (sensitive to pH), and equilibrium constants are 100 to 200. Methylene bridges are also formed only at acid pH. Direct analysis of methylene ether bridges has also been performed by NMR [226]. A simpliﬁed model (no reaction reversibility) of melamine–formaldehyde formation based on Tomita’s kinetic scheme [225] has been presented [227] and afterwards extended to reaction in a CSTR [228], also considering reaction reversibility. There is no experimental validation, but it is noteworthy for the use of a program to calculate the CLD of a nonlinear reversible polycondensation in order to overcome the astronomical number of reaction possibilities in the rates of formation of individual oligomers.

3.3 Polycondensation Processes in Detail

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

resols, which are highly branched, low molecular weight (150–1500) polymers with a formaldehyde/phenol stoichiometric ratio between 1.2 to 3, formed at alkaline pH; novolacs, made at acid pH, with a formaldehyde/phenol stoichiometric ratio between 0.5 and 0.8, which have a diﬀerent and much less branched structure than resols. They are low molecular weight (500–5000) thermoplastics, further crosslinked with hexamethylenetetramine. Phenolic resins are mainly used as wood adhesives, laminates, molded parts, insulating varnishes, abrasives, and rigid foams. Novolacs can be made using either strong acid catalysts (sulfuric acid is preferred) or at pH 4 to 7 using carboxylates of divalent metals (such as Zn, Mn, Mg). These catalysts complex phenol and methanediol and lead to formation of omethylolphenol in a ﬁrst step. Subsequent addition steps may also be orthodirected (Scheme 3.17), or more randomly distributed (in the case of Zn). H

O

M2+ OH CH2 OH

OH

OH - M2+

CH2OH

- H2 O

- M2+

OH CH2

...

- H2 O +

OH

Formation of ortho-directed methylene bridges.

Scheme 3.17.

Novolacs prepared with acid catalysts have a more random structure. The branching density is low because nonterminal rings are less reactive. This is caused by molecular coiling, which is especially important in high ortho-novolacs. Phenol groups tend to associate through hydrogen bonds and change the molecular conformation (Scheme 3.18). Nonterminal units are often assumed to stay

CH2 O O

H O

H H

CH2

Scheme 3.18.

O

CH2 H O H H O

CH2

CH2

Hypothetical intramolecular hydrogen bonds in novolac resins.

107

108

3 Polycondensation

preferentially inside the molecular coils and decrease their reactivity because of this. Mathematical models describing formation of novolacs both in batch and continuous reactors have been developed by Frontini et al. [230] and Kumar et al. [231]. They consider the existence of at most one methylol group per molecule, and lump together all isomers with the same unit counts. A Monte Carlo method [232] can also be used in order to obtain a more detailed description at molecular level. The initial addition of formaldehyde to phenol in alkaline media could for the ﬁrst time be successfully described, thanks to Zavitsas and collaborators [233]. Resol formation in further reactions is a complex process, owing to the several diﬀerent aromatic reaction sites and substitution eﬀects [234]; a total of 19 can be distinguished [235, 236]. Concentrations of fragments have been computed, assuming irreversible reactions. Number-average and weight-average molecular weights have been predicted using the ‘‘recursive’’ approach. Experimental determination of the necessary structural and kinetic parameters is a huge task, which requires extensive use of 13 C NMR and synthesis of model compounds. The research information [237, 238] allows modeling of these reacting systems to be elaborated with better chemical support. Fairly good agreement of model and experimentally measured functional group concentrations in resol formation at various stoichiometric ratios has been claimed [239] and it is expected that a trustworthy quantitative description of these systems may eventually be achieved. Validation of these predictions is plagued by experimental diﬃculties, and has mainly been carried out through measurement of individual oligomer and functional group concentrations. Reaction is exothermal (DH ¼ 80 kJ mol1 ). Heat of reaction is removed using water reﬂux, sometimes with a small amount of inert solvent (aromatics are inert only if they carry deactivating groups), and relatively small batch reactors ( 2 to 10 m 3 ) are usually preferred. Resol reactors have been the subject of studies, such as Ref. 240, concerning operation in accident situations. In novolac production, formaldehyde can be fed continuously in order to increase safety. 3.3.5

Epoxy Resins

The three-membered cyclic ether group oxirane, a 1,2-epoxide, or an epoxy group reacts with substances containing an active hydrogen group, such as amines, phenols, and carboxylic acids, or can be polymerized with anion or cation initiators, therefore yielding a great variety of potentially useful polymers. Nearly all of them are thermosetting, and used as coatings and adhesives. Their cure processes, which should be considered in close connection with the method of producing the ﬁnal material, will not be discussed in this section. Instead, a brief review of processes to make the macromonomers containing epoxy groups, the so-called epoxy resins, will be presented. The most widely used epoxy resins are formed through the reaction between epichlorohydrin (ECH; 3 in Scheme 3.19) and bisphenol A.

3.3 Polycondensation Processes in Detail

ClCH2CH CH2 O 3

+

CH3 C CH3

CH2 CHCH2O O Scheme 3.19.

CH3 C CH3

HO

-NaCl

OH

OCH2CHCH2O n

CH3 C CH3

OCH2CH CH2 O

Formation of epoxy resin by reaction of bisphenol A with epichlorohydrin.

The so-called ‘‘taﬀy process’’ consists in the two-phase reaction of an alkaline solution of bisphenol A with ECH in stoichiometric excess [241]. The main reaction as described above is accompanied by side reactions, such as hydrolysis and alcoholysis of chlorine and epoxides in ECH. These reactions create molecules with functionality one or even zero, and must of course be minimized. The kinetics has been studied by Enikolopyan et al. [242] and Gao [243], among others. Branching formation occurs to a low extent and can usually be neglected; the reaction can be described as a linear irreversible polycondensation AXA þ BYC, with A ¼ aOH, B ¼ aCl, and C ¼ epoxide (ECH and oligomers have diﬀerent reactivities). The similar epoxidation of novolacs with ECH has been additionally studied by Oyanguren and Williams [244]. In this system, intramolecular ring formation has been measured. The so-called ‘‘advancement process’’ consists in the melt reaction of bisphenol A (or a similar monomer) with a bifunctional epoxy resin in the presence of a catalyst, with the goal of producing a higher molecular weight, bifunctional, epoxy resin. This process leads to branching, due to the reaction of the pendent hydroxyl group with epoxide, and eventually gelation occurs. Its kinetics has recently been studied by Smith and Ishida [276]. The activation energy of the branching reaction was found to be higher (20 as compared to 18 kcal mol1 ) than that of chain extension, and both constants have been determined both for the catalyzed and noncatalyzed reaction. 3.3.6

Polyurethanes and Polyureas

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

R NCO

+ HO

R'

R

N

O C O R' H

Scheme 3.20.

Addition of alcohols to isocyanates.

4

109

110

3 Polycondensation

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

R NCO

+ H2N

R

R'

N

O C NH R'

5

H Scheme 3.21.

Addition of primary amines to isocyanates.

Reaction with water produces an amine and carbon dioxide through an unstable carbamic acid intermediate (Scheme 3.22).

R NCO

R

+ H2O

N

O C O H

H R

N

Scheme 3.22.

+ CO2

H

H Reaction of water with isocyanates.

As the amine reacts again with isocyanate, this reaction will lead to branching, as will consecutive reactions with carbamates, leading to allophanates 6 (Scheme 3.23) and with ureas, leading to biurets 7 (Scheme 3.24).

R NCO + R

N

O C O R'

R

N

H

C N O

Scheme 3.23.

R

6

H

Reaction of urethanes with isocyanates leading to allophanates.

R NCO + R

N

O C NH R'

R

N

H

O C NH R' C N

O Scheme 3.24.

O C O R'

R

7

H

Reaction of ureas with isocyanates leading to biurets.

Trimerization of isocyanates, leading to the thermostable isocyanurate ring 8, occurs with basic catalysts (multifunctional amines, carboxylates, alkoxides, and so on) through allophanate intermediates [252–254] (Scheme 3.25). Isocyanate groups also react to form uretdiones 9. This is an equilibrium reaction, which is mainly important at high isocyanate concentration (Scheme 3.26). Most urethane polymers are thermosets [247, 248]. They are mainly used in the production of foams, taking advantage of the reaction of water and of the precipitation of insoluble ureas, which stabilize the foam even before the system

3.3 Polycondensation Processes in Detail

R

N

O C O C N

O

R' R'

+ R NCO

R

N

H

O C

O O

C N O

R

R

R

H N C O

R O

N

N N R

R + R' OH O

8 Scheme 3.25.

Formation of isocyanurates.

O C 2 R NCO

R N

N R C O

Scheme 3.26.

111

9

Formation of uretdiones.

gels chemically. Amine- or hydroxyethyl-capped branched polyols of molecular weight of the order of a few thousands are preferred. They are most often starshaped poly(oxypropylene), in combination with shorter polyols such as glycerine, additives such as surfactants and demolding agents, blowing agents, and water. Rigid foams use more branched polyols than do ﬂexible foams. Another important use is the fabrication of plastics by RIM (reaction injection molding) (see Figure 3.11), which exploits the very fast reaction which can be

Fig. 3.11. Scheme for an RIM machine with a jet impingement mix-head (on a very exaggerated scale), in its recirculation and injection modes.

112

3 Polycondensation

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

3.4 Modeling of Complex Polycondensation Reactions

reactors for very low molecular weight or solvent-based processes, to tubular reactors with static mixers or extruders. Owing to the strong exothermicity of the main reaction, cooling has to be used in order to prevent side reactions from becoming too important. Many processes at low temperature and in homogeneous phase can nevertheless be analyzed through the methods described in Section 3.4.4, as they now stand.

3.4

Modeling of Complex Polycondensation Reactions 3.4.1

Overview

Rate equations allowing the prediction of concentrations of reactive groups, including more complex molecular fragments (monads, dyads and so on) from mass balance equations are established in Section 3.4.2 for irreversible reactions or systems with at most FSSEs if reversible reactions exist. Stoichiometric coeﬃcients are introduced in order to obtain a fairly general formalism, later exploited in Section 3.4.4. Knowledge of the distributions of numbers of bonds connecting repeating units makes it possible to describe molecular structure at chemical equilibrium. In Section 3.4.3, the main results concerning molecular weights and network properties of chemical systems verifying FSSEs are presented, using the theory of branching processes. The presence of rings is also allowed. The results of Section 3.4.2 are useful in order to predict average numbers of bonds for each monad, needed for predicting average molecular weights. Irreversible polycondensations can be tackled quite easily using a general kinetic approach developed in Section 3.4.4, allowing prediction of average molecular weights before or after gelation, and even molecular weight distributions (lumping together isomers with the same numbers of groups). Prediction of molecular weight distributions for reversible, linear, alternating polycondensation is discussed in Section 3.4.5. Mathematical diﬃculties grow considerably in the presence of SSSEs. There seems to be no alternative to Monte Carlo methods for dealing with reversible nonlinear polycondensations or even linear polycondensations where more than two kinds of bonds are present. 3.4.2

Description of Reactions in Polycondensations of Several Monomers with Substitution Eﬀects

The goal of this section is to present a general nomenclature of chemical entities and reactions which provides a concise form for writing the rate equations and mass balances of chemical species. The existence of substitution eﬀects seriously complicates the task, because the same molecular entity has to be labeled in a dif-

113

114

3 Polycondensation

ferent way, not because it has intrinsically changed, but because its neighbors have reacted. We introduce the following nomenclature for the species (monomer units or functional groups) and vectors h; g; e storing their indices: Xh i A gi ; A giþ We i Z iR Zei ; Zeþi

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

The NA functional (or end) groups A i will be distinguished (even if chemically similar) according to the monomer unit Xh i where they are attached. h is yet another vector of indices, with size NA . NR reactions involving pairs of functional groups create connections between monomer units, possibly (with the well-known exceptions of epoxides and isocyanate reactions) also forming by-products. A total number NW of by-products Wi will be considered for the sake of generality. By-product We i is formed by the reaction between functional groups A gi and A giþ creating the bond ZiR . Vectors e; gþ and g have sizes NR . This deﬁnition allows for more than one possible kind of bond between two given monomer units, as happens for instance if a carboxylic acid reacts with glycerol, which possesses distinguishable primary and secondary hydroxyls. In some of these NR reactions, such as in the case of self-condensations of silanols in silicone formation and of methylols in formaldehyde polymerizations, an end group may react with itself. The number of such reactions in this subset will be deﬁned as NRs . For each bond, it is useful to deﬁne a positive sense in the direction of the unit with higher or equal index, which will be coincident for the NRs reactions considered above. So, there is a total number of NZ ¼ 2NR NRs kinds of directed bonds Z i , incident on monomer units Xzi ; the set of monomer units and bonds are the vertices and edges of a directed graph (or digraph). Vector z, of size NZ , therefore contains the indices of the repeating units to which each directed bond points. There is a one-to-one correspondence between directed bonds and half-bonds hanging from the repeating units at each side. The directed bonds associated with ZiR will be named Zei and Zeþi , using yet another pair of vectors of indices, eþ and e , of sizes NZ . Hence, the vectors z and h, deﬁning respectively their incident and adjacent monomer units, will be related through hgiþ ¼ zeþi and hgi ¼ zei . In order to avoid multiple levels of indexing in equations, the notation given by Eqs. (62) (loosely inspired by indirect addressing of computer assembly languages) will be used hereafter. A giþ 1 A½iþ

Zeþi 1 Z½iþ

We i 1 W½i

Xh i 1 X½iA

Xzi 1 X½iZ

ð62Þ

3.4 Modeling of Complex Polycondensation Reactions

For the NR reactions which create new bonds from functional groups and also (very often) a by-product, we will introduce apparent second-order rate constants of the forward reaction (ki , i ¼ 1; NR ), as well as apparent ﬁrst-order rate constants of the backward reactions (kiZ ), related to the former through the equilibrium ratios Ki and the concentration of the corresponding by-product, if it exists, according to Eq. (63). kiZ ¼

½W½i ki Ki

ð63Þ

This may look rather artiﬁcial, but it helps in situations such as the thermal decomposition of urethanes or ureas, which are ﬁrst-order reactions. By convention, in such cases we will introduce a nil by-product W0 (assuming that the indices are counted starting from one) with a constant unit concentration. If the groups react independently, by deﬁnition there is no substitution eﬀect. The condensation reaction creating a bond (the same as two half-bonds) can be written as Eq. (64). ki

A½i þ A½iþ T Z½i þ Z½iþ þ W½i

ð64Þ

kiZ

To account for ﬁrst-shell substitution eﬀects (FSSEs), a more general expression [Eq. (65)] can be written, using stoichiometric coeﬃcients n: NW NA NZ X X X ðninA þ ninAþ ÞAn þ nW ðninZ þ ninZþ ÞZn ¼ 0 in Wn þ n¼1

n¼1

ð65Þ

n¼1

ninA ; ninZ correspond, respectively, to functional groups An and oriented bonds Zn coming out of the monomer unit in which stood A½i , and a similar convention is used for ninAþ ; ninZþ. Aþ Zþ Z No FSSE means that, for every reaction i, nigA ¼ n þ ¼ 1, nie ¼ n þ ¼ 1, and, igi iei i i moreover, all other stoichiometric coeﬃcients are nil. Likewise, it is useful to introduce the stoichiometric coeﬃcients of by-products, nW in . This trick only works with FSSEs. Higher-order substitution eﬀects (see Sections 3.1.5 and 3.4.5) are much more diﬃcult to describe. In the absence of FSSEs, the exchange of bonds ZiR and ZjR , necessarily forming the same by-product (e i ¼ ej ), can be described through Eqs. (66). kijE

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

X A½iþ þ X

þ

Z½ j Z½ jþ X

þþ

kijEþ

ð66Þ

! X Z½i Z½iþ X Eþ

þ

þ A½ jþ X

þþ

kji

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

115

116

3 Polycondensation NA X

AE AEþ AEþþ ðnijn þ nijn þ njin ÞAn þ

n¼1

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

ð67Þ NA X

NZ X AEþ AEþþ AEþþþ ZEþ ZEþþ ZEþþþ ðnijn þ nijn þ nijn ÞAn þ ðnijn þ nijn nijn ÞZn ¼ 0

n¼1

n¼1

AE ZE Stoichiometric coeﬃcients nijn ; nijn are the changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X where either AEþþ ZEþþ AEþþþ ZEþþþ ; nijn ; nijn ; nijn are the changes in A½i or A½iþ were attached, and nijn numbers of functional groups An and bonds Zn , respectively, connected to the root unit X þþ where stood the living group A½ j or A½ jþ. The other stoichiometric coefAEþ ZEþ AEþþ ZEþþ ; nijn ; nijn ; nijn are the changes in the numbers of groups ﬁcients nijn attached to root unit X þ which gets connected to the unit where stood the attacking group, and they are nil if there are no substitution eﬀects. The above reactions modify the counts of functional groups of similar chemical nature (for example, distinguishable kinds of amides/amines/carboxylic acids). If there is only a single kind of bond, there is no net creation or destruction of functional groups or bonds, as shown by the cancellation of the stoichiometric coeﬃcients in Eq. (67), but a reshuﬄing of pieces of the reacting molecules takes place. Rates of production by chemical reaction of the various groups are obtained using Eqs. (68)–(70).

R An ¼

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

þ

NR X NR X

e

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

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

R Zn ¼

ð68Þ

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

þ

NR X NR X

e

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

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

RWn ¼

NR X

Z k i nW in ð½A½i ½A½iþ ki ½Z½iþ Þ

ð69Þ ð70Þ

i¼1

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

3.4 Modeling of Complex Polycondensation Reactions

estimated as the sum of products of the molar volume of by-products by their rate of elimination by phase change. The mass balances of the functional groups in a batch reactor can thus be written as Eqs. (71). d½An ¼ RAn RV ½An dt

ð71Þ

d½Zn ¼ RZn RV ½Zn dt A convenient way of computing the concentrations of groups at chemical equilibrium consists in integrating the system of ODE [Eq. (71)] until close to steady state. 3.4.3

Equilibrium Polycondensations with Several Monomers

Instead of the elegant but often error-prone Gordon’s notation, we will introduce a more straightforward description, hopefully easier to translate into computer programs. The goal is to obtain a set of formulae for predicting average molecular weights, molecular weight distributions, and other polymer properties, valid for generic chemical systems. These computations assume there is some way of predicting how molecular fragments (usually monads, unless higher-order substitution eﬀects have to be tackled) are mutually connected. More speciﬁcally, it is necessary to know the distributions of the numbers of bonds connecting the fragments, for each kind of fragment. In fact, one may need only some of the moments of the aforementioned distributions for making a few calculations. A kinetic method may be used for this prediction, but mainly as a means to avoid solving equations derived from mass action laws for concentrations of fragments. This has been one of our motivations for presenting the formalism in Section 3.4.2. Each directed bond Z i is supposed to start a pendent chain Vi ðxZ ; xA Þ with counts of end groups and directed bonds xA and xZ , respectively. Notice that the molecular graphs have to be considered as digraphs, otherwise xZ would be meaningless: it would be impossible to know the counts of the monomer units according to their chemical nature. All isomeric trees with the same counts of groups are lumped into the same chemical species leading to vector count distributions with NZA ¼ NZ þ NA independent variables. Vectors of dummy Laplace variables sA and sZ will be associated with the counts of unreacted groups and directed bonds. Variables sA and sZ will be often ranged together as subvectors of a vector s, of size NZA .

117

118

3 Polycondensation

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

ð72Þ

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

1

if j ¼ zi

0

if j 0 zi

ð73Þ

The chemical system is further described through knowledge of the molar fractions of monomers or monomer units, yX i , (summing to 1) and of the molecular weights of monomer units, unreacted groups, and half-bonds, respectively MX i ; MA i and MZ i . Hence, the molecular weight M½Vi ðxZ ; xA Þ of a tree Vi ðxZ ; xA Þ can be computed through Eq. (74).

M½Vi ðxZ ; xA Þ ¼

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

ð74Þ

j¼1

X i ðxZ ; xA Þ is, according to the concept introduced in Section 3.1.5, a monad with vectors of group counts xZ and xA . Its concentration, normalized by the concentration of repeating units X i , can be thought of as a probability: the probability that a certain repeating unit is attached to those counts of groups, as stated in Eq. (75). PfX i is connected to xZ ; xA groupsg ¼ ½X i ðxZ ; xA Þ/½X i

ð75Þ

Let F X i ðsZ ; sA Þ; F A i ðsZ ; sA Þ and F Z i ðsZ ; sA Þ be the probability generating functions (pgf ) of the counts of the diﬀerent kinds of connecting and unreacted functional groups directly linked to a unit X i , an unreacted group A i or a directed bond Z i [Eq. (76)]. y X

F J ðsZ ; sA Þ ¼

xZ1 ¼0 y X

y X

y X

xZN ¼0 xA1 ¼0 Z

y X xAN ¼0 A

xZ

xAN

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

xAN ¼0 A

J ¼ Xi; Ai; Zi

ð76Þ

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

F ðsÞ ¼

s1 Aj

qF X½ jA q log sA j

qF X½ jA q log sA j js¼1

X

X

½ jA ½ jA ¼ s1 A j LA j ðsÞ/lA j NZA

ð77Þ

3.4 Modeling of Complex Polycondensation Reactions

F Z j ðsÞ ¼ s1 Zj

qF X½ jZ q log sZ j

qF X½ jZ q log szj js¼1

X

X

½ jZ ½ jZ ¼ s1 Z j LZ j ðsÞ/lZ j

ð78Þ

NZA

Notice the use of L with lower indexes for the derivatives of the pgf values with respect to the logarithms of dummy Laplace parameters, as well as of l for their moments, a useful convention which will be encountered often in the rest of this chapter. If a pgf relative to the count of monomer units is desired, for a vector of dummy Laplace variables sX , it can be found by obtaining the pgf with respect to the counts of directed bonds with Eq. (79). sZ ¼ ZZX sX

ð79Þ

The average numbers of bonds Z j ; lZX ij , and of unreacted functional groups A j ; lAX ij , attached to a monomer unit X i , will be often needed, and can be obtained using Eqs. (80) and (81). lZX ij ¼ lAX ij ¼

qF X i q log sZ j js¼1

NZA

qF X i q log sA j js¼1

NZA

ð80Þ

ð81Þ

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

MP ¼

NX X i¼1

" y X i MX i þ

NZ X

lZX ij MZ j

þ

j¼1

NA X

# lAX ij MA j

ð82Þ

j¼1

For the classic self-polycondensation of XA f , with equal and independent groups A; p, the conversion of A groups, is introduced in Eqs. (83). F X ¼ ½ð1 pÞsA þ psZ f F Z ¼ F A ¼ ½ð1 pÞsA þ psZ f 1 lZX ¼ f p

lAX ¼ f ð1 pÞ

ð83Þ

MP ¼ MX þ f pMZ þ f ð1 pÞMA Expressions for the less trivial case XA f þ YBg C illustrated diagrammatically in Figure 3.12, A reacting with B or C, B not reacting with C (such as adipic acid þ glycerol, f ¼ g ¼ 2), with constant reactivity of end groups A; B, or C, are presented in Table 3.3.

119

120

3 Polycondensation

Fig. 3.12.

Example of proposed notation: polycondensation XA2 þ YB2 C.

Tab. 3.3. Probability generating functions F describing polycondensation XA f þ YBg C and its gelation condition. " # " # ½ð1 pA ÞsA þ pAB sZAB þ pAC sZAC f F XX X F ¼ ½ð1 pB ÞsB þ pB sZBA g ½ð1 pC ÞsC þ pC sZCA F XY

2

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

2

Gelation condition

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

F AC

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

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

1 f 1

Group counts have multinomial distributions for these simple systems and can be easily related to conversions of functional groups (which are the probabilities of reaction). The theory of branching processes leads to a system of NZ algebraic equations [Eqs. (84)] for the pgf of pendent trees of the diﬀerent kinds: Vi ðsZ ; sA Þ ¼ sZ i F Z i ½VðsZ ; sA Þ; sA i ¼ 1; . . . ; NZ

ð84Þ

The vector v of the NZ probabilities of extinction (the fractions of ﬁnite pendent chains) v ¼ Vð1NZA Þ contains the solutions of system (84) for s ¼ 1NZA . Gelation occurs when system (84) has a double root v ¼ 1NZA for s ¼ 1NZA, implying that its Jacobian becomes nil [Eq. (85)]. " # qF Z i I ¼ j½LzZj i ð1NZA Þ Ij ¼ 0 qnj

ð85Þ

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

3.4 Modeling of Complex Polycondensation Reactions

pg ¼

1 f 1

ð86Þ

Prediction of the gel point for XA f þ YBg C is also presented in Table 3.3. The pgf values of trees starting with a prescribed monomer unit X i , an unreacted group A i , or a directed bond Z i , are obtained from the theory of branching processes through Eqs. (87) and (88). GYi ðsZ ; sA Þ ¼ sYi F Yi ½VðsZ ; sA Þ; sA Y ¼ Z; A

ð87Þ

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

ð88Þ

The chain length or molecular weight distribution of polymer is usually described primarily with the help of the molar concentration of species with x monomer units, ½Px , in which x is the degree of polymerization. Since we are dealing with several kinds of units, an overall degree of polymerization can be deﬁned, which is the sum of the degrees of polymerization corresponding to the various monomer units (the components of vector xX ). Notice that vector xZ with the counts of directed bonds has more information about the molecular composition. There is a basic diﬃculty when trying to use Eqs. (87) and (88): they provide pgf values of monomer units or groups, not directly molar concentrations of polymer molecules, even as generating functions. These latter will have to be computed relative to the molar concentrations of the various groups or monomer units, and thus will come multiplied also by the number of those groups in the distributions. For instance, if there is only one kind of monomer unit and so just one kind of directed bond, G X as computed by Eq. (88) will provide the generating function of x½Px /½X with respect to x; for an alternating polycondensation of two monomers, G X 1 ðsZ1 ; sZ2 Þ as computed by Eq. (88) will provide the generating function of x1 ½Pðx1 ; x2 Þ/½X1 with respect to x1 and x2 . Fractions of units and groups of the various kinds in ﬁnite molecules (in sol), after gelation, ySX i ; ySZ i and ySA i , can be computed by replacing s ¼ 1NZA in the above equations to give Eq. (89). ySYi ¼ F Yi ðv; 1NA Þ Y ¼ X; Z; A

ð89Þ

Prediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions (EANJs) and chains (EANCs), respectively me and ne [260, 261]. EANJs are the intersection of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13). These concentrations can be easily predicted, given the probabilities of extinction and the moments with respect to the numbers of pendent chains as previously deﬁned. Deﬁning xZy as the count of inﬁnite pendent chains and the correspondent dummy Laplace variable as sZy , its pgf for the chains stemming out of a monomer unit X i is F X i ½v þ sZy ð1NZ vÞ; 1NA , and so me and ne can be computed using Eqs. (90) and (91).

121

122

3 Polycondensation

Finite pendent chain

Infinite pendent chain EANJ Junction

EANC EANJ

Chain connecting to gel Fig. 3.13.

Elastically active and inactive junctions and chains in a polymer network.

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

i¼1

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

ð90Þ

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

NZ X NZ X

) nj nk ð1 nj Þð1

nk ÞLZXjiZk ðv; 1NA Þ

ð91Þ

j¼1 k¼1

For instance, assuming the so-called ‘‘phantom network’’ model, shear modulus Ge would be predicted for gaussian chains to be given by Eq. (92). Ge ¼ RTðne me Þ

ð92Þ

In the presence of gel, it is convenient to introduce the pgf of ﬁnite pendent chains, named V^i ðsZ ; sA Þ, and the pgf of counts of ﬁnite pendent chains connected to units or groups, which can be found using Eqs. (93) and (94).

3.4 Modeling of Complex Polycondensation Reactions

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

ð93Þ

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

ð94Þ

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

Y ¼ X; Z; A

ð95Þ

After computing the probabilities of extinction, the moments in Eq. (96) can be evaluated. l^YZij ...Zk ¼ LYZi j ...Zk ðv; 1NA Þ/ySYi

Y ¼ Z; A

ð96Þ

Pgf values of ﬁnite pendent chains with respect to molecular weight, for which the dummy Laplace variable associated with molecular weight is sM , can be found from Eq. (97) (notice the conventional use of a power of a scalar to a vector): M

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

þMX ½NZ Z

MA

MAN

; sM 1 ; . . . ; sM

A

M þMX z

Þ ¼ V^i ðsM z

MA ; sM Þ ð97Þ

Prediction of average molecular weights is now possible by introducing GM ðsM Þ, the pgf values of the mass fractions of the polymer molecules in the sol, wi , with respect to their molecular weight Mi , deﬁned below; the index i in the inﬁnite sum sweeps all ﬁnite polymer molecules. GM ðsM Þ ¼

y X

X

Mi sM wi ¼

NY X

MY MA ^ M ðsM Þ; sM wYi sM i F^Yi ½V

ð98Þ

Y ¼X; Z; A i¼1

i¼1

The mass fractions of the units and groups in Eq. (98) above are relative to the mass of the sol. The weight fraction of the sol wS , relative to the overall mass of the polymer computed by Eq. (82), is therefore given by Eqs. (99)–(102). wS ¼

NX X

y X i ySX i MX i þ

j¼1

i¼1

wX i ¼

y X i ySX i MX i MP wS MZ i

wZ i ¼

NZ X

NX X

l^ZX ij MZ j þ

NA X

!, l^AX ij MA j

MP

ð99Þ

j¼1

ð100Þ

X y X j ySX j l^Z ji

j¼1

MP wS

ð101Þ

123

124

3 Polycondensation

MA i

NX X

X y X j ySX i l^A ji

j¼1

wA i ¼

ð102Þ

MP wS

Before carrying out the evaluation of distributions and average molecular weights, a special reasoning must be carried out in order to compute number-average molecular weight and degrees of polymerization. When it is taken into account that, with the reaction of every pair of end groups in ﬁnite molecules, one polymer molecule is consumed, the number of moles of polymer molecules per mole of monomer units before gelation is given by Eq. (103).

yP ¼ 1

NX NZ X 1X yXi lZX ij 2 i¼1 j¼1

ð103Þ

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

yP ¼

NX X

y X i ySX i

i¼1

NZ 1X 1 l^X i 2 j¼1 Z j

! ð104Þ

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

Mn ¼

y X i ySX i MX i þ

NZ X

MZ i l^ZX ij

j¼1

i¼1 NX X

y X i ySX i

i¼1

þ

NA X j¼1

NZ 1X 1 l^X i 2 j¼1 Z j

!

! MA i l^AX ij ð105Þ

An expression for the number-average degree of polymerization of the sol follows from Eq. (105) by setting the molecular weights of monomer units equal to one and the molecular weights of bonds and unreacted groups equal to zero [Eq. (106)]. NX X

xn ¼

y X i ySX i

i¼1 NX X i¼1

y X i ySX i

NZ 1X l^X i 1 2 j¼1 Z j

!

ð106Þ

The moments with respect to molecular weight lM ; lMM , and so on, can be now obtained through diﬀerentiation of Eq. (98) with respect to log sM and setting sM ¼ 1. First of all, the systems of linear algebraic equations (107)–(113) must be solved.

3.4 Modeling of Complex Polycondensation Reactions j

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

¼

j ½di

ð107Þ

l^ZZ ij 1 ½l^ZAji ½MA i

ð108Þ

" ½miZZ

¼

j ½di

l^ZZ ij 1

2ðMZ i þ MX½iZ Þ

NZ X

mZl l^ZZli

þ

¼

j ½di

" ¼

j ½di

ð109Þ

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

l^ZZ ij 1

NZ X NA X

ð110Þ

l¼1 m ¼1

m¼1

½miAA

# Z ^Z i mZl mm lZl Zm

l ¼1 m¼1

l ¼1

½miZA

NZ X NZ X

mlA MAm l^ZZliAm

þ

l¼1 m¼1

NA X NA X

# MAl MAm l^AZliAm

ð111Þ

l¼1 m¼1

j

ð112Þ

½miMA ¼ ½di l^ZZ ij 1 ½MA2 i

ð113Þ

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

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

lM ¼

Y ¼X; Z; A i¼1

j¼1

ð114Þ

j¼1

lMM ¼ Mw MZ ¼

X Y¼X; Z; A

(

NY X

" w Yi

MY2i

j¼1

i¼1

þ

þ 2MYi

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

!

j¼1

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

þ

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

þ2

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

þ

NA X NA X j¼1 k¼1

MA j MAk l^AYjiAk þ

NA X

#) MA2 j l^AYji

ð115Þ

j¼1

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

125

126

3 Polycondensation

The weight-average and z-average degrees of polymerization, x w and x z , are obtained in the same way as for number-average degrees of polymerization: the molecular weights of the monomer units are set equal to one and the molecular weights of bonds and unreacted groups are set equal to zero. For the polycondensation of a single monomer XA f , this leads to Eqs. (116) and (117). xw ¼

1 þ pð2v 1Þ 1 p½1 þ vð f 2Þ

xz ¼ 1 þ

þ

ð116Þ

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

ð117Þ

ð1 p þ 2pvÞ½1 pð f 1Þð1 p þ pvÞ f 2 2

The number-average degree of polymerization x n is obtained through a stoichiometric reasoning as previously discussed [Eq. (118)]. xn ¼

1 p þ pv 1 p þ pvð1 f /2Þ

ð118Þ

Values of average degrees of polymerization versus conversion of end groups p shown for f ¼ 2 and f ¼ 3 in Figure 3.1 have been computed using the above expressions. Pgf values of the various distributions with respect to the counts of monomer units, bonds, or unreacted functional groups can be obtained from Eq. (98) by setting equal to one the molecular weight of the species in question and to zero the molecular weights of all the other species. Analytical inversion by computing derivatives with respect to dummy Laplace variables on s ¼ 0 is feasible with the simplest chemical systems, for which the resulting recurrence formulae are not too complex, as happens with the Stockmayer distribution. Numerical inversion of generating functions of the concentration distributions is y P s x ½Px usually a better way to predict them. Unless the cost of evaluating GðsÞ ¼ x ¼0

is too high and more sophisticated methods based upon Laplace transform inversion are needed, an accurate evaluation can be obtained using the method independently developed by Mills [262] and ourselves [263] (see also Ref. 264 for a thorough analysis of round-oﬀ and truncation errors of similar approaches). It consists in computing the inversion contour integral on a circle C in a complex plane centered on the origin with radius jsj slightly below 1, using the trapezium rule and Fast Fourier Transform for evaluating the sums [Eq. (119)].

½Px ¼

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

ð119Þ

3.4 Modeling of Complex Polycondensation Reactions

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

y X

b n sCn ½sA ð1 ar Þ þ ar V nð f 2Þ1

n¼1

G ¼ ð1 yc ÞsC0 ½sA ð1 ar Þ þ ar V f þ

y X

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

n¼1

b0 ¼

f ð1 yc Þ ; f 2yc

bn ¼

ð f 2Þ y Xc ðnÞ f 2yc

ð120Þ

A pgf for the counts of repeating units X will result from substituting sC0 in Eq. (120) by sX and sCn by sXn. Variable ar in the expressions (120) is the probability of

127

128

3 Polycondensation

reaction of the unreacted functional groups after ring formation in stage 1. It is necessary to eliminate it, and to introduce the mass action laws for the rings. This last step is not straightforward, as it requires the application of graph theory in order to compute the concentrations of linear polymer molecules ½L n in Eq. (23). The ﬁnal result, which reduces to the distribution found by Jacobson and Stockmayer in their classic paper [21] for f ¼ 2, is Eq. (121). ½Px ¼ ½X ð f 2yc Þ y Xc ðnÞ ¼ y X n¼1

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

nK c ðnÞ n q ; ½X

nK c ðnÞq n ¼ ½X yc ;

q¼

f ð f 1Þar ð1 yc Þ f 2yc

yc ¼

f ð p ar Þ 2ð1 ar Þ

ð121Þ

A solution is also known for the analogous system of the alternating polycondensation XA f þ YBg [267]. Dilution with an inert solvent will make [X] decrease without aﬀecting the cyclization constants much, and the fraction of rings will increase, until it becomes practically unity, a phenomenon which has experimental support obtained using polysiloxanes as model polymers. In bulk systems, the fraction of rings is usually only a few per cent. With nonlinear polycondensations of aliphatic monomers, gel points in bulk are aﬀected by a few per cent due to cyclizations, and elastic properties are also affected (that eﬀect using TBP has been modeled by Dusˇek et al. [269]). Application of the approach described above could lead to improved modeling where small numbers of rings are present. Taking cyclizations into account raises the question of introducing information about the spatial location of atoms in models of network formation. The classic gelation theory described here considers a uniform distribution in space of all the chemical properties. Stauﬀer, one of the main contributors to the progress of percolation theory, has strongly criticized this view, claiming that Gordon’s theory is inapplicable in the vicinity of gel point [270]. Gordon has not accepted that argument [271], claiming his theory to be universal and capable of describing gel formation in any homogeneous system. A new theory, starting with classic gelation theory and using TBP as a particular case, has been developed by Kuchanov [272, 273], and considers the molecular graphs to be embedded in ordinary three-dimensional space – not in a lattice, as is usually done in percolation theory. Generating functions are replaced by generating functionals of the ensemble of positions of the functional groups and repeating units. Dependence of space coordinates is eventually eliminated by averaging, so the algebraic equations become integral equations. In spite of its power, it has not become a widely used instrument for dealing with ‘‘real’’ complex chemical sys-

3.4 Modeling of Complex Polycondensation Reactions

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

Kinetic Modeling of Irreversible Polycondensations

In his pioneering study of nonlinear polycondensation [7], Stockmayer has already checked his statistical solution [Eq. (5)] by solving the mass balance equations in a batch reactor for the concentrations of functional groups A and the set of isomeric polymer molecules Px with x repeating units X [Eqs. (122)]. ( x1 d½Px 1X ¼k ½ yð f 2Þ þ 2½ðx yÞð f 2Þ dt 2 y¼1 þ 2½Py ½Pxy ½A½xð f 2Þ þ 2½Px d½A ¼ k½A 2 ; dt

½Px jt¼0 ¼ ½X ;

) ð122Þ

½Ajt¼0 ¼ f ½X

The two solutions are identical. Hence, for a long time no importance was attributed to the use of a kinetic approach for describing batch polycondensations starting from monomers, and the statistical approach was preferred. Of course, chemical engineers had to deal with semi-batch and continuous stirred tank reactors where the statistical approach, although possible, is cumbersome and error-prone, so a few papers appeared in the 1960s dealing with kinetically controlled linear polycondensations [274–276]. In reality, Kuchanov [277, 278] has shown that,with polycondensations presenting FSSEs, kinetic and statistical approaches give distinct results for average molecular weights and gel points. Dusˇek [279] has pointed out that this behavior is even more visible when dealing with polyadditions (linear polyaddition leads to a Poisson CLD, whereas a simplistic use of a statistical approach would lead to a geometrical/Schulz–Flory CLD) and has conﬁrmed this result using Monte Carlo simulation of XA f polycondensation with FSSEs [280]. Sarmoria and Miller [35] have tried to extend the ‘‘recursive approach’’ to systems presenting FSSE by considering network building starting from dyads and larger fragments. But chemical systems can be found in which this latter idea does not provide useful results [281], so that use of statistical approaches outside the description of chemical equilibrium now seems more like a waste of time. Kuchanov’s kinetic approach divides polymer molecules into classes PðxÞ having a vectorial count of groups x. In this approach, ‘‘groups’’ An should include not only the unreacted functional groups, but also the bonds and repeating units, and even larger molecular fragments when needed. We will use NXZA as the number of kinds of groups in that generalized sense.

129

130

3 Polycondensation

It is possible to obtain a rate equation for the members of each class by adding the contributions of the various condensation reactions, leading to a version of Smoluchowski’s coagulation equation. Ring-forming reactions involving functional groups in the same monomer can be described by a simple extension of the preceding FSSE model, just by considering unimolecular reactions and new fake functional groups, which are pairs of groups [Eq. (123)]. ki

A½i !

N XZA X

nin An

i ¼ 1; NR

ð123Þ

n¼1

The rate of formation of groups requires a modiﬁcation of Eq. (68), since it is no longer possible to include breakage or exchange reactions. Adding the unimolecular reactions deﬁned above, the new general rate equation becomes Eq. (124).

R An

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

ð124Þ

i¼1

The multiple sums in the rate of formation of PðxÞ, which will not be presented, are simpliﬁed through consideration of its generating function [Eqs. (125), (126)] NR X þ GRP ðsÞ ¼ ki G ni Gni i¼1

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

þ

NR X i¼1

ki

qG ðG 1Þ q log s½i ni

ð125Þ

where

J

Gni ¼

N XZA Y

n

J

snin

J ¼ þ; ;

ð126Þ

n¼1

Equation (125) replaces a similar expression in Ref. 282 with the advantage of considering only the reactions actually taking place, and not every combination of pairs of unreacted groups, which does not make sense now, as repeating units are also An moieties. Insertion of Eq. (125) into mass balance equations, such as a continuous stirred tank reactor (CSTR) with constant volume, leads to a nonlinear ﬁrst-order PDE [Eq. (127)].

3.4 Modeling of Complex Polycondensation Reactions

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

þ

NR X

ki

i¼1

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

GðsÞjt¼0 ¼ G0 ðsÞ

ð127Þ

Solution of the above equation by the method of characteristics [283] is described in Ref. 282, earlier examples being found in Refs. 284–286. They will not be reproduced here, for the sake of brevity. If GðsÞ is to be evaluated, in order to take advantage of the numerical inversion formula Eq. (119), or if average degrees of polymerization in the presence of gel have to be predicted, numerical solution of Eq. (127) leads to a two-point boundary solution problem with twice as many unknowns as the number of derivative terms log sn (the number of active groups in the polymer). A much simpler problem, as Galina was apparently the ﬁrst to remark [287], is P , in the prediction of the moments of ½Px with respect to the counts of groups, lmn... the absence of gel. Diﬀerentiation of Eq. (126) with respect to log sn and setting s ¼ 1NZXA leads to an ODE system with known initial conditions, which has a straightforward numerical solution [Eq. (128)]. dljkP dt

¼

NR X

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

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

þ

NR X

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

m¼1

ljkPF ljkP t

RV ljkP

ð128Þ

As there is no gel, a rate equation for the overall concentration of polymer ½P (zeroth-order moment) is obtained from Eq. (126), setting s ¼ 1NZXA [Eqs. (129)]. NR X ½Pf ½P q½P ¼ ki ½A½i ½A½iþ RV ½P þ t qt i¼1

½Pjt¼0 ¼ ½P0

ð129Þ

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

131

132

3 Polycondensation N ZXA X

Mn ¼

N ZXA N ZXA X X

MAn ½An

n¼1

Mw ¼

½P

m¼1 n¼1 N ZXA X

P MAm MAn lmn

ð130Þ MAn ½An

n¼1

This approach can only deal with ring-forming reactions either for a limited number of the smallest rings, or alternatively, for linear polycondensations. The important practical case of the irreversible polycondensation of AXA þ BYB þ BYC (C being an inert group) leads to the rate laws in Eqs. (131) for the molecules with the six possible combinations of end groups PnAA ; PnAB ; . . . PnCC and rings Cn , where index n counts the most frequent kind of repeating units in the molecule: n1 X

RPnAA ¼ k 4

RPnBB ¼ k 4

AA ½PmAA ½Pnm

þ2

m¼1

m¼1

n1 X

n1 X

BB ½PmBB ½Pnm

þ

m¼1

" RPnAB ¼ k

n1 X

n X

BB 4 ½PmAA ½Pnmþ1 m¼1

! AB ½PmAA ½Pnm

2½PnBB ½A

½PnAB ð½A

!

AB 2 ½PmBB ½Pnm m¼1

þ

2½PnAA ½B

#

n1 X

AB ½PmAB ½Pnm m¼1

þ ½BÞ kc ðnÞ½PnAB

RCn ¼ kc ðnÞ½PnAB RPnAC ¼ k 2

n X

BC ½PmAA ½Pnmþ1

m¼1

RPnBC ¼ k 2

n1 X

n1 X

! AC ½PmAB ½Pnm

½PnAC ½B

m¼1

AC ½PmBB ½Pnm

m¼1

RPnCC ¼ k

þ

n1 X

þ

n1 X

! BC ½PmAB ½Pnm

½PnBC ½A

m¼1

BC ½PmAC ½Pnm

m¼1

RA ¼ RB ¼ k½A½B

y X

kc ðnÞ½PnAB

ð131Þ

n¼1

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

3.4 Modeling of Complex Polycondensation Reactions

equation of Eqs. (131) may be done accurately. A more elegant method uses generating functions [288] and avoids the problem of the lack of closure of the above equations for any ﬁnite N. When the kinetic approach was at an early stage, it was thought that it could provide no information about polymer or network properties. More recently, a description of batch polycondensation using TBP which is rigorously equivalent to the one obtained by the kinetic approach was found [289], taking into account the times of birth of molecules, so that the fundamental restriction does not hold. Through more elementary reasonings, it is nevertheless possible to estimate probabilities of extinction and thus network elastic properties [282] or average radius of gyration [290]. 3.4.5

Kinetic Modeling of Linear Reversible Polycondensations

The reversible alternating polycondensation with FSSEs in both monomers (see Section 3.1.5), disregarding exchange reactions, is a convenient case study for discussing problems of modeling this kind of systems. It can be described by the rate laws of Eqs. (132) and (133). RP1AA ¼ 4k1 ½P1AA ½P1BB 2k2 ½P1AA ð½P1AB þ ½ZB Þ þ k1Z ½P1AB þ k2Z ½ZA RP1BB ¼ 4k1 ½P1AA ½P1BB 2k3 ½P1BB ð½P1AB þ ½ZA Þ þ k1Z ½P1AB þ k3Z ½ZB RP1AB ¼ 4k1 ½P1AA ½P1BB ½P1AB ½2k2 ½P2AA þ 2k3 ½P1BB

ð132Þ

þ k4 ð2½P1AB þ ½ZA þ ½ZB Þ k1Z ½P1AB þ 2k2Z ½P2AA þ 2k3Z ½P2BB þ 2k4Z ð½ZA þ ½ZB 2½P2AA 2½P2BB Þ n b 2:

AB RPnAA ¼ 2k2 ½P1AA ½Pn1 þ 2k4

n2 X

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

m¼1

"

½PnAA ½2k2Z

þ

2k4Z ðn

2Þ þ

k3Z ½PnAB

þ

k4Z

½ZA

n X

# ð½PmAB

þ

2½PmAA Þ

m¼2

AB þ 2k4 RPnBB ¼ 2k3 ½P1BB ½Pn1

n2 X

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

m ¼1

"

½PnBB ½2k3Z

þ

2k4Z ðn

2Þ þ

k2Z ½PnAB

þ

k4Z

½ZB

n X m¼2

# ð½PmAB

þ

2½PmBB Þ

133

134

3 Polycondensation

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

þ k4

AB ½PmAB ½Pnm

þ4

n1 X

! BB ½PmAA ½Pnmþ1

m¼2

m¼1

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

" þ k4Z ½ZA þ ½ZB 2

n X

½PnAB 2

m¼2

nþ1 X

# ð½PmAA þ ½PmBB Þ

ð133Þ

m¼2

It may be observed from these expressions that, in general, it is not possible to obtain a closed ﬁnite set of rate equations for the ﬁrst oligomers – rate equations for oligomers with chain length n always depend on concentrations of oligomers with chain length n þ 1. Only when the rate constants of reverse equations are equal do the above expressions simplify (FSSE for reverse reaction), and the abovementioned diﬃculty disappears. On introduction of generating functions of the CLD, deﬁned in Eq. (134), the concentrations of end groups, trimers, and tetramers conform with Eqs. (134) and (135).

GAA ¼

y X

s n2 ½PnAA GBB ¼

n¼2

y X

s n2 ½PnBB GAB ¼

n¼2

y X

s n2 ½PnAB

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

ð134Þ

n¼2

½P2AB ¼ GAB ð0Þ

ð135Þ

The rate equations in terms of those generating functions become Eqs. (136). RGAA ¼ 2k2 ½P1AA ðsGAB þ ½P1AB Þ þ k4 s 2 GAA GAB 2GAA ½2k3 ½P1BB þ k4 ð½P1AB þ ½ZB Þ 2k2Z GAA 2k4Z s þ k4Z

qGAA þ k3Z GAB qs

½ZA 2GAA GAB 1s

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

½ZB 2GBB GAB 1s

qGBB þ k2Z GAB qs

3.4 Modeling of Complex Polycondensation Reactions

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

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

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

GAA ½P2AA GBB ½P2BB þ 2ðk3Z k4Z Þ s s

ð136Þ

Rate laws for ½ZA and ½ZB become Eq. (137): RZA ¼ 2k2 ½P1AA ð½ZB þ 2½P1AB Þ 2k3 ½P1BB ½ZA þ k4 ð2½P1AB 2 ½ZA ½ZB Þ k2Z ð½ZA þ 2½P2AA Þ k3Z ð½ZB 2½P2BB Þ 5 þ k4Z ½X þ ½Y þ ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2 RZB ¼ 2k2 ½P1AA ½ZB þ 2k3 ½P1BB ð½ZA þ 2½P1AB Þ þ k4 ð2½P1AB 2 ½ZA ½ZB Þ k2Z ð½ZA 2½P2AA Þ k3Z ð½ZB þ 2½P2BB Þ 5 þ k4Z ½X þ ½Y þ ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2

ð137Þ

A rate law for by-product [Eq. (138)] is usually required. RW ¼ 4k1 ½P1AA ½P1BB þ 2k2 ½P1AA ½ZA þ 2k3 ½P1BB ½ZA þ k4 ½ZA ½ZB k1Z ½P1AB k2Z ½ZA k3Z ½ZB k4Z

y X

½ð2n 4Þð½PnAA þ ½PnBB Þ þ ð2n 3Þ½PnAB

n¼2

¼ 4k1 ½P1AA ½P1BB þ 2k2 ½P1AA ½ZA þ 2k3 ½P1BB ½ZA þ k4 ½ZA ½ZB k1Z ½P1AB k2Z ½ZA k3Z ½ZB 3 k4Z ½X þ ½Y ð½ZA þ ½ZB Þ ½P1AA ½P1BB 2½P1AB 2

ð138Þ

Insertion of these rate laws in mass balances of ideal reactors (batch/plug ﬂow or transient CSTR) leads to systems of semi-linear, ﬁrst-order, partial diﬀerential equations, with a single family of characteristics [Eq. (139)]. ds ¼ 2sk4Z dt

ð139Þ

The relationship of Eq. (138) allows integration along characteristics, relating initial values GAA ð0; s 0 Þ; GAB ð0; s 0 Þ, and GBB ð0; s 0 Þ to GAA ðt; sÞ; GAB ðt; sÞ, and GBB ðt; sÞ,

135

136

3 Polycondensation

but a serious problem remains: integration along characteristics and prediction of concentration of ﬁrst oligomers have to be done at the same time. Only the solution with equal reactivity has been published [291]. A sketch of an iterative method to solve this more complex problem is described below (no actual calculations have been published at the time of writing). 1. Two auxiliary characteristics sþ ðtÞ ¼ s ðtÞ, starting at sþ ¼ es js0 j and s ¼ es js0 j, respectively, for t ¼ 0, are introduced in order to provide an accurate numerical estimation of oligomer concentrations through Eq. (140). GI ð0Þ ¼

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

ð140Þ

Parameter es should be a small value (say 106 to 108 ). Owing to Eq. (139), it is easy to relate the auxiliary characteristics to the ‘‘main’’ characteristic sðtÞ starting with s ¼ s0 through Eq. (141). sþ ðtÞ ¼ s ðtÞ ¼ es sðtÞ

ð141Þ

2. The resolution of a system of ODE comprising Eq. (139), the balances of end groups ZA , ZB , by-product W and diﬀerential equations for GAA ðt; sÞ, GAB ðt; sÞ, GBB ðt; sÞ, GAA ðt; sþ Þ, GAB ðt; sþ Þ, GBB ðt; sþ Þ, GAA ðt; s Þ, GAB ðt; s Þ and GBB ðt; s Þ, will eventually allow the evaluation of all unknown variables of this problem. As s0 is to be found iteratively in order that the characteristic starting with s ¼ s0 will attain the prescribed value of s at the prescribed time t, this is a boundary value problem, but very straightforward to solve. If there are more than two functional groups per repeating unit, the approach described above is no longer useful, as there is no simple way of describing formation of chemical species through breaking of larger ones, owing to their branched structure. Nevertheless, in most situations resembling industrial practice, such as CSTRs in series, the CLD of these chemical systems will stay close to equilibrium (see Ref. 291). The distributions of monads depend only on the equilibrium ratios and it is not necessary to use models with SSSEs for the reverse reactions (the source of the most challenging mathematical problems), as the same distributions will be found using equal values of the rate constants of the reverse reaction.

Notation

1N av b

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

Notation

bA Cd Cg D db dR dT Ec ei

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

137

138

3 Polycondensation

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

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

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

Notation

M Mi

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

i¼1

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

i¼1

i¼1

empirical parameter in Eq. (56) [mol1 m 3 ] mA number of functional (or end) groups NA Avogadro–Lo¨schmidt number [mol1 ] NAL Nb number of bubbles ﬂux of component j [mol m2 s1 ] N_ j number of distinctive chemical bonds NR number of symmetrical chemical bonds NRs number of kinds of monomer units NX number of by-products NW number of volatile (or precipitating) components NY number of half-bonds NZ NZA ¼ NZ þ NA overall number of kinds of functional groups and half-bonds NZXA ¼ NZ þ NX þ NA overall number of kinds of functional groups, monomer units, and half-bonds n_ screw rotational speed (as rotations per unit time) [s1 ] Avrami exponent ncr pressure inside a bubble [Pa] Pb local pressure over a bubble [Pa] Pme vapor pressure of W [Pa] PW S PW vapor pressure of pure W [Pa] p conversion of reference functional groups A conversion of functional groups A having given rise to rings pc Q volumetric ﬂow rate of polymer [m 3 s1 ] volumetric ﬂow rate of gas [m 3 s1 ] Qg R ideal gas constant [J mol1 K1 ] Re Reynolds number for the rising movement of a bubble rate of formation by chemical reaction of species j [mol m3 s1 ] Rj RV combined relative rate of change of reaction volume by chemical reaction and transfer of by-product [s1 ] r stoichiometric ratio in alternating polycondensations radius of a bubble [m] rb s vector of dummy Laplace variables associating sA and sZ (deﬁned below)

139

140

3 Polycondensation

sA

vector of dummy Laplace variables associated with the counts of unreacted functional groups dummy Laplace variables associated with the counts of rings (includsCn ing sC0 for the count of units X in the chains connecting rings) for the equilibrium polycondensation of a single monomer XA f sM dummy Laplace variable associated with molecular weight vector of dummy Laplace variables associated with the counts of halfsZ bonds dummy Laplace variable associated with xZy s Zy sZl dummy Laplace variable associated with the counts of bonds not belonging to rings for the equilibrium polycondensation of a single monomer XA f T absolute temperature [K] reference temperature [K] T0 t time [s] exposure time [s] tf exposure time of liquid in the pool of a vented extruder [s] t fP bubble rising speed [m s1 ] ub average velocity in directions x; z [m s1 ] ux ; uz V volume of reacting mixture [m 3 ] volume of a bubble [m 3 ] Vb Vi ðsZ ; sA Þ pgf of pendent chains Vi V^i ðsZ ; sA Þ pgf of ﬁnite pendent chains Vi Vj molar volume of species j [m 3 mol1 ] ^ pgf of ﬁnite pendent chains Vi with respect to molecular weight VMi ðsM Þ overall melt volume [m 3 ] Vm volume of liquid in compartment j in a staged model of an RDC devoVmj latilizer [m 3 ] v vector of the NZ probabilities of extinction (the fractions of ﬁnite pendent chains) average transverse velocity of the screw dragging the ﬁlm [m s1 ] vT wcr weight fraction of crystalline material weight fraction of j wj weight fraction of the sol, relative to the overall mass of the polymer wS x degree of polymerization, the number of repeating units in a polymer molecule x coordinate along the perimeter, when discussing mass transfer from a liquid ﬂowing over a cylindrical wall [m] vector storing the counts of functional groups xA xi degree of polymerization of generic polymer species i, where i is an arbitrary y index y P P xn ¼ x i ½Pi ½Pi number-average degree of polymerization of polymer i¼1 i¼1 y y P P xw ¼ xi2 ½Pi x i ½Pi weight-average degree of polymerization of polymer i¼1

i¼1

Notation

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

ZZX z zi

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

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

141

142

3 Polycondensation

ﬁrst and second moments of molecular weight of distribution of polymer m dynamic viscosity [N m1 s1 ] concentrations of elastically active network junctions (EANJ) me [mol m3 ] n stoichiometric coeﬃcient (the change in number of moles caused by chemical reaction) stoichiometric coeﬃcients of functional groups An and oriented bonds ninA ; ninZ Zn coming out of the monomer unit in which A½i was situated for the reaction forming Z iR Aþ Zþ nin ; nin stoichiometric coeﬃcients of functional groups An and oriented bonds Zn coming out of the monomer unit in which A½iþ was situated for the reaction forming Z iR AE ZE nijn ; nijn changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X where either A½i or A½iþ were attached, for the exchange reaction forming Z iR at the expense of Z jR AEþþ ZEþþ AEþþþ ZEþþþ nijn ; nijn ; nijn ; nijn changes in numbers of functional groups An and bonds Zn , respectively, connected to the root unit X þþ where the living group A½ j or A½ jþ was situated, for the exchange reaction forming Z iR at the expense of Z jR AEþ ZEþ AEþþ ZEþþ nijn ; nijn ; nijn ; nijn changes in numbers of groups attached to the root unit X þ which gets connected to the unit where the attacking group was situated, for the exchange reaction forming Z iR at the expense of Z jR W nin stoichiometric coeﬃcients of by-products for the reaction forming Z iR nin stoichiometric coeﬃcient of group An for the ith intramolecular reaction concentration of elastically active network chains [mol m3 ] ne density of j [kg m3 ] rj s surface tension [N m1 ] t space time based upon feed ﬂow rate, Vm /Q F for a CSTR [s] tr structural relaxation time [s] parameter in Eq. (11) [s] t0 tortuosity for by-product diﬀusion in a solid polymer particle tD (dimensionless) volume fraction of crystalline material fcr volume fraction of j fj fraction of channel extruder ﬁlled with liquid fL w Flory–Huggins parameter (binary mixture) (dimensionless) weight fraction activity coeﬃcient of W WW lM ; lMM

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

Notation

Aj

active component (molecular species or fragment) j; nonvolatile, nonprecipitating ring molecule with n bonds Cn linear polymer molecule with n 1 bonds and a pair of active end Ln groups length of tube LT generic polymer species, where i is an arbitrary index Pi IJ Pn linear polymer molecule with end groups I and J and n repeating units of the most frequent kind ½U molar concentration of species U Vi ðxZ ; xA Þ pendent chain starting with a half-bond Z i and containing vector counts of bonds and functional groups, respectively, xZ and xA W by-product of condensation reaction We i 1 W½i by-product of condensation reaction leading to bond Z iR X; Y monomer units Xh i 1 X½iA monomer unit carrying functional group A i Xz i 1 X½iZ monomer unit to which each half-bond Z i points volatile (or precipitating) component j Yj Z bond group; Z iR is the ith distinctive bond in the chemical system bond group in fragment AXZA YZ (polycondensation AXA þ BYB) ZA bond group in fragment BYZB XZ (polycondensation AXA þ BYB) ZB bond group in dimer AXZD YB (polycondensation AXA þ BYB) ZD Zei 1 Z½i ; Zeþi 1 Z½iþ ‘‘half-bonds’’ of bond Z iR ZP bond group in fragment ZXZP Z (polycondensation AXA þ BYB) Subscripts aq b blk c cr F g org sat

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

Superscripts * x

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

Acronyms BHET BPA

bis(2-hydroxyethyl) terephthalate bisphenol A

143

144

3 Polycondensation

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

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

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2

3 4 5

6

7

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

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

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

17 18 19 20 21 22 23 24

25

26

27 28 29 30 31 32 33 34 35 36 37

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

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

260 261 262 263 264 265

266 267

268 269

270 271 272 273

274 275 276

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

277 S. I. Kuchanov, Methods of Kinetic

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282 283

284 285 286 287 288

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291

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

151

153

4

Free-radical Polymerization: Homogeneous1 Robin A. Hutchinson

Free-radical polymerization (FRP) is one of the most important commercial processes for preparing high molecular weight polymers. It can be applied to almost all vinyl monomers under mild reaction conditions over a wide temperature range and, although requiring the absence of oxygen, is tolerant of water. Multiple monomers can be easily copolymerized via FRP, leading to the preparation of an endless range of copolymers with properties dependent on the proportion of the incorporated comonomers. This chapter will provide an overview of the kinetics and mechanisms, and the techniques used to construct mathematical representations of bulk and solution FRP. The description will also serve as a good base for the suspension and emulsion FRP chapters that follow.

4.1

FRP Properties and Applications

Polymers produced via free-radical chemistry include the following major families:

Low-density polyethylene (LDPE) and copolymers, used primarily in ﬁlms and packaging applications. LDPE has density of <0.94 g cm3 , and is produced via high-pressure free-radical polymerization; polyethylenes of higher density (and polypropylene) are produced via transition metal catalysis, as described elsewhere (see Chapter 8). Poly(vinyl chloride) and copolymers, used primarily to produce pipe and ﬁttings, ﬂooring material, and ﬁlms and sheet. Polystyrene and its co- and terpolymers with acrylonitrile and butadiene. Homopolymer is used for packaging and containers, while the acrylonitrile-containing polymers are used for various molded products in the appliance, electronics, and

1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

154

4 Free-radical Polymerization: Homogeneous

automotive industries. Styrene–butadiene is the most widely used synthetic rubber. Acrylic- and methacrylic-based polymers. Poly(methyl methacrylate) (PMMA), due to its transparency and weatherability, is used extensively in signs, lighting ﬁxtures, and windows. Polyacrylates and copolymers are used extensively in the adhesives and coatings markets, and are also combined with acrylonitrile to make acrylic ﬁbers. Poly(vinyl acetate) and copolymers, used extensively in adhesives, coatings, and paper and textile treatment. Fluoropolymers including polytetraﬂuoroethylene and copolymers, used widely in the wire and cable industry. They also have many specialty applications as coatings due to their inertness and low-surface tension. In addition to these major product families, there exist many other smaller-volume (but often high-value) polymeric products synthesized via free-radical chemistry. These products are manufactured via heterogeneous (emulsion, suspension) or homogeneous (bulk, solution) polymerization in a wide range of reactor conﬁgurations ranging from tubular to well-mixed tanks (and everything in between) in processes that may be continuous, batch, or semi-batch. FRP kinetics together with reactor design and operating conditions controls the composition and architecture of the polymer produced. As with other chemistries, the ﬁnal product consists of a mixture of polymer chains with varying length, composition, and structural properties such as branch points and branch lengths. The processing and end-use properties of the polymer depend upon the distributions of these characteristics, not just the average values. Thus it is necessary to develop methods to relate the fundamental kinetic mechanisms of free-radical polymerization to these distributed characteristics.

4.2

Chain Initiation

Free-radical polymerization, like other chain growth mechanisms, involves the sequential addition of vinyl monomers to an active center. The deﬁning characteristic of FRP is that the active centers are free radicals. Each growing polymeric radical increases in size rapidly; a typical chain is initiated, grows to high molecular weight, and is terminated in the time scale of, at most, a few seconds. When the macromolecule stops growing it cannot generally react further (barring side reactions), and is considered ‘‘dead’’. These dead chains have a residence time of minutes or hours in the polymerization reactor, and the ﬁnal polymer product is an intimate mixture of chains formed under time and/or spatially varying conditions. The free radicals that initiate polymerization are in most cases generated by thermal or photochemical homolytic cleavage of covalent bonds. Commercial initiators include azo and peroxy compounds. The driving force for the dissociation of azo

4.2 Chain Initiation

155

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

N N

H3C

N2

CH3

+

2 C

CH3

CH3

CH3

CH3 Scheme 4.1.

CN

CN

CN

Decomposition of AIBN.

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

R

O

CH3

O O

+O

(a) CH3

CH3

CH3

O R

O

(b) CH3

. R

CH3

+ CO2 +

CH3

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

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

O

CH3 CH3

156

4 Free-radical Polymerization: Homogeneous

polymer) in the system, or recombine to form inactive compounds. The relative rates of these processes are dependent on both the nature of the primary radicals and the monomer system. The mechanistic pathway followed will inﬂuence endgroup structures of the polymeric products, aﬀecting ﬁnal properties such as thermal stability. Further details on initiator decomposition kinetics and mechanistic pathways can be found in the excellent monograph by Moad and Solomon [3]. The above discussion highlights the fact that the simple initiation process of most polymer texts is the exception rather than the rule, with the fraction of primary radicals that initiate a new polymer chain a complex function of the reaction system. Nonetheless, the kinetic treatment is usually simpliﬁed by the introduction of a fractional initiator eﬃciency ( f ), formally deﬁned by Eq. (1), where n is the number of moles of primary radicals generated per mole of initiator; n ¼ 2 for most common initiators.

f ¼

Initiation Rate of Propagating Chains nðRate of Initiator DisappearanceÞ

ð1Þ

The initiator eﬃciency is normally in the range 0.4–0.9, with a low value indicating ineﬃcient usage of the initiator and potentially a high formation rate of undesired by-products. The kinetic descriptions in this chapter are developed for thermal unimolecular scission of a compound to yield two radicals, as this is the most common means of generating radicals in industrial systems. Thermal initiation of monomers is an additional mechanism capable of forming primary radicals at higher temperatures, as discussed for styrene (see Section 4.3.1.3). Other initiation systems are also available, and bear a brief mention at this point. Initiators with multiple peroxide linkages ðn > 2Þ have been the subject of recent academic study [4]. Photoinitiators that produce radicals by ultraviolet irradiation are commonly used to initiate crosslinking and curing reactions in polymeric systems, as the rate of initiation can be controlled through the intensity and location of the light source. And ﬁnally, a redox (reduction–oxidation) process is often used to initiate chains in emulsion polymerization (see Chapter 6).

4.3

Polymerization Mechanisms and Kinetics

It is important to achieve an understanding of how the basic FRP mechanisms control polymerization rate and average polymer chain length. This section starts with the derivation of appropriate kinetic expressions for a single monomer system. Complicating (but industrially important) secondary reactions are then introduced, followed by the extension to multi-monomer systems. Dispersed throughout are up-to-date estimates for important free-radical polymerization rate coeﬃcients, and descriptions of how they are obtained experimentally.

4.3 Polymerization Mechanisms and Kinetics

4.3.1

Homopolymerization

Many commercial polymers, including polystyrene, PMMA, and LDPE, are synthesized via homogeneous free-radical polymerization of a single monomer. Homopolymer properties are controlled by average chain length and chain-length distribution as well as, in some cases, structural characteristics such as branching level. Basic Mechanisms The basic set of FRP mechanisms includes initiation, propagation, termination, and transfer to monomer and solvent or transfer agent, as shown in Scheme 4.3. 4.3.1.1

k

Initiator Decomposition

d I  →2 f I ∗

Chain Initiation

i I ∗ + M  → P1

Chain Propagation

Pn + M → Pn +1

k

kp

Chain Termination k

By Combination

tc Pn + Pm  → Dn+m

By Disproportionation

td Pn + Pm  → Dn + Dm

k

Chain Transfer k mon

To Monomer

tr Pn + M  → Dn + M *

k mon

i M * + M  → P1

k sol

To Solvent or Agent Scheme 4.3.

tr Pn + S → Dn + S *

k sol

i S * + M → P1

Basic free-radical homopolymerization mechanism.

The subscript n denotes the number of monomeric units in growing polymer radicals (Pn ) and dead polymer chains (Dn ). Each reaction has an associated kinetic rate law expression and a speciﬁc rate coeﬃcient. The free-radical initiator (I) unimolecularly decomposes (with rate coeﬃcient kd ) to form two primary radicals (I ) with eﬃciency f. Chain initiation occurs when the primary radical adds to monomer M, and chain growth continues via successive addition of monomer units to the radical center (chain propagation, with rate coeﬃcient k p ). Bimolecular coupling of two growing chains results in the loss of two radicals from the system and the formation of either one (termination by combination, k tc ) or two (termination by disproportionation, k td ) dead polymer chains. Chain stoppage may also occur via a transfer mechanism, where the growing radical abstracts a weakly bonded

157

158

4 Free-radical Polymerization: Homogeneous

atom (usually hydrogen) from monomer or other molecules (solvent or chaintransfer agent, denoted by S) in the system to generate a dead polymer chain as well as a new radical that initiates another polymer chain. A key assumption implicit in the formulation of Scheme 4.3 is that the rates of propagation, transfer, and termination reactions are independent of n, the length(s) of the radical(s) involved. It is known that propagation and likely transfer reactions involving very short chains (n ¼ 1; 2; 3) are faster by a factor of 10 than addition to long-chain radicals [5], but this eﬀect diminishes rapidly with chain length and has a negligible eﬀect on the overall kinetics of the system. Chain termination, the coupling of two polymeric radicals, is a very fast chemical process that is controlled by the rate at which the two radicals ﬁnd each other in the reaction system. The nature of this diﬀusion control makes termination the most complex reaction in the polymerization process since the apparent rate coeﬃcient can vary greatly with system conditions such as monomer conversion and solution viscosity (see Section 4.3.3). Although termination may also exhibit some chainlength dependence, most engineering treatments of FRP neglect this complex dependence. For further discussion of the individual mechanisms of Scheme 4.3 and their rate coeﬃcients, see Section 4.3.1.2. The set of rate laws that can be written from Scheme 4.3 is given by Eqs. (2)–(6). Initiator Decomposition

R d ¼ kd ½I

ð2Þ

Chain Initiation

R init ¼ 2f kd ½I

ð3Þ

Chain Propagation

R p ¼ k p ½M½Ptot

ð4Þ 2

Chain Termination

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

Chain Transfer

R tr ¼ ðktrmon ½M þ ktrsol ½SÞ½Ptot

2

ð5Þ ð6Þ

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

½Ptot ¼

y X ½Pn

ð7Þ

n¼1

In some literature the right-hand side of the termination rate expression [Eq. (5)] is written as 2k t ½Ptot 2 . The mode of termination – combination or disproportionation – has no eﬀect on the overall termination rate, and thus the two events can also be expressed by the nomenclature of Eq. (8), where d is the fraction of the termination events that occur by disproportionation. k t ¼ k tc þ k td ;

d¼

k td k tc þ k td

ð8Þ

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

4.3 Polymerization Mechanisms and Kinetics

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

½Ptot ¼

R init kt

1=2

¼

2f kd ½I kt

1=2 ð9Þ

The consumption of monomer by chain-initiation or transfer events is negligible compared to that by propagation. This result, called the long-chain hypothesis (LCH), must be true if high molecular weight polymer is being produced. Thus the rate of polymerization (disappearance of monomer) can be taken as equal to the rate of propagation (R pol ¼ R p ) with the rate of heat generation proportional to the rate of this exothermic reaction.

Under these (generally valid) assumptions, the classic expressions for rate of polymerization (R pol ), kinetic chain length (u, the average number of monomer units on a living chain), and instantaneous degree of polymerization (DPninst , the average number of monomer units on a dead polymer chain formed at any instant) are given in Eqs. (10)–(12), respectively.

1=2

R pol

2f kd ½I ¼ k p ½M½Ptot ¼ k p ½M kt

u¼

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

DPninst ¼

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

ð10Þ

ð11Þ

ð12Þ

The diﬀerence between Eqs. (11) and (12) arises because termination by combination yields a single polymer chain such that the chain length of dead polymer formed (DPninst ) is greater than the chain length of polymer radicals (u) in the system at the same instant. Table 4.1 lists the range of concentration and rate coeﬃcient values typically encountered in homogeneous free-radical polymerization systems at low conversion. These can be combined with Eqs. (9)–(12) to illustrate the tradeoﬀs involved be-

159

160

4 Free-radical Polymerization: Homogeneous Tab. 4.1. Typical values of coeﬃcients and concentrations in low-conversion homogeneous FRP systems.

Coeﬃcient/Concentration

Typical range

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

106 –104 0.4–0.9 10 2 –10 4 10 6 –10 8 106 –104 106 –103 104 –102 1–10 1–10

[a] At

low conversion.

tween the desire for high throughput (R pol ) and the need to produce high MW (DPn ) polymer.

The denominator of Eq. (12), the rate of dead chain formation, must be of the order 105 –108 chains L1 s1 in order to produce polymer with a DPn of 10 2 –10 4 . Individual polymer radicals exist on average only for a fraction of a second, as calculated by the expression u=ðk p ½M). Thus after the ﬁrst few seconds of polymerization, the concentration of dead polymer chains is higher than that of polymeric radicals, and by the end of a typical polymerization the concentration of dead chains is orders of magnitude higher than [Ptot ]. Final polymer MW and MWD (molecular weight distribution) are controlled by how the concentrations and kinetic coeﬃcients in Eq. (12) vary with polymer conversion. The theoretical MW limit for a system is controlled by transfer events, and is calculated by setting [Ptot ] to zero in Eq. (12). For bulk FRP with no solvent, limiting values of DPn are 10 4 –10 6 . However, this theoretical limit can only be approached in homogeneous FRP by reducing rates of polymerization to extremely low levels. For most systems both termination and transfer events play an important role in controlling polymer MW. To produce high MW polymer (DPn of 10 2 –10 4 ) it is necessary to keep total radical concentration low, so that [Ptot ] is of the order 108 –106 mol L1 . This dictates the choice of initiator such that R init (the product of kd and [I]) is also of order 108 –106 mol L1 s1 . Transfer can occur to monomer, solvent or any other species in the system. In some cases, chain-transfer agents are added deliberately to limit and control polymer DPn . These agents are generally chosen such that the rate of abstraction is much higher than that which occurs with monomer or solvent (k tr =k p ¼ 103 – 10 0 ) and thus can be added in trace quantities (f 1 mol L1 ). The use of transfer agents allows for independent manipulation and control of R pol and DPn , but is only possible if the desired MW is less than that achieved for the corresponding transfer-free reaction.

4.3 Polymerization Mechanisms and Kinetics

Initial rates of polymerization (monomer consumption rates) at low conversion are of order 104 –102 mol L1 s1 . Approximately 10 3 –10 5 s are required to take a batch system to complete conversion. Faster rates can be achieved by increasing R init , but at the expense of decreased polymer MW. Achievable values of R pol are also often limited by the heat removal capabilities of the reactor system, as the heat released by monomer addition is of the order 50–100 kJ mol1 .

It should be cautioned that these statements are generalities for a typical FRP system. Rate coeﬃcients vary with monomer, initiator, and solvent choice (see Section 4.3.1.2) as well as polymerization conditions, and the kinetic treatment is complicated by the occurrence of side reactions (Section 4.3.1.3) and the variation of k t with conversion and other system conditions (Section 4.3.3). These factors necessitate the use of more-powerful modeling techniques to quantitatively describe FRP systems (Section 4.4). Nonetheless, Eqs. (9)–(12) provide an idea of the factors controlling rate and MW, and are very useful for a qualitative examination of FRP systems. Kinetic Coeﬃcients The coupling of polymer MW and polymerization rate is further illustrated via rearrangement of Eq. (12) to give Eq. (13). 4.3.1.2

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

0:5k tc R pol kp2 ½M 2

þ

k td R pol kp2 ½M 2

þ

ktrmon ktrsol ½S þ k p ½M kp

ð13Þ

R pol [Eq. (10)] and DPn are both dependent on k 2p =k t , with DPn also a function of mode of termination (disproportionation versus combination) and chain transfer. Although R pol and DPn are easily measured experimentally, it is not possible to resolve the quantities into estimates for individual rate coeﬃcients. Even the estimation of the ratio k 2p =k t from R pol requires independent knowledge of initiator characteristics f and kd , and the assumption that radicals are not being consumed or retarded by adventitious impurities in the system. These factors have led to considerable scatter in rate coeﬃcients reported in the literature (for example, Ref. 7), especially for individual rate coeﬃcients [8, 9]. Yet individual values, and knowledge of how they vary with temperature, are required for model development and an accurate representation of multi-monomer systems. The emergence of specialized experimental techniques since 1988 has greatly improved this situation and led to an improved understanding of free-radical polymerization kinetics. The following discussion highlights some of these advances, as well as summarizing key FRP rate coeﬃcients as expressed by the Arrhenius law [Eq. (14)]. k i ¼ A i expððEi þ 0:1DVi PÞ=RTÞ

ð14Þ

161

162

4 Free-radical Polymerization: Homogeneous

Activation energies (Ei ) and volumes (DVi ) are reported with units of kJ mol1 and cm 3 mol1 respectively, with T in K and P in bar. All second-order rate coeﬃcients are reported with units of L mol1 s1 . Initiation Thermal scission of an initiator is the most common means of generating radicals in FRP (see Section 4.2). This unimolecular reaction is characterized by a ﬁrst-order rate coeﬃcient (kd , s1 ) so that, for a constant-volume batch system, Eq. (2) may be integrated to yield Eq. (15), with ½I0 the initial concentration at t ¼ 0.

½I ¼ ½I0 expðkd tÞ

ð15Þ

The decomposition kinetics is often expressed by the half-life of the initiator, the time needed for the concentration to decrease to half of its initial value [Eq. (16)]. t 1=2 ¼ ðln 2Þ=kd

ð16Þ

The requirements for an initiator can vary widely: an initiator with a half-life of 10 h might be used in an academic study so that [I] does not change signiﬁcantly during the course of an experiment, while an initiator used for high-pressure ethylene polymerization typically has a half-life on the order of a few seconds. Activation energies for thermal homolysis of peroxide and azo compounds are in the range of 100–150 kJ mol1 , and thus decomposition rates are very temperature-sensitive: the t 1=2 of benzoyl peroxide drops from 13 h at 70 C to 0.4 h at 100 C. The dependence on pressure is much less, in the range of 0 to 15 cm 3 mol1 [1, 2], but still important for high-pressure LDPE systems. Special care must be taken in handling and transport of thermal initiators, especially those with faster decomposition rates. Values of kd at a particular temperature and pressure can be determined by measuring initiator concentration as a function of reaction time using a technique such as infrared spectroscopy. The experimental diﬃculty increases as half-life shortens, where special care must be taken to eliminate transient nonisothermal eﬀects. Decomposition kinetics are summarized for a wide range of initiators in the Polymer Handbook [7] and in trade literature available from commercial suppliers. Of special note is the recent work of Buback and co-workers that systematically examines not only how Ed and DVd vary with alkyl substituent for the peroxyester family (Scheme 4.2), but also how the substituent choice aﬀects the decomposition pathway and initiator eﬃciency [1, 2]. Propagation In general, chain growth or propagation proceeds in a highly selective manner to yield a polymer chain consisting of head-to-tail linkages (Scheme 4.4). In order for high polymer to be formed the propagating radical must not be too stable: addition must occur at a suﬃciently high rate in comparison with competing transfer and termination events.

4.3 Polymerization Mechanisms and Kinetics

R1 CH2

R1

+

C R2

Scheme 4.4.

R1 CH2 C

H2C R2

R2

R1 CH2

C R2

Free-radical chain propagation.

A number of nonsteady polymerization rate techniques have been introduced to measure k p , many prone to signiﬁcant error [8, 9]. The introduction of a new method, that couples pulsed-laser-induced polymerization (PLP) with size exclusion chromatographic (SEC) analysis of the resulting polymer [10], has greatly improved the reliability of k p data. In this technique, a mixture of monomer and photoinitiator is illuminated by short laser pulses separated by a time of t 0 , typically 0.01–0.2 s. Propagation and termination (but no initiation) occur between pulses, with the radical concentration and the rate of termination both decreasing with time according to Eq. (5). Those growing macroradicals formed by a laser pulse which escape termination will all have the same chain length (within a narrow Poisson distribution) that increases with time. There is a high probability that these surviving radicals will be terminated at t 0 , when a new population of short radicals are generated by the next laser pulse. Thus, a signiﬁcant fraction of dead chains formed have a chain length DP0 corresponding to a lifetime of t 0 seconds [Eq. (17)]. DP0 ¼ k p ½Mt 0

ð17Þ

Since radicals have a certain probability of surviving the laser ﬂash and of terminating at a later pulse, the relative concentration of polymer with chain lengths 2DP0 ; 3DP0 ; . . . is also increased. As a result, PLP produces a well-structured MWD with peaks at chain lengths of DP0 and its multiples, as shown in Figure 4.1. With known values for t 0 and [M] (kept constant, as samples are pulsed only long enough to allow a conversion of 1–3%), Eq. (17) yields a direct estimate of k p from the experimentally-determined value of DP0 . PLP-SEC has proven to be a very simple and robust experimental technique for determining k p and its temperature dependence, provided adequate care is taken with SEC analysis. Data collected from various research laboratories around the world have been compiled in a series of papers for styrene [11] and methacrylates [12–14]. These papers provide benchmark k p data for these monomer systems, illustrate the good agreement between facilities (typically 10–20%), and make recommendations for best experimental practices. Table 4.2 summarizes the Arrhenius parameters for free-radical propagation of a wide range of monomers, as determined by PLP-SEC. An extensive review by Beuermann and Buback [15] contains further discussion and data for additional monomers. The data for acrylates are limited, due to measurement diﬃculties arising from additional secondary mechanisms that occur (see Section 4.3.1.3). What

163

4 Free-radical Polymerization: Homogeneous 1.2

2DP0

1 wt log(MW)

164

0.8

DP0

0.6 0.4 0.2 0 3.5

4

4.5

5

5.5

6

6.5

Log MW

A typical PLP-generated MWD, as measured by SEC for poly(butyl methacrylate) produced by PLP of bulk butyl methacrylate at 25 C and a laser repetition rate of 10 Hz. DP0 is determined from the inﬂection point of the peak, obtained by diﬀerentiating the distribution. Fig. 4.1.

is striking is not only the large variation in k p and activation energies between monomer families, but also the similar behavior within a monomer family. All methacrylates exhibit a similar temperature (Ep of 21 to 23 kJ mol1 ) and pressure (DVp of 15 to 17 cm 3 mol1 ) dependence, and the k p values for alkyl methacrylates at 50 C increase with increasing size of the alkyl ester group (methyl to dodecyl) by a factor of less than 2. The alkyl acrylate family exhibits exactly the same trends, although the activation energies (17 to 18 kJ mol1 ) and volumes (10 to 12 cm 3 mol1 ) are lower than for methacrylates, and the values of k p at 50 C

Tab. 4.2.

Arrhenius k p parameters for various monomers determined by PLP-SEC.[a]

˚

Monomer

Ep [kJ molC1 ]

DVp [cm 3 molC1 ]

Ap [L molC1 sC1 ]

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

34.3 32.5 22.4 22.9 21.0 22.9 23.0 20.8

27.0 12.1 16.7 16.5 16.0 15.0 16.2 n.d.

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

54 238 648 757 995 1230 1204 1504

20.7 17.7 17.4 17.0

10.7 11.7 n.d. 11.7

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

6625 22 900 27 900 32 000

[a] All

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

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

4.3 Polymerization Mechanisms and Kinetics

are 30 to 40 times greater. The propagation behavior is aﬀected more signiﬁcantly by electron-donating or -withdrawing substituents present in functional monomers such as glycidyl and 2-hydroxypropyl methacrylates. The PLP-SEC method has also been utilized to show that there is little to no solvent inﬂuence on propagation kinetics of most monomers [15]. However, the propagation kinetics of acrylic acid in water is a strong function of concentration, a result that may be explained by a change in local monomer concentration around the propagating radical center [16]. With accurate estimates of k p now available, the possibility of correlating the coeﬃcient with the structural characteristics of the propagating radicals and monomers is receiving attention. While some progress has been made in relating activation energy to reaction enthalpy corrected for polar and resonance factors [17], more work is required for quantitative predictions. Termination With independent measures of k p available, k t can now be estimated from the lumped ratio of k 2p =k t . Specialized techniques involving pulsed-laserinduced polymerization have also been developed to yield accurate estimates of the ratio k p =k t [15]. Termination rates in FRP are always diﬀusion-controlled so that the apparent value of k t depends on the conditions under which it has been measured, including the lengths of the radicals involved in the reaction. Nonetheless, the assumption of chain-length independence is usually made for modeling of commercial FRP systems, since the errors introduced are not large. For a more detailed discussion of the diﬀusion-controlled nature of the termination reaction, including its strong conversion dependence, see Section 4.3.3. Most measurements of k t have been carried out at low monomer conversion, and thus it is useful to tabulate available data in this regime. Signiﬁcant scatter, as much as an order of magnitude, is found in the data contained in the Polymer Handbook [7], as reﬂected in the vinyl acetate k t data in Table 4.3. The uncertainty

Tab. 4.3.

Arrhenius kt parameters for various monomers.[a]

˚

Monomer

Et [kJ molC1 ]

DVt [cm 3 molC1 ]

At [L molC1 sC1 ]

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

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

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

15.6 14 15 15[b] 15[b] – 20 16 21

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

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

[a] All

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

165

166

4 Free-radical Polymerization: Homogeneous

arises from a number of measurement and interpretation factors [18]. The rest of the data in Table 4.3 are based on PLP studies, and have a higher level of accuracy (within a factor of 2); these are taken from the recent review by Beuermann and Buback [15], adjusted to conform to the convention used in Eq. (5) for the termination rate. Note the large diﬀerence in magnitude between values for styrene, methyl methacrylate, and methyl acrylate (k t of 1 10 8 –5 10 8 L mol1 s1 ) and values for dodecyl acrylate and dodecyl methacrylate (@0:5 10 7 –1 10 7 L mol1 s1 ). This has been attributed to diﬀerences in segmental diﬀusion rates and/or steric hindrance of the large ester side groups. The reported activation energies and volumes are consistent with the diﬀusion-controlled nature of the reaction. The mode of termination is also of importance, as it aﬀects the molecular architecture of the polymer formed, and thus some of its properties. Polymer polydispersity (Mw/Mn ) is 1.5 if all chains are terminated by combination of two radicals, and 2 if chains are terminated by disproportionation or chain transfer. As shown in Scheme 4.5, termination by combination results in head-to-head linkages in the polymer chain, whereas disproportionation results in the formation of an unsaturated end group that may undergo further reaction.

R1

R1 CH2

+

C R2

C R2

CH2

R1

R1

CH2 C

C

R2

R2 Scheme 4.5.

R2 R1

R1 CH2 CH

CH2

+

C

CH

R2

Free-radical chain termination by combination and disproportionation.

The relative importance of termination by disproportionation versus combination expressed by d [Eq. (8)] is diﬃcult to measure. The value depends largely on the structure of the monomer: d is in the range of 0.5–0.8 for a-methylvinyl monomers such as the methacrylates and 0.05–0.2 for styrene and acrylates [3]. A weak temperature dependency has been proposed, but is diﬃcult to verify within the scatter of the data [19]. Chain transfer Chain transfer in radical polymerization involves the transfer of the radical center from a polymeric radical to another molecule. It can occur to all of the substances present in the polymerization system, and always causes a reduction in u and DPninst . Scheme 4.3 includes only transfer to monomer and solvent (or added transfer agent), but transfer to initiator and dead polymer can also occur. The former is often neglected since [I] is small relative to the concentration of other species, while the latter can be an important reaction impacting MWD and

4.3 Polymerization Mechanisms and Kinetics

ﬁnal polymer properties (see Section 4.3.1.3). As well as reducing chain length, the fragments from the transfer reactions (I and M in Scheme 4.3) are incorporated as end groups in the ﬁnal polymer product. The rate and MW equations in Section 4.3.1.1 were derived assuming that the radicals formed by transfer reinitiate new polymeric radicals quickly, within about the same time period as that required for a propagation step (k imon A k isol A k p ). This assumption is valid for transfer to most species, and can be veriﬁed by examining whether addition of the transfer agent has an eﬀect on polymerization rate. If the low-conversion polymerization rate is signiﬁcantly decreased, the species is classiﬁed as a retarding agent or inhibitor, and Eqs. (9)–(13) no longer describe the kinetics of the system; see Section 4.3.1.3. The chain-transfer ability of a compound can be studied by varying its concentration while holding all other variables ﬁxed. By stopping the reaction at very low conversion such that concentrations and diﬀusion-controlled k t values (and thus R pol and DPninst ) are kept constant, Eq. (13) can be rearranged as Eq. (18). 1 1 k sol ½S ¼ þ tr DPn ðDPn Þ0 k p ½M

ð18Þ

A plot of 1=DPn against ½S=½M should yield a straight line with slope k trsol =k p and intercept 1=ðDPn Þ0 , where ðDPn Þ0 is the average chain length measured in the absence of transfer agent. The form of this equation leads to the deﬁnition of a chaintransfer constant C for each species, including monomer, in Eq. (19). Ctrsol ¼

ktrsol ; kp

Ctrmon ¼

ktrmon kp

ð19Þ

It is these ratios and their temperature dependence that are tabulated in standard references (for example, Ref. 7). Solvent/transfer agent For some homogeneous FRP systems, solvent is added not for its chain-transfer ability, but as a diluent to control viscosity and/or heat transfer. In other systems, a small amount of chain-transfer agent (CTA) is added speciﬁcally to control and reduce the MW of the polymer. Thus values for Ctrsol may vary widely from a low of 106 –104 to a high of 101 –10 1 , depending on the number of weakly bonded atoms (generally hydrogen or halogen) on the transfer agent and their ease of abstraction. For very active compounds (for example, thiols and halogenated compounds) with Ctrsol > 101 it is necessary to account for the consumption of CTA during the course of the polymerization, since a changing ratio of ½S=½M will cause a corresponding drift in polymer MW. In such cases careful control of CTA addition to the system is required. If it is added at higher concentrations, telomerization (formation of low-MW species) will occur rather than polymerization. Table 4.4 summarizes values for a few typical solvents and CTA compounds at 50 C. Most organic compounds have low transfer rates (Ctrsol of 106 –104 ) so that

167

168

4 Free-radical Polymerization: Homogeneous Tab. 4.4.

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

Benzene Toluene Ethyl acetate Triethylamine CCl 4 1-Butanethiol

Styrene

Methyl methacrylate

Methyl acrylate

Vinyl acetate

Ethylene[b]

2 106 1 105 5 104 5 104 1 102 20

4 106 2 105 1 105 8 104 2 104 0.6

2 105 1 104 6 105 4 102 2 104 1.5

1 104 2 103 2 104 4 102 0.8 50

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

values at 50 C [7]; there can be an order of magnitude range in literature data. [b] Ethylene values at 130 C and 1360 atm [20]. [a] Representative

transfer is important only when they are present in high concentration (that is, as solvent). They tend to be more reactive toward a radical such as ethylene or vinyl acetate than a resonance-stabilized radical such as that of styrene. For a given radical type, aliphatic compounds that yield tertiary radicals will be more eﬀective transfer agents than those that produce secondary radicals. Halomethanes and thiols, used as MW-controlling agents due to their high transfer rates, react faster with nucleophilic radicals such as styrene or vinyl acetate than with (meth)acrylates. Abstraction reactions from organic solvents generally have higher activation energies than propagation, with (Etr Ep ) in the range of 20–50 kJ mol1 [21, 22]. The Polymer Handbook [7] and the monograph by Moad and Solomon [3] provide a summary of available data for a wider range of monomer–CTA (solvent) pairings. Monomer Transfer to monomer cannot be avoided, and the maximum upper limit of chain length that can be achieved in a polymerization is Ctrmon , assuming the absence of all other transfer and termination events. The details of the mechanism depend upon the speciﬁc monomer involved. For monomers that contain aliphatic hydrogens such as vinyl acetate and (meth)acrylates, the transfer process involves H-atom abstraction to form an unsaturated new radical, as shown for vinyl acetate in Scheme 4.6. The polymer chain that grows from this radical thus contains an unsaturated end group that may undergo further reaction. Transfer to monomer rates are generally very low, and are diﬃcult to measure experimentally since the ratio with propagation (R trmon =R p ) is independent of [M]. Table 4.5 summarizes

O

CH2

O

O

C CH3

C

+

H Scheme 4.6.

O H2C

C

O O

C CH3 CH2

O

C CH3

CH2

+

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

O H2C

C H

C CH2

4.3 Polymerization Mechanisms and Kinetics Tab. 4.5.

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

Styrene Methyl methacrylate Butyl acrylate Vinyl acetate Ethylene[b]

C trmon

E tr C E p [kJ molC1 ]

Reference

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

– 23.7 15.2 – 43.9

7 23 24 7 25

values at 50 C, unless otherwise noted; values reported in Ref. 7 can show an order of magnitude range. [b] At 250 C and 2000 bar. [a] Representative

the range of values found in the literature. Ctrmon increases with temperature, with (Etrmon Ep ) in the range of 10–40 kJ mol1 . Additional Mechanisms The basic mechanisms of Scheme 4.3 are common to and occur in every FRP system. Other mechanisms are more system speciﬁc, dependent not only on the choice of monomer but also the process operating conditions. These additional mechanisms complicate the kinetic analysis and often play an important role in controlling polymerization rate and polymer structure under typical industrial operating conditions. 4.3.1.3

Thermal initiation Primary radicals in most FRP systems are generated by scission of an added initiator. Free-radical polymerizations can also be initiated by the monomer itself, or by reactions involving trace impurities in the system. Generally the rate of radical generation by these processes is very low – negligible compared with R init from the added initiator. Styrene, however, exhibits signiﬁcant thermal polymerization at temperatures of 100 C or more. Indeed, it is not necessary to add initiator to styrenic systems at high temperatures, as the rate of thermal initiation is suﬃcient to make an industrially viable process [26]. Acrylates and methacrylates have also been reported to undergo thermal polymerization, but at a significantly slower rate than styrene. The mechanism behind the thermal polymerization of styrene is still under debate, but is believed to involve the reversible formation of a dimeric species by a Diels–Alder reaction, followed by the subsequent hydrogen transfer to a third styrene molecule to form two radicals that can initiate polymerization (Scheme 4.7). This complex mechanism can be approximated by a third-order dependency on styrene concentration [Eq. (20)] [27]. k therm

3M ! 2I R therm ¼ k therm ½M 3 ;

k therm

L2 mol 2 s

¼ 2:2 10 5 expð114:8=RTÞ

ð20Þ

169

170

4 Free-radical Polymerization: Homogeneous

+

CH CH

+

+

Scheme 4.7.

Thermal initiation of styrene.

This expression from the mid-1970s has stood the test of time, although the preexponential factor has been increased by a factor of 2–3 to ﬁt 260–340 C data obtained in a more recent study [26]. Retardation and inhibition Some substances retard or suppress free-radical polymerization by reacting with primary or polymer radicals to yield nonradical products or radicals that are too stable to add further monomer. By decreasing the concentration of reactive radicals in the system, polymerization rate is slowed (retardation) or stopped completely (inhibition). Phenolic inhibitors (for example, hydroquinone, monomethylhydroquinone) are commonly added to monomers at parts-per-million levels in order to prevent polymerization during transport and storage by rapidly and eﬀectively scavenging any radicals (Eq. (21), in which R represents I or Pn ) that may form. kinhib

R þ Z ! dead products;

R inhib ¼ kinhib ð½Ptot þ ½I Þ½Z

ð21Þ

Ideally an inhibitor (Z) stops all polymerization until completely consumed (R inhib g R p ), at which time polymerization proceeds at a normal rate, as shown schematically in Figure 4.2. In many academic studies, monomer is distilled or passed over a column to remove inhibitor before polymerization. This puriﬁcation is not done in industry if the concentration of inhibitor in the monomer is much lower than the concentration of initiator added to the system, since the excess of initiator-generated radicals quickly consumes the inhibitor. Retardation, the slowing of polymerization rate by consumption of radicals, can take diﬀerent forms and thus must be considered on a case-by-case basis. If R inhib is of similar magnitude to R p , polymerization will proceed at a lower rate until all of the retarding species is consumed (curve a in Figure 4.2). The kinetic expressions (Section 4.3.1.1) cannot be applied since termination is no longer the

4.3 Polymerization Mechanisms and Kinetics

Conversion

Normal Polymerization

Inhibition

a

b

Retardation

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

sole mechanism of radical consumption; application of radical stationarity yields R init ¼ R term þ R inhib . A diﬀerent form of retardation occurs when a radical species formed from transfer (S in Scheme 4.3) reinitiates at a slow rate. In addition to the slower reaction rate with monomer to form a polymer radical, the termination of S with other radicals in the system may also need to be considered (Scheme 4.8). Explicit balances must be written for S , and the extra mechanisms must be included when deriving expressions for [Ptot ], R pol , and DPn . As solvent/transfer agent is generally not completely consumed, the retardation eﬀect will last the duration of the polymerization (curve b in Figure 4.2). The degree of retardation depends on the value of k isol , which can vary with monomer type; many carbon-centered radicals show much lower reactivity toward vinyl esters (for example, vinyl acetate) than (meth)acrylates [3]. k sol

tr Pn + S → Dn + S *

k sol

i S *+ M → P1

k sol

t S *+ Pn → Dn

Scheme 4.8.

Retardation of polymerization by solvent or transfer agent.

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

171

172

4 Free-radical Polymerization: Homogeneous

Depropagation The addition of a radical to a double bond – the propagation reaction – is potentially a reversible process [Eq. (22)]. kp

Pn þ M Ð Pnþ1 ; kdep

R dep ¼ kdep ½Ptot

ð22Þ

The relative importance of the reverse reaction is governed by the free energy change, DGp ¼ DHp TDSp , where DHp represents the enthalpy and DSp represents the entropy change upon propagation; polymerization can only occur spontaneously when DGp is negative. With most common monomers, depropagation is negligible at typical reaction temperatures. For some systems, however, the enthalpy and entropy terms for propagation are delicately balanced so that depropagation has a signiﬁcant eﬀect on polymerization rate (R pol ) at higher temperatures [Eqs. (23)]. R pol ¼ R p R dep ¼ ðk p ½M kdep Þ½Ptot ¼ kpeﬀ ½M½Ptot kpeﬀ ¼ k p

ð23Þ

kdep ½M

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

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

ð24Þ

This equation can also be rearranged to Eq. (25), to deﬁne the ceiling temperature Tc at which, for a given monomer concentration, the polymerization rate becomes zero: Tc ¼

DHp DSp þ R ln½M

ð25Þ

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

DSp ¼ R ln

Ap A dep

þ R lnð½MÞ

ð26Þ

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

4.3 Polymerization Mechanisms and Kinetics Depropagation behavior of common monomers.

Tab. 4.6.

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

˚

CDH p [kJ molC1 ] [a]

Tc [ C][b]

35 73 56 80 88 102

19 335 194 394 460 577

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

[b] Calculated

tions. Styrene has a slightly lower ceiling temperature and depropagation must be considered above 250 C. This temperature range, while higher than that generally employed for styrene polymerization, is used for some commercial processes [26]. The addition of a methyl group to a monomer greatly reduces Tc , as seen by comparing values for a-methylstyrene to styrene and methacrylates to acrylates. eﬀ Figure 4.3 is a plot of k p measured for dodecyl methacrylate (bulk monomer) by eﬀ the PLP-SEC technique [28]. At 180 C, the highest temperature examined, k p is 1 1 1 1 4000 L mol s , less than half of the k p value of 9400 L mol s extrapolated from the lower-temperature data. With known Arrhenius coeﬃcients for propagation (Table 4.2), the temperature dependence of kdep may be estimated directly from

4 3.8

log[kpeff (L/mol-s)]

3.6 3.4 3.2 3 2.8 2.6 2.4 2.0

2.2

2.4

2.6

2.8

3.0

1000/T (K -1)

Fig. 4.3. Eﬀective propagation rate coeﬃcient for bulk dodecyl methacrylate measured by PLP. The line indicates the forward rate coeﬃcient (k p ) calculated according to the Arrhenius parameters of Table 4.2. Experimental data are taken from Ref. 28.

3.2

3.4

3.6

173

4 Free-radical Polymerization: Homogeneous

174

these data. The corresponding values for DHp (54 kJ mol1 ) and DSp (123 J mol1 K1 ) agree well with thermodynamic estimates. The depropagation behavior of many methacrylate monomers is similar [28], and depropagation must be considered at temperatures above 120 C, especially for systems with low monomer concentration [29]. Long-chain branching All of the mechanisms presented to this point do not change the basic linear architecture of the polymer chains, with each repeat unit linked to two others. Branched polymers, those in which the repeat units are not linked solely in a linear array, can have signiﬁcantly diﬀerent physical properties than their linear counterpoints, depending upon the number and distribution of the branches along the polymer backbone as well as their length. Understanding the mechanisms by which these branches are formed, therefore, is a key component to manipulation and control of polymer structure. Branches can be formed either by intramolecular reactions, discussed in the next subsection, or intermolecular reactions. The reaction of a polymeric radical with another polymer chain, or long-chain branching, can occur by three distinct mechanisms. The common feature is the re-activation of a dead polymer chain. As well as creating branched structures, these reactions signiﬁcantly broaden the polymer chain-length distribution. Intermolecular transfer to polymer The transfer of a radical center from a polymeric radical to another polymer chain is shown in Scheme 4.9. Addition of monomer to the mid-chain radical produces a polymer with a branch point, with the length controlled by the number of propagation events before chain transfer or radical– radical termination occurs. An additional subscript can be added to track the number of long-chain branches formed [Eq. (27)]. pol

mk tr

Pn; b þ Dm; c ! Dn; b þ Pm; cþ1 ;

pol

pol

R tr ¼ k tr ½Ptot m1

ð27Þ

This reaction does not change the number of monomer units that have been polymerized or the number of polymer chains in the system, and thus has no eﬀect on DPn . The rate expression in Eq. (27) is written assuming that all repeat units on all

R1

R1 CH2

+

C

CH2 C

CH2

CH2

CH2

+

H

H

R1

R1 CH2 C

R1

R1

CH2

+

H2C

CH R1

CH2 C

CH2

H2C R1 CH

Scheme 4.9.

Long-chain branch formation by chain transfer to polymer.

CH2 C

CH2

4.3 Polymerization Mechanisms and Kinetics

dead polymer chains have an equal probability of reaction, as represented by m1, the total concentration of polymerized monomer units in the system [Eq. (28)].

m1 ¼

y X y X

n½Dn; b

ð28Þ

n¼1 b¼0

Note that the reaction is not proportional to the number of chains in the system, but to the number of repeat units contained in the chains. This has two important consequences. First, the importance of transfer to polymer increases with polymer conversion (x p ) in the system, as seen in Eq. (29) by looking at the ratio of branching to monomer consumption. pol

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

ð29Þ

Thus polymerizations operating at high monomer conversion will have signiﬁcantly higher branching than low-conversion systems. Second, longer polymer chains – those with more repeat units – are more likely to participate in a branching reaction than short chains. Since the re-activated chains increase in length through subsequent propagation, this leads to a broadening of the molecular weight distribution reﬂected by an increase in the weight-average MW (Mw ); even low levels of branching can increase polydispersity values (Mw/Mn ) to 5–15 compared to 2–3 for linear polymers. As well as conversion, the importance of transfer to polymer depends upon the monomer system. The reaction can be important in systems with very reactive radicals such as ethylene [30–32], vinyl acetate [33–35], and acrylate [36, 37] polymerizations, but seldom occurs in styrene and methacrylate systems. Transfer to polymer usually occurs via abstraction of a methine hydrogen as shown in Scheme 4.9, but may also involve other easily abstracted H-atoms, such as the acetate methyl pol pol hydrogens on poly(vinyl acetate). Transfer constants to polymer (Ctr ¼ k tr =k p ) are not as readily determined as other transfer constants because the process does not decrease DPn . Long-chain branching (LCB) levels are usually quite low, less than 2 per 1000 repeat units, making it diﬃcult to employ NMR. Indirect methods such as multi-detector SEC [32, 38] are often used, leading to a signiﬁcant scatter pol in reported Ctr values [7]. Like other transfer events, the relative importance increases with temperature. Reaction with unsaturated polymer chains Termination by disproportionation, transfer to monomer, and chain scission (discussed later in this section) create polymer chains with terminal unsaturation (denoted in Eq. (30) by D ¼ ). These reactive chains, sometimes called macromonomers, can add to a growing radical to form a long-chain branch. pol

kp

¼ Pn; b þ Dm; c ! Pnþm; bþcþ1 ;

pol ¼ R pol p ¼ k p ½Ptot ½Dtot

ð30Þ

175

4 Free-radical Polymerization: Homogeneous

176

R1 CH2

+

C

R1

R1 H2C

CH2 C

C

R1 CH2

C

R2

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

Scheme 4.10.

This reaction is fundamentally diﬀerent than transfer to polymer in several respects. It is an addition rather than a transfer (H-abstraction) reaction (Scheme 4.10). The reaction rate is dependent on the number of unsaturated chain ends rather than the number of repeat units in the chains so that its importance relative ¼ =½M according to Eq. (31). to propagation is controlled by the ratio ½Dtot pol

pol

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

ð31Þ

Unlike transfer to polymer, the mechanism combines two polymer chains (and all of their repeat units) into one, aﬀecting both DPn and DPw . Reaction with terminally unsaturated chains can be important in vinyl acetate [33] and highertemperature methacrylate [3] polymerizations. Reaction with multifunctional monomers Another way to introduce branching is through addition of a multifunctional monomer to the polymer system. An example is the addition of small levels of ethylene glycol dimethacrylate (EGDMA) to methyl methacrylate (MMA), as shown in Scheme 4.11. Reaction of the ﬁrst dou-

O C

OCH3

C

+

C

CH2

O

O CH2CH2 O

C O

C CH2

H2C C

C

CH3

H3C

H3C

CH3 C CH2

O

CH2

O CH2CH2 O

C

O

C H3C CH3

CH3 C

O C CH2

C H3C

OCH3

+

O C H2C C

C CH2

C

C O O CH2 CH2 O

O O CH3 C C

CH2

H3C

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

Scheme 4.11.

CH2

O O H2C CH2 O O C C CH3

CH2

4.3 Polymerization Mechanisms and Kinetics

177

ble bond incorporates EGDMA in the PMMA chain, with the second reactive site remaining as a pendent double bond. Reaction of this pendent bond with another polymer radical creates a branched structure termed a crosslink, since the branch is tetrafunctional rather than trifunctional in nature. The addition of a small amount of difunctional monomer such as EGDMA to MMA [39, 40] or divinylbenzene to styrene is a means to increase polymer MW without decreasing radical concentration. Addition of higher levels leads to an interconnected branched, or network, polymer. Short-chain branching (intramolecular transfer to polymer) Intramolecular H-atom abstraction, often called back-biting, occurs via the formation of a six-membered ring, as shown in Scheme 4.12 for butyl acrylate (nBA). Monomer addition to the resulting interior radical leaves a short-chain branch (SCB) consisting of two repeat units. This mechanism has long been known important for high-pressure LDPE production at 150–300 C [30], where the number of short-chain branches is in the range of 20–50 per 1000 ethylene repeat units. This level of SCB signiﬁcantly decreases the polyethylene crystallinity and gives LDPE some of its unique properties; LDPE density is 0.92 g cm3 compared to 0.98 g cm3 for linear polyethylene. Multiple back-biting events lead to other SCB structures; in addition to the common butyl branch in LDPE, ethyl and 2-ethylhexyl branches have been identiﬁed using 13 C NMR [30, 41, 42].

COOBu COOBu COOBu R CH

CH

CH

C

COOBu COOBu COOBu

COOBu R

CH H HC COOBu

CH

COOBu CH

C H CH

COOBu

Formation of a mid-chain radical by intramolecular chain transfer to polymer. Monomer addition to the new radical structure creates a short-chain branch in the polymer.

Scheme 4.12.

In ethylene/nBA copolymerization it is observed that the methine hydrogens on nBA units in the polymer chain are much more susceptible to abstraction than hydrogen abstraction from a CH2 unit in the backbone [43]. This result, along with the observed high concentration of branch points formed at conditions of very low polymer concentration during nBA homopolymerization [36], indicates that the intramolecular (back-biting) mechanism is also important in acrylate polymerizations. There is strong evidence to suggest that monomer addition to the resulting mid-chain radical, due to its higher stability, is much lower than addition to a chain-end radical [44]. Thus the mechanism aﬀects nBA polymerization rate as well as polymer structure [37]. The same type of back-biting reaction has also

178

4 Free-radical Polymerization: Homogeneous

been shown to occur during styrene polymerization at high temperatures (260– 340 C) [26]. Chain scission The radical structure formed by intra- or intermolecular transfer to polymer is less reactive than a chain-end radical. Under higher-temperature conditions, the radical can undergo b-fragmentation (chain scission) as shown in Scheme 4.13 for butyl acrylate [45]. The scission can occur in either direction, yielding a short-chain radical or a short-chain unsaturated trimer species. As well as lowering polymer MW, the scission event produces an unsaturated chain end that can react further, as discussed previously (see Scheme 4.10).

COOBu COOBu COOBu CH2 C CH CH

COOBu COOBu COOBu CH

CH

COOBu

+

HC

COOBu CH2

COOBu CH

C H

COOBu COOBu

CH COOBu

CH

CH

+

COOBu COOBu H2C

C

HC

COOBu CH2

b-Scission of butyl acrylate mid-chain radical.

Scheme 4.13.

Scission events can occur in any system where mid-chain radicals are formed. However, scission is more temperature-activated than H-abstraction and thus becomes important only at elevated temperatures. The reaction is not believed to occur during butyl acrylate polymerization at 75 C [37], but is shown to be important at 140 C [29, 45]. Scission is a dominant mechanism in styrene polymerizations at 260–340 C [26], and also occurs during LDPE production [30]. Scission of midchain radicals formed via intermolecular transfer to polymer can have a signiﬁcant eﬀect on the breadth and the shape of polymer MWD [46]. Kinetic treatment of these more complex mechanisms is often diﬃcult. Equations (32)–(34) are a network of reactions developed to treat intramolecular transfer, short-chain branch formation, and b-scission for butyl acrylate polymerization [47]. kbb

Pn ! Q n ;

R bb ¼ kbb ½Ptot

kptert

Q n þ M ! Pnþ1 þ SCB;

ð32Þ RSCB ¼

kptert ½M½Q tot

ð33Þ

0:5kb

¼ Q n ! Dn2 þ P2 0:5kb

Q n ! Pn3 þ D3¼ ;

R b ¼ kb ½Q tot

ð34Þ

The kinetic coeﬃcients are estimated by measuring the level of quaternary carbons (equated to short-chain branch level) and terminal unsaturations (D ¼ ) by NMR.

4.3 Polymerization Mechanisms and Kinetics

Termination of the mid-chain radical (Q n ) is also considered in the scheme. While the mechanism provides improved understanding of this complex system, many questions still remain: does the mid-chain radical terminate at the same rate as chain-end radicals; what is the reactivity of the unsaturated chain end; and does the mid-chain radical scission with equal probability in each direction? The reaction engineering challenge is to consider the set of mechanisms required to represent the basic rate behavior and polymer architecture of the system without introducing unneeded complexity. 4.3.2

Copolymerization

Reaction of two or more monomers by free radical polymerization is an eﬀective way of altering the balance of properties of commercial products. Addition of the polar monomer acrylonitrile to styrene (or methyl acrylate to ethylene) produces a polymer that combines the strength of the base homopolymer with much improved oil and grease resistance. The adhesive and cohesive properties of coatings resins are balanced by controlling the mix and relative proportions of monomers in the recipe. FRP leads to the formation of statistical copolymers, where the arrangement of monomers within the chains is dictated purely by kinetic factors. However, reactivity of a monomer in copolymerization cannot be predicted from its behavior in homopolymerization. Vinyl acetate polymerizes about 30 times more quickly than styrene (see Table 4.2), yet the product is almost pure polystyrene if the two monomers are copolymerized together in a 50:50 mixture. a-Methylstyrene cannot be homopolymerized to form high-MW polymer due to its low ceiling temperature (see Table 4.6), yet is readily incorporated into copolymer at elevated temperatures. These and other similar observations can be understood by considering copolymerization mechanisms and kinetics. Basic Mechanisms In copolymerization, the presence of more than one type of monomer adds an extra degree of complexity to the kinetics. The diﬀerent monomers form diﬀerent radical structures, and the relative rates of chain growth depend on the structure of both monomer and radical. It is these propagation mechanisms that control polymer composition (the relative amounts of each monomer unit incorporated into the copolymer) and sequence distribution (the way in which these monomer units are arranged within the chain). Developing a set of mechanisms to describe how radical structure aﬀects termination and transfer rates is required to represent copolymer chain length and molecular weight distributions. The most common treatment of free radical copolymerization kinetics assumes that the reactivity of the polymer radical depends solely on the nature of its terminal monomer unit; that is, that the identity of the penultimate unit on the radical does not aﬀect its reactivity. This assumption provides a good representation of polymer composition and sequence distribution, but not necessarily polymeriza4.3.2.1

179

180

4 Free-radical Polymerization: Homogeneous

tion rate (see Section 4.3.2.2). This so-called terminal model is widely used to model free-radical copolymerization according to the set of mechanisms in Scheme 4.14. k

Initiator Decomposition

d I  →2 f I ∗

Chain Initiation

j I ∗ + M j  → P1 j

Chain Propagation

ij Pni + M j → Pn +j 1

ki

kp

Chain Termination ktc

By Combination

ij Pni + Pmj → Dn + m

By Disproportionation

ij Pni + Pmj → Dn + Dm

ktd

Chain Transfer To Monomer

ktrmon

ij Pni + M j  → Dn + P1 j

ktrsol

To Solvent or Agent

i Pni + S → Dn + S *

k isol

j S * + M j → P1 j

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

Scheme 4.14.

In this scheme monomer-j (denoted by Mj ) adds to the initiator primary radical to form a polymer radical of type-j and unit length. The dead polymer and radical-i chains of length n (Dn and Pni ) are made up of a mixture of the monomer types in the system, with their relative amounts governed by the copolymer composition equation developed below. Chain growth occurs by addition of Mj to radical-i (Pni ) with the propagation rate coeﬃcient k pij dependent on both radical and monomer type. The rate coeﬃcients for transfer and termination reactions can also be dependent on the nature of the radical center, as indicated by subscripts. However, since radical–radical termination is a diﬀusion-controlled reaction, the rate coeﬃcient cop can usually be assumed to be independent of radical type, such that kt ¼ k tij for all i and j. Most of the kinetic coeﬃcients in Scheme 4.14 are binary parameters, dependent on the radical and monomer type. Thus the polymerization behavior of three or more monomers can be estimated reliably from knowledge of the corresponding binary copolymerizations. For a two-monomer system assuming the long-chain hypothesis, the consumption rates of the two monomers are written as in Eqs. (35). 1 2 ½M1 þ k p21 ½Ptot ½M1 R pol1 ¼ k p11 ½Ptot 1 2 R pol2 ¼ k p12 ½Ptot ½M2 þ k p22 ½Ptot ½M2

ð35Þ

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

i ½Ptot ¼

y X ½Pni

ð36Þ

n¼1

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

¼

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

ð37Þ

Application of the quasi-steady-state assumption yields the ratio of the radical types as in Eq. (38), and substitution and rearrangement leads to the well-known polymer composition equation [Eq. (39)], where f1 and f2 are the mole fractions of M1 and M2 in the monomer mixture, and monomer reactivity ratios r1 and r2 are deﬁned as k p11 =k p12 and k p22 =k p21 . 1 ½Ptot k p21 ½M1 2 ¼ k ½M ½Ptot p12 2

¼ Fpinst 1

ð38Þ

r1 f12 þ f1 f2 þ 2 f1 f2 þ r2 f22

ð39Þ

r1 f12

Equation (39) deﬁnes the composition of the copolymer formed at any instant during polymerization, and is dependent only on the ratios of the propagation rate coeﬃcients and not on their absolute values. Copolymer properties depend on the distribution of the monomer units along the chain as well as the average composition. Reactivity ratios also control copolymer sequence distribution. The probability P11 that an M1 unit follows an M1 unit in the copolymer is equal to the rate of M1 M1 formation divided by the sum of the rates of all additions to radical-1 [Eq. (40)].

P11 ¼

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

¼

r1 f1 r1 f1 þ f2

ð40Þ

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

f2 r1 f1 þ f2

ð41Þ

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

181

182

4 Free-radical Polymerization: Homogeneous

P22 ¼

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

P21 ¼ 1 P22 ¼

¼

r2 f2 r2 f2 þ f1

f1 r2 f2 þ f1

ð42Þ

ð43Þ

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

n 1

NðM2 ; nj Þ ¼ P22j P21

ð44Þ

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

1 1 ¼ ; 1 P11 P12

N2 ¼

1 1 ¼ 1 P22 P21

ð45Þ

Implicit in these expressions is the approximation, valid for long-chain polymer, that the number of M1 M2 linkages in a chain is equal to the number of M2 M1 linkages. Thus the ratio of M1 to M2 units in the chain must equal the ratio of the respective average sequence lengths [Eq. (46)]. Fpinst 1 Fpinst 2

¼

N1 N2

ð46Þ

Substitution and rearrangement of this equation yields the polymer composition equation, Eq. (39). Thus it is possible to estimate reactivity ratios for binary copolymers from triad distributions measured by NMR analysis. While copolymer composition and sequence distribution are only functions of the reactivity ratios, the same is not true for polymerization rate. The overall rate of monomer consumption is given by Eq. (47), where [Mtot ] indicates the total monomer concentration (½M1 þ ½M2 ). 1 2 1 2 R pol ¼ k p11 ½Ptot ½M1 þ k p21 ½Ptot ½M1 þ k p12 ½Ptot ½M2 þ k p22 ½Ptot ½M2 ! 2 2 XX ¼ k pij fri fj ½Ptot ½Mtot

ð47Þ

i¼1 j¼1

The fraction of radical-i in the system, fri [Eq. (48)] can be calculated from Eq. (38).

4.3 Polymerization Mechanisms and Kinetics

fri ¼

i Ptot Pi ¼ tot 2 þ Ptot Ptot

1 Ptot

ð48Þ

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

½Ptot ¼

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

ð49Þ

The form of Eq. (47) is analogous to the homopolymerization rate expression [Eq. (10)], with a copolymer-averaged rate coeﬃcient for propagation (generalized for a system with Nmon diﬀerent monomers) deﬁned in Eq. (50). k pcop ¼

Nmon X Nmon X

k pij fri fj

ð50Þ

i¼1 j¼1

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

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

ð51Þ

Using Eqs. (47)–(51), the copolymerization reaction rate can be analyzed as for cop homopolymerization (see Section 4.3.1.4), with k p now a function of monomer composition. Kinetic Coeﬃcients The traditional method for determining reactivity ratios involves determination of copolymer composition for a range of monomer feeds at very low conversion; that is, measuring Fp1 as a function of f1 . NMR measurement of sequence distributions provides additional information about chain microstructure, but suﬀers from greater experimental noise and signal assignment uncertainty from tacticity effects. There is a large body of published r1 –r2 data for monomer pairings, summarized in Ref. 7. The scatter for these ratios is much less than found in k p and k t data, but care still must be exercised when extracting values from this and similar compilations. Some error can arise from the methodology used to estimate r1 and r2 from Fp1 versus f1 data. The estimation is best accomplished by nonlinear least squares techniques [48, 49], and a statistical analysis also provides a guide to the optimal monomer compositions at which experimentation should be performed to improve the estimates [48]. Error can also occur if polymer conversion is suﬃciently high for f1 to deviate signiﬁcantly from the zero-conversion value (in which case an integrated form of Eq. (39) must be used [50]), or if the experimental system does not remain homogeneous (often observed with acid monomers). 4.3.2.2

183

184

4 Free-radical Polymerization: Homogeneous Tab. 4.7.

Monomer reactivity ratios at 50 C[a], r1 is tabulated horizontally and r2 vertically.

Monomer-1

Monomer-2

Styrene Alkyl methacrylate Alkyl acrylate Vinyl acetate [a] Representative

Styrene

Methacrylate

Acrylate

Vinyl acetate

– 0.4 0.2 0.02

0.6 – 0.4 0.03

0.8 2.2 – 0.03

40 20 6 –

values Ref. 7.

There are only a few systems for which the terminal model upon which Eq. (39) is based does not provide a good description of copolymer composition. The resulting polymers exhibit an alternating structure (for example, styrene–maleic anhydride), or the observed reactivity ratios in the system vary with monomer concentration or solvent choice. Since they often include a polar monomer with strong electron-withdrawing or electron-donating properties, alternative kinetic models that include the formation of monomer complexes have been developed to represent these systems [3, 51]. The majority of systems, however, are well behaved and well represented by the terminal model. Table 4.7 summarizes values of reactivity ratios for styrene, alkyl methacrylate, alkyl acrylate, and vinyl acetate systems. Reactivity ratios for alkyl methacrylates and acrylates (for example, methyl, butyl, dodecyl) exhibit a family type behavior, with the composition data of various systems ﬁtted well by a single curve [52]. Values for ethylene copolymer systems at typical production conditions are summarized in Table 4.8: While r1 (1 ¼ ethylene) values for alkyl methacrylates, alkyl acrylates, acrylic acid, and methacrylic acid could not be distinguished within experimental error [53], there were signiﬁcant diﬀerences in the r2 values [54]. It is informative to consider some of the implications of these values. In Figure 4.4 the relationship between polymer and monomer composition is plotted for various copolymer systems:

Tab. 4.8. Monomer reactivity ratios for ethylene (monomer-1) systems at 240 C and 2000 bar [53, 54].

Monomer-2

r1

r2

Alkyl methacrylates Methacrylic acid Alkyl acrylates Acrylic acid Vinyl acetate [a]

0.058 0.058 0.058 0.058 1.0

7 11 4 8 1.0

[a] From

Ref. 7.

4.3 Polymerization Mechanisms and Kinetics 1 0.9 0.8 0.7

Fp 1

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

0.2 0.1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f1 Fig. 4.4. Relationship between monomer composition f1 and corresponding instantaneous polymer composition Fpinst for 1 various monomer pairings (MMA ¼ methyl methacrylate, Sty ¼ styrene, MA ¼ methyl acrylate, VAc ¼ vinyl acetate, Eth ¼ ethylene).

For the case where r1 ¼ r2 ¼ 1:0 (ethylene–vinyl acetate; methacrylate– methacrylate), the monomers have equal reactivity to propagation and will be incorporated into polymer at the same ratio as they are in the monomer phase (Fp1 ¼ f1 ). Where r1 > 1 and r2 < 1, the copolymer will always be richer in monomer-1 than in the monomer phase, so that monomer-1 will become depleted in a batch polymerization. The further the reactivity ratios deviate from unity, the greater the deviation between polymer and monomer composition. Systems that exhibit this behavior include methacrylate–acrylate polymerizations, and styrene, methacrylates, or acrylates polymerized with vinyl acetate or ethylene. With both r1 and r2 less than unity (styrene–acrylate, styrene–methacrylate), cross-propagation is favored over homopropagation and the copolymer tends toward an alternating structure. The system has an azeotropic composition at which the copolymer composition is exactly equal to the monomer composition.

Reactivity ratios exhibit a weak temperature dependence that is often diﬃcult to measure. With increasing temperature, the ratios tend to approach unity as demonstrated for styrene–butyl acrylate [55], butyl acrylate–methyl methacrylate [56], and ethylene copolymer systems [53, 54]. The temperature dependence of the latter agrees well with activation energies reported for addition of monomers to small radicals [57].

185

4 Free-radical Polymerization: Homogeneous

1000.0

cop

(L/mol-s)

10000.0

kp

186

100.0

10.0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

f 1 (MMA) Fig. 4.5. Relationship between monomer composition f1 and corresponding copolymer-averaged propagation rate coeﬃcient, cop k p . Lines are calculated assuming terminal-model kinetics and points are experimental data for: MMA–nBA (—, e) [62] and MMA–styrene (---, C) [59] at 20 C.

While the terminal model eﬀectively represents copolymer composition for most systems, it does not provide as good of a description of polymerization propagation behavior. Rate abnormalities have previously been attributed to termination reactions and represented by introducing physically unrealistic cross-termination mechanisms [58]. Careful experimental work by Fukuda and co-workers [51], however, demonstrated that the rate deviations are due to the inadequacy of the terminal model to describe propagation, with further evidence obtained by applicacop tion of the PLP-SEC technique to measure k p . Many common systems show deviation from terminal propagation kinetics, including styrene–methacrylate [51, 59], styrene–acrylate [60, 61], and acrylate–methacrylate [52, 62] systems. As cop shown in Figure 4.5, the measured k p values can be higher or lower than the terminal model predictions, with the deviation substantial in some cases. The ‘‘implicit penultimate unit eﬀect’’ model, which accounts for the inﬂuence of the penultimate monomer unit of the growing polymer radical on the propagation kinetics [51], provides a good representation of this behavior [Eqs. (52)]. k pcop ¼ k p11

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

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

k p22

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

ð52Þ

4.3 Polymerization Mechanisms and Kinetics

The extra parameters s1 and s2 , called radical reactivity ratios, capture the eﬀect of the penultimate unit on the addition rate of monomer [Eqs. (53)]. k p211 k p111

s1 ¼

s2 ¼

k p122 k p222

ð53Þ

A value of greater than unity for si indicates that a comonomer unit-j in the penultimate position increases the addition rate of monomer-i to radical-i compared to the homopolymerization case. The kinetics of diﬀusion-controlled termination in copolymerization is also diﬃcult to study. The original interpretation of low-conversion rate data was based on a chemically controlled model utilizing a cross-termination factor [Eq. (54)]. cop

kt

¼

Nmon X Nmon X i¼1 j¼1

k tij fri frj ;

F¼

k t 12 k t 11 k t22

ð54Þ

Assuming terminal propagation kinetics, the best ﬁt for F was found to be much greater than unity for styrene–methacrylate, styrene–acrylate, and other systems cop such that kt was greater than either homotermination value. When the deviation of propagation kinetics from the terminal model is taken into account, however, cop the estimates for kt become well behaved and bounded by the homotermination values [51]. A penultimate model [51, 63] that accounts for the inﬂuence of polymer composition on segmental diﬀusion is required to ﬁt recent acrylate– cop methacrylate low-conversion kt data [63, 64]. In order to use this representation (Eq. (55), with the cross-termination coeﬃcients k t 12; 12 and k t21; 21 ﬁtted to experimental data), it is necessary to track the four possible penultimate radical types in the system, with frij ði; j ¼ 1; 2Þ the penultimate radical fraction. cop

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

ð55Þ

While the terminal model and reactivity ratios provide a good description of copolymer composition, the kinetic studies summarized in this section indicate that adcop ditional parameters and penultimate radical fractions are required to represent k p cop and kt . (For a discussion of transfer to monomer in copolymer systems, see Ref. 65.) These mechanistic complexities are often not considered when developing FRP models for polymer reaction engineering applications. It is expected that this situation will change as more data become available. Additional Mechanisms The secondary mechanisms presented for homopolymerization in Section 4.3.1.3 also occur in multi-monomer systems. The kinetics of depropagation, chain transfer to polymer, and chain scission are strongly inﬂuenced by not only the nature of the monomer and terminal radical, but also the penultimate unit on the polymer radical. 4.3.2.3

187

4 Free-radical Polymerization: Homogeneous kp

11 → ~~~ M 1M 1 . ~~~ M 1 ⋅ + M 1 

 →

188

kdep 11 kp12

~~~ M 1 ⋅ + M 2 → ~~~ M 1M 2 . kp

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

kp

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

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

Depropagation It is not possible to produce high MW homopolymer close to the ceiling temperature of the monomer (see Eq. (25) and Table 4.6). Addition of a non-depropagating monomer (M2 ) to the system disrupts this behavior, as illustrated by Scheme 4.15. Depropagation of radical-1 becomes a competitive process with addition of monomer-2, with monomer-1 units at the radical end becoming irreversibly trapped in the growing chain as soon as monomer-2 adds. Furthermore, depropagation of monomer-1 will occur only if an M1 unit is also in the penultimate position; depropagation to the less-stable M2 radical can be assumed to be unlikely (kdep21 A 0). Thus MMA can be successfully copolymerized with ethylene at 290 C [57], and a-methylstyrene with various comonomers at temperatures well above its ceiling temperature [66, 67]. Depropagation in copolymerization aﬀects polymer composition, sequence probabilities, and polymerization rate. With Scheme 4.15, the relative rates of monomer consumption are given by Eq. (56) (compare with Eq. (37) in the absence of depropagation).

Fpinst 1 Fpinst 2

R pol1 ¼ ¼ R pol2

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

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

ð56Þ

The ratio fr11 =ð fr11 þ fr21 Þ accounts for the fraction of radical-1 that ends in a 11 diad. The eﬀect of depropagation becomes larger as total monomer concentration decreases and as the fraction of the depropagating monomer in the system increases. Lowry [68] ﬁrst derived the composition expressions for the situation where only one monomer depropagates, and general expressions have been developed for the situation where all four of the propagation reactions are reversible [66, 69]. Depropagation must be considered when examining the kinetics of starved-feed semi-batch copolymerization involving methacrylates at the highertemperature conditions typically used to produce acrylic coatings [29, 70]. Chain transfer to polymer The rate of chain transfer to polymer is dependent on both the reactivity of the radical and the abstractability of the hydrogen atom on the monomer unit in the polymer chain. For the case of intermolecular chain j transfer (long-chain branching), this is represented by Eq. (57), where m1 represents the total concentration of polymerized monomer-j units in the system and

4.3 Polymerization Mechanisms and Kinetics

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

mj k tr

j

ij

pol

Pn;i b þ Dm; c ! Dn; b þ Pm; cþ1 ;

j

m1 ¼

y X y X

nj ½Dn; b ;

n¼

n¼1 b¼0

Nmon X

R tr ¼

Nmon X Nmon X

j

pol

i k trij ½Ptot m1

ð57Þ

i¼1 j¼1

ð58Þ

nj

j¼1

Active radicals (ethylene, acrylate, vinyl acetate) are more likely to abstract from a polymer chain than styrenic or methacrylate radicals, and acrylate and vinyl acetate monomer units on a chain are more likely to have an H-atom abstracted. Thus it is not uncommon for the intermolecular transfer to polymer rate of one pairing (for example, acrylate radical to acrylate monomer unit) to dominate the system, with pol the overall transfer rate R tr decreasing rapidly with increasing content of the lessreactive monomer. The situation is more complicated in the case of intramolecular transfer, which occurs through the formation of a six-membered ring. In the case of acrylate (1)/ methacrylate (2), it can be assumed that the methacrylate radical is not reactive enough to back-bite and that the acrylate radical can only abstract hydrogen if the antepenultimate unit on the chain is also an acrylate unit. Thus back-biting can occur only for two monomer sequences (M1 M1 M1 and M1 M2 M1 ) at the radical end, as shown in Scheme 4.16 [70]. The overall back-biting rate must be corrected for the sequence probabilities [Eqs. (40)–(43)] at the chain end, according to Eq. (59). COOBu COOBu C

C

R

H

COOBu

COOBu COOBu

CH

CH

C

C R

HC

H CH

COOBu

COOBu COOBu C

C

R

H

H3C

COOBu C

COOBu

COOBu COOBu C R

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

Scheme 4.16.

COOBu

C

H3C H

COOBu C

CH COOBu

189

190

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

ð59Þ

Back-biting mechanisms have also been examined for styrene/acrylate [71] and ethylene copolymer systems [43]. Chain scission There is evidence that the rate of b-fragmentation of a mid-chain radical is aﬀected by the nature of the units adjacent to the mid-chain radical. Harada et al. [72] studied the copolymerization of cyclohexyl acrylate and methyl methacrylate at high temperatures. As in Scheme 4.16, H-abstraction only happens to the acrylate unit. If the adjacent units are acrylate and methacrylate, two types of fragmentation with diﬀerent reaction rates are possible, as illustrated in Scheme 4.17. It was observed that the number of the unsaturated end groups per chain is increased by increasing the methacrylate content in the polymerization. Thus it was deduced that fragmentation greatly favors the generation of tertiary (methacrylate chain end) radical species, a result in agreement with the high rate of fragmentation observed in methacrylate macromonomer systems [73].

CH3

CH3

C CH2

HC

C CH2

fast

CH2

C

+

CH2

H2C

CO2CH3

CO2CH3

CO2CH3 CO2CH3 CO2CH3

CH2 C

HC CO2CH3

CH3

slow

C CH2

C CH2

CO2CH3

CH2 CO2CH3

+

HC CO2CH3

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

Scheme 4.17.

Fragmentation after intramolecular transfer results in the formation of a longchain radical and trimer species or a dimer radical and an unsaturated dead chain (Scheme 4.13). Consideration of all possible pathways and structures becomes complex, but the resulting model requires no additional parameters from the homopolymerization back-biting/scission case and provides a good representation of high temperature acrylate–methacrylate copolymerization [70]. 4.3.3

Diﬀusion-controlled Reactions

The kinetic schemes in this chapter have been written assuming that k t is independent of the sizes of the radicals involved in the termination reaction. This is not

4.3 Polymerization Mechanisms and Kinetics 1.0E+09

1.0E+08

kt (L/mol-s)

1.0E+07

1.0E+06

1.0E+05

MMA nBA DA

1.0E+04

1.0E+03

1.0E+02 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Conversion

Fig. 4.6. Typical variation of k t with conversion for methyl methacrylate (MMA), butyl acrylate (nBA), and dodecyl acrylate (DA). Based upon data from [15].

strictly true, since the termination rate is limited by the rates at which the radical ends can encounter each other. For a diﬀusion-controlled reaction, the apparent rate coeﬃcient is aﬀected not only by pressure and temperature, but also by system viscosity (a function of solvent, polymer concentration, and MW) and the lengths of the two terminating radicals. This complex behavior, as well as experimental difﬁculties in measuring k t , has led to a large scatter in reported values, even at low conversion [7]. Through the application of pulsed-laser experimental techniques [15] and a critical examination of available data [18], the situation is starting to improve. For most commercial free radical polymerization, the errors involved by neglecting the dependence of k t on radical chain length are not large. The change in k t with conversion (increasing viscosity), however, cannot be neglected. Figure 4.6 shows the three to four orders of magnitude decrease in k t typically observed. The shape of the curve reﬂects the changing nature of the rate-controlling diﬀusion mechanism. The usual division is as follows:

Low conversion: the system viscosity is still low, and the two chains diﬀuse together quickly. The rate of reaction is controlled by segmental diﬀusion, the internal reorganization of the chain that is required to bring the reactive ends together. In this region k t is of the order of 10 8 L mol1 s1 for many common monomers (Table 4.3), with the value remaining relatively constant up to 10– 20% conversion. Solvent choice can have a signiﬁcant eﬀect on the value [18].

191

192

4 Free-radical Polymerization: Homogeneous

Lower values of 10 6 L mol1 s1 for dodecyl (meth)acrylate is attributed to shielding of the radicals by the long-chain dodecyl ester groups; for these monomers k t remains relatively constant up to 60% conversion [64]. Even lower k t values are measured for termination during polymerization of sterically hindered monomers such as the itaconates [74] and acrylate trimer species [44]. The variation of low-conversion k t with polymer composition in copolymer systems can be represented by Eq. (55). Medium to high conversion: the large increase in system viscosity with polymer formation leads to a change in the controlling mechanism. The rate of reaction is controlled by how quickly the two chains ﬁnd each other among the tangle of dead polymer chains in the system. This so-called center-of-mass or translational diﬀusion mechanism is complex, aﬀected by the lengths of the reacting chains as well as the system viscosity. The value of k t can drop by several orders of magnitude in this regime. Very high conversion: at high conversion, the system may become so viscous that the polymer radicals move more quickly through propagation (addition of new monomer units) than by translation. This phenomenon, called reaction diﬀusion, leads to a second plateau region in the k t versus conversion plot, with k t proportional to k p ð1 x p Þ. If the glass transition temperature of the reaction mixture exceeds the reaction temperature, the propagation reaction and apparent initiator eﬃciency [75] may also become diﬀusion-controlled. The overall behavior of k t with conversion is often modeled as a composite of the various diﬀusional processes [Eq. (60)]. 1 1 1 ¼ þ k t k t; SD k t; TD þ k t; RD

ð60Þ

The subscripts SD; TD, and RD refer to segmental diﬀusion, translational diﬀusion, and reaction diﬀusion. k t; SD can be set to the low conversion values summarized in Table 4.3, and k t; RD is set proportional to propagation, with proportionality coeﬃcient CRD ﬁtted to experimental data [Eq. (61)]. k t; RD ¼ CRD k p ð1 x p Þ

ð61Þ

Many semi-empirical approaches have been used to model k t; TD as a function of system viscosity, conversion, or free volume. The latter treatment relates the chain diﬀusivity to the system free volume vf by Eq. (62), with parameters Ci ﬁtted to experimental data. k t; TD z C1 expðC2 vf Þ

ð62Þ

This eﬀect of polymer MW on system viscosity may be captured by expressing C1 as a function of Mw . It is observed experimentally that addition of a powerful chain-transfer agent to MMA lowers polymer MW and system viscosity, thereby in-

4.4 Polymer Reaction Engineering 1 0.9 0.8

Conversion

0.7 0.6 0.5 0.4 0.3 0.2

MMA nBA

0.1 0 0

500

1000

1500

2000

time (s) Fig. 4.7. Typical time–conversion plots for methyl methacrylate and butyl acrylate batch polymerizations. The sharp increase in rate seen for MMA, known as the gel eﬀect, is due to the large decrease in k t with conversion.

creasing k t , while addition of a small amount of EGDMA crosslinking agent has the opposite eﬀect [39]. Details and variations of this modeling approach can be found in the literature [40, 76–79]. A good model for k t is necessary to capture the time–conversion behavior in homogeneous batch FRP systems. The large decrease in k t at intermediate conversion results in an increase in radical concentration [Eq. (9)] and a corresponding increase in R pol [Eq. (10)] that causes an upward curvature in the time–conversion plot (Figure 4.7). This accelerated rate is accompanied by a large heat release that can be diﬃcult to remove from the viscous reaction system. The severity of the gel eﬀect is directly related to the magnitude of the decrease in k t , as seen by comparing the nBA and MMA rate proﬁles in the ﬁgure. The decrease also leads to an increase in DPninst [Eq. (13)] for systems where MW is controlled by termination.

4.4

Polymer Reaction Engineering

The design of an industrial polymerization process begins with a clear understanding of objectives and an appreciation of constraints. Design and operation requirements are very diﬀerent for a process manufacturing several grades of a high-volume commodity homopolymer, and one that produces dozens of diﬀer-

193

194

4 Free-radical Polymerization: Homogeneous

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

4.4 Polymer Reaction Engineering

cooling system is able to maintain control, albeit at a higher than desired temperature. However, large variations in the temperature proﬁle can translate into diﬀerences in the ﬁnal product properties, particularly molecular weight distribution. The monograph by Biesenberger and Sebastian [80] provides an excellent discussion of thermal eﬀects, including thermal runaway. Mixing Mixing can directly aﬀect the kinetics, molecular weight, and composition in radical polymerizations by homogenizing local concentration gradients, but it can also indirectly play an important role through its role in reducing thermal gradients in a reactor. In small-scale experiments, most transport phenomena (heat transfer, mixing) occur suﬃciently fast for overall behavior to be dictated primarily by reaction kinetics. However, as scale increases, kinetic and transport eﬀects become increasingly coupled. Homogeneous FRP reactions oﬀer a particularly challenging problem because of the large increase in viscosity accompanying the conversion from bulk monomer (@1 cp for liquids) to polymer (> 10 5 cp). The increase in viscosity can greatly aﬀect the reaction kinetics (see Section 4.3.3) as well as the heat removal and quality of mixing in the system. Some processes are designed to not require mixing. For example, PMMA can be polymerized in large sheets. By having large surface areas available for heat transfer, adequate temperature control is achieved without the need to provide mixing during polymerization. Other bulk polymerizations (for example, styrene) employ more than one reactor in series, since diﬀerent reactor conﬁgurations and agitators will be required as the viscosity increases. Solution polymerizations oﬀer low viscosity but the trend in industry is to eliminate (or greatly reduce) the use of organic solvents. Mixing can also be an issue at lower viscosities such as those found in LDPE systems, where the high-temperature conditions make for very fast reactions (for example, initiator half-life of <1 s) on the order of characteristic mixing times. 4.4.1

Types of Industrial Reactors

There are three major classiﬁcations of chemical processes, categorized by the method by which the reactants are added to the reaction vessel. Varying the contacting pattern can dramatically alter the local reaction conditions (for example, concentrations of individual species, including monomers, initiators, chaintransfer agents, and so on), and is therefore a potentially powerful design tool for controlling properties such as molecular weight distribution, copolymer composition distribution, and degree of branching. Because of the ability to manipulate local monomer concentrations, the rate of polymerization can also be controlled, thereby providing safer and more robust operation. Batch Processes All of the reactants are added to the reactor prior to starting the polymerization. No material is added to or removed from the reactor during operation. When the polymerization is complete, the contents are discharged and the reactor prepared for 4.4.1.1

195

196

4 Free-radical Polymerization: Homogeneous

the next batch. Batch polymerizations are the simplest to run, but oﬀer the least control over the polymerization. For polymerizations with more than one monomer, the relative consumption rates of the diﬀerent monomers will be governed by their respective reactivities, possibly resulting in broad copolymer composition distributions and inhomogeneous product. Another feature of batch polymerizations is that reactant concentrations change throughout the polymerization. Molecular weight distribution drift is therefore a common phenomenon and can lead to very broad distributions in the ﬁnal product. From an economic perspective, batch polymerizations suﬀer from downtime between batches, although much progress has been made in automating many of the reactant weighing, charging, and discharging steps to minimize this interval. Automation has also improved reproducibility of batch reactions. For operations where changes to the formulation or the polymerization conditions are common, batch processes have the advantage of being ﬂexible and readily adaptable to new products. Semi-batch Processes These processes (also called semi-continuous) are similar to batch processes, except that reactants can be added and/or products removed during the polymerization. Usually only a portion of the total reactant charge is initially fed into the reactor. The polymerization is then started, and reactants are added during reaction in order to control a desired property (for example molecular weight distribution, copolymer composition distribution) or the reaction rate. Any reactant can be fed, and it is common practice to add monomer(s), initiators, and/or chain-transfer agents. Two of the most common applications for semi-batch operation in homogeneous FRP are control of copolymer composition distribution and control of reaction rate. In a batch reaction, copolymer composition drifts according to the inherent reactivities of the monomers. However in semi-batch operation, drift can be substantially reduced by maintaining a (near) constant concentration ratio of the respective monomers in the reactor. Production of low molecular weight co- or terpolymers (for example coatings) is also readily done using this type of approach. Initiator and monomers are continuously fed in the desired ratio to provide composition and MW control while maintaining starved conditions (low monomer concentration) in the reactor. This mode of operation also ensures that at any time the monomer concentration in the reactor is low, and therefore the maximum potential hazard in the event of a thermal runaway reaction is minimized. A potential concern with operating in a starved mode is that polymer concentrations are high, resulting in higher rates of transfer to polymer and branching reactions. 4.4.1.2

Continuous Processes Reactants are fed, and products and unconsumed reactants are removed, continuously. The process may take place in a single reactor or in a train (series) of reactors in which the monomer conversion is gradually increased. Most continuous processes are operated at ‘‘steady-state’’ conditions, meaning all reactant concentrations and process conditions (temperature, pressure, and so on) are time-invariant. This can be an enormous advantage for certain types of properties if the reactor is 4.4.1.3

4.4 Polymer Reaction Engineering

also well mixed (no spatial variations); because concentrations are constant, there is no molecular weight distribution drift, and no composition distribution drift. Thus, narrower molecular weight distributions can be produced in a well-mixed continuous reactor for linear polymer systems than in batch systems. Branching reactions, however, broaden the MWD in a well-mixed continuous reactor to a greater extent than in a batch system due to the residence time distribution [80, 81]. For large-volume polymers with a limited number of variations to the polymerization conditions (for example, formulation changes), continuous processes are favored because of their low operating cost, high throughput rates, more uniform product quality, and simplicity of operation. 4.4.2

Mathematical Modeling of FRP Kinetics

Mathematical modeling is a powerful methodology to improve the understanding and operation of polymer processes. A good process model can be used to predict the inﬂuence of operating conditions on reaction rate and polymer properties, to guide (along with appropriate experimentation) the selection and optimization of standard operating conditions for existing and new polymer grades, to guide process development from lab to pilot-plant to full-scale production, to help discriminate between kinetic and physical (for example, heat and mass transfer) eﬀects, to perform design and safety studies, to train plant personnel, and to understand and optimize transitions and other dynamic behavior (that is, process control). The modeling approach and level of detail should be dictated by the application. Whereas an empirical model linking measured inputs and outputs may be the best solution for control of an existing industrial reactor, it would be totally inappropriate for design of a new process or to choose operating conditions for a new polymer grade. On the other hand, it makes little sense to develop a model that can predict detailed polymer architecture for control purposes when the only measure of polymer structure is a melt index value obtained from the lab two hours after the sample was produced. While empirical modeling has its uses, the focus of this section is the development of fundamental models based upon ﬁrst-principles descriptions of chemical and physical phenomena. Although a perfect description of an actual process is, in the end, an unattainable goal, the attempt often leads to valuable insights that can aid process and product development, scale-up, and optimization. Techniques to model the FRP mechanisms and kinetics of Section 4.3 are presented ﬁrst, followed by a discussion of issues related to reactor modeling. Wherever possible, examples have been selected to emphasize industrial application. Although a few historical references are included, the main focus is on advances made since the mid-1990s. The objective of kinetic modeling is to build a description of how polymer architecture and polymerization rate depend on reaction conditions (temperature, pressure) and species concentrations from a deﬁned set of kinetic mechanisms; a dynamic model is required to examine how properties change as a function of time. The mechanisms to be included in a model depend upon its end-use. For simple

197

198

4 Free-radical Polymerization: Homogeneous

mass and energy balances, it is only necessary to consider those that consume monomer, initiator, and radicals – initiation, propagation, and radical–radical termination. To track polymer molecular weight, all mechanisms that include radical transfer must also be included. Additional balances are needed to follow other molecular properties, such as the density of short- or long-chain branches, end-group functionality, and the creation and reaction of terminal double bonds. Method of Moments The equations in Section 4.3, while useful for examining rate of polymerization and instantaneous chain length, are not written in an appropriate form for substituting into a generalized reactor model. In addition, they do not provide a means of tracking higher MW averages that are strongly aﬀected by long-chain branching reactions. One of the challenges in modeling polymerization systems is how to reduce a very large number of individual species (living and dead chains with lengths from 1 to >10 5 , often with other distributed attributes such as the number of branch points) to a tractable solution. The classical approach to this problem is to reduce the system of equations through deﬁnition of the principal moments of the various distributions [82]. Construction of moment balances allows the tracking of average polymer properties: for molecular weight this would be Mn (number-average), Mw (weight-average), and possibly Mz , and for branched systems it is possible to track the number-average (B n ) and weight-average (B w ) number of branches per chain. Only the basics of the mathematical treatment for a homopolymerization system will be given here: more details can be found in recent comprehensive reviews [83, 84]. Consider a free radical system that includes the basic set of mechanisms shown in Scheme 4.3 as well as long-chain branching [Eq. (27)]. The moments for the radical (l j ) and dead (zj ) polymer distributions are deﬁned in Eqs. (63) and (64). 4.4.2.1

lj ¼

y X

n j ½Pn

ð63Þ

n j ½Dn

ð64Þ

n¼1

vj ¼

y X n¼1

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

mj ¼

y X

n j ð½Dn þ ½Pn Þ

ð65Þ

n¼1

With ½Dn g ½Pn there is little diﬀerence in magnitude between mj and zj . Its introduction, however, eliminates the moment closure problem created by the LCB mechanism [40, 81, 85]. Many of the moments have precise physical meanings.

4.4 Polymer Reaction Engineering

The zeroth live moment, l 0 , is the concentration of polymer radicals in the system (denoted by [Ptot ] in Section 4.3), and the ﬁrst live moment, l1 , is the concentration of monomer units contained in all growing radicals. Similarly, m 0 is the concentration of all polymer chains in the system, and m1 is the concentration of monomer units bound in all polymer chains. These moment deﬁnitions collapse the inﬁnite set of equations for polymeric species into a manageable subset used to calculate MW averages, where m w is the molecular weight of the monomeric repeat unit [Eqs. (66)]. Mn ¼ m w

m1 ; m0

Mw ¼ m w

m2 ; m1

Mz ¼ m w

m3 m2

ð66Þ

For this example, equations for the kinetic expressions for m 0 ; m1, and m 2 will be developed for the calculation of Mn and Mw . The ﬁrst step is to formulate balances for live radicals, dead chains, and total chains of length n, accounting for all of the consumption and generation terms from the kinetic mechanisms [Eqs. (67)–(69)]. ( RPn ¼

2f kd ½I þ

ktrmon ½M

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

j¼1

þ k p ½Mð½Pn1 ½Pn Þ ( ktrmon ½M

þ

ktrsol ½S

) y X þ ðk td þ k tc Þ ½Pj ½Pn j¼1

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

( R Dn ¼

ktrmon ½M pol

þ k tr ½Pn ( RPn þDn ¼

þ

ktrsol ½S

y X

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

j½Dj k tr n½Dn

j¼1

2f kd ½I þ

ð67Þ

j¼1

y X

½Pj

ð68Þ

j¼1

ktrmon ½M

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

j¼1

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

ð69Þ

The origin of the various terms in these balances should be evident by looking at the mechanisms of Scheme 4.3 and Eq. (27). The Kronecker delta function [dðxÞ ¼ 1 if x ¼ 0 and sðxÞ ¼ 0 if x 0 0 ] accounts for the generation of new polymeric radicals (P1 ), and the terms for transfer to polymer account for the probabil-

199

200

4 Free-radical Polymerization: Homogeneous

ity that transfer to a certain chain Dn is proportional to chain length n. The expression for termination by combination accounts for the possibility of creating Dn from any combination of two smaller radical fragments whose lengths sum to n. The next step in the procedure is to substitute these species balances into the moment deﬁnitions in Eqs. (63)–(65). The use of generating functions [82, 86, 87] eliminates the tedium (and possible errors) of performing the required series summations, and leads to Eqs. (70)–(78) for the moments. Live moments: Rl 0 ¼ 2 f kd ½I ðk td þ k tc Þl02 Rl1 ¼ 2f kd ½I þ

ktrmon ½Ml 0

þ

ð70Þ ktrsol ½Sl 0

þ k p ½Ml 0 pol

fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl1 þ k tr ðl 0 v2 l1 v1 Þ Rl2 ¼ 2f kd ½I þ

ktrmon ½Ml 0

þ

ktrsol ½Sl 0

ð71Þ

þ k p ½Mðl 0 þ 2l1 Þ pol

fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl2 þ k tr ðl 0 v3 l2 v1 Þ

ð72Þ

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

ð73Þ pol

R v1 ¼ fktrmon ½M þ ktrsol ½S þ ðk td þ k tc Þl 0 gl1 k tr ðl 0 v2 l1 v1 Þ R v2 ¼

fktrmon ½M

þ

ktrsol ½S

þ ðk td þ k tc Þl 0 gl2 þ

k tc l12

pol k tr ðl 0 v3

ð74Þ l2 v1 Þ

ð75Þ

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

ð76Þ

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

ð77Þ

Rm 2 ¼ 2f kd ½I þ ktrmon ½Ml 0 þ ktrsol ½Sl 0 þ k p ½Mðl 0 þ 2l1 Þ þ k tc l12

ð78Þ

The set of moment expressions to be considered consists of either the live and dead moments [Eqs. (70)–(75)] or the live and bulk moments [Eqs. (70), (71) and (76)– (78)], substituting m1 and m 2 for z1 and z2 in Eq. (71). Choice of the former, while it is common practice in the literature, suﬀers from the problem that the equations for l2 and z2 depend on z3 . The Hulburt and Kutz [88] method is often used to approximate z3 , assuming that the molecular weight distribution can be represented by a truncated series of Laguerre polynomials by using a gamma distribution weighting function [Eq. (79)]. v3 ¼

v2 ð2v0 v2 v12 Þ v0 v 1

ð79Þ

4.4 Polymer Reaction Engineering

Using the bulk moments not only eliminates the need for this approximation, but also reduces the set of equations by one, since Eq. (72) is not required to solve for Mw . An additional balance [Eq. (80)] can be added to either set of equations to track the concentration of LCB formed by the transfer to polymer mechanism, pol

RLCB ¼ k tr l 0 m1

ð80Þ

Thus, a set of six equations [Eqs. (70), (71), (76)–(78), and (80)] can be used to collapse the molecular weight distribution into its principle averages to calculate Mn , Mw , and LCB density. The set of moment equations developed here can be extended to include additional complex mechanisms, such as reactivity of terminal double bonds [Eq. (30), Scheme 4.10] [81], crosslinking (Scheme 4.11) [40], and chain b-scission following intermolecular H-abstraction [89]. The methodology can also be extended to copolymerization systems, either by deﬁning copolymer-averaged rate coeﬃcients as in Eqs. (50) and (54) [6, 85–87], or by deﬁning additional moment quantities (for examples, see Refs. 40, 81, 85). Furthermore, since it is easy to implement as part of larger-scale reactor modeling, it is the standard methodology used in process simulation packages (see, for example, Refs. 90, 91). For discussion regarding the ﬁnal step of model development, substitution of the kinetic expressions for the moments into reactor balances, see Section 4.4.3. Modeling of Distributions The major limitation of models based on the method of moments is that they only track average quantities. While adequate for most situations, more detail may be needed if the objective of the study is to improve our knowledge of the kinetics – for example, to examine the combined eﬀect of chain-scission and long-chain branching on polymer architecture, or to incorporate chain-length dependent termination kinetics into the mechanistic scheme. In such cases, the kinetic and modeling assumptions can be tested more rigorously through a detailed comparison with full molecular weight distributions (MWDs) measured experimentally. Recent advances in modeling tools now make it possible to simulate the complete MWD, as well as how a second distributed quantity (for example, LCB) varies with chain length. The modeling of complete MWDs has long been possible for linear polymer systems, that is, those without any branching. In the absence of long-chain branching and employing the QSSA, a recursive relationship can be derived for Pn and Dn from Eqs. (67) and (68), leading to Eqs. (81) and (82) for the weight-fraction distribution of polymer formed at any instant in time [92]. 4.4.2.2

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

t¼

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

b¼

1 1þbþt

k tc ½l 0 k p ½M

n ð81Þ

ð82Þ

201

202

4 Free-radical Polymerization: Homogeneous

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

1 ; t þ 0:5b

DPwinst ¼

2t þ 3b ðt þ bÞ 2

ð83Þ

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

4.4 Polymer Reaction Engineering

The commercial software Predici2 package uses yet another numerical technique, calculating MWDs using a discrete Galerkin technique with variable grid and variable order [97]. In 2000, the package was extended to follow branch point concentrations as a function of chain length through the introduction of balance equations [98]. The possibility of performing these tasks – calculation of complete MWD as well as LCB distribution – in a commercial software package is especially noteworthy because it makes it possible for a wider range of practitioners to perform detailed kinetic modeling. These new modeling capabilities, in combination with improved characterization techniques, will hasten progress to a better understanding and representation of complex polymer architecture [46, 99]. 4.4.3

Reactor Modeling

The kinetic expressions of Section 4.4.2 are substituted into overall material and energy balances to construct a model to represent an FRP process. For a wellmixed reactor system, Eqs. (84)–(89) comprise the general system of equations for homopolymerization. dðV½MÞ ¼ ðqin ½Min Þ ðqout ½MÞ VR pol dt

ð84Þ

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

ð85Þ

dðV½SÞ ¼ ðqin ½Sin Þ ðqout ½SÞ VRtrsol dt

ð86Þ

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

ð87Þ

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

ð88Þ

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

ð89Þ

In these balances, which must be accompanied by appropriate initial conditions and a volume balance, V is the reactor volume, qin and qout are the inlet and outlet volumetric ﬂow rates, r and c p are the density and heat capacity of the mixture, and Q is the rate of heat removal from the reactor system. The various source terms have been presented earlier: Eq. (10) for the monomer and energy balance, Eq. (2) for the initiator balance, Rtrsol ¼ ktrsol ½Sl 0 in the solvent balance, Eqs. (70) and (71) for the live moment balances, and Eqs. (76)–(78) for the bulk moment balances.

203

204

4 Free-radical Polymerization: Homogeneous

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

dV V dr ¼ dt r dt

ð90Þ

Volume contraction due to the diﬀerence in polymer and monomer densities can be as high as 20% for bulk polymerization, and should not be neglected when solving the material balances. Solution of the initiator balance and substitution into the monomer balance lead to a diﬀerential equation for conversion, Eq. (91), where the volume contraction factor is deﬁned as e ¼ ðrﬁnal r0 Þ=rﬁnal where rﬁnal is the system density at 100% conversion of monomer to polymer and r0 is the initial density of the system. dx p ¼ dt

kp 1=2

kt

!

2f kd ½I0 1 ex p

!1=2

kd t ð1 x p Þ exp 2

ð91Þ

Continuous Polymerization Equations (84)–(89) can be solved to provide a description of the dynamic behavior of a continuous well-stirred system. Analytical solutions may be derived for the steady-state case by setting all derivative terms to zero. Assuming no inﬂow of radicals, Eq. (92) results for l 0. 4.4.3.2

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

ð92Þ

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

½l 0 ¼

2f kd ½I 1=2 kt

ð93Þ

[I] can be calculated from the steady-state solution of Eq. (83), namely Eq. (94), where y ¼ V=qout is the reactor residence time and e ¼ ðrout rin Þ=rout is the fractional density change between inlet and outlet streams.

½I ¼

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

ð94Þ

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

4.4 Polymer Reaction Engineering

x p ¼ yk p l 0 ¼ y

kp 1=2 kt

!

2f kd ½Iin ð1 eÞð1 þ ykd Þ

1=2 ð95Þ

Assuming no inﬂow of polymer into the reactor and the long-chain hypothesis, the pol steady-state values for DPn and DPw in the absence of LCB (k tr ¼ 0) are given by Eq. (83). LCB and reaction with terminal double bonds broaden the MWD signiﬁcantly [81].

Complex Flowsheets These are often constructed to represent systems with nonideal mixing and fast reaction. A classic example is the high-pressure high-temperature free radical production of ethylene copolymers, generally conducted in a homogeneous phase consisting largely of supercritical ethylene monomer. These conditions make for very fast reactions (for example, initiator half-life of <1 s), promote numerous side reactions (long-chain branching, short-chain branching, and chain scission), and introduce the potential of thermal runaway. Recent models of these systems combine a detailed description of polymerization kinetics with a complex ﬂowsheet of continuous well-mixed tanks in series with recycle to represent mixing in a multizone, multifeed autoclave reactor [100–102]. Models for multifeed tubular systems also include heat transfer and pressure drop along the length of the system [103]. The general strategy is to ‘‘tune’’ the model (based upon a set of proposed kinetic mechanisms captured by the method of moments) by ﬁtting kinetic coeﬃcients and mixing and heat transfer parameters to a set of industrial data, and then to use the model to interpret and optimize industrial operating conditions. Advances in computing power have allowed the complexity of these models to increase. 4.4.3.3

Computational Fluid Dynamics (CFD) CFD simulation is emerging as an alternative and more fundamental approach to examining polymerization systems with complex mixing and reaction. Once again, much of the work is focused on high-pressure ethylene polymerization systems. A major challenge is incorporating both macromixing (turbulent diﬀusion and convection) as well as micromixing (molecular diﬀusion) into the representation [104]. The ﬁrst eﬀorts in this area [104, 105] concern themselves with the prediction of temperature and conversion proﬁles in the reactors; to simplify the calculational load they consider only initiation, propagation, and termination reactions. More recently, Kolhapure and Fox [106] incorporated a more complete kinetic scheme to allow the prediction of polymer MW, polydispersity, and average branching number. These CFD studies can point the way to improved reactor design and operation; for example, by examining the importance of initiator distribution at the injection point, and deﬁning conditions for stable reactor operation [106]. An article in 2000 discussed the implementation of CFD calculations within a process simulation package [107]. Although not yet applied to polymerization systems, this advancement shows enormous promise. 4.4.3.4

205

206

4 Free-radical Polymerization: Homogeneous

Model-based Control Model-based control of polymerization systems has also garnered its share of attention. The goal of these eﬀorts is the development of robust strategies to guide and control the manufacture of polymer safely and reproducibly in the face of unmeasured disturbances and frequent product grade transitions. The main challenge in controlling polymerization systems is the lack of on-line measurements of polymer structure. A review by Congalidis and Richards [108] provides a good summary of literature focusing on this diﬃcult issue; for further discussion, see Chapter 12 of this book. In most cases, the implementation of detailed fundamental models is not warranted for control application. However, simple models can often be combined with limited on-line measurements (for example, in Refs. 109–112) to improve control performance. Fundamental models can also be used to test empirical models developed for control purposes [113]. 4.4.3.5

4.5

Summary

Polymer reaction engineering issues can pose major challenges for industrial-scale polymer synthesis. As reactor size increases, transport phenomena such as heat and mass transfer become more diﬃcult, and large-scale homogeneous FRP processes are often limited by transport eﬀects. In many cases, the ﬁnal reaction rate, molecular weight, and copolymer composition are determined by the coupled effects of reaction kinetics and transport phenomena. Mathematical modeling, in conjunction with a strong experimental program, is a powerful means to improve our mechanistic and process understanding. The use of empirical models has utility in the control of polymerization reactors, and can be an important contributor to product quality and process robustness. Continuing development of new measurement techniques, together with an ability to relate measurements to functional properties, will be a critical area of future research. While many current free-radical polymerization processes have been in industrial use for years, the next several years may see the emergence of new industrial technologies. Promising technologies for homogeneous FRP include living radical polymerization chemistries and polymerization in supercritical carbon dioxide. The adoption of new technology requires identifying an application or product for which the new technology is clearly advantaged, and successfully overcoming the numerous scaleup challenges in converting the process to industrial scale. These processes will require a detailed understanding of polymerization kinetics combined with fundamental reaction engineering principles. Notation

Ai CtrXXX

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

Notation

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

207

208

4 Free-radical Polymerization: Homogeneous

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

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

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

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

References

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

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

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

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

99 M. Busch, Macromol. Theory Simul.

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

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

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5

Free-radical Polymerization: Suspension1 B. W. Brooks 5.1

Key Features of Suspension Polymerization

Many important polymers are made commercially via suspension polymerization of vinyl monomers. These include poly(vinyl chloride), poly(methyl methacrylate), expandable polystyrene, styrene–acrylonitrile copolymers and a variety of ionexchange resins and specialist materials. The annual polymer production from suspension processes is very high. 5.1.1

Basic Ideas

There are two key requirements for any commercial free radical polymerization process. The polymerization rate must be reasonably high and the polymer product must have the correct molecular weight distribution. The conditions that are necessary to achieve those goals can be predicted from the kinetics of homogeneous free radical polymerization (see Chapter 4). But those conditions cannot be obtained easily in large-scale production when bulk processes are used. Uniform mixing and temperature control are diﬃcult to achieve because the high heat of polymerization is combined with a large viscosity of the reaction medium. As most polymers are poor thermal conductors, heat transfer from large reactors is usually poor because the ratio of surface area to volume decreases as the reaction mass increases. In suspension polymerization, drops of a monomer-containing phase are dispersed in a continuous liquid phase. Monomer solubility in the continuous phase is often low and polymer is produced inside the drops. Although the drop viscosity increases with monomer conversion, the eﬀective viscosity of the suspension remains low and eﬃcient agitation is possible. The ratio of surface area to volume for small drops is relatively high and local heat transfer is good. If the continuous 1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

214

5 Free-radical Polymerization: Suspension

phase is aqueous and well mixed, heat transfer from the reactor is also good. That permits eﬀective control of temperature and of those variables which depend on temperature, which include reaction rates, polymer molecular weight, and copolymer composition. Suspension stability is maintained by good agitation and by the use of drop stabilizers. Removal of the stabilizing agents, after polymerization, may not be complete and some contamination of the ﬁnal product is sometimes inevitable. Suspension polymerization usually requires larger reactor volumes than bulk processes because the vessels often contain about 50% of the continuous phase. Suspension polymerization has been reviewed previously by Munzer and Trommsdorﬀ [1], Bieringer et al. [2], Warson [3], Brooks [4], Yuan et al. [5], Vivaldo-Lima et al. [6], and Arshady [7]. 5.1.2

Essential Process Components

The chemical events that occur inside drops of the dispersed phase are similar to those found in bulk polymerization. The drops contain monomer (or monomers), radical generators (often called initiators), and polymer. Sometimes chain-transfer agents are added also. The continuous phase is often regarded as chemically inert, but drop stabilizers are usually present in it and, in some cases, those stabilizers participate in the polymerization process. For a discussion of stabilizer types, see Section 5.2.1. 5.1.3

Polymerization Kinetics

Higher conversions of monomers can be accommodated more readily in suspension processes than in bulk processes because suspensions are more mobile than molten polymers. Therefore, simple rate expressions may not be applicable because the values of some rate coeﬃcients diminish at high polymer concentrations (see Section 5.3.2). It is often assumed that the polymerization chemistry which occurs in the dispersed phase is identical to that which occurs in the equivalent bulk process. That assumption may be valid if the monomers and initiator are virtually insoluble in the continuous phase. Then, polymerization rates, molecular weight distributions, and copolymer compositions can be predicted from conventional kinetic schemes. But drop stabilizers may react with species inside the drops (for example, to form graft copolymers). When all the monomers in a suspension polymerization are virtually immiscible with the continuous phase, then the instantaneous copolymer composition can be predicted from idealized relationships which apply to homogeneous systems. However, the use of those relationships is not straightforward if one, or more, of the monomers is partially soluble in the continuous phase, because the actual composition of the drops may then be unknown. The eﬀective monomer concentrations,

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

Monomer 1 Monomer 2

AS S

AN MA

VC VA

MMA MAA

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

1.18 0.85 1.0 1.0 60

1.02 0.7 0.75 1.54 50

1.68 0.23 2.47 1.99 70

0.35 1.63 0.63 1.07 69

Abbreviations: AS, p-acetoxystyrene; S, styrene; AN, acrylonitrile; MA, methyl acrylate; VC, vinyl chloride; VA, vinyl acetate, MMA, methyl methacrylate; MAA, methacrylic acid.

which should be used in these relationships, might be predicted if the appropriate partition coeﬃcients for the two phases can be obtained. That is not often possible but models that allow for water solubility of monomers have been developed for the copolymerization of vinyl chloride and vinyl acetate [8] and for the copolymerization of styrene and acrylonitrile [9]. Apparent reactivity ratios obtained directly from suspension polymerization experiments may not be identical to those expected from the equivalent bulk processes if some monomer migrates to the continuous phase. Ashady et al. [10] found values for reactivity ratios that were not expected from results observed in bulk or solution copolymerization. Izumi and Kitagawa [11] showed that reactivity ratios for suspension copolymerization, of acrylonitrile and methyl acrylate, were diﬀerent from those obtained from either solution or emulsion polymerization. Table 5.1 compares reactivity ratios obtained from solution copolymerization with those observed in suspension copolymerization. If the initiator in suspension polymerization is slightly soluble in water, then simultaneous emulsion polymerization may occur when free stabilizer remains in the continuous phase [12, 13]. 5.1.4

Fluid–Fluid Dispersions and Reactor Type

Batch, or semi-batch, reactors are often used for suspension polymerization on an industrial scale. Dispersions in tubular ﬂow reactors are diﬃcult to maintain and a continuous stirred tank would produce drops containing partially polymerized material that would coalesce in the receiving equipment. However, new types of ﬂow reactors are being developed for suspension polymerization (see Section 5.4.1). Many of the published studies on drop behavior in agitated liquid–liquid dispersions are applicable to suspension polymerization. But sometimes their use is limited because they do not account for changing physical properties of drops or for the presence of drop stabilizers.

215

216

5 Free-radical Polymerization: Suspension

5.1.5

Uses of Products from Suspension Polymerization

Advantages of suspension polymerization are not restricted to temperature control. Some polymers, such as poly(vinyl chloride) (PVC), are immiscible with their monomers. Subsequent polymer coagulation makes bulk processes diﬃcult to control. In suspension polymerization, however, that problem is avoided. There, coagulation is largely conﬁned to the drop interiors and aggregation of polymerizing drops is restricted. That is why suspension polymerization is used for the largescale production of PVC. In that case, the initial drop diameters, and the ﬁnal particle sizes, range between 10 and 100 mm. If a polymer product is required in particulate form, then suspension polymerization is especially suitable. The energy required to disperse monomer drops is lower than that required to break up the ﬁnished polymer. Also, drop size control before polymerization is easier to achieve than particle size manipulation of granulated polymer. Suspension polymerization provides a good route to functionalized particles such as those used in ion-exchange resins. Expandable polystyrene beads are also made by suspension polymerization. When a product is to be used in ‘‘bead form’’, initial drop diameters can be as large as 1–2 mm [1].

5.2

Stability and Size Control of Drops

For many applications, the size range of the ﬁnal product particles is very important. For example, bead diameters aﬀect ﬂow rates through ion-exchange columns. But particle size can also be important when the polymer is to be converted to a macroscopic object. Heat transfer rates to polymer particles during extrusion and mass transfer rates of plasticizers in particulate polymers both depend on particle size. During suspension polymerization, drop size depends on the physical properties of the two phases, the phase ratio, the nature of the suspension ﬂow, and the condition of the phase interface. Interfacial tension and drop stability depend largely on the nature of the drop stabilizer. If no stabilizer were used to protect the drops, the suspension would be unstable and the ﬁnal polymer particles would reach an undesirable size. The adsorption of stabilizer molecules at the interface between monomer and the continuous phase reduces the interfacial tension and hence reduces the energy required to form drops. Drop stability against coalescence depends largely on the ability of the stabilizer to form a thin protective ﬁlm at the interface. That gives the drops better elastic properties [14]. The eﬀect of the elastic properties is enhanced by increasing the concentration of the suspending agent, until a certain surface coverage of the drops is reached. At that point, a ‘‘critical surface coverage’’ is established and a further increase in the suspending agent concentration will have a very little eﬀect on the drop stability [15].

5.2 Stability and Size Control of Drops

5.2.1

Stabilizer Types

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

217

218

5 Free-radical Polymerization: Suspension

solid is a stabilizer or a destabilizer in suspension polymerization. Solids that provide a relatively large contact angle (such as aluminium hydroxide) would be stabilizers in aqueous media. In contrast, those which have a relatively small contact angle (such as carbon black) would tend to be destabilizers. In practical operation, the contact angle will probably vary with any given inorganic solid if diﬀerent monomers are used. In many cases it is not easy to measure contact angles. Two theories, screen theory and coverage theory, have been suggested to explain the mechanism of drop stabilization by inorganic solids [21]. According to the screen theory, ﬁnely divided inorganic solids which are dispersed in water form a screen. Dispersed monomer drops smaller than the mesh of the screen are free to move through the meshes, while those bigger than the mesh cannot pass through the meshes and are stopped from coalescing further. In the coverage theory, it is suggested that dispersed inorganic solids cover the surfaces of monomer drops and form a layer which prevents drop coalescence. Both theories are plausible but many workers believe that the stabilization is obtained mainly by a coverage eﬀect. However, not all inorganic solids are adsorbed by the monomer drops. When Wolters et al. [23] used hydroxyapatite and calcium carbonate in the suspension polymerization of styrene, the adsorption equilibrium was found to be far on the side of desorption. Also, Wang and Brooks [24] noticed that many of their stabilizing particles settled on the bottom of the vessel when stirring of a liquid–liquid dispersion ceased. If some particles remain in the continuous phase, it is not feasible for the coverage theory to provide a description for every case of suspension polymerization in which an inorganic solid is used as a stabilizer. Wang and Brooks suggested a crowding theory which takes account of the dynamic eﬀects of stabilizer particles [24]. In that model, particles are not required to be permanently adsorbed on drop surfaces but they become eﬀective when two drops come close together. Although crowding theory does not explain all the observations, it is compatible with a wide range of experimental results. The addition of surfactants often improves the dispersion action of the ﬁne particles, but it is possible to stabilize dispersions solely with mono-sized spherical colloidal particles [25]. 5.2.2

Drop Breakage Mechanisms

In an agitated suspension, the dispersed phase can be broken into small drops when its surface is disrupted. That disruption can be caused either by frictional forces (via viscous shear) or by inertial forces (via turbulence) [26]. The ratio of the external disrupting force to the interfacial tension force is often expressed as the Weber number, We. Drop deformation increases as We increases. When We exceeds a critical value, a drop will break into smaller drops. The ﬂuids in agitated vessels are often turbulent. If the turbulence in local regions can be regarded as isotropic, a criterion for the drop breakage mechanism can be developed [26, 27]. In turbulent ﬂow, random eddies are superimposed on the main ﬂow. Eddy sizes are inﬂuenced by the location of the vessel walls and are

5.2 Stability and Size Control of Drops

restricted by the impeller diameter [28]. Kinetic energy is transferred to smaller eddies in a sequential fashion, until it reaches the smallest ones. This transfer is assumed to occur without energy dissipation. However, when the kinetic energy reaches the small eddies, it is dissipated as heat to overcome the viscous forces. In theories of local isotropy, it is assumed that the small eddies are statistically independent of each other. Velocity ﬂuctuations are determined by the local rate of energy dissipation per unit mass of ﬂuid (e) and by the kinematic viscosity (n). Kolmogorov [29], by dimensional reasoning, derived an expression [Eq. (1)] for the length of the smallest eddy (h). h ¼ n 3=4 e1=4

ð1Þ

Rushton et al. [30] suggested that local isotropy could exist when the Reynolds number is >10 4 . The macroscale of turbulence, L (approximately equal to the impeller diameter), is then much larger than the microscale of turbulence. Here, the Reynolds number (Re) is deﬁned by Eq. (2), where N is the impeller speed, D is the impeller diameter, and n is the kinematic viscosity of the dispersion. Re ¼ ND 2 =n ¼ ND 2 r=m

ð2Þ

Pressure ﬂuctuations can deform the drops and they may break if the inertial forces exceed the interfacial tension forces. Kolmogoroﬀ [27] and Hinze [31] derived an expression for the maximum drop diameter, d max , that should be observed when turbulence is isotropic. Here, d g h and the viscous forces may be neglected in comparison with the inertial forces; d max , can then be related to a critical Weber number, We crit , by Eq. (3). We crit ¼ rc u 2 ðdÞd=s

ð3Þ

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

ð4Þ

Rushton et al. [30] showed that Eq. (5) holds, so that Eqs. (6) [32] and (7) follow. e ¼ C3 N 3 D 2 d max ¼ C4 ðs=rc Þ

ð5Þ 3=5

N

6=5

d max =D ¼ C4 ðWeÞ0:6

4=5

D

ð6Þ ð7Þ

In order to use this expression, it is often necessary to relate d max (which is diﬃcult to measure) to the Sauter mean diameter, d32 (which is more readily determined). A linear relationship between d32 and d max has been demonstrated by Sprow [33], Coulaloglou and Tavlarides [28], Kuriyama et al. [34], and Zerfa and Brooks [35].

219

5 Free-radical Polymerization: Suspension

30 Volume fraction of dispersed phase

4

d 32 /D ( x 10 )

220

0.01 0.05 0.1 0.2 0.3 0.4

20

10

0 0

10 0.027(1 + 3.06

20 )We

- 0.6

30 4

( x 10 )

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

Fig. 5.1.

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

ð8Þ

Equation (6) has been veriﬁed with a large number of liquid–liquid dispersions where the volume fraction of the dispersed phase is not high, that is, in noncoalescing systems [28]. But, unlike many laboratory studies, real suspension polymerizations are operated with a high volume fraction of dispersed phase, j. Then, drop coalescence and breakage happen simultaneously and damping of the turbulent ﬂuctuations may occur. To allow for this, a factor f ðjÞ is often introduced into Eq. (7). Experimental expressions for f ðjÞ are reviewed by Yuan et al. [5], and by Zerfa and Brooks [35]. The latter workers showed that, in the suspension polymerization of vinyl chloride, d32 was given by Eq. (9). d32 =D ¼ 0:027ðþ3:06jÞWe0:6

ð9Þ

Equation (9) applied when values of j ranged between 0.01 and 0.4, as shown in Figure 5.1. An expression with similar form is given by Borwankar et al. [15]. When drop coalescence is signiﬁcant, Shinnar [32] showed that there will be a minimum drop size, d min , that is proportional to N 0:75 . The extent of drop coalescence is reduced by addition of an appropriate drop stabilizer. If the surface coverage of the drops by the stabilizer exceeds a critical value then the dispersion can remain stable after agitation ceases [15]. It must be remembered that correlations between average drop sizes and We only apply at steady state. The steady-state drop size distribution (DSD) can take some time to become established [18, 36–38]. That may not be a serious problem with nonreacting dispersions but, in suspension polymerization, the physical prop-

5.2 Stability and Size Control of Drops

erties change with time so that the DSD can continue to change during the process [37]. When the diameter of the drops is less than the Kolmogorov length h, stresses from viscous shear will be much larger than those from inertial eﬀects. Drop breakage is then the result of viscous shear [26]. Here, again, a drop will break if the deformation variable, sometimes described as a generalized Weber group [31], exceeds a critical value [39]. Taylor showed [40] that the extent of drop distortion, before breakup, depended on the ratio of phase viscosities (Eq. (10), where md and mc are the viscosities of the dispersed and the continuous phase, respectively). We crit ¼ mc ðqu=qrÞðd=sÞ ¼ f1 ðmd =mc Þ

ð10Þ

Shinnar and Church [26, 32] demonstrated that for locally isotropic ﬂow where ðqu=qrÞ 2 ¼ e=n, d max is given by Eqs. (11) and (12). d max ¼ ðsnc1=2 =mc e 1=2 Þ f ðmd =mc Þ

ð11Þ

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

ð12Þ

When viscosity-increasing agents were present in the continuous phase for the suspension copolymerization of styrene and divinylbenzene, Jegat et al. [41] found that drop breakage could occur via viscous shear when ﬂow in the suspension was turbulent. In that case, the Kolmogorov length would have been larger that that found when low-viscosity aqueous solutions are used. The form taken by f ðmd =mc Þ depends on the nature of the ﬂow. In laminar ﬂow, the contributions of rotational and elongational components can be expressed via the value of a parameter a, which ranges between 0 and 1. For simple shear a ¼ 0, and for pure elongational (hyperbolic) ﬂow a ¼ 1 [43]. With low a values, We crit (now equivalent to the critical capillary number Cacrit ) passes through a minimum as md =mc increases. After the minimum is reached, relatively small increases in md =mc are accompanied by large increases in We crit . When a ¼ 0, We crit tends to an inﬁnitely high value if md =mc > 4 so that drop breakage no longer occurs [42, 43]; see Figure 5.2. Correlations that are found in idealized laboratory studies only provide rough guides to events that occur in commercial reactors. Local ﬂow in large stirred vessels is often ill deﬁned. Also, drop breakage can produce more than two new drops. Chatzi and Kiparissides [44] suggested that the formation of a daughter drop is accompanied by the formation of a number of smaller satellite drops. Drop stabilizers inﬂuence turbulence near drop surfaces and non-Newtonian behavior leads to further complications. Therefore, it is not surprising that the many published studies on drop behavior in liquid–liquid suspensions lead to a variety of diﬀerent results. Leng and Quarderer [39] found that d max depended on N 0:8 . By using data from the literature, for the suspension polymerization of methyl methacrylate, they also showed that d max depended on mc0:5 when drop breakage occurred by viscous shear. Presumably, the drop size was not able to respond to changes in md =mc that

221

5 Free-radical Polymerization: Suspension 1.6

A

1.2

Ca crit

222

0.8 0.4

B

0 0

1

2

3

4

log10 (viscosity ratio) + 3 Fig. 5.2. Critical capillary number against log10 of viscosity ratio (md =mc ) for laminar ﬂow. (A) a ¼ 0 (simple shear ﬂow); (B) a ¼ 1 (hyperbolic ﬂow). Data from Refs. 42 and 43.

would have occurred during polymerization. However, average drop size was found to be a linear function of interfacial tension, as expected, when viscous shear was important. Borwankar et al. [15] suggested that the relationship between d max and agitator speed may depend on agitator type. When polymer-containing drops are broken, their elastic properties must be taken into account. Arai et al. [45] derived a correlation for drop breakup from a Voigt model to represent the elastic properties. The validity of the correlation was conﬁrmed experimentally using a dispersed phase with a wide range of viscosity. Wang and Calabrese [46] carried out experiments with drops made from wellcharacterized model ﬂuids. They showed that the inﬂuence of interfacial tension on drop breakage decreased as the dispersed-phase viscosity increased. 5.2.3

Drop Coalescence

The volume fraction of drops in commercial suspension polymerization reactors is usually high and drop coalescence cannot be ignored. In liquid–liquid dispersions the drop size distributions (DSDs) depend on the breakage and on coalescence processes. From experiments in which both drop breakage and coalescence occurred, Kuriyama et al. [34] found that drop sizes reached a steady value within an hour, when the initial drop viscosity was low. But with a high drop viscosity, the drop size reduction continued for longer periods of time and the ﬁnal drop size was higher. Although model, nonreacting, ﬂuids were used for those experiments, the results are relevant to suspension polymerization. In their study of drop coalescence in the suspension polymerization of styrene, Konno et al. [47] found that the Sauter mean diameter increased as the polymer viscosity increased. They also concluded that the stabilizer does not eﬀectively prevent the coalescence of drops with diameters larger than d max . The overall rate of drop coalescence is related to the collision frequency of the drops and to the coalescence eﬃciency. By comparing drop collisions in agitated

5.2 Stability and Size Control of Drops

dispersions with molecular collisions in homogeneous ﬂuids, previous workers have developed expressions for collision rates between drops of diﬀerent sizes [48, 49]. If drops adhere for suﬃcient time to allow them to deform, and to permit drainage of the continuous phase that is trapped between them, then coalescence may occur [26]. By taking account of these events, expressions can be obtained for the coalescence eﬃciency [50]. Expressions, for coalescence and for collision rates are not easy to use because they often contain parameters that are diﬃcult to quantify. Alvarez et al. [50] constructed a model for drop breakage and coalescence, in the suspension polymerization of styrene, which takes account of viscosity eﬀects. That model assumes that breakage of a drop, exposed to a turbulent ﬂow ﬁeld, is a result of ﬂuctuations with a wavelength equal to the drop diameter. Fluctuations with wavelengths that are smaller or larger than the drop diameter do not lead to drop breakage. 5.2.4

Drop Size Distributions

Even when expressions for drop breakage and coalescence rates are available, their successful use must allow for variations of turbulence intensity within the reactor. Maggioris et al. [51] describe a two-compartment population balance model for an agitated suspension polymerization reactor. That model distinguishes between the impeller region and the remainder of the reactor. Both regions are assumed to be well mixed and CFD simulations predicted drop size distributions that were compatible with experimental results from nonpolymerizing model liquid–liquid dispersions. For some combinations of stabilizer type and stirrer speed, bimodal drop size distributions were predicted by the model and found in the experiments. Yang et al. [52] showed how the size distribution of nonreacting styrene drops changed with agitation time. Bimodality developed and the relative size of the two peaks in the distribution depended on the amount of drop stabilizer that was used. Calabrese et al. [46, 53] showed that, in dilute agitated suspensions for which coalescence is negligible, the equilibrium drop size distribution broadened considerably as dispersed-phase viscosity increased. If no chemical reaction occurs, then the rates of drop breakage and drop coalescence eventually become equal and a stable DSD is obtained [54]. Considerable periods of time might be required for that to occur [18], especially when the drops have a high viscosity [55]. But, in the case of suspension polymerization, the physical properties of the drops change with monomer conversion. In a batch process, early increases in drop viscosity reduce the rates of breakage and coalescence before a steady-state DSD can be established. Then, the DSD continues to change, and the average drop size increases, as the polymer content of the drops increases. Consequently, the ﬁnal particle size distribution (PSD) is quite diﬀerent from the steady-state DSD that would be expected from nonpolymerizing drops. That was shown to be the case by Jahanzad et al. [37] in the suspension polymerization of methyl methacrylate. There, the DSDs of polymerizing drops were broader than that of the DSDs in the corresponding nonpolymerizing monomer drops. But the

223

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5 Free-radical Polymerization: Suspension

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

Drop Mixing

Most of the coalescence events described in Section 5.2.3 occur when drops have a uniform chemical composition, but sometimes it is necessary to add material to a

5.2 Stability and Size Control of Drops

Drop number

200

A

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

B

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

C

150 100 50 0 15

35

55

75

95

Drop diameter (microns) Fig. 5.3. Change in drop size with stirrer speed with PVA 72.5% hydrolyzed. (A) N ¼ 350 rpm (5.8 s1 ); (B) N ¼ 650 rpm (10.8 s1 ); (C) N reduced from 650 to 350 rpm. Data from Ref. 18.

reactor in which a suspension already exists. That can happen if control of copolymer composition is important. In batch operation, copolymer composition usually changes with overall conversion because monomers react at diﬀerent rates [58]. This drift in copolymer composition may be limited by adding one of the mono-

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5 Free-radical Polymerization: Suspension

Drop number

200

A

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

Drop number

200

B

150 100 50 0 15

35

55

75

95

Drop diameter (microns)

200

Drop number

226

150

C

100 50 0 15

35

55

75

95

115

Drop diameter (microns) Change in drop size with stirrer speed with PVA 55% hydrolyzed. (A) N ¼ 350 rpm (5.8 s1 ); (B) N ¼ 650 rpm (10.8 s1 ); (C) N reduced from 650 to 350 rpm. Data from Ref. [18]. Fig. 5.4.

mers to the reactor incrementally (that is, by using a semi-batch procedure). But, when new dispersible material is added to an existing suspension, the new material and existing drops can remain segregated for a signiﬁcant period of time. Then, new drops may form with a monomer composition that diﬀers from that of

5.2 Stability and Size Control of Drops

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

227

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5 Free-radical Polymerization: Suspension

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

Fig. 5.5.

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

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

5.3

Events at High Monomer Conversion

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

5.3 Events at High Monomer Conversion

5.3.1

Breakage of Highly Viscous Drops

In batch reactors drop breakage and coalescence are aﬀected by the polymerization process because the viscosity of the polymerizing ﬂuid often increases. Many aspects of the interaction of drop behavior with the polymerization process have been discussed already (see Section 5.2). 5.3.2

Polymerization Kinetics in Viscous Drops

Free radical polymerization kinetics has received much attention and many aspects of the process are well understood (see Chapter 4). Most academic investigations have been carried out in ‘‘idealized’’ conditions where the extent of monomer conversion is low. The classical expression for the rate of polymerization (R p ), in a single-phase reaction, is Eq. (13), where k p is the propagation rate coeﬃcient, CM is monomer concentration, R i is the initiation rate and k t is the termination rate coeﬃcient [63]. R p ¼ k p CM ðR i =k t Þ 1=2

ð13Þ

If radicals are generated by the thermal decomposition of an added initiator then, at steady-state conditions, Eq. (14) applies, where f is an eﬃciency factor, kd is the initiator decomposition rate coeﬃcient and CI is the initiator concentration. R i ¼ 2f kd CI

ð14Þ

Chain termination is often diﬀusion-controlled and the value of k t diminishes substantially as the polymer concentration increases. At high polymer concentrations, k p decreases also [64]. The reduction in k t leads to an increase in the polymerization rate, a phenomenon often described as a ‘‘gel eﬀect’’. Although most workers agree that the chain termination reaction is diﬀusioncontrolled, reliable quantitative relationships between the rate coeﬃcients and measurable properties of the reaction medium are not generally available. Radical diﬀusion can depend on solution viscosity, polymer volume fraction and polymer molecular weight. The latter three entities are interrelated in complicated ways; meaningful experiments, which may relate them to the rate coeﬃcients, are not easy to devise [65]. However, Brooks et al. showed that eﬀects of viscosity changes on polymerization rate could be distinguished from the eﬀects of polymer volume fraction [66]. In some cases, the value of f may also depend on polymer content [67]. The value of kd can often be determined independently but traditional steadystate experiments give values for k p =k t0:5 and do not provide separate values for k p and k t . This restriction becomes a problem for reactor modeling because, in the presence of polymer, both k p and k t diminish (even in isothermal conditions).

229

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5 Free-radical Polymerization: Suspension

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

Consequences of Polymer Precipitation Inside Drops

Polymers that are insoluble in their monomers will precipitate during polymerization. The resulting fouling problems that occur in bulk polymerization are greatly reduced by using suspension polymerization. This is one of the reasons for choosing a suspension process for PVC manufacture, as discussed previously (see Section 5.1.5). Manipulation of polymer precipitation, inside the drops, during

5.3 Events at High Monomer Conversion

Fig. 5.6.

Schematic representation of polymer formation inside vinyl chloride drops.

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

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5 Free-radical Polymerization: Suspension

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

CPA (g DOP/100g PVC)

5.3 Events at High Monomer Conversion 100 90 80 70 60 50 40 30 20 0

20

40

60

80

100

Polymerisation conversion (%) Fig. 5.7. Eﬀect of polymerization conversion on the cold plasticizer absorption (CPA) of PVC. Reproduced by permission of John Wiley & Sons, Inc. From Ref. 82.

of initiator addition [62]. High conversions of VCM are not always desirable because the porosity of PVC particles usually decreases linearly with monomer conversion, as shown in Figure 5.7 [60, 82]. When a nonionic surfactant is used as an SSA, in conjunction with PVA, the porosity of PVC increases as the concentration of nonionic surfactant increases. Here, increased porosity may be the result of incomplete drop coalescence creating voids at the sites where droplets are not well contacted [82]. Nonionic surfactant with a lower HLB value (hence with a higher aﬃnity for VCM and a higher solubility in VCM) is more eﬀective in raising the product porosity. The surface of primary particle aggregates becomes coarser as surfactant is added [82]. This could be the consequence of an altering interfacial tension between the PVC-rich phase and the monomer. That might be expected to decrease the contact deformation of the primary particles and increase the pore space. When nonionic SSAs are used, PVC porosity decreases linearly with an increase of polymerization temperature [82]. The porosity increases as the concentration of nonionic surfactant increases, and a surfactant with a lower HLB value is more eﬀective in raising the porosity. The increase in porosity may be caused by a combination of increased coalescence of VCM/PVC droplets and the nonionic surfactant’s steric eﬀect inside the droplets [82]. Bao et al. showed that particle morphology and PVC properties can be controlled by using blends of PVA suspending agents with diﬀering degrees of hydrolysis [94]. The use of PVA, with a low degree of hydrolysis, as an SSA increased particle porosity in the suspension copolymerization of vinyl chloride and propylene [95]. In that case, the SSA was more soluble in the organic phase than the primary stabilizer (a cellulose ether). In vinyl chloride polymerization, particle porosity facilitates the subsequent uptake of plasticizers, but in other cases particle porosity is induced to enhance access to functional groups within the polymer. That is useful when polymer beads are required for use in ion-exchange columns, or in analytical instruments. A diene comonomer can be added to a monomer, to promote crosslinking and phase sepa-

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5 Free-radical Polymerization: Suspension

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

5.4

Reaction Engineering for Suspension Polymerization

The discussion above (see Sections 5.2 and 5.3) provides guidelines for the design of suspension polymerization processes on an industrial scale. However, extrapolating from idealized small-scale studies to full-scale commercial operation is not straightforward. Maintenance of uniform reactor conditions cannot be guaranteed inside large reactors and controlled heat transfer from reactors can be diﬃcult. 5.4.1

Dispersion Maintenance and Reactor Choice

Many suspension polymerization processes employ stirred vessels. Oldshue et al. [100] showed that agitator type and vessel geometry have an important inﬂuence on internal liquid ﬂow. This must be taken into account in the design of large vessels for suspension polymerization. Maggioris et al. [51] and Vivaldo-Lima et al. [101] devised two-compartment population balance models to account for the large spatial variations of the local turbulent kinetic energy. These permit the prediction of drop size evolution in a suspension polymerization reactor with a high volume fraction of dispersed phase. Basic expressions were modiﬁed to allow for evolving physical properties of the suspension. Maggioris et al. [51] showed that it was feasible to estimate the volume ratio of the impeller and circulation regions, the ratio of turbulent dissipation rates, and the exchange ﬂow rate of the two compartments at diﬀerent agitation rates and continuous-phase viscosities. Although many industrial suspension polymerizations are described as ‘‘batch’’ processes, they are not genuine batch operations because some material enters the reactor after polymerization has started. In some cases, substantial reﬂux of monomers occurs and condensed monomer returns to the reactor continuously. In others, a semi-batch process is used to control product composition. In both these situations, unconverted monomer with low viscosity is being mixed with drops of higher viscosity. The complexities of these mixing processes have been discussed above; see Section 5.2.5. Reactor conﬁgurations other than conventional stirred tanks have been proposed for suspension polymerization. Draft tubes, or ‘‘internal loops’’, can be used for suspension polymerization [102], but drop size changes can occur [103] and ﬂow patterns may be complicated [104]. Tanaka et al. [105] used a loop reactor for the suspension polymerization of styrene. They employed a double agitation method to control the transient droplet diameter distribution and the ﬁnal particle size distribution. Ni et al. [106] developed an oscillatory baﬄed reactor for batch suspension polymerization of methyl methacrylate. Fluid mixing was achieved by eddies that are generated when a ﬂuid passes through a set of equally spaced, stationary, oriﬁce baﬄes that are located inside a tube. Periodically formed vortices were con-

5.4 Reaction Engineering for Suspension Polymerization

trolled by choosing appropriate values for oscillation frequency, oscillation amplitude, baﬄe diameter, and baﬄe spacing. The ﬁnal particle size distribution and polymer molecular weight were comparable with those of a product obtained from a conventional stirred tank reactor. Although most industrial polymerization processes are batch or semi-batch operations, some continuous-ﬂow processes have been proposed [107]. Use of ﬂow reactors may aﬀect the nature of the drop size distribution and the molecular weight distribution. If ﬂow through a back-mixed region occurs, the drops will be subject to a distribution of residence times and the nature of the mixing for the dispersed phase is important. Baade et al. [108] and Taylor and Reichert [109] showed that complete segregation of drops, in the suspension polymerization of vinyl acetate, produced an increase in polymer molecular weight. Also, the distribution of molecular weights became broader than that expected in a batch reactor. Complete mixing of the drops gave a molecular weight distribution which was narrower than that found in batch reactors. An oscillatory baﬄed reactor can also be used for continuous-ﬂow suspension polymerization. Ni et al. [110] describe the eﬀect of superimposed oscillations on the main ﬂow through a tube. Radial mixing of the ﬂuid occurs but near plug-ﬂow is obtained. A constant level of turbulence intensity in the reactor can lead to particle size distributions similar to those expected from a batch reactor. 5.4.2

Agitation and Heat Transfer in Suspensions

In single-phase processes, reactor agitation inﬂuences the rate of heat transfer to, and from, the surroundings and determines the quality of mixing. With suspension polymerization reactors, the agitation must also be good enough to generate, and to maintain, a two-phase dispersion. When conventional stirred tanks are used, a ‘‘standard’’ reactor geometry may be adequate to promote convective heat transfer. Here, the height of the total suspension (H), the impeller diameter (D) and the clearance underneath the impeller (G) are related to the tank diameter (T) as follows:

HAT D A T=3 G A T=3

The use of baﬄes limits nonuniformity in the turbulence and restricts vortex formation. Vortices are undesirable because the centrifugal eﬀect favors drop congregation and may promote unwanted drop coalescence. That can lead to polymer deposition either on the agitator or the reactor walls (depending on the relative density of the aqueous and nonaqueous phases). When vertical baﬄes are close to the walls their width (B) is often given by:

B A T=10

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5 Free-radical Polymerization: Suspension

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

5.4 Reaction Engineering for Suspension Polymerization

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

Scaleup Limitations with Suspension Polymerization

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

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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

239

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5 Free-radical Polymerization: Suspension

some cases a redox initiator system is used. Redox reactions produce free radicals at relatively low temperatures, which is advantageous when the polymerization is very fast at higher temperatures. This type of polymerization is not always strictly analogous to ‘‘simple’’ conventional suspension polymerization. In some circumstances, there is evidence to suggest that the polymerization rate is diminished when large amounts of an oil-soluble surfactant are used to stabilize the monomer drops, and that a small amount of polymerization occurs in the oil phase [120]. The polymerization kinetics can be complicated when redox systems are used and the mechanism of radical generation may depend on the speciﬁc reductant– oxidant pair that is chosen [121]. 5.5.2

Drop Mixing with Redox Initiators

At least one of the two major components of a redox initiator (reductant or oxidant) must be segregated from the monomer while the suspension is being formed. Otherwise, polymerization would begin before the desired DSD was attained. Often, drops of an aqueous solution of monomer and oxidant are initially dispersed in an oil and stabilized with an appropriate agent. Then aqueous reductant is added to start the reaction [122, 121]. Therefore, the initial suspension has two types of aqueous drops which must become mixed before polymerization can begin. Liu and Brooks [123] used a freeze–fracture technique with electron microscopy to show how ‘‘large’’ reductant drops became mixed with ‘‘small’’ monomercontaining oxidant drops in the early stages of acrylic acid polymerization (Figure 5.8). The actual volumes and concentrations used for the two aqueous solutions affect the rate of viscosity increase in the drops and the polymerization rate [124].

5.6

Future Developments

Future work on suspension polymerization will be concerned with new products and with new processes. Much of the new work on recipes and product properties is material-speciﬁc and will continue to appear in the patent literature. The development of new processes requires further fundamental investigation. Improvements to existing processes need closer attention to ways in which reactors are operated. These should include the following aspects. 5.6.1

Developing Startup Procedures for Batch and Semi-batch Reactors

Suspension polymerization can rarely be described as a conventional batch process because uniform physical conditions cannot be created instantly. The way in which the ingredients are brought together can aﬀect the polymer properties. Therefore, improvements in the control of product quality may depend on developing better

5.6 Future Developments

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

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

241

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5 Free-radical Polymerization: Suspension

a large surface area and they can distort the distribution of drop stabilizer during the initial drop development stage. 5.6.2

Maintaining Turbulence Uniformity in Batch Reactors

Turbulence in conventional stirred vessels is far from uniform and prediction (or control) of DSD is diﬃcult. Thus, the ﬁnal PSD is often very wide. That is undesirable when the ﬁnal particles are used directly (for example, in ion-exchange resins). In those cases, a narrow PSD is often required. New reactor geometries and agitation methods could ensure the correct values for the overall Reynolds and Weber numbers and also give more uniform distribution of turbulence conditions. That will lead to a lower proportion of ‘‘oﬀ-speciﬁcation’’ material. Here, the use of new reactors, such as the oscillatory baﬄed reactor, could be helpful. 5.6.3

Developing Viable Continuous-ﬂow Processes

Continuous-ﬂow reactors are often developed to produce high tonnages with low reactor maintenance but, with suspension polymerization, continuous-ﬂow reactors could oﬀer other advantages. If ‘‘near plug-ﬂow’’ conditions could be attained, the drop formation and polymerization stages could be separated to give better control of the DSD. Also, temperature programming would become feasible so that variation of drop viscosity with monomer conversion could be manipulated to achieve better control of drop coalescence and of the ﬁnal PSD. For a continuous-ﬂow reactor to operate successfully the following conditions must be attained simultaneously: a suitable residence time (often a few hours); small regions of localized turbulence; restricted overall back-mixing; and good heat transfer. New designs of oscillatory baﬄed reactors [106] may make such processes feasible on an industrial scale. 5.6.4

Quantitative Allowance for the Eﬀects of Changes in the Properties of the Continuous Phase

In suspension polymerization, the continuous phase is often regarded as ‘‘inert’’. Consequently, events that occur in that phase are sometimes neglected, even when they are potentially important. The overall Reynolds number is often high enough to ensure that drop breakage is via turbulence and that ‘‘conventional’’ Weber number correlations apply. But when the volume fraction of drops is high, or when a signiﬁcant fraction of the drop stabilizer remains in the continuous phase, the eﬀective viscosity of the suspension becomes high. Then, the Kolmorgorov length may exceed the drop diameter, and drop breakage via viscous shear becomes important. More work is required to determine the relationships between drop sizes and agitation conditions when the ﬂuid ﬂow is not fully turbulent. Maintenance

Notation

of uniform viscous shear (as opposed to maintenance of uniform turbulence) throughout the reactor will bring a new set of challenges. As more complex copolymers are developed, the range of monomer properties in any one recipe will widen. The chances of all monomers being ‘‘insoluble’’ in the continuous phase are small. Therefore the eﬀects of monomer distribution between the phases, especially on the copolymer composition, will be important. Prediction of phase distribution is not easy in systems that are thermodynamically nonideal and it may become necessary to develop new sensors that monitor the composition of the continuous phase. 5.6.5

Further Study of the Role of Secondary Suspending Agents

The porosity and structure of polymer particles are important for many product applications. Then, the use of SSAs is often important. However, to gain a better understanding of their role, it is necessary to determine what fraction of those stabilizers is inside drops of the dispersed phase and how they aﬀect structure inside the particles. The importance of grafting of SSAs to the polymer should be clariﬁed. 5.6.6

Further Characterization of Stabilizers from Inorganic Powders

Particulate stabilizers will continue to have a place in suspension polymerization because they can often be removed after polymerization, but our understanding of their action is far from complete. Some of the theories do not always agree with experimental results. Further systematic studies are required to determine the effect of the size and surface condition of the stabilizer particles on their ability to stabilize drops. Also, the location of those particles at the end of polymerization is important. In some cases, it may be advantageous to treat stabilizer particles so that they remain with the polymer product in order to confer some desirable property.

Notation

B C1 ; C3 ; C4 CI CM d d max d min d32 D

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

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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

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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

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

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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

251

252

6 Emulsion Polymerization

Fig. 6.2.

Intervals in the batch emulsion polymerization.

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

6.2 Features of Emulsion Polymerization

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

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6 Emulsion Polymerization

quently, the monomer concentration in the polymer particles is roughly constant during Interval II. The transport of reactants (monomers, chain-transfer agents) with water solubility lower than that of the styrene from monomer droplets to polymer particles may be diﬀusionally limited. Because of the polymerization and monomer transport, the polymer particles grow in size, and after some time the monomer droplets disappear. This marks the end of Interval II. The monomer conversion at which Interval II ends depends on the capability of the polymer particle to be swollen by monomer. The higher the maximum swelling, the earlier the monomer droplets disappear. In general, the more water-soluble the monomer, the higher the maximum swelling, and hence the lower the monomer conversion at the end of Interval II. Thus, the transition from Interval II to Interval III occurs at about 40% conversion for styrene and at about 15% conversion for vinyl acetate. This means that most monomer polymerizes in Interval III (Figure 6.2(d)). In this interval, monomer concentration in the polymer particles decreases continuously. The ﬁnal product is a waterborne, concentrated (50–60 wt.% solids) dispersion of tiny (80–500 nm in diameter) polymer particles called latex. In a semicontinuous reactor in which monomers, surfactant, initiator, and water may be continuously fed into the reactor, emulsion polymerization does not follow the sequence of events described above. Thus, slow monomer feed and fast surfactant feed may lead to a system composed of polymer particles and micelles (Figure 6.3(a)). The system will contain only monomer-swollen polymer particles if both monomer and surfactant are fed slowly (Figure 6.3(b)). On the other hand, a fast monomer feed and a low surfactant feed will lead to a system containing monomer droplets and polymer particles (Figure 6.3(c)). 6.2.2

Radical Compartmentalization

From a mechanistic point of view, the compartmentalization of radicals among a huge number of polymer particles is likely to be the most distinctive feature of emulsion polymerization. This refers to the fact that the radicals are distributed among the diﬀerent particles, and hence radicals in diﬀerent particles cannot terminate between them. Consequently, the overall concentration of radicals is higher than in homogeneous (bulk and solution) free-radical polymerization, leading to a higher polymerization rate. In addition, the compartmentalization of radicals in a large number of particles results in a longer lifetime of the radicals, which leads to higher molecular weights. 6.2.3

Advantages of Emulsion Polymerization

Free-radical polymerization is a very exothermic process, liable to suﬀer thermal runaways. Compared with bulk and solution polymerization, emulsion polymeriza-

6.2 Features of Emulsion Polymerization

Fig. 6.3.

Species present in semicontinuous emulsion polymerization.

tion is advantageous because the low viscosity of the latex allows high heat removal rates. In addition, because of its high heat capacity, the water absorbs large amounts of heat with only a moderate increase in temperature. This allows combination of high polymerization rates and good temperature control. Compared with solution polymerization, emulsion polymerization is environmentally friendly, as water is used as the reaction medium. In addition, the low viscosity of the system allows easy removal of both unreacted monomer and VOCs. Because of the compartmentalization of the radicals, emulsion polymerization overcomes the limita-

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6 Emulsion Polymerization

tions imposed by the kinetics of bulk and solution polymerization, and high polymerization rates and high molecular weights can be achieved simultaneously. Some valuable products with applications in paper coating, leather treatment, binders for nonwoven fabrics, additives for paper, textiles and construction materials, impact modiﬁers for plastic matrices, and diagnostic tests and drug delivery systems, can only be produced by emulsion polymerization. In addition, when needed (for example, for rubber for tires) latexes are easy to process into dry polymer. The main disadvantage of emulsion polymerization is that the product contains emulsiﬁer and residues of initiator, which give it water sensitivity.

6.3

Alternative Polymerization Techniques

Dispersed polymers are also achieved by inverse emulsion polymerization, miniemulsion polymerization, dispersion polymerization, and microemulsion polymerization. Inverse emulsion polymerization is used in the preparation of solventbased dispersions of polymers produced from water-soluble monomers. The process is similar to conventional emulsion polymerization but the dispersed phase is an aqueous solution of the monomer and the continuous phase is an organic solvent [10]. In miniemulsion polymerization [11], the size of the monomer droplets is substantially reduced (dd ¼ 50–1000 nm) by combining a suitable emulsiﬁer and an eﬃcient emulsiﬁcation apparatus, and stabilizing the resulting monomer miniemulsion against diﬀusional degradation by using a costabilizer (hydrophobic, low molecular weight compound). The available surfactant adsorbs on the large surface area of the droplets, and hence no micelles are formed. When the initiator is added to the system, the radicals enter the monomer droplets, which become polymer particles. Droplet nucleation minimizes the diﬀusional limitations encountered in conventional emulsion polymerization and allows the incorporation of water-insoluble compounds (monomers, polymers, catalysts, catalytic chain-transfer agents, inorganic materials, agents for controlled radical polymerization) in the reaction loci. In the last few years, plenty of new applications have been discovered. These applications include:

production of high-solids, low-viscosity latexes; stable operation in continuous polymerization reactors; controlled radical polymerization in dispersed media; catalytic polymerization; encapsulation of inorganic solids; incorporation of hydrophobic monomers; production of hybrid polymer particles; miniemulsion polymerization in nonaqueous media; anionic polymerization; step polymerization in aqueous dispersed media;

6.3 Alternative Polymerization Techniques

Fig. 6.4.

Dispersion polymerization.

production of low molecular weight polymers in dispersed media; preparation of latexes with special particle morphology.

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

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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.

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6 Emulsion Polymerization

Morton–Flory–Huggins equation [21] being the most commonly applied. For a multimonomer system, the parameters of the Morton–Flory–Huggins are not usually available, and it has been shown [22] that for solids contents typical of commercial latexes (> 50 wt.%) the use of the Morton–Flory–Huggins equation does not provide signiﬁcant advantages over the use of partition coeﬃcients. In the latj ter case, the system of algebraic equations (5) and (6) needs to be solved, where Ki is the partition coeﬃcient of monomer i between phase j and the aqueous phase, j fi the volume fraction of monomer i in phase j, fww the volume fraction of water in the aqueous phase, fpp the volume fraction of polymer in the polymer particles, Vp the volume of polymer particles, Vd the volume of monomer droplets, Vw the volume of the aqueous phase, and Vi ; Vpol , and W are the volumes of monomer i, polymer, and water, respectively. Equilibrium equations: j

j

Ki ¼

fi fiw

j ¼ polymer particles; droplets

ð5Þ

Material balances: fpp þ

X

p

fi ¼ 1

i

fww þ

X

fiw ¼ 1

i

X

fid ¼ 1

ð6Þ

i p

Vp fi þ Vd fid þ Vw fiw ¼ Vi Vw fww ¼ W Vp fpp ¼ Vpol Eﬃcient methods for solving Eqs (5) and (6) are given elsewhere [19, 20]. 6.4.2

Average Number of Radicals per Particle

The average number of radicals per particle n is given by Eq. (7). X

nNn n¼ X Nn

ð7Þ

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

6.4 Kinetics of Emulsion Polymerization

Fig. 6.5.

Processes controlling the average number of radicals per particle.

initiators. Often, these radicals are too hydrophilic and cannot directly enter the hydrophobic phases (polymer particles, monomer droplets, micelles). Therefore, they must stay in the aqueous phase until they polymerize, thus adding a number of monomer units and becoming hydrophobic enough to enter the organic phases. As the concentration of monomer in the aqueous phase is low, the period needed to reach the critical length for entry may be long and a substantial fraction of the radicals may terminate in the aqueous phase, resulting in a low initiation eﬃciency. In conventional emulsion polymerization, the total area of the polymer particles is much larger than that of the monomer droplets; therefore most of the radicals are captured by the polymer particles. There is some debate about the mechanism for radical entry; diﬀusional [23] and propagational [24]. The diﬀusional model is more general, whereas the propagational model is a simpler approach applicable to many homopolymerizations. The rate of radical entry can 3 also be expressed as Eq. (8), where ka is the entry rate coeﬃcient [maqueous phase mol1 s1 ] and ½Rw the concentration of radicals in the aqueous phase [mol m3 aqueous phase ]. Rate of entry ¼ ka ½Rw

ðradicals/particle sÞ

ð8Þ

Equation (8) is a simpliﬁcation because it assumes that all radicals are able to enter the polymer particles, but the radicals directly produced from inorganic initiators are too hydrophilic to be able to enter a hydrophobic phase. On the other hand, the radicals generated from radical desorption (see below) are hydrophobic and able to enter the polymer particles regardless of their length. Once inside the polymer particles, the radicals undergo the classical mecha-

261

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6 Emulsion Polymerization

nisms of free-radical polymerization: propagation, termination, chain transfer, and so on. Obviously, in order to suﬀer a bimolecular termination reaction, the particle should contain two or more radicals, as radicals in diﬀerent particles cannot terminate between them. The rate of termination in a particle with n radicals is given by Eq. (9). 2kt nðn 1Þ ¼ 2cnðn 1Þ Rate of Termination ¼ vp NA

radicals particle s

ð9Þ

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

ð10Þ

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

ð11Þ

The rate of radical exit from particles with n radicals is the product of kdðnÞ and n [Eq. (12)], and kdðnÞ is given by Eq. (13). Rate of Exit ¼ kdðnÞ n ðradicals/particle sÞ 0 1 n lN p A gNA @ n kdðnÞ ¼ l 1 lNp þ k p ½Mw þ 2k tw ½Rw hm

ð12Þ

ðs1 Þ

ð13Þ

In Eq. (13), the term n=n takes account of the fact that in particles with n radicals, the desorption rate is proportional to n but the re-entry of newly desorbed small radicals is proportional to n. Because radical entry, exit, and termination are stochastic events, particles have a diﬀerent number of radicals and the number of radicals in a given particle varies stochastically with time. Equation (14) gives the population balance of particles

6.4 Kinetics of Emulsion Polymerization

with n radicals; it includes the concentration of radicals in the aqueous phase, and hence the material balance for these radicals is needed from Eq. (15), where f is the eﬃciency factor of the initiator radicals, kI the rate coeﬃcient for initiator decomposition [s1 ], [I] the concentration of the thermal initiator in the aqueous phase [mol m3 aqueous phase ], and k tw the termination rate in the aqueous phase 3 1 1 s ]. [maqueous phase mol dNn ¼ ka ½Rw Nn1 þ kdðnþ1Þ ðn þ 1ÞNnþ1 þ cðn þ 2Þðn þ 1ÞNnþ2 dt ðka ½Rw Nn þ kdðnÞ nNn þ cnðn 1ÞNn Þ n ¼ 0; 1; 2; 3 . . .

ð particles/sÞ

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

ð14Þ 3 ðmol/maqueous phase sÞ ð15Þ

For most practical cases, the pseudo steady-state assumption can be applied to the radicals in the polymer particles and in the aqueous phase. Therefore, Eqs. (14) and (15) are converted into algebraic equations by making the left-hand sides equal to zero. The exact solution for n, under pseudo steady-state conditions, has been given in terms of Bessel functions [26], but their use is not friendly. A simpler and still accurate equation for n is Eq. (16), with C deﬁned in Eq. (17). [27]. n¼

C¼

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

ð16Þ

ð17Þ

Equations (16) and (17) should be solved together with Eq. (15). The solution of this system of algebraic equations includes the three limiting cases of the pioneering work of Smith and Ewart [28]. When the exit rate of radicals is zero (kd ¼ 0) and the termination of the entering radical and that existing in the polymer particle is instantaneous, the average number of radicals per particle is n ¼ 0:5. This is known as Smith–Ewart Case 2 and corresponds to systems in which (1) there is no chain transfer to small molecules (that is, monomers and CTAs) or these small molecules are highly water insoluble, and (2) the polymer particles are relatively small (typically dp < 200 nm). Because for Smith–Ewart Case 2 n is constant, the polymerization rate is directly proportional to the number of polymer particles. In systems in which the entry rate per particle is limited and the rate of radical desorption is relatively high (large kd ), n f 0:5. This is known as Smith–Ewart Case 1 and corresponds to systems with (1) relatively water-soluble monomers or relatively water-soluble CTAs, (2) small particles (dp < 100 nm), (3) a low rate of generation of radicals from the initiator, and (4) a large number of particles. Under these circumstances, n can be easily calculated by means of Eq. (18).

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6 Emulsion Polymerization

n¼

ka ½Rw 2ka ½Rw þ kd

ð18Þ

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

ð19Þ

For a given solids content under Smith–Ewart Case 3 kinetics, n is inversely proportional to the number of polymer particles, and hence the polymerization rate is independent of the number of polymer particles if aqueous-phase termination is negligible. Otherwise, the polymerization rate increases with Np . 6.4.3

Number of Polymer Particles

Particle nucleation may occur through heterogeneous nucleation, homogeneous nucleation, and coagulative nucleation (Figure 6.6). The rate of particle generation depends on the mechanism considered. Heterogeneous Nucleation The rate of formation of polymer particles by heterogeneous nucleation is expressed by Eq. (20), where R nuc is the nucleation rate, kam the rate coeﬃcient for radical entry into the micelles, and Nm the number of micelles in the reactor given by Eq. (21). 6.4.3.1

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

ðSw CMCVw ÞNA nm

ð particles/m 3 sÞ

ðmicellesÞ

ð20Þ ð21Þ

In Eq. (21), CMC is the critical micellar concentration [mol m3 aqueous phase ] and hence CMC Vw is the amount of surfactant dissolved in the aqueous phase; nm is the aggregation number (average number of molecules of surfactant per micelle); and Sw [mol] is the amount of surfactant that is in the aqueous phase forming micelles and dissolved in such a phase. Sw can be calculated by means of the overall material balance for the surfactant, Eq. (22), where ST [mol] is the total

6.4 Kinetics of Emulsion Polymerization

Fig. 6.6.

Particle nucleation mechanisms in emulsion polymerization.

amount of surfactant in the reactor, as [m 2 mol1 ] the parking area – that is, the area of the saturated surface of the polymer particles covered by 1 mol of surfactant – and A p [m 2 ] the total surface area of the polymer particles given by Eq. (23). Sw ¼ ST Ap ¼

Ap as

4:83Np0:33

ð22Þ Vpol fpp

!0:66 ð23Þ

265

266

6 Emulsion Polymerization

Assuming no termination of radicals in the aqueous phase and that during nucleation n ¼ 0:5, Smith and Ewart [28] solved Eqs. (20)–(23), obtaining Eq. (24) for the dependence of the number of particles on surfactant and initiator concentrations, where rv [m 3 s1 ] is the volumetric growth rate of one polymer particle.

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

ð24Þ

The fulﬁllment of this equation, in particular the 0.6th power dependence of Np with respect ST , is often considered as a proof of the occurrence of micellar nucleation. However, one should be aware of the assumptions used in the derivation of Eq. (24). Actually, when radical desorption is taken into account, the solution of Eqs (20)–(23) leads to Eq. (25) [29, 30], where z approaches unity as the water solubility of the monomer increases.

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

0:6 a z a 1

ð25Þ

Homogeneous Nucleation In homogeneous nucleation, particles are formed by precipitation of the growing oligoradicals once they exceed the critical length that makes them insoluble in water. The water solubility of the oligoradicals depends on their composition. In this context, the critical length of the oligoradicals formed from an inorganic water-soluble initiator such as potassium persulfate (that is, one containing an inorganic fragment) will be longer than that of the oligoradicals formed from desorbed radicals that are much more hydrophobic. Therefore, the rate of formation of particles by homogeneous nucleation, R nuc , is the rate of polymerization of oligoradicals of critical length as expressed by Eq. (26), where ½Mw is the concentration of monomer in the aqueous phase, and R jcrit and Micrit are the number of oligoradicals of critical length (with jcrit > icrit ) formed from the initiator and from desorbed radicals, respectively. 6.4.3.2

R nuc ¼ k p ½Mw

ðR jcrit þ Micrit Þ V

ð particles/m 3 sÞ

ð26Þ

R jcrit and Micrit are calculated from the balances of radicals of type R and M in the aqueous phase assuming that pseudo steady-state conditions apply [Eqs. (27) and (28)]. jcritdþ1 d1 a2

R jcrit ¼ a1 Micrit ¼

kd nNp icrit a k p ½Mw 1

2f kI INA k p ½Mw

ðradicalsÞ

ðradicalsÞ

ð27Þ

ð28Þ

6.5 Molecular Weights

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

a1 ¼

k p ½Mw þ 2k tw ½Rw þ ka a2 ¼

Np Nm þ kam NA Vw NA Vw

k p ½Mw k p ½Mw þ 2k tw ½Rw

ð29Þ

ð30Þ

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

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

1 ð particles/m 3 sÞ V

ð31Þ

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

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

ð32Þ ð33Þ

6.5

Molecular Weights

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

267

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6 Emulsion Polymerization

which diﬀerent particles may have a diﬀerent number of radicals. In addition, the number of radicals in a given particle varies with time. Therefore, the length of the macromolecules formed in a given particle depends on the number of radicals in the particle at the moment in which the macromolecule was formed. Furthermore, the architecture of the polymer formed depends on both the formulation and the kinetics of the process. Thus, for monofunctional monomers with a polymerization scheme that does not include chain transfer to polymer, linear polymers are obtained. On the other hand, branched and crosslinked polymers are obtained when multifunctional monomers are included in the formulation and when chain transfer to polymer is operative. Mathematical models for the calculation of the molecular weight distribution of linear [36] and nonlinear [37–40] polymers are available, but a detailed discussion of this issue is out of the scope of the present chapter. Instead, some simpliﬁed equations for the calculation of the molecular weights of linear polymers in the limiting cases of Smith and Ewart [28] will be presented. 6.5.1

Linear Polymers

As discussed above, the MWD depends on the number of radicals per particle. Smith and Ewart [28] distinguished three limiting cases: In Case 1 n f 0:5; in Case 2 n ¼ 0:5; and in Case 3 n g 0:5. In Cases 1 and 2, the probability of having particles with more than one radical is almost negligible, and hence the system may be considered to be formed by particles with no radicals and particles with one radical (zero–one system). In Case 3, the average number of radicals is large and the kinetics is close to bulk polymerization. Zero–One System (Smith–Ewart Cases 1 and 2) In a zero–one system, the inactive polymer chains are formed in particles containing one radical by chain transfer to monomer and by instantaneous termination upon entry of one radical. The balance for inactive chains of length m in particles with one radical (N1 ) is given by Eq. (34), where R m is the total number of radicals of length m in particles N1 . 6.5.1.1

dMm ¼ k tr; M ½Mp R m þ ka ½Rw R m dt

m ¼ 1; 2; 3; . . .

ðmacromolecules/sÞ

ð34Þ

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

y X m¼1

m k Mm

ð35Þ

6.5 Molecular Weights

Combination of Eqs. (34) and (35) yields Eq. (36), where Ruk is the generation rate of the kth-order moment of the distribution of inactive chains and mk is the kthorder moment of the distribution of active chains.

Ruk ¼ ðk tr; M ½Mp þ ka ½Rw Þ

mk V

ð36Þ

The balance for active chains of length m in particles with one radical (N1 ) is obtained from Eq. (37). X dR m ¼ ka ½Rw N0 þ k tr; M ½Mp R m kd N1 dm¼1 þ k p ½Mp ðRm1 R m Þ dt k tr; M ½Mp R m ka ½Rw R m ¼ 0

ð37Þ

The ﬁrst moments of the distribution of active polymer chains can be calculated from Eq. (37) by applying the pseudo steady-state assumption to the active chains [Eqs. (38)].

m0 ¼

m1 ¼

X

R m ¼ N1 ¼ nNp

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

m 2 ¼ m1 1 þ

ka ½Rw þ k tr; M ½Mp ! 2k p ½Mp

ð38Þ

ka ½Rw þ k tr; M ½Mp

The cumulative number-(Mn ) and weight-(M w ) average molecular weights can be calculated from the moments of the MWD by means of Eqs. (39), where PM is the average molecular weight of the repeating unit in the polymer chain.

Mn ¼

n1 PM ; n0

Mw ¼

n2 PM n1

ð39Þ

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

Mni ¼

Rv1 PM ; Rv0

Mwi ¼

Rv2 PM Rv1

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

ð40Þ

269

270

6 Emulsion Polymerization

M ni ¼

A

M wi ¼

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

PM

ð41Þ

2k p ½Mp m2 PM ¼ 1 þ PM A 2M ni m1 ka ½Rw þ k tr; M ½Mp

For the Smith–Ewart Case 2, ka ½Rw g k tr; M ½Mp and then Eq. (42) holds. M ni A

k p ½Mp ka ½Rw

PM

ð42Þ

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

ka ½Rw ¼ 2f kI ½I M ni A

NA Vw Np

k p ½Mp Np PM 2f kI ½INA Vw

ð43Þ

ð44Þ

Consequently the molecular weight, as well as the polymerization rate, increases with the number of polymer particles, and hence it is possible to increase both Rp and the molecular weights at the same time by conveniently adjusting the formulation to increase the number of polymer particles. This is a speciﬁc feature of emulsion polymerization that cannot be achieved in any homogeneous (bulk or solution) free-radical polymerization. For the Smith–Ewart Case 1, k tr; M ½Mp g ka ½Rw and then Eq. (45) applies. M ni A

kp PM k tr; M

ð45Þ

Therefore the molecular weights are controlled by chain-transfer reactions and are independent of Np . Pseudo Bulk System (Smith–Ewart Case 3) For Smith–Ewart Case 3, the number of radicals per particle is large and the kinetics approaches bulk polymerization. In this case, the concentration of radicals in the polymer particles is given by Eq. (19) and the molecular weight is mainly controlled by chain transfer and bimolecular termination. The rate of generation of polymer of length m in particles with i radicals is given by Eq. (46). 6.5.1.2

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

ðmacromolecules/sÞ ð46Þ

From Eq. (46) the moments of the inactive chains can be calculated according to Eqs. (47), and the moments of the active chains can be calculated by applying the pseudo steady state in the material balance of the active radicals [Eqs. (48) and (49)]. Rn0 ¼ ð2cd þ cc Þ

m02 m þ k tr; M ½Mp 0 Np V V

Rn1 ¼ ð2cd þ 2cc Þ

m 0 m1 m þ k tr; M ½Mp 1 Np V V

Rn2 ¼ ð2cd þ 2cc Þ

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

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

ð47Þ

ð48Þ

m1 ¼

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

m2 ¼

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

ð49Þ

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

Mni ¼

Mwi

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

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

ð50Þ

271

272

6 Emulsion Polymerization

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

k p ½Mp Np ð2cd þ cc Þm 0

PM

M wi

m cc m1 2cd þ cc cc 1þ ¼ 2þ PM A Mni PM m1 ðcd þ cc Þm 0 cd þ cc 2ðcd þ cc Þ

ð51Þ

Therefore: M wi ¼ 2Mni

if bimolecular termination is only by disproportionation; and

3 M wi ¼ M ni 2

if bimolecular termination is only by combination:

Because for a given polymer content cc and cd are proportional to Np , the molecular weights are independent of the number of particles. On the other hand, if chain-transfer reactions are predominant over bimolecular termination, the instantaneous molecular weights are given by Eqs. (52). M ni ¼

kp PM k tr; M

ð52Þ

M wi ¼ 2Mni For this case, the molecular weights are also independent of Np and depend only on the ratio k p =k tr; M , and polydispersity is equal to 2. 6.5.2

Nonlinear Polymers: Branching, Crosslinking, and Gel Formation

In contrast to linear polymers, which, once they are formed, preserve their structure (molecular size) during the rest of the process, the molecular weight of the nonlinear polymers evolves throughout the whole process because they may become active and inactive several times during that process. Thus, inactive nonlinear chains formed earlier in the process may become active through either a chain-transfer reaction to polymer or a propagation to pendent double bonds, further polymerizing and modifying their structure. The microstructure of these nonlinear polymers is more complex than that of linear polymers, and besides the MWD of the polymer, other properties such as the level of branching and crosslinking density are required to fully characterize the polymer. In addition, large polymer networks (gel) are formed in some cases. The accurate calculation of the structure of these nonlinear polymers is not an easy task in homogeneous systems [41, 42], and it is even more complex in emul-

6.6 Particle Morphology

sion polymerization, because of the compartmentalized nature of the system. Nevertheless, some solutions have been reported [37–40, 43–46]. The discussion of these approaches is outside the scope of this chapter. Branches are formed by chain transfer to polymer (either intermolecular or intramolecular) or by propagation to pendent double bonds (PDBs). Terminal double bonds may be formed by chain transfer to monomer, termination by disproportionation, and b-scission at relatively high temperatures. PDBs may also be present if multifunctional monomers are used in the formulation. Intermolecular chain transfer to polymer and propagation to PDBs produces long-chain branches (LCBs), and intramolecular chain transfer to polymer, so-called backbiting, produces short-chain branches (SCBs). Chain transfer to polymer occurs through hydrogen abstraction either from the backbone or from a side group. A crosslink point is a tetrafunctional unit that links two polymer chains, namely an H-shaped structure. The mechanisms leading to tetrafunctional units are: intermolecular chain transfer to polymer followed by bimolecular termination by combination of branched radicals, or propagation of a growing branched radical on a terminal or pendent double bond, or polymerization of formulations including multifunctional monomers (often called crosslinkers).

In the free-radical polymerization of vinylic monomers, gel is formed if the following conditions are fulﬁlled: (1) a mechanism leading to LCBs is operative, for example chain transfer to polymer, and (2) a mechanism that links polymer chains is also operative, for example termination by combination or propagation to pendent double bonds. Obviously, if multifunctional monomers are homopolymerized or copolymerized with vinylic monomers, gel is also formed. In this case, the predominant reaction linking polymer chains and leading to polymeric networks is the propagation to pendent double bonds of the multifunctional monomers. Under these circumstances, polymer chains of increasing density of branching and crosslinking are formed and their molecular weight will rapidly increase, leading to the formation of gel polymer.

6.6

Particle Morphology

Latexes made out of composite polymer particles, that is, particles containing different phases, present deﬁnitive advantages in many applications. Thus, particles formed by an elastic core and a hard shell are used as impact modiﬁers for polymer matrices. Hard core–soft shell particles are particularly useful for paints because they have a low minimum ﬁlm formation temperature and are not sticky at higher temperatures. Hollow particles are eﬃcient opaciﬁers, and hybrid polymer– polymer particles, such as epoxy–acrylic polymer particles, combine the properties

273

274

6 Emulsion Polymerization

Fig. 6.7.

Development of particle morphology.

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

6.7 Living Polymerization in Emulsion

portional to the interfacial tensions. The ﬁnal morphology heavily depends on the kinetics of cluster migration [47–49]. Metastable morphologies can be achieved by working under starved conditions (high internal viscosity of the particles) and promoting grafting reactions (low interfacial tensions). Equilibrium morphologies may be attained if the internal viscosity of the particle is low, and if the polymers are very incompatible (high interfacial tensions resulting in high van der Waals forces). The equilibrium morphology is the one that minimizes the interfacial energy of the system; it depends on the polymer–polymer and polymer–water interfacial tensions [47].

6.7

Living Polymerization in Emulsion

Free-radical polymerization (FRP) is a widespread technique for preparing longchain polymers (see Chapter 4). Its success is mainly due to the broad range of monomers that can be readily polymerized by this technique, to its tolerance toward functional monomers, and to the mild reaction conditions required to run the polymerization. Moreover, as discussed in the preceding sections in this chapter, emulsion polymerization (and in general every polymerization reaction carried out using water as the continuous medium) has alleviated all the residual inconveniences of FRP. In fact, the corresponding processes do not involve a large demand for organic solvents, they easily achieve large monomer conversions, and they exhibit reduced problems of heat removal, thus making the process intrinsically safer. On the other hand, careful control of the polymer microstructure (in the broad sense deﬁned in Section 6.1) is still an issue. With reference to the macromolecule structure (that is, average molecular weight, polydispersity, composition, monomer sequences, chain architecture, degree of crosslinking, and chain-end functionalization), a novel process has recently been proposed with much greater potential in terms of control capacity: living radical polymerization (LRP). By this speciﬁc process it becomes possible to obtain block copolymers by FRP, ﬁlling the gap between the free-radical polymerization technique and other polymerization techniques such as living anionic or cationic polymerization. 6.7.1

Chemistry of LRP

In conventional FRP, termination mainly by bimolecular combination limits the chain lifetime to a small fraction of the entire process time. Therefore, changes in the operating conditions (such as monomer concentration and composition, viscosity, temperature, and so on) aﬀect the structure of the polymer chains produced at diﬀerent stages of the process, thus increasing the product heterogeneity. In a living polymerization, instead, these changes are equally distributed among all the polymer chains, which grow uniformly during the whole reaction. Moreover, as

275

276

6 Emulsion Polymerization

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

6.7 Living Polymerization in Emulsion

erage DP of the polymer is given by the ratio between the converted monomer, M0 XT (where XT is the monomer conversion and M0 the initial amount of monomer), and the total number of polymer chains. Note that some irreversibly terminated chains are produced anyhow, since these reactions are always taking place in competition with the activation/deactivation reactions. If the fraction of terminated chains remains negligible compared to the initial amount of initiator, RX, this last value corresponds also to the concentration of the dormant chains in the system and the DP grows linearly with conversion (DP ¼ M0 XT =RX). Moreover, if the number of active periods each chain experiences is large enough (that is, if the number of monomer units added per active period is small enough), the polymer growth is distributed throughout the duration of the process and all the chains grow uniformly, leading to low polydispersity of the chain length distributions. Diﬀerent living mechanisms are available and the main ones are brieﬂy enumerated in the following sections. Nitroxide-mediated Polymerization (NMP) The ability of stable nitroxide radicals to react with carbon-centered radicals and to act as radical inhibitors was already known at the beginning of the 1980s [54], when Solomon and co-workers showed that the reversible reaction of nitroxides with growing polymer chains can be used to produce low-DP polymers [55]. But it is only in the 1990s with the work of Georges and co-workers [56, 57] that this novel polymerization technique, and in general LRP, received the attention they deserved. This living mechanism [Eq. (a)] consists of the reversible combination of the growing radical chains, R n , and the so-called ‘‘persistent radical species’’, X (the nitroxide radical group), to form dormant polymer chains, R n X: 6.7.1.1

kde

Rn þ X S RnX kac

ðaÞ

Today, many diﬀerent routes are known which use diﬀerent persistent radicals [54–58]. Among these, TEMPO is by far the most widely used, even though it suffers very limited applicability to monomers unlike styrene, and requires high operating temperatures (about 120–140 C). More recent studies were aimed to reduce the operating temperature and to broaden the monomer applicability so as to enlarge the spectrum of block copolymers accessible by this technique [58]. Despite these eﬀorts, the range of application remains quite limited. Atom-transfer Radical Polymerization (ATRP) Atom-transfer radical polymerization (ATRP) was ﬁrst reported by Kato et al. [59] and by Wang et al. in 1995 [60, 61]. This mechanism is based on the so-called atom-transfer radical addition reaction, and it is catalyzed by a metal: the homolytic cleavage of the bond in an organic halide occurs through transfer of the halogen to the metal complex, accompanied by oxidation of the metal atom. The catalytic cycle is closed by back-transfer from the transition metal to the ﬁnal adduct of the halogen. It is clear that, if the radical produced can undergo a few propagation steps 6.7.1.2

277

278

6 Emulsion Polymerization

before participating in the back-transfer, and if this product is still able to undergo another transfer cycle, this reaction can be used to produce the same exchange between active and dormant states as is found in NMP. The resulting reversible reaction is represented by Eq. (b), where X indicates the halogen atom, MeðnÞ the metal with the corresponding oxidation state and Li the ligand. kde

R n þ X –Meðnþ1Þ =Li S R n X þ MeðnÞ =Li kac

ðbÞ

ATRP owes most of its success to its high compatibility with many diﬀerent monomers, such as styrene, acrylates, methacrylates, (meth)acrylamides, and acrylonitrile, which made this technique readily available for the production of several new block copolymers [62]. Even though the majority of the work has been done with copper as the transition metal, styrene ATRP has been carried out using Fe-, Ru-, Ni-, Pd-, and Co-based systems [62]. Note that ATRP does not require the high reaction temperature typical of NMP and this is also part of the success of this polymerization technique. Diﬀerent ligands have been used to solubilize the copper atom and it has been noticed that they not only prepare the copper for the reaction, but can also modify the reactivity of the metal toward both the activation and deactivation reactions. Actually, the presence in the system of a metal, the need for complex ligands to solubilize it, and the deep color typically imparted by this complex to the ﬁnal polymer (if not removed) represent the major drawbacks of this process. Thus far, chlorine and bromine have been used successfully as the halogen atoms, whereas iodine gives rise to side reactions [63]. Degenerative Transfer (DT) As mentioned above, in both NMP and ATRP the exchange between the active and the dormant state is based on a reversible (although diﬀerent) termination mechanism. Therefore, the exchange directly aﬀects the radical concentration. In LRP by DT, instead, this exchange is carried out by direct transfer of the o-end group between an active and a dormant chain. When an iodine atom is used as end group, the reaction can be summarized by Eq. (c), where R n I indicates the generic dormant species with iodine. 6.7.1.3

kex

Rn þ Rm I ! Rm þ RnI

ðcÞ

Therefore, the main diﬀerence from the previous two systems is that this living mechanism does not form new radicals and a conventional initiator is needed to start and ‘‘sustain’’ the reaction. The initial amount of this species has to be properly selected. As a matter of fact, since the living reaction of Eq. (c) is not aﬀecting the radical concentration, the ﬁnal concentration of the chains terminated by bimolecular combination will be half of the initial concentration of initiator. Therefore, the initial concentration of the species carrying the iodine group (in the following simply called the ‘‘transfer agent’’) determines the ﬁnal DP of the polymer, provided that the initiator concentration is small compared to that of the transfer agent.

6.7 Living Polymerization in Emulsion

Only a few papers have appeared in the literature dealing with LRP by DT [64– 66], and the applications are almost completely limited to the homopolymerization of styrene. In this case, it was possible to obtain good control of the ﬁnal CLD, with polydispersity values as low as 1.3–1.4. Better performances are diﬃcult to obtain with styrene, mainly because of the limited transfer activity of the iodine atoms. This is the main reason for the very poor results obtained when applying this process to the polymerization of acrylates (for example, n-butyl acrylate), and for the complete lack of control reported for other monomers [64–66]. Reversible Addition–Fragmentation Transfer (RAFT) Polymerization The RAFT process can be regarded as a special case of degenerative transfer. As shown in Eq. (d), the reaction proceeds through the direct interaction of an active and a dormant chain with the formation of a reaction intermediate involving both chains [67, 68]. At this stage, the reaction can either go back, forming the initial radical again, or proceed forward with the transfer of the Y ¼ CðZÞ–Y moiety from the dormant to the active chain, which is now identiﬁed as the transfer species. 6.7.1.4

kadd

ðþÞ

kfrag

R m YC . ðZÞYR n ! R m þ R n YCðZÞY R n þ R m YCðZÞY !

ðdÞ

ðÞ

kfrag

Note that the best results have been reported when using a sulfur atom as the Y group [67–70]. The process is started by introducing into the system a so-called RAFT agent, the structure of which can be described as R–Y –CðZÞ ¼ Y. The correct choice of the R group, or leaving group, is of paramount importance, not only because this is going to be one polymer chain end (the other end being occupied by the RAFT group), but mainly because it inﬂuences the initial reactivity of the RAFT agent. From Eq. (d), it is in fact clear that in the ﬁrst addition reaction an intermediate species is formed, having the R group on one side and the polymer radical on the other. The fate of such a species – that is, the probability that the following fragmentation reaction proceeds backward or forward – depends mainly upon thermodynamic considerations, namely upon which, the polymer radical or the leaving-group radical, is the more stable. In other words, if the leaving group generates a radical that is too unstable compared to the polymer radical, the RAFT is always going to proceed backward, and thus no control is actually achieved. Similar considerations apply in block polymerization of two monomers, where the ﬁrst block can be regarded as a special case of the R group. Accordingly, care must be given to the correct choice of which monomer has to be polymerized ﬁrst. Even though the most satisfactory results have been obtained by RAFT polymerization of styrene (Figure 6.8 shows an example of the application of RAFT to bulk styrene polymerization, indicating the good control of polymer growth), the process is also very eﬀective for many other monomers, such as acrylates and methacrylates [67–70]. Moreover, the operating temperatures used to carry out this polymerization are usually close to those typical of conventional radical polymerization

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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

6.7 Living Polymerization in Emulsion

by diﬀusion, deactivation soon becomes the favored reaction path anyway. This is often referred to as the ‘‘persistent radical eﬀect’’ [72, 73]. To explain this behavior, let us ﬁrst point out that, because of the nature of NMP, where new radical chains can be created from dormant ones by activation, this mechanism does not require a radical initiator. Accordingly, when the reaction starts, dormant chains start to activate, building up a radical concentration. At the same time, the two bimolecular termination reactions, termination and deactivation, start to operate. But, while deactivation simply produces a dormant chain again, termination subtracts polymer chains from the activation/deactivation equilibrium and, according to simple stoichiometric arguments, trapping radicals (nitroxides) are accumulated. The ultimate eﬀect of this accumulation of trapping radicals is that deactivation becomes faster and faster, thus shortening the active lifetime of the active radicals and their concentration, but also decreasing the ﬁnal amount of irreversibly terminated chains. Fischer et al. derived Eq. (53) to evaluate the concentration of radicals for an NMP [72], where ðRXÞ0 is the initial concentration of dormant chains. R¼

kac ðRXÞ0 3kde kt t

1=3 ð53Þ

This equation conﬁrms that the radical concentration steadily decreases in time. Albeit the living action in ATRP takes place by a diﬀerent reaction mechanism, the same arguments presented above for NMP can be used. It can be easily veriﬁed that the concentration of the metal in the reduced form, MeðnÞ [compare the corresponding reaction scheme, Eq. (b)], remains roughly constant during the whole process, and therefore the activation process can be approximated as a monomolecular process as in NMP. On the other hand, degenerative transfer and RAFT are characterized by completely diﬀerent kinetics when the polymerization is performed in a homogeneous system. Given the nature of the living mechanism of these two systems, based on a transfer reaction, radical concentration is not aﬀected by the living system. Thus, the persistent radical eﬀect is totally absent and the kinetics of DT and RAFT is identical to that of a conventional nonliving system. Accordingly, an initiator is necessary to sustain the reaction and the concentration of radical chains is set by the equilibrium between initiation and bimolecular termination; that is, the well known formula in Eq. (54), where Ri represents the rate of radical generation, holds in this case also. R ¼ ðRi =kt Þ 1=2

ð54Þ

No matter which is the living mechanism under consideration, it is fundamental to keep the radical concentration as low as possible to ensure a ﬁnal fraction of dead chains which is negligible with respect to the dormant polymer. In a bulk or solution polymerization, the ﬁnal concentration of dead chains is a function of the radical concentration only: high polymerization rates correspond to high dead

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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.

6.7 Living Polymerization in Emulsion

Fig. 6.9. Expected kinetics for LRP in emulsion: (a) RAFT/DT; (b) NMP/ATRP. i ¼ number of radicals per particle; r; fc ; fac ; fde ¼ frequencies of radical entry, bimolecular termination, activation, and deactivation, respectively.

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

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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

6.7 Living Polymerization in Emulsion

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

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6 Emulsion Polymerization

Fig. 6.10. Example of application of RAFT to miniemulsion polymerization for the formation of poly(methyl methacrylate)b-polystyrene [74]. Left: conversion proﬁle; right: evolution of molecular weight distribution as measured by GPC.

6.8

Emulsion Polymerization Reactors

Emulsion polymers are ‘‘products-by-process’’ whose microstructure and properties are determined during the polymerization. Therefore, the reactor type, the operation mode, and the control strategy play a key role in achieving an eﬃcient, safe, and consistent production of high-quality emulsion polymers. 6.8.1

Reactor Types and Processes

In principle, both stirred-tank reactors and tubular reactors, and combinations thereof, may be employed. Stirred-tank Reactors The stirred-tank reactor is the reactor most commonly used in emulsion polymerization. This reactor may operate in batch, semibatch, or continuous mode. A batch reactor is a closed system in which time is the only independent variable. The batch operation can be used for small-scale production of homopolymers from monomers with a relatively low heat of polymerization. However, the drawbacks associated with this type of operation limit its industrial use. These drawbacks are: 6.8.1.1

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

6.8 Emulsion Polymerization Reactors

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

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6 Emulsion Polymerization

most important modiﬁed tubular reactors include loop reactors, pulsed-ﬂow reactors, wicker-tube reactors, and Couette–Taylor ﬂow reactors. The continuous-loop reactor is probably the only tubular reactor used in commercial production of emulsion polymers [93, 94] and its use is limited to production of vinyl acetate homopolymers and copolymers (with ethylene and Veova 10) [95–97]. A continuous-loop reactor consists of a tubular loop that connects the inlet and the outlet of a recycle pump. The macromixing of such reactors is between that of plug-ﬂow and well-mixed reactors. Commonly, the loop recirculation rate is signiﬁcantly greater than the feed rate. Under these circumstances, the residence time distribution is very close to that of a CSTR [98]. Its high heat-transfer area/reactor volume ratio allows eﬃcient heat removal and hence high conversions in short residence times can be achieved. This results in a substantial reduction in the reactor volume. The main disadvantage of this reactor is that highly mechanically stable formulations are required to prevent shear-induced coagulation at high recycling rates. Pulsed-ﬂow reactors consist basically of a column in which a periodic external pulsation is provided. The goal of the pulsed ﬂow regime is to achieve highly eﬀective mixing, minimizing phase segregation, wall fouling, and tube plugging [99, 100]. The introduction of sieved plates, Raschig rings, or baﬄes is reported to improve mixing [101, 102]. However, these internal elements may be sources of coagulation, and cleaning coagulum from these reactors will be very diﬃcult. The wicker-tube reactor consists of a coiled tube which meanders between solid, ﬁxed, cylindrical supports. The heat removal capacity is high and it is claimed that the multiple changes in ﬂow direction allow the production of a polymer dispersion with a very low coagulum content [103, 104]. The Couette–Taylor reactor consists of two concentric cylinders of which the outer one is ﬁxed and jacketed, while the inner one rotates as a stirrer. In this way, the reaction takes place in the annulus formed between the two cylinders. For a speciﬁc conﬁguration, if the inner cylinder exceeds a certain speed the ﬂuid inside the gap develops counter-rotating toroidal vortices. The boundaries of the vortices represent a barrier for axial mixing, while radial mixing inside each vortex is good. Consequently the Couette–Taylor reactor may be modeled as a hydrodynamically formed train of continuous stirred tank reactors. Good radial mixing, which reduces phase segregation and plugging, and high heat removal capacity are the main characteristics of these reactors [105–108]. 6.8.2

Reactor Equipment

Despite the availability of diﬀerent reactors, commercial emulsion polymerization is mostly carried out in stirred-tank reactors or in reactors that have a similar macromixing behavior, such as the loop reactor. The mixing problems associated with tubular reactors, together with a lack of ﬂexibility in controlling product properties, lowers the attractiveness of tubular reactors for commercial production. Therefore the discussion will be limited to stirred-tank reactors.

6.8 Emulsion Polymerization Reactors

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

289

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6 Emulsion Polymerization

Empirical correlations for the key aspects of mixing, such as power consumption, emulsiﬁcation, liquid circulation, and mixing time, can be found elsewhere [111]. The advances in computational ﬂuid dynamics (CFD) combined with new techniques of measuring the local ﬂow pattern are likely to transform the whole ﬁeld. Heat Transfer Polymerization reactions are highly exothermic and the heat generated must be removed in order to control reactor temperature. In commercial emulsion polymerization, the heat removal rate from the reactor is often the factor limiting productivity. Both safety and product quality depend on the heat removal capacity of the reactor. For industrial reactors, it is often not suﬃcient to operate with a simple cooling jacket and other devices must be incorporated in the design [112]: 6.8.2.2

improved jacket design to increase turbulence, and hence the heat-transfer coeﬃcient (dimpled, half-pipe); cooled baﬄes (the use of internal coils is not an option because they tend to increase fouling and coagulation); external loop heat exchangers for the reaction medium – in this case, a mechanically stable formulation is required; external heat exchangers can be used also to cool the feedstreams in continuous and semicontinuous operations; reﬂux condensers. Increasing the agitation speed to increase the internal heat-transfer coeﬃcient is not an option because a high impeller speed can lead to coagulum formation. In any case, heat-transfer requirements are largely determined by the operation mode. Batch is the most critical operation because high polymerization rates are achieved due to the high monomer concentration. In semibatch mode, the heat generated can be easily controlled by the monomer feed rate. In these reactors, extra cooling is provided by the cold feed. In continuous mode, the continuous cold feed facilitates the control of the reactor temperature, particularly when the reactor temperature is high.

6.9

Reaction Engineering

Independently of the operation mode (batch, semibatch, or continuous), in wellmixed stirred-tank reactors the properties do not vary signiﬁcantly with the position in the reactor, and time is the only independent variable. Therefore, the necessary balances for the reactor design may be made at macroscopic level – that is, for the reactor as a whole.

6.9 Reaction Engineering

6.9.1

Mass Balances

Considering inlet and outlet streams in the reactor, the mass balance [Eq. (56)] for any species i results, dNi ¼ Fie Fis þ ðRi ÞV dt

ð56Þ

where Ni [mol] is the total amount of compound i in the reactor; Fie [mol s1 ] the inlet molar ﬂow rate of component i; Fis [mol s1 ] the outlet molar ﬂow rate of i, (Ri ) [mol s1 m3 ] the net generation rate of i in the reactor, and V [m 3 ] the reactor volume. Equation (56) applies for the particular combination of monomers, initiator, water, and emulsiﬁer, and for amounts of each polymerized monomer. Equation (56) also applies to the number of polymer particles and of precursor particles and the moments of the molecular weight distribution, although in those cases units other than moles should be used. The speciﬁc forms of the net generation rates are discussed below. Usually, the polymerization rate of monomer j is expressed as the rate of monomer consumption (Rpj ), and hence ðRMj Þ ¼ Rpj . For batch processes, Fie ¼ Fis ¼ 0, and for semicontinuous operation Fis ¼ 0. For the continuous operation Fis can be calculated from Eq. (57), where V is constant in a continuous operation and Q s is the volumetric outlet ﬂow rate. Fis ¼ Q s

Ni V

ð57Þ

In emulsion polymerization, the density of the reaction medium does not change signiﬁcantly. Therefore, the volumetric inlet and outlet ﬂow rates can be considered to be the same. The advance of the polymerization is usually given in terms of the conversion Xi of monomer i [Eq. (58)]. Here Polymeri and Monomeri are either in grams or in moles. Xi ¼

Polymeri Polymeri þ Monomeri

ð58Þ in the reactor

When more than one monomer is polymerized, an overall conversion can be calculated from Eq. (59), where XT can be gravimetric or molar depending on the units used for Polymeri and Monomeri (grams or moles, respectively). 2

3 Polymeri 5 X XT ¼ 4 X Polymeri þ Monomeri X

ð59Þ in the reactor

291

292

6 Emulsion Polymerization Tab. 6.3.

Usual conversion deﬁnitions.[a]

Conversion

Fractional conversion of monomer i Overall fractional conversion Overall global conversion

Batch

Ni0 Ni Ni0 P ðNi0 Ni Þ P Ni0

Semicontinuous Ð Ni0 þ Fie dt Ni Ð Ni0 þ Fie dt Ð P ðNi0 þ Fie dt Ni Þ Ð P ðNi0 þ Fie dtÞ Ð P ðNi0 þ Fie dt Ni Þ P NTi

Continuous (steady state) Fie Fis Fie P ðFie Fis Þ P Fie

[a] N

i0 ¼ number of moles of monomer i at time zero; NTi ¼ the total amount of monomer i [mol] to be fed in a semicontinuous operation. In order to calculate the gravimetric conversions, it is necessary to multiply each term by the molecular weight of the corresponding monomer.

The calculation of the monomer conversion depends on the operation mode. Table 6.3 summarizes the expressions for the diﬀerent reactors. The instantaneous copolymer composition refers to the composition of the copolymer that is being formed at a given time. Referred to monomer 1 this composition is given by Eq. (60), where Rpi is the polymerization rate of monomer i. Rp1 y1i ¼ X Rpi

ð60Þ

The cumulative composition is the average composition of the copolymer formed up to a given time, as stated in Eq. (61), where Npoli is the amount [mol] of monomer i polymerized. Npol1 y1cum ¼ X Npoli

ð61Þ

In continuous operations under steady-state conditions, y1i ¼ y1cum during the whole process. 6.9.2

Heat Balance

Assuming that the energy balance for the reacting systems is essentially an enthalpy balance, this reduces to Eq. (62), where cpi and cpie [J mol1 K1 ] are the heat capacity of compound i in the reactor and under the entry conditions, respectively. Rpi [mol m3 s1 ] is the polymerization rate of monomer i, (DHri ) [J mol1 ]

6.9 Reaction Engineering

is the polymerization heat of monomer i under the reactor conditions, T [K] is the reactor temperature, Te [K] the temperature of the feeds, Q loss [J s1 ] the heat losses to the surroundings (for example, through the reactor lid), Q stirring [J s1 ] the heat produced by the agitator, and Q transfer [J s1 ] the heat removed through the heat removal devices (cooling jacket, cooling baﬄes, external heat exchanger and reﬂux condenser). X

Ni cpi

X dT X ¼ Rpi ðDHri ÞV Fie cpie ðT Te Þ dt þ Q transfer þ Q loss þ Q stirring

ð62Þ

For heat removal through the cooling jacket Q transfer is given by Eq. (63), where U [J m2 s1 K1 ] is the overall heat-transfer coeﬃcient, A [m2 ] the total heattransfer area and DTml the logarithmic mean temperature diﬀerence between the cooling ﬂuid and the reaction medium. Q transfer ¼ UADTml

ð63Þ

DTml is given by Eq. (64), where Twe and Tws [K] are the inlet and outlet temperatures of the cooling ﬂuid (normally water) in the jacket. If Twe A Tws then DTml A ðT Tw Þ. DTml ¼

ðT Twe Þ ðT Tws Þ ðT Twe Þ ln ðT Tws Þ

ð64Þ

The overall heat-transfer coeﬃcient includes several resistances in series, but the internal resistance usually controls the heat-transfer rate (hi A U). The internal heat-transfer coeﬃcient is a function of several factors such as the impeller type and dimensions, the impeller speed, the reactor diameter, and physical properties of the ﬂuid. Empirical correlations based on dimensionless groups can be used. Equation (65) presents the usual form of these expressions [111], where Nu; Pr and Re are the Nusselt, Prandtl, and Reynolds numbers, j and jw the viscosity of the reaction medium at the reactor and wall temperatures respectively, and a; b; c, and d are constants. Nu ¼ a Re b Pr c

j jw

d ð65Þ

Due to changes in the properties of the reaction media (for example, viscosity) the overall heat-transfer coeﬃcient changes during the process. In semicontinuous operation, the heat-transfer area varies during the operation.

293

294

6 Emulsion Polymerization

A practical method to determine the heat transferred in any cooling device is to measure the ﬂow rate and the inlet and outlet temperatures of the cooling ﬂuid, which are related by Eq. (66) where m_ w [kg s1 ] is the mass ﬂow rate and cpw [J kg1 K1 ] the heat capacity. Q transfer ¼ m_ w cpw ðTwe Tws Þ

ð66Þ

Combination of Eqs. (62) and (63) or (66) allows the estimation of the polymerization rate from temperature measurements. This method, which is called reaction calorimetry (see Section 6.10.1.6), is a powerful noninvasive on-line monitoring technique and it has been extensively applied to polymerization reactors [113, 114]. 6.9.3

Polymer Particle Population Balance (Particle Size Distribution)

Particle size distribution strongly aﬀects rheology [115], which in turn inﬂuences heat removal rate, mixing, mass transfer, and stability of the latex. On many occasions all of these aspects determine the scaleup and the feasibility of the operation. In addition, particle size distribution aﬀects ﬁlm formation and some application properties [116, 117]. There are several ways in which the PSD can be represented, and often this depends on the method used to measure it. Thus, histograms are used when the PSD is determined by transmission electron microscopy. However, the mathematical analysis is simpler if the PSD is deﬁned in terms of the number density of 3 ], nðvÞ. The units of nðvÞ are the polymer particles of unswollen volume v [mparticle number of particles per unit of unswollen volume of particle. From this deﬁnition, the number of particles with unswollen volumes between v1 and v2 is given by Eq. (67), and the total number of particles by Eq. (68). Np ðv1 ! v2 Þ ¼

ð v2

nðvÞ dv

ð67Þ

v1

Np ¼

ðy

nðvÞ dv

ð68Þ

0

Equation (69) gives the macroscopic population balance for a CSTR, where the lefthand side accounts for the accumulation of particles in the reactor, the ﬁrst term on the right-hand side accounts for the entry of particles into the reactor, the second for the exit of particles from the reactor, the third for the formation and loss of particles of unswollen volume v due to particle growth, the fourth for the loss of particles by coagulation with other particles and the ﬁfth term accounts for the formation of particles of unswollen volume v by particle coagulation. In Eq. (69) nðvÞ and ne ðvÞ are the reactor and inlet number density of polymer particles, Q s [m 3 s1 ] 3 s1 ] the voluis the volumetric ﬂow rate, V [m 3 ] the reactor volume, rv ðvÞ [mparticle metric growth rate of each particle of volume v, kðv; v 0 Þ the coagulation rate con-

6.9 Reaction Engineering

stant for particles of volumes v and v 0 , and v0 the volume of particles formed by nucleation. ð qnðvÞ Q s Qs qrv ðvÞnðvÞ nðvÞ y kðv; v 0 Þnðv 0 Þ dv 0 ne ðvÞ nðvÞ ¼ V V qt qv V v0 ð 1 vv0 þ kðv v 0 ; v 0 Þnðv v 0 Þnðv 0 Þ dv 0 2V v0

ð69Þ

This equation is a partial-diﬀerential-integral equation and the nucleation term is best incorporated as a boundary condition at volume v0 as in Eq. (70) [118], where R nuc is the nucleation rate given by Eqs. (20), (26), (31), and (33). nðv0 Þ ¼

R nuc V rv ðv0 Þ

ð70Þ

The ﬁrst and second terms on the right-hand side of Eq. (69) should be removed for batch reactors, as well as for semicontinuous reactors to which no particles are fed. On the other hand, the coagulation terms may be neglected for stable formulations. Equations (69) and (70) are conveniently solved by using orthogonal collocation [119, 120]. 6.9.4

Scaleup

The main objective of scaleup is to reproduce the laboratory results in commercialscale reactors in such a way that the end-use polymer properties can be maintained. In this way, a successful scaleup takes place when the latex is produced on a large scale at planned rates, at the projected manufacturing cost, and to the desired quality standards [121]. To achieve this goal, knowledge of the mechanisms that control the reactor behavior and the product properties is required [122]. Then, the design variables that aﬀect these controlling mechanisms must be maintained constant from the smaller scale to the commercial equipment. In theory, this could be done by applying the principle of similarity by means of dimensionless groups characterizing the phenomena of interest. However, when several mechanisms are involved in the process, it is impossible to maintain all the dimensionless groups constant simultaneously. In this case, it is very important to know which similarities to keep and which to sacriﬁce. The ﬂow into the vessel, together with heat and mass transfer, must be considered in emulsion polymerization reactors because all of these mechanisms can have a great inﬂuence on polymer properties and reactor behavior. Unfortunately, the mixing and heat-transfer parameters do not scale equally. In Table 6.4 several scale factors are shown for a change from a 50 L pilot-scale vessel to a full-scale vessel of 50 m 3 . The calculated values assume geometric similarity with a constant impeller/tank diameter ratio and a constant height/tank diameter ratio. This table

295

296

6 Emulsion Polymerization Tab. 6.4.

Several scale factors during scaleup.

Parameter [a]

Pilot scale (50 L)

Plant scale of 50 m 3 (scaleup procedure)

P (power) N (impeller rpm) Q (impeller ﬂow) P=V (power/volume) q=V (heat transfer rate/volume) ND (tip speed) Q=Q S (impeller ﬂow/feed rate) Q=H (impeller ﬂow/head) Re (Reynolds number)

1 1 1 1 1 1 1 1 1

10 3 0.21 215 1 0.08 2.1 0.21 46.8 21.5

[a] Units

10 8 10 10 4 10 5 1 100 10 1 10 3

100 0.1 100 0.1 0.05 1 0.1 100 10

10 5 1 10 3 100 0.21 10 1 10 100

0.1 0.01 10 0.0001 0.01 0.1 0.01 10 3 1

are as given in the Notation section.

shows, in diﬀerent columns the change in selected parameters when one of them is held constant. It is often essential to maintain the same rate of heat transfer in the large-scale unit. Nevertheless this scaleup criterion is impractical because it demands excessive impeller tip speeds and high power costs. For this reason, to compensate the negative eﬀect in the heat-transfer rate when another scaleup criterion is selected, additional cooling devices are necessary in most cases [123]. With respect to mixing characteristics, it is very important to identify the parameters that have a major eﬀect on the polymer properties. This task is diﬃcult in emulsion polymerization, where mixing aﬀects several polymer properties. Monomer droplet and particle size, mass transfer between the diﬀerent phases, coagulation and kinetics may be aﬀected by mixing parameters. The power per volume or the tip speed are the more usual scaleup criteria, but an adequate combination of several mixing parameters can be better to maintain product quality. In some cases, changes in the formulation can be utilized to maintain dynamic similarity. Changes in viscosity during the process aﬀect mixing, and scaleup becomes more diﬃcult. Intermediate pilot plant experiments are required in most cases to make a successful scaleup. Computational ﬂuid dynamics will probably transform scaleup in the near future.

6.10

On-line Monitoring in Emulsion Polymerization Reactors

In the production of dispersed polymers, the main objectives to be fulﬁlled are:

Safety: The reactor temperature must be kept under safe limits to avoid thermal runaways. In addition, violation of environmental regulations both in the plant environment and in the ﬁnished products must be avoided.

6.10 On-line Monitoring in Emulsion Polymerization Reactors

Production rate: The goal is to maximize the production rate of the available reactors. Product quality: The required quality is given by the end-use properties such as viscosity, scrub resistance, tensile strength, ﬂexibility, elasticity, toughness, and glosss. In order to implement the process conditions that lead to the required speciﬁcations of safety, production rate, and product quality, it is necessary to develop suitable control strategies. Reaction control strategies rely on both eﬃcient monitoring techniques and state estimation and ﬁltering techniques. In this section the main focus is on the instrumentation available for monitoring emulsion polymerization reactors. Detailed reviews for on-line monitoring techniques for polymerization reactors are available [124–126]. Polymer latexes are ‘‘products-by-process’’, and therefore the required structural and morphological properties of the polymer that yield the adequate end-use properties are produced in the reactor. In polymer latexes the properties that one would like to control during the polymerization include: monomer(s) conversion, copolymer composition, MWD, PSD, particle morphology, branching and crosslinking, gel content, and particle size distribution. Not all of these properties can be monitored on-line, although some can be made observable by combining available measurements and mathematical models. Examples of observable properties are copolymer composition by means of calorimetric or density measurements [127– 129] and instantaneous molecular weights from measurements of unreacted monomer and CTAs (gas chromatography [130] or reaction calorimetry [131] in linear polymers). Other properties, such as molecular weight distribution of nonlinear polymers and particle morphology, are currently neither measurable nor observable. 6.10.1

On-line Sensor Selection

One of the issues when monitoring an emulsion polymerization reactor is selection of the most appropriate technique [124, 126]. For instance, monomer conversion and copolymer composition can be monitored on-line by means of densimetry, refractive index, gas chromatography, calorimetry, ultrasound, ﬂuorescence, ultraviolet reﬂection, and other spectroscopic methods such as Raman, mid-range infrared, and near-infrared. Figure 6.11 can be used as a guide for the selection of a technique for monitoring monomer conversion and copolymer composition. The main idea behind this plot is the fact that the amount of information provided by the available techniques is diﬀerent and furthermore, the implementation of these techniques involves different degrees of diﬃculty (including here robustness in a harsh environment, maintenance, and know-how required). The ideal technique would be located in the upper left-hand corner, but unfortunately no such technique is currently available.

297

298

6 Emulsion Polymerization

Fig. 6.11.

Guideline chart for sensor selection.

In the rest of Section 6.10.1, a brief description of the monitoring techniques shown in Figure 6.11 will be presented, emphasizing the advantages and disadvantages of each one, and ﬁnally examples of application for emulsion polymerization reactors will be given. Latex Gas Chromatography In this technique, a latex sample from the reactor is ﬁrst diluted and then injected into the GC to analyze unreacted monomers and CTAs [132, 133]. Although successful implementation of this technique for latexes with a high solids content has been reported [134], the setup is liable to suﬀer clogging. Additional disadvantages of this monitoring technique include the time delays associated with the analysis itself when multicomponent copolymerizations are monitored. This drawback can be partially alleviated by using capillary columns and modern gas chromatography equipment with reduced analysis time. 6.10.1.1

Head-space Gas Chromatography This GC-based monitoring technique circumvents the lack of robustness of the latex GC technique by analyzing the reactor head-space [135, 136]. However, the estimation of the monomer concentrations in the polymer particles requires that equilibrium between the latex and gas phase is attained. In addition, the values of the equilibrium parameters are required, which introduces an additional uncertainty to the measurements. Note also that formulation ingredients such as CTAs are not measurable by this means. 6.10.1.2

Densimetry This technique is based on the density change that occurs when monomer is converted to polymer. This diﬀerence, which is obviously a maximum in bulk polymerization, allows emulsion polymerizations of relatively high solids content to be accurately monitored [137]. The main disadvantage of on-line densimetry is that, as for latex GC, a sample of the reaction medium must be introduced in the thermostated measurement cell and the system is liable to suﬀer clogging. A further dis6.10.1.3

6.10 On-line Monitoring in Emulsion Polymerization Reactors

advantage of densimetry is that it provides only overall conversion measurements for copolymerization, and hence partial conversions and copolymer compositions must be inferred using a model [129]. An advantage of densimetry in comparison to GC is that there is no delay in the measurement of density, which can be monitored almost continuously. Ultrasound The principle of ultrasound relies on the propagation of the ultrasonic wave pressure. The measurable properties are the ultrasonic velocity and the attenuation of the wave, which are a function of the medium where the wave propagates and hence of the density, viscosity, and compressibility. The attenuation measurements are not very reliable for on-line monitoring because the attenuation of the wave in dispersed systems is complex and depends on the dispersion medium, the viscous losses within the particles and at the interface between the particle and the continuous phase, thermal losses, sound scattering in dispersed media, and the dynamic relaxation of the polymeric material [138, 139]. Furthermore, there are technical problems in measuring the attenuation at high solids and the presence of gas bubbles can make the measurements diﬃcult. Sound propagation velocity is better suited to monitor emulsion polymerization reactors and it has been proven in systems with high conversion and high solids content [140–142]. In principle, the main advantages of this technique are that it is noninvasive in nature (although care should be taken with emitter–receiver transducers located inside the reactor), it is fast and cheap, and, in addition to monomer conversion, other latex properties such as particle size, monomer solubilities, and critical micellar concentration can also be measured [141]. The main disadvantage is that to exploit the information on the sound velocity and attenuation fully, calibration is required to obtain accurate predictions of monomer conversion or other properties such as particle size because the theoretical models linking those characteristics to sound propagation velocity are not predictive. Nevertheless, on-line sound velocity is one of the more promising techniques for monitoring industrial-size polymerization reactors because of the fast and robust measurement and easy maintenance at a relatively low cost. 6.10.1.4

Spectroscopic Techniques Since the mid-1990s, there has been plenty of activity regarding the use of spectroscopic techniques for on-line evaluation of polymer properties [143–146]. This has been possible due to the recent development of ﬁber-optic probes, which allow insitu measurements in remote and harsh environments (high temperatures, pressures, toxic environments, and so on). An additional advantage is that a ﬁber-optic probe can be installed in an existing reactor within a short time without expensive modiﬁcations. Fluorescent, ultraviolet (UV), infrared (IR), near-infrared (NIR), mid-infrared (MIR) and Raman spectroscopic techniques can be used for polymerization reaction monitoring. These can be divided between absorption- and emission-based techniques. IR, NIR, and MIR are absorption-based. 6.10.1.5

299

300

6 Emulsion Polymerization

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

Water is a weak scatterer and hence Raman is well suited to monitor polymerization in dispersed media. It is very sensitive to CbC bonds. This makes it possible to follow, with high accuracy, the disappearance of monomer by polymerization. Low-cost ﬁber-optic technology can be used to monitor polymerization reactors remotely. The penetration depth of the Raman signal is greater than for NIR and hence in emulsion polymerization interference by monomer droplets can be avoided.

6.10 On-line Monitoring in Emulsion Polymerization Reactors a)

Weight Fraction MMA

0.006 0.005 0.004 0.003 0.002 0.001 0 0

0.2

0.4

0.6

0.8

1

Conversion b)

Weight Fraction BA

0.05

0.04

0.03

0.02

0.01

0 0

0.2

0.4

0.6

0.8

1

Conversion Fig. 6.12. On-line monitoring of a seeded semibatch emulsion copolymerization of MMA/BA. Evolution of (a) MMA; (b) BA.

Figure 6.12 shows an example of monitoring a semibatch emulsion polymerization of MMA/BA of high solids content (55 wt.%) by means of on-line FT-Raman spectroscopy [150]. It should be pointed out that this monomer system was challenging because the chemical structures of the monomers are very similar, and hence most of the bands overlap. Chemometric (partial least squares, PLS) analysis was neces-

301

6 Emulsion Polymerization c)

Solids Content (%)

302

60

50

40

30

20

10 0

0.2

0.4

0.6

0.8

1

Conversion Fig. 6.12.

(c) solids content. g, gravimetry and gas chromatography; , FT-Raman.

sary to quantitatively determine the concentrations of monomers and the solids content. Furthermore, in some cases nonlinear calibration techniques are necessary because PLS, a multivariate linear calibration, may not be suﬃcient if the degree of nonlinearity between the spectra and the properties of interest is signiﬁcant. Reaction Calorimetry Reaction calorimetry is probably the cheapest, easiest, and most robust monitoring technique for polymerization reactors, due to the large enthalpy of polymerization of most monomers. The technique is noninvasive (basically, only temperature sensors are required), and it is industrially applicable [151, 152]. It yields continuous information on the heat released by polymerization and hence it is also very useful for safety issues. The main drawback is that only overall polymerization rates can be obtained. Consequently, the determination of the individual rates requires estimation techniques [114, 153–155]. Reaction calorimetry is based on the energy balance in the reactor, given by Eq. (71), 6.10.1.6

X

Ni cpi

dT ¼ Q r þ Q feed þ Q transfer þ Q loss þ Q stirring dt

ð71Þ

where the term on the left-hand side is the heat accumulated in the reactor, Q r is the heat generation rate due to polymerization, Q feed is the sensible heat genera-

6.10 On-line Monitoring in Emulsion Polymerization Reactors

tion rate due to the feeding of reagents into the reactor, Q transfer is the heat ﬂow across the reactor wall, and Q loss and Q stirring represent the rate of heat losses and of heating due to stirring, respectively. The generation rate of the heat of reaction, Q r , can be calculated from the other terms, provided that these can be calculated with suﬃcient accuracy. In emulsion polymerization reactors, the largest of these terms is Q transfer . In heat-ﬂow calorimetry, Q transfer is calculated from the measurements of the reactor (T) and jacket (Tw ) temperatures by applying Eq. (72), where U is the overall heat-transfer coeﬃcient and A the heat-transfer area. Q transfer ¼ UAðT Tw Þ

ð72Þ

The implementation of heat-ﬂow calorimetry requires knowledge of the evolution of U. This is the weakest point of this technique. In heat-balance calorimetry, eqs. (71) and (72) are coupled with the energy balance in the jacket to produce Eq. (73), where m w is the mass of cooling ﬂuid in the jacket, m_ w the mass ﬂow rate of cooling ﬂuid in the jacket, cpw its speciﬁc heat capacity, Twe the inlet jacket temperature and Tws the outlet jacket temperature. m w cpw

dTws ¼ UAðTws TÞ þ m_ w cpw ðTwe Tws Þ dt

ð73Þ

Heat-balance calorimetry allows the simultaneous estimation of U and Q r , provided that Twe Tws could be accurately measured. Therefore, heat-balance calorimetry is best suited for large-scale industrial reactors because a signiﬁcant diﬀerence between the jacket inlet and outlet temperatures is necessary, and this is the case in industrial reactors. The advantage of this approach is that a-priori information of the overall heat-transfer coeﬃcient is not necessary and hence it is more robust and reliable than heat-ﬂow calorimetry. Oscillation calorimetry also allows simultaneous determination of the heat of reaction and the overall heat-transfer coeﬃcient from temperature measurements of the reactor and the jacket. This is done by taking advantage of the diﬀerent dynamics of heat transfer (fast) and heat of the reaction (slow) when an oscillation of either the reactor or jacket temperature is created artiﬁcially. The analysis of the oscillatory temperature signals allows calculatation of the UA term and Q r simultaneously [156, 157]. The oscillation of the jacket and reactor temperatures can be achieved in diﬀerent ways, but perhaps the most practical one is by imposing a sinusoidal oscillation in the set-point of the reactor temperature. This approach can only be applied to small reactors (< 20 L) with high ﬂow rates of the cooling ﬂuid, because the oscillating signal is strongly attenuated as the reactor size increases. The higher reactor time-constants make the estimation of Q r and UA very uncertain [158]. The amount of free monomer and the copolymer composition in emulsion polymerization reactors can be inferred from measurement of the heat of reaction, Q r

303

6 Emulsion Polymerization

a) 350 Free Monomer (g)

Gravimetry

300

Chromatography Calorimetry

250 200 150 100 50 0 0

50

100

150

200

250

300

350

Time (min)

b) 1.0 ________

Terpolymer Composition

304

Calorimetry

Full Points GC

0.8

Open Points NMR

0.6 BA

0.4 MMA

0.2

VAc

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Conversion Fig. 6.13. On-line monitoring of emulsion polymerization reactor by means of calorimetric measurements: (a) free monomer in VAc/BA/AA semibatch emulsion polymerization; (b) terpolymer composition in the VAc/MMA/BA semibatch emulsion polymerization.

[127, 128]. Figure 6.13(a) shows the evolution of the free monomer concentration as inferred from on-line reaction calorimetry compared with the oﬀ-line measurement (gravimetry and gas chromatography) for a VAc/BA/AA high solids content semibatch emulsion polymerization. In Figure 6.13(b), the estimation of the terpolymer composition from calorimetric measurements is compared with NMR and GC measurements for a VAc/MMA/BA emulsion terpolymerization [159].

6.11 Control of Emulsion Polymerization Reactors

6.11

Control of Emulsion Polymerization Reactors

The objective of reactor control is to achieve an eﬃcient, safe, and consistent production of high-quality materials. The quality of the emulsion polymers is given by its end-use properties such as mechanical strength; tack, peel, and shear resistances; brightness; impact strength; weatherability; drawability; and elastic modulus. These end-use properties depend on the molecular and morphological properties of the latex, which are the variables that can be controlled in the reactor. Therefore, the relationship between the structural and morphological properties of the polymer latex and the end-use properties is required to establish which are the characteristics of the polymer latex to be produced/controlled in the reactor (Figure 6.1). The development of ﬁrst-principle structure–properties relationships is very challenging as the basis is not yet well established. Thus, the current industrial practice is a trial and error methodology. In order to reduce this time-consuming methodology semiempirical approaches can be used. An example of this type of approach was recently reported to relate the MWD of an acrylic latex of a given composition and the adhesive properties [160, 161]. However, this relationship only provides a way to estimate the end-use properties from a given microstructural property (MWD in the example); in practice, one would like to know the microstructure that will yield a given set of end-use properties (see Figure 6.14). This can be achieved by inverting the model using appropriate techniques of optimization or neural networks [160, 161]. Once the desired molecular and morphological properties of the latex are established, a closed-loop control strategy can be implemented. A typical closed-loop control strategy used to control copolymer composition and molecular weight distribution is shown in Figure 6.14. In the control scheme, x are state variables of the polymerization reactor, y are measurable properties of the polymer latex, ymeas are properties that are monitored on-line, and u are the manipulated variables of the process (basically ﬂow rates of monomers and chain-transfer agents).

Fig. 6.14. Closed-loop control strategy for optimal control of composition and MWD in emulsion polymerization reactors.

305

306

6 Emulsion Polymerization

The closed-loop control strategy requires calculation of the set-point, that is, the trajectories or proﬁles of the state variables as a function of a measured variable (overall conversion). These proﬁles can be calculated by means of an optimization algorithm. In what follows, a brief description of the calculation of the optimal trajectories for copolymer composition and the MWD control is presented. The goal of the optimization algorithm is to calculate the set-point trajectories of the state (controlled) variables that ensure the production of an emulsion polymer of the desired copolymer composition and MWD in the minimum process time. To achieve this goal, the objective function to be minimized is as expressed in Eq. (74), where Rp is the polymerization rate and XT is the overall conversion. Min

ð 1

½M p

0

1 dXT Rp

ð74Þ

The minimization of Eq. (74) is subjected to the following constraints: Constraint 1: product composition and MWD The polymer produced must have the desired copolymer composition and the ﬁnal MWD. The condition to produce a latex with a given copolymer composition is that the ratio of the monomer concentrations in the polymer particles must be kept at the value that ensures the production of the desired composition. This comonomer ratio can be calculated from the Mayo–Lewis equation, Eq. (75), where r1 and r2 are the reactivity ratios and y1i is the instantaneous composition referred to monomer 1.

½M1 p ½M2 p

¼

ð2y1i 1Þ G fð2y1i 1Þ 2 4r1 ð y1i 1Þ y1i r2 g 1=2 2r1 ð y1i 1Þ

ð75Þ

The calculation of the condition to produce a latex with a given MWD is based on the fact that for linear polymers produced by free-radical polymerization, the polymer chains do not suﬀer any modiﬁcation once they are formed. This opens the possibility of decomposing the desired ﬁnal MWD in a series of instantaneous MWDs to be produced at diﬀerent stages of the reaction [130]. When chain transfer to a CTA is the main termination event, each of those MWDs can be characterized by the number-average chain length, X ni , according to Eq. (76). X ni ¼

k p1 ½M1 p þ k p2 ½M2 p ktr; CTA ½CTAp

ð76Þ

Therefore the problem reduces to calculating the sequence of the values of X ni that provide the desired ﬁnal MWD. In order to calculate the X ni values that should be produced at each value of XT , the ﬁnal MWD is discretized as in Eq. (77), where XTf is the ﬁnal overall conversion; X nij is the instantaneous number-average chain length produced in the conversion increment j, Wj ðnÞ is the instantaneous MWD

6.11 Control of Emulsion Polymerization Reactors

produced in the conversion increment j, and k is the number of increments into which XTf is divided. Wc ðnÞ

k k 1 X DXT X n n ¼ Wj ðnÞDXTj ¼ exp XTf j¼1 XTf j¼1 X nij X nij

! ð77Þ

Note that in the discretization the most probable distribution is used because, if chain transfer to CTAs is the main termination event, the instantaneously formed polymer obeys this distribution (polydispersity index ¼ 2). For a given number of conversion increments, the required values of X nij can be calculated by minimizing the Eq. (78), where Wcd ðnÞ and Wc ðnÞ are the desired and the calculated MWDs. X ½Wcd ðnÞ Wc ðnÞ 2 Min X ni

ð78Þ

n

This is a nonlinear optimization in which the number of values of n should be greater than the number of conversion increments. A priori any MWD, Wcd ðnÞ, with polydispersity index equal or greater than 2 can be prepared by this method. Strictly speaking, the maximum molecular weight achievable with this technique is that produced with the minimum amount of CTA that ensures the termination by chain transfer to CTAs is the main termination event. In practice, this is very close to the molecular weight obtained without CTA. On the other hand, MWDs containing very low molecular weights may require the use of amounts of CTA that exceed the maximum allowable quantities (usually lower than 1 wt.% based on monomer) used in industrial practice. The minimization of Eq. (78) provides the values of X nij to be produced at diﬀerent DXT [Eq. (79)]. X ni o

½M1 p þ ½M2 p ½CTAp

¼ f ðXT Þ

ð79Þ

Equations (75) and (79) are used as the constraints in the optimization algorithm to produce a copolymer of constant instantaneous composition Y1i and a given MWD, Wcd ðnÞ. Constraint 2: Safety considerations Polymerizations are exothermic processes that can cause runaways, so the maximum amounts of the monomer that can be present in the reactor should be limited for safety reasons. Therefore, to design safe processes an analysis of the risk parameters must be made in order to obtain the limits in reaction conditions for safe operation: namely, the limits in monomer concentration and temperature that ensure that the pressure buildup in the reactor will not exceed the maximum pressure that the reactor can withstand. The risk parameters are the onset temperature, the adiabatic temperature increase, and the maximum temperature and pressure that may be reached during a polymerization

307

308

6 Emulsion Polymerization

process under adiabatic conditions. To assess risk parameters, adiabatic calorimeters of low thermal inertia (phi-factor, F, less than 15%) are used. The onset temperature of the polymerization reaction, Tonset , is deﬁned as the temperature at which the self-heating rate is equal to a given arbitrary onset criterion. This criterion depends on the sensitivity of the equipment and is used as a reference to calculate the adiabatic temperature increase, DT, which depends on the total amount of monomer in the formulation, MiTOT , the heat of polymerization (DHri ), and the heat capacity of the reaction medium according to Eq. (80), where Ni and cpi are the amount and the heat capacity, respectively, of each reagent i present in the reaction medium. X DT ¼

i

MiTOT ðDHri Þ X Ni cpi

ð80Þ

i

The adiabatic temperature increase, DT, is calculated for the experiments carried out in the adiabatic calorimeters by subtracting the Tonset from the maximum temperature achieved, Tmax . DT ¼ Tmax Tonset

ð81Þ

Thus, the dependence of the risk parameters on process variables such as the monomer concentration in the polymer particles, particle size, solids content and initiator/monomer ratio are of paramount importance to establish the safe regions of operation of an emulsion polymerization reactor, and furthermore to develop optimal control strategies under safe conditions. A typical result obtained in an adiabatic calorimeter for an emulsion polymerization reaction is shown in Figure 6.15. From the evolution of the temperature, the onset of the adiabatic reaction and the maximum temperature achieved can be determined. Therefore, adiabatic temperature increases under these conditions can be easily calculated. The evolution of the pressure provides an indication of the maximum pressure reached. Figure 6.16 shows the evolution of the risk parameters as a function of the polymer/monomer ratio for the emulsion polymerization of VAc/BA/AA (78.5:18.5:3) as obtained in a VSP2 reactor (Fauske & Associates). The particle size of the seed and the initiator/monomer ratio were the same in all the experiments. The plot shows that the onset temperature decreases as the monomer concentration in the polymer particles increases. Tmax ; DT, and pressure (the latter not shown) increase upon increasing the monomer content. The analysis can be extended for other process variables such as solids content, particle size, and initiator/monomer ratios. An example of the safety limits for the VAc/BA/AA emulsion polymerization system is shown in Figure 6.17. The plot, which is based on the data displayed in Figure 6.16, is constructed assuming that the polymerization will be carried out at

6.11 Control of Emulsion Polymerization Reactors

a) 120

T (°C)

100

80

60

40

20 0

50

100

150

200

250

300

200

250

300

time (min)

b) P (atm)

4 3,5 3 2,5 2 1,5 1 0,5 0

50

100

150

time (min)

Fig. 6.15. Time evolution of (a) temperature; (b) pressure during emulsion polymerization of VAc/BA/AA (78.5:18.5:3) with initial monomer/polymer ratio ¼ 50:50; initiator/monomer ratio ¼ 0.002 wt.%; ﬁnal solids content ¼ 50 wt.%; seed particle size ¼ 156 nm.

80 C (Twork ) and that the maximum temperature allowed to run the process, Tlimit , is 100 C. Note that this temperature must be calculated on the basis of the maximum pressure that the reactor can withstand, and is also a function of the process variables. The graph shows that the region of polymer/monomer ratio below 3:1 (concentration of monomer in the polymer particles higher than ½Mp ¼ 2:90 mol L1 ) will not be safe. In other words, if higher monomer concentrations are used in the process and the cooling system fails, there is a risk of exceeding the Tlimit tempera-

309

6 Emulsion Polymerization

Temperature (°C)

140 120

Tonset ∆T

100

Tmax

80 60 40 20 0 0

2

4

6

8

10

Polymer/Monomer Fig. 6.16. Evolution of the risk parameters as a function of the polymer/monomer ratio for a VAc/BA/AA emulsion polymerization of high solids content.

140 Temperature (°C)

310

T

120

work

+ ∆T T

limit

100 80 T

work

60 SAFE REGION

40 ∆T

20 0 0

2 DANGER

4

6

8

Polym er / M onom er [M ]

p

Fig. 6.17. Safety regions for a VAc/BA/AA emulsion polymerization. Polymerization temperature ¼ 80 C; maximum temperature allowed for the process ¼ 100 C.

10

6.11 Control of Emulsion Polymerization Reactors

ture (Twork þ DT > Tlimit ) and the runaway will take place. Above the limit polymer/ mononor ¼ 3 (½Mp < 2:90 mol L1 ) the process can be operated safely because in none of the cases does the [Twork þ DT] operating line exceed the limit of the safe operation temperature. Constraint 3: Nonremoval of monomer and CTA The monomers and CTA already charged in the reactor cannot be removed (Eqs. (82)–(84), where the subscripts f and pol stand for free and polymerized amounts, respectively).

d½M1f þ M1pol b0 dXT

ð82Þ

d½M2f þ M2pol b0 dXT

ð83Þ

d½CTA f þ CTA pol b0 dXT

ð84Þ

Constraint 4: Limitation on monomer and CTA addition The maximum amounts of the monomer and CTA that can be added to the reactor are the total amounts of these compounds in the formulation, M1TOT ; M2TOT , and CTATOT , respectively [Eqs. (85)–(87)].

M1f þ M1pol a M1TOT

ð85Þ

M2f þ M2pol a M2TOT

ð86Þ

CTA f þ CTA pol a CTATOT

ð87Þ

The optimization provides the amounts of monomers and CTAs in the reactor at any overall conversion. These proﬁles are independent of the kinetics of the process and can be regarded as master curves. Once the trajectories of the amounts of monomers and CTAs as a function of the conversion are calculated, the implementation of the closed-loop strategy (Figure 6.14) reduces to tracking these proﬁles. To do so, on-line measurements of the overall conversion and of the free amount of monomers and CTA are necessary. Reaction calorimetry plus state estimation is probably the easiest, cheapest, and most robust option from an industrial perspective. At each sampling time a nonlinear controller calculates the values of the manipulated variables u (ﬂow rates of monomer and CTA) that must be added during the sampling interval to ensure tracking of the master trajectories and hence to produce the desired polymer. A number of nonlinear controllers have been reported in the literature for this purpose: nonlinear model predictive controllers (NMPCs) [162], nonlinear geometric controllers (NGCs) [163], and internal model controllers (IMCs) [164] being the ones that have gained more attention in the specialized literature.

311

6 Emulsion Polymerization 0.015

2 Butyl acrylate Styrene

1.5

0.012

CTA

0.009 1 0.006 0.5 0.003

0

0

0.2

0.4

0.6

0.8

1

Total amount of CTA (mol/L)

Total amounts of monomers (mol/L)

312

0

Overall Conversion

Optimal trajectories for the amount of monomer and CTA to produce S/n-BA ¼ 50:50 copolymer with a bimodal MWD. Fig. 6.18.

An example of the performance of an on-line control strategy like the one depicted above is shown in Figures 6.18 and 6.19 [165]. A copolymer with constant composition, styrene/n-butyl acrylate (S/BA) ¼ 50:50 and a bimodal MWD with two peaks (50 wt.% of the polymer in each peak) of diﬀerent polydispersities were sought: Mw1 ¼ 1:05 10 6 and PI1 ¼ 2:5 and Mw2 ¼ 1:15 10 5 and PI2 ¼ 3. Figure 6.18 presents the optimal proﬁles of styrene, n-butyl acrylate, and tert-dodecyl mercaptan (TDM) to produce the copolymer with the properties mentioned. It can be seen that to maintain the comonomer ratio the required amount of n-BA was always greater than that of styrene. Moreover, during the production of the ﬁrst 50% of the polymer, the amount of TDM required was low since the mode with the high molecular weight was prepared ﬁrst; at 50% conversion, a sudden addition of TDM was required to start producing the low molecular-weight mode. Figure 6.19 presents a comparison between the desired and obtained copolymer composition and MWD during the controlled experiment.

Notation

A Ap as c cc

total heat-transfer area [m 2 ] total surface area of the polymer particles [m 2 ] saturated surface of the polymer particles covered by 1 mol of surfactant [m 2 mol1 ] overall termination rate coeﬃcient in the polymer particles [Eq. (9)] [s1 ] rate coeﬃcient for bimolecular termination by combination [s1 ]

Notation

a)

Copolymer Composition

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Conversion

b) Desired

w

df/dlog(M )

0.6

X=0.18

0.5

X=0.47 X=0.60

0.4

X=0.77

0.3

X=0.96

0.2 0.1 0.0 3.0

3.5

4.0

4.5

5.0

5.5 6.0 log (M )

6.5

7.0

w

Experimental results of the on-line controlled emulsion polymerization of S/n-BA: (a) cumulative copolymer composition; (b) MWD produced at diﬀerent conversions. Fig. 6.19.

cd CMC cpi cpie cpw ½CTAp CTA f

rate coeﬃcient for bimolecular termination by disproportionation [s1 ] critical micellar concentration [mol m3 ] heat capacity of compound i in the reactor [J mol1 K1 ] heat capacity of compound i under the entry conditions [J mol1 K1 ] heat capacity of the cooling ﬂuid [J kg1 K1 ] concentration of CTA in polymer particle [mol m3 ] free amount of CTA in the reactor [mol]

313

314

6 Emulsion Polymerization

CTA pol CTATOT D dd dp f fac fc fde Fie Fis H hi (DHri Þ I [I] icrit jcrit j Ki kðv; v 0 Þ ka kac kadd kam kd kd ðnÞ kde kex ðÞ kfrag ðþÞ

kfrag kI kp k pi kpji kt kt

ktr; CTA k tr; M k tw

amount of CTA polymerized [mol] total amount of CTA in the formulation [mol] impeller diameter [m] diameter of the monomer droplets [nm] diameter of the polymer particles [nm] eﬃciency factor of the initiator radicals frequency of activation in living polymerization [s1 ] frequency of bimolecular termination [s1 ] frequency of deactivation in living polymerization [s1 ] inlet molar ﬂow rate of component i [mol s1 ] outlet molar ﬂow rate of component i [mol s1 ] velocity head [J kg1 ] internal heat-transfer coeﬃcient [J m2 s1 K1 ] polymerization heat of monomer i under the reactor conditions [J mol1 ] amount of initiator [mol] concentration of initiator in the aqueous phase [mol m3 ] critical length of the oligoradicals formed from desorbed radicals critical length of the oligoradicals formed from initiator partition coeﬃcient of monomer i between the phase j and the aqueous phase. coagulation rate constant for particles of volumes v and v 0 [m 3 particle1 s1 ] entry rate coeﬃcient [m 3 mol1 s1 ] activation rate coeﬃcient in NMP or ATRP [s1 or m 3 mol1 s1 ] addition rate coeﬃcient in RAFT [m 3 mol1 s1 ] rate coeﬃcient for radical entry into the micelles [m 3 mol1 s1 ] rate coeﬃcient for radical exit [s1 ] rate coeﬃcient of radical exit from particles with n radicals [s1 ] deactivation rate coeﬃcient in NMP or ATRP [m 3 mol1 s1 ] exchange rate coeﬃcient in DT [m 3 mol1 s1 ] fragmentation rate coeﬃcient in RAFT, backward reaction [s1 ] fragmentation rate coeﬃcient in RAFT, forward reaction [s1 ] rate coeﬃcient for initiator decomposition [s1 ] propagation rate constant [m 3 mol1 s1 ] average propagation rate constant of monomer i in copolymerization [m 3 mol1 s1 ] propagation rate constant of radicals with terminal unit j with monomer i [m 3 mol1 s1 ] termination rate coeﬃcient in the polymer particles [m 3 mol1 s1 ] reversible termination rate coeﬃcient in living polymerization [m 3 mol1 s1 ] average chain transfer to CTA rate constant in copolymerization [m 3 mol1 s1 ] chain transfer to monomer rate coeﬃcient [m 3 mol1 s1 ] termination rate coeﬃcient in the aqueous phase [m 3 mol1 s1 ]

Notation

m Micrit Mif Mipol ½Mi p MiTOT Mm Mmi Mn Mni ½Mp mw m_ w Mw Mwi ½Mw N n n NA nðvÞ ne ðvÞ Ni Ni0 nm Nm Nn Np Npoli Npr NTi Nu P Pj PM Pr q Q Q feed Q loss Qr

partition coeﬃcient of small radicals between polymer particles and the aqueous phase [Eq. (11)] number of oligoradicals of critical length formed from desorbed radicals amount of free monomer i in the reactor [mol] amount of monomer i polymerized [mol] concentration of monomer i in the polymer particles [mol m3 ] total amount of monomer i in the formulation [mol] number of inactive chains of length m number of inactive chains of length m in particles with i radicals cumulative number-average molecular weight instantaneous number-average molecular weight monomer concentration in the polymer particles [mol/m 3 ] mass of the cooling ﬂuid in the jacket [Kg] mass ﬂow rate of the cooling ﬂuid [Kg/s] cumulative weight-average molecular weight instantaneous weight-average molecular weight monomer concentration in the aqueous phase [mol/m 3 ] impeller rpm number of radicals in a particle average number of radicals per particle Avogadro’s number number of polymer particles per unit of unswollen volume of particle [particles m3 particle ] inlet number density of polymer particles [particles m3 particle ] total amount of compound i in the reactor [mol] moles of monomer i in the reactor at time zero [mol] aggregation number of surfactant [molecules/micelle] number of micelles number of polymer particles with n radicals number of polymer particles moles of monomer i polymerized [mol] number of precursor particles total amount of monomer i to be feed in a semicontinuous operation [mol] Nusselt dimensionless number impeller power consumption [J s1 ] time-averaged probability of ﬁnding an active chain with ultimate unit of type j average molecular weight of the repeating unit in the polymer chain Prandtl dimensionless number heat transfer rate [J s1 ] impeller ﬂow [m 3 s1 ] sensible heat generation rate due to feeding into the reactor [J s1 ] rate of heat losses to the surroundings [J s1 ] heat generation rate by polymerization reaction [J s1 ]

315

316

6 Emulsion Polymerization

Qs Q stirring Q transfer Re Ri ri R jcrit Rm Rmi R nuc Rp Rpi Rpp RX rv rv ðvÞ Rnk ½Rw ST Sw t T DT Te Tg Tlim Tmax DTml Tonset Tw Twe Tws U V Vi Vd Vw Vp vp Vpol W Wc ðnÞ Wcd ðnÞ

volumetric ﬂow rate [m 3 s1 ] rate of heat production by the agitator [J s1 ] heat removal rate [J s1 ] Reynolds dimensionless number net generation rate of component i [mol m3 s1 ] reactivity ratio of monomer i number of oligoradicals of critical length formed from initiator number of radicals of length m number of radicals of length m in particles with i radicals nucleation rate of polymer particles [particles m3 s1 ] overall polymerization rate [mol m3 s1 ] polymerization rate of monomer i [mol m3 s1 ] polymerization rate per polymer particle [mol particles1 s1 ] amount of the ‘‘capping’’ species in controlled polymerization [mol] volumetric growth rate of one polymer particle [m 3 s1 ] volumetric growth rate of a particle of volume v [m 3 s1 ] generation rate of the kth moment of the distribution of inactive chains concentration of radicals in the aqueous phase [mol m3 ] total amount of surfactant in the reactor [mol] amount of surfactant in the aqueous phase [mol] reaction time [s] reactor temperature [K] adiabatic temperature increase temperature of the feed [K] glass transition temperature [K] maximum temperature achievable for a safe operation [K] maximum temperature obtained during an adiabatic polymerization reaction [K] logarithmic mean temperature diﬀerence temperature at which the polymerization start under adiabatic conditions [K] jacket temperature [K] inlet temperature of the cooling ﬂuid in the jacket [K] outlet temperature of the cooling ﬂuid in the jacket [K] overall heat-transfer coeﬃcient [J m2 s1 K1 ] reactor volume [m 3 ] volume of monomer i [m 3 ] volume of the droplet phase [m 3 ] volume of the aqueous phase [m 3 ] volume of the polymer particles [m 3 ] volume of a swollen polymer particle [m 3 ] volume of polymer [m 3 ] volume of water [m 3 ] cumulative weight MWD desired cumulative MWD

References

Wj ðnÞ [X] Xi X ni X nij XT XTf DXTj y1cum y1i

instantaneous weight MWD of the polymer formed at conversion increment j concentration of ‘‘living agent’’ in controlled polymerization [mol m3 ] conversion of monomer i instantaneous number-average chain length instantaneous number-average chain length of the polymer formed in conversion increment j overall conversion ﬁnal overall conversion jth overall conversion increment cumulative copolymer composition referred to monomer 1 instantaneous copolymer composition referred to monomer 1

Greek a1 a2 g d h l mk nk r t j F P fM j fi P fp fww C jw

parameter of Eq. (29) parameter of Eq. (30) generation rate of small radicals by chain transfer [Eq. (11)] [mol m5 ] critical length for entry of radicals generated from the initiator consumption rate of small radicals generated by chain transfer [Eq. (11)] [m2 ] overall mass-transfer rate coeﬃcient [m 3 s1 ] kth-order moment of the distribution of active chains kth-order moment of the distribution of inactive chains frequency of radical entry [s1 ] space time [s] viscosity of the reaction medium at the reactor temperature [kg m1 s1 ] fraction of the heat of reaction used to heat the reactor walls volume fraction of monomer in the polymer particles volume fraction of monomer i in phase j volume fraction of polymer in the polymer particles volume fraction of water in the aqueous phase parameter of Eq. (17) viscosity of the reaction medium at the wall temperature [kg m1 s1 ]

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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

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

324

7 Ionic Polymerization

ﬂuenced by the nature of the surroundings, nor by the solvent nor by penultimate units of the chain; at least, the inﬂuences of penultimate units [8] or solvents [9] are quite small. In contrast, ionic polymerization is site-based: that is, the nature of the active center strongly depends on the nature of the initiator. Propagation is by monomer insertion at an ion pair [Eq. (1) or (2)], where the counter ion comes from the initiator system. þ X @@Pnþ X þ M ! @@Pnþ1

ð1Þ

or þ @@Pn Y þ þ M ! @@P nþ1 Y

ð2Þ

So the kinetic scheme and parameters not only depend on M, but they may diﬀer considerably according to the initiator used. Furthermore, the active centers are not uniform as in free-radical polymerization, but because of their ionic nature there usually exists a complex chemical equilibrium between diﬀerent species, even if the initiator has a unique structure. This equilibrium [Eq. (3)] between free ions, solvent-separated ion pairs, contact ion pairs, covalent polarized bonds, and between associated and non-associated species, as well as the concentration of these species strongly depend on the polarity or solvating power of the solvent system, the solvent itself, and the presence of other salts. X þ Y þ $ X ==Y þ $ X ; Y þ $ X d Y dþ $ 1=n ðXYÞn

ð3Þ

All these species can in principle participate in all the reactions with diﬀerent rate coeﬃcients. These facts make ionic polymerizations diﬃcult to access for kinetic modeling from a practical point of view in an industrial environment. There is another important diﬀerence between free-radical and ionic polymerization. In free-radical polymerization, there are system-immanent, unavoidable termination reactions, the bimolecular disproportionation and combination between two radicals. Because these termination reactions are very fast (k t ¼ 10 7 –10 9 M1 s1 ) compared to propagation (k p ¼ 10 0 –10 3 M1 s1 ) and radical formation by initiator decomposition (kd ¼ 103 –101 s1 ) the pseudo-steady-state hypothesis can be applied. In ionic systems, in general, there are no such system-immanent termination reactions between species carrying the kinetic chain and, if there are reactions which terminate the active species, they often occur on a similar time scale to initiation and propagation, or they are even slower. Examples of such reactions, which irreversibly terminate the kinetic chain, are elimination reactions where the eliminated species is not able to re-initiate the polymerization, such as in the elimination of LiH or NaH during anionic polymerization of styrene or butadiene [10–13]. In cationic polymerization, the collapse of ion pairs to covalent species, as in reaction (4) where the CaF bond is too strong for the ion pair to be reformed again, is a true termination reaction.

7.2 Anionic Polymerization

@@CH2 Cþ HPh þ PF 6 ! @@CH2 CHPhaF þ PF5

ð4Þ

Many other similar eliminations are not true termination reactions, but are transfer reactions, when the eliminated species can re-initiate a new chain, as in reaction (5). @@CH2 aCþ ðCH3 Þ2 ;

SO3 H ! @@CH2 aCðCH3 ÞbCH2 þ H2 SO4

ð5Þ

So, ionic polymerization oﬀers the possibility, by proper choice of the reaction conditions, of running as a living polymerization. Living polymerization in its pure form is deﬁned not only by the absence of any termination reaction destroying the active center, but also by the absence of any transfer reaction, so that the kinetic chain length and the length of the individual chain are the same and increase linearly with conversion. However, especially in many cationic systems, the active center may be preserved throughout the reaction, but transfer reactions to other molecules such as monomers, polymers, solvent, and so on may stop the individual chain and initiate a new growing chain [Eqs. (6)–(8)]. - - -Pn þ M ! - - -Pnþ1

- - -Pn

þ T ! - - -Pn þ T

T þ M ! - - -P1

propagation

ð6Þ

transfer

ð7Þ

re-initiation

ð8Þ

So, the growth of the individual chains may be terminated by transfer reactions. The active center itself is preserved, but a new chain is started. Thus, transfer reactions change the molecular weight of the polymer, but they do not change the concentration of active centers. The conversion kinetics is only changed by transfer reactions if the re-initiation is much slower than the propagation step. Another diﬀerence between free-radical and ionic polymerization that must also be kept in mind is economic rather than scientiﬁc. Most of the monomers can be polymerized by radical or ionic mechanisms (see Table 7.1), but in most cases the requirement for purity of monomers and solvents in ionic polymerization is much higher, and initiators are more expensive, so ionic polymerization is chosen only if the monomers do not polymerize by a radical pathway, or if ionic polymerization oﬀers other advantages, such as access to molecular structures like block copolymers or specially designed molecular weight distributions.

7.2

Anionic Polymerization

Anionic polymerization involves a wide variety of reactions leading to high molecular weight molecules with the participation of an anionic species. The most important systems are those for the anionic polymerization of diene rubbers [14–

325

326

7 Ionic Polymerization Tab. 7.1.

Monomers with diﬀerent initiation mechanisms.

Monomers

Ethylene a-Oleﬁns 1,1-Dialkyl oleﬁns 1,3-Dienes Styrene, a-methylstyrene Halogenated oleﬁns Vinyl esters Acrylates, methacrylates Acrylonitrile, methacrylonitrile Acrylamide, methacrylamide Vinyl ethers N-Vinylcarbazole N-Vinylpyrrolidone Aldehydes, ketones

Type of initiation Radical

Cationic

Anionic

þ þ þ þ þ þ þ þ þ þ

þ þ þ þ þ þ þ þ

þ þ þ þ þ þ þ

16], styrenic polymers, and block copolymers of styrene and dienes [17–21] as well as the ring-opening polymerization of cyclic ethers, lactones, or lactams. Here especially, polymers from ethylene oxide and propylene oxide are important as base materials for polyurethanes [22–24]. 7.2.1

Anionic Polymerization of Hydrocarbon Monomers – Living Polymerization

There are two main reasons why anionic polymerization of hydrocarbon monomers such as styrene, a-methylstyrene, butadiene, isoprene, and so on has gained scientiﬁc and industrial importance:

These monomers may be polymerized under such conditions that termination and transfer reactions are absent, resulting in a so-called ‘‘living’’ polymerization. The microstructure of the polydienes so produced can be varied over a wide range. So the microstructure of dienes strongly depends on the counter ion (see Table 7.2.) and the solvent or the presence of additives, which are capable of shifting the equilibrium in Eq. (3) from the right to the left (see Table 7.3.). An attempt to rationalize the inﬂuence of polar additives on the microstructure is given in Ref. 25. Generally, the vinyl content increases with the fraction of free ions. Association Behavior/Kinetics There are numerous publications [1, 5, 6] dealing with the equilibrium of Eq. (3) between free ions as one extreme possibility for the structure of the initiating and/ 7.2.1.1

7.2 Anionic Polymerization Tab. 7.2. Microstructure of polybutadienes produced in hydrocarbon solvent with diﬀerent counter ions (from [25a]).

Alkali metal

cis-1,4 [ %]

trans-1,4 [ %]

1,2 [ %]

Li Na K Rb Cs

35 10 15 7 6

52 25 40 31 35

13 65 45 62 59

or polymerizing species, and aggregation of more or less covalently bonded species as the other extreme. All these species may participate in the polymerization, and the rates for the same reaction may diﬀer by several orders of magnitude depending on the species involved. For example, the propagation rate coeﬃcient for anionic polymerization of styrene in THF at 25 C is 8 10 4 M1 s1 for the free ion, 200 M1 s1 for the ion pair [26], and values of 0.0155 and 0.024 M0:5 s1 are reported [27, 28] for the overall propagation rate coeﬃcient k p K 1=2 of the equilibrium system from monomeric and dimeric polystyryllithium in benzene and cyclohexane. In particular, the question of association in lithium-based polymerization in nonpolar solvents, one of the most important industrial systems, and the determination of the kinetic scheme and parameters, are still under debate. In general, the association behavior is formulated as in Eq. (9), where n gives the association number, which generally is between 2 and 6, depending on the structure of P (monomer or initiator), solvent, and temperature (see Table 7.4). K ass

ðPaLiÞn ! nPaLi

ð9Þ

Tab. 7.3. Vinyl content of polybutadienes produced in hydrocarbon solvent with butyllithium as initiator with diﬀerent additives (from [28a]).

Additive

Molar ratio additive/lithium

1,2 [ %]

Diethyl ether Diethyl ether THF THF THF Diglyme Diglyme Diglyme

10:1 5:1 6:1 3:1 1:1 4:1 2:1 1:1

16 10 43 25 17 87 85 78

327

328

7 Ionic Polymerization Tab. 7.4.

Association number for diﬀerent Li organyls.[a]

Li alkyl

Solvent

n

n-BuLi

benzene cyclohexane THF cyclohexane benzene THF benzene hexane cyclohexane benzene THF benzene cyclohexane hexane benzene hexane cyclohexane cyclohexane benzene hexane

6–6.3 6 2–2.8 4 4 1.1 4 4 2 2 1 2 2 2 2 2 2–4 2–4.3 2–3.7 2

sec-BuLi

t-BuLi Menthyl-Li Benzyl-Li Poly(styryl)-Li

Poly(isoprenyl)-Li

Poly(butadienyl)-Li

[a] Compilation

of data given in Ref. 6, pp. 16, 20, 138.

With the assumption that association is high and the concentration of the monomeric species is low, the concentration of the monomeric species is given by Eq. (10), where PaLi is either the initiating lithium alkyl or the propagating species. ½PaLi ¼

K ass ½PaLi0 n

1=n ð10Þ

Assuming that the associated species are not reactive, or at least they are much less reactive than the non-associated one, the reaction rates for initiation and propagation are given by Eq. (11), where k is either ki or k p. K ass 1=n 1=n 1=n r ¼ k½PaLi½M ¼ k ½PaLi0 ½M ¼ kobs ½PaLi0 ½M n

ð11Þ

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

7.2 Anionic Polymerization

systems, is used here as an example to point out which aspects may become important for reactor and process layout. One reason for the debate is the high sensitivity of ionic systems to impurities, which may cause experimental errors leading to various interpretations of the data. However, from a mechanistic point of view also there are arguments that this simple correlation does not seem to be a general rule. For styrene with t-butyllithium initiation is reported to be independent of monomer concentration [34]; for dienes various orders are reported with diﬀerences between fractional reaction order and degree of aggregation (see Table 7.1 in Ref. 6). Furthermore, cross-aggregation between initiating and propagating species is likely to occur and must be considered, as well as intermediate equilibria between hexamers, tetramers, dimers, and monomers. In Ref. 35 this problem is addressed for the cross-association of styryland butadienyllithium. There are a number of recent publications reporting much higher association numbers of up to 100 and more [36–39], which are under discussion [40, 41]. Bywater discussed [42, 43] the diﬃculties in separating K ass and k p . Nevertheless, there are some reports on the separate determination of K ass , [44–47], which however are contradictory. With the exception of those in Ref. 47, most of the data are from investigations at rather low temperatures (< 30 C) compared to industrial conditions, which are at considerably higher temperatures (60– 100 C). So, extrapolating the association behavior from low to higher temperatures may cause errors. In addition it must be stated that Eq. (10) is only valid if the concentration of non-associated species is low. Otherwise the equilibrium of Eq. (9) must be considered a priori: for example, for n ¼ 2 in the case of styrene, the concentration of non-associated PaLi is given by Eq. (12). s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K ass K ass 2 K ass ð12Þ þ þ ½PaLi0 ½PaLi ¼ 4 4 2 Figure 7.1 gives the temperature dependence of K ass and k p for styrene from Ref. 44. When the fraction a of non-associated polystyryllithium is calculated with the ˙ 10000000

1000

-1

K ass / M

kp / M s

10

10

-1

10000

0.01

0.1

0.00001 0.00000001 0.0025

0.003

0.0035

0.001 0.004

1/T / 1/K

Fig. 7.1.

Association equilibrium constant K ass and propagation rate constant k p as f ðTÞ [44].

329

7 Ionic Polymerization 10

1

α

330

0.1

0.01

0.001 0.000001

0.0001

Kass / M

0.01

1

Fraction a of non-associated species as a function of K ass for dimerization for various polystyryllithium concentrations (b, 0.0001 M; , 0.001 M; þ, 0.01 M). Broken lines: exact solution according to Eq. (12); full lines: approximate solution according to Eq. (10). Fig. 7.2.

simpliﬁcation in Eq. (10) and the correct solution in Eq. (12), Figure 7.2 shows that there may be considerable deviations for values of K ass ¼ 105 M and higher. So, for industrial temperatures the fraction of associated species may be rather low and a ﬁrst-order kinetic law with respect to lithium concentration may result [10, 33]. Table 7.5 gives some examples of the temperature dependence of the propagation rate coeﬃcient for Li-initiated styrene polymerization according to Eq. (11) with n ¼ 2. From Figure 7.3 it can be seen that there are deviations of one order of magnitude between the results of diﬀerent authors, even when considering the same solvent and a rather limited temperature range of 4–60 C. In Ref. 48 data for styrene and butadiene are given up to 70 C assuming n ¼ 2 for styrene and n ¼ 2 or 4 for butadiene. More diﬃculties will arise when extrapolating to higher temperatures and tackling the problem of temperature dependence of the association behavior. In Ref. 10 an attempt is made to describe the propagation over a wider tem-

Literature values for propagation rate coeﬃcient kp0 ¼ k p ðK=2Þ 0:5 for Li-initiated styrene polymerization in hydrocarbon solvents. Tab. 7.5.

˚

kOp; 0 [MC0:5 sC1 ]

Ea [kJ molC1 ]

Solvent

Temperature [ C]

Ref.

4:5 10 8 2:6 10 9 2:9 10 11 5:2 10 8 5:1 10 7 6:8 10 8

60.3 64.0 78.6 60.3 56.9 63.6

none none cyclohexane benzene toluene toluene

20–50 4–21 30–50 10–30 20–50 30–60

49 50 51 27 52 53

7.2 Anionic Polymerization 1 acc. to [49] acc. to [50] acc. to [51] acc. to [27] acc. to [53] acc. to [52]

0.01

1/2

kp K / M s

-1 -1

0.1

0.001

0.0001 0.003

0.0031

0.0032

0.0033

0.0034

1/T / K

-1

0.0035

0.0036

0.0037

Fig. 7.3. Comparison of literature values for propagation rate coeﬃcient kp0 ¼ k p ðK=2Þ 0:5 for lithum-initiated styrene polymerization in hydrocarbon solvents.

perature range, from 10 C up to 100 C, with k p ¼ 1 10 11 e7900=T M1 s1 and K ¼ 3:2 10 38 e27600=T M according to Eq. (12), which collapses to a simple ﬁrstorder kinetic with respect to initiator above 40 C. There is no judgment on the quality of the various results, but this overview can serve as an example for the diﬃculties in ionic polymerization kinetics in general, and careful checking of literature kinetic data is strongly recommended before they are used for reactor layout. This somewhat extensive discussion on the association behavior of lithiuminitiated polymerization, the best-investigated ionic system, shows the diﬃculties in kinetic modeling of ionic polymerization. Contrary to free-radical polymerization, the existence of all possible diﬀerent species must be considered for every system under investigation. Molecular Weight Distribution of Living Polymers The living nature of the anionic polymerization of hydrocarbon monomers has been revealed by Szwarc [54, 55]. There is still an ongoing debate on the exact definition of living polymerization [5, 6, 55–57]. For the scope of this chapter we will refer pragmatically to living polymers if the active end groups of the individual chains ‘‘retain the propensity of growth for at least as long a period as needed for the completion of the intended synthesis’’ [5]: that is, initiation and propagation [(Eqs. (13) and (14)] are the only reactions, irreversible termination and transfer reactions being absent. 7.2.1.2

331

332

7 Ionic Polymerization

I þ M ! P1

initiation

ð13Þ

P1 þ M ! Piþ1

propagation

ð14Þ

In the sense of this deﬁnition, the associated species discussed above can also be called living, even if they do not participate in a propagation reaction, but they are in equilibrium with the active non-associated species on a much shorter time scale than the overall reaction. For simplicity, in the following discussion the association is not considered explicitly, but if it is known quantitatively it can be considered by making the rate coeﬃcients a function of [PaLi]. From the reactions in Eqs. (13) and (14), the set of diﬀerential equations (15)– (20) can be derived.

d½I ¼ k i ½I ½M dt

ð15Þ

y X d½M ¼ k i ½I ½M þ k p ½M ½Pi ¼ RP dt i¼1

ð16Þ

d½P1 ¼ k i ½I ½M þ k p ½P1 ½M dt

ð17Þ

d½Pi ¼ k p ½Pi1 ½M þ k p ½Pi ½M dt

ð18Þ

y X

½Pi ¼ ½I 0 ½I

ð19Þ

i½Pi ¼ ½M0 ½M

ð20Þ

i¼1 y X i¼1

For fast initiation (k i g k p ), all initiator is transferred to growing chains immediately, so monomer conversion and the kinetic chain length are given by Eqs. (21)– (23). ½M ¼ ½M0 ek p ½I 0 t

ð21Þ

½M0 ½M ½M0 ½M0 ekP ½I 0 t ½M0 ¼ ¼ xM ½I 0 ½I 0 ½I 0

ð22Þ

½Pi ¼ ½I 0

ð23Þ

n¼ with y X i¼1

7.2 Anionic Polymerization

The resulting frequency distribution hðiÞ of chain lengths i can be derived by solving the system of diﬀerential equations sequentially starting with ½P1 t¼0 ¼ ½I 0 and the deﬁnition of the kinetic chain length given above [Eq. (24)]. ½Pi ½Pi nði1Þ en ¼ hðiÞ ¼ X ¼ ði 1Þ! ½Pi ½I 0

ð24Þ

The weight distribution is given by Eq. (25). wðiÞ ¼

nði1Þ en ði 1Þ!ðn þ 1Þ

ð25Þ

The average number- and weight-average degrees of polymerization and the polydispersity for this Poisson distribution are given by Eqs. (26)–(28). Pn ¼

½M0 ½M ¼1þn ½I 0

Pw ¼ 1 þ n þ D¼

ð26Þ

n A 1 þ Pn 1þn

ð27Þ

Pw A1 Pn

ð28Þ

If, however, k i g k p does not hold, meaning if the initiation rate coeﬃcient is in the order of k p or even less, there is no immediate conversion of initiator to growing chains and the simpliﬁcation of Eq. (23) becomes invalid and must be replaced by Eq. (19). The solution for this general case is given in Refs. 58 and 59. From Eqs. (15) and (16), the variation of monomer conversion as a function of initiator conversion is given by Eq. (29) with r ¼ k p =k i > 1. ½M ½M0 ¼ ð1 rÞð½I ½I 0 Þ þ r½I 0 ln

½I ½I 0

ð29Þ

The variation of M and I with time cannot be solved analytically in this case, but must be found numerically. The frequency and weight distributions of the socalled Gold distribution are given by Eqs. (30) and (31), with R ¼ r 1, q ¼ r=R and u ¼ ðr 1Þ lnð½I 0 =½I Þ. The averages of this distribution are given by Eqs. (32) and (33).

hðiÞ ¼

½Ni ¼ y X ½Ni i¼1

i y r qu X uj e R j! j¼i rð1 eu=R Þ

ð30Þ

333

7 Ionic Polymerization

wðiÞ ¼

y i½Ni r i ru=ðr1Þ X uj ru u=R e Þ þ ¼ ð1 rÞð1 e y X r1 R j! j¼i i½Ni

ð31Þ

i¼1

ð1 rÞð1 eu=ðr1Þ Þ þ Pn ¼

ru r1

ð32Þ

ð1 eu=ðr1Þ Þ

ru 1 ru þ 2ru þ ð2r 1Þðr 1Þð1 eu=ðr1Þ Þ r 1 r 1 Pw ¼ ru ð1 rÞð1 eu=ðr1Þ Þ þ r1

ð33Þ

A Poisson distribution will also result when using bifunctional initiators [60], with the peak maximum at 2½M0 =½I 0 . Figure 7.4 gives a comparison of the Poisson distribution for n ¼ 50 (k i > k p ), the Gold distribution for (k i < k p , r ¼ k p =k i ¼ 100), and the Schulz–Flory or most probable distribution resulting from step growth polymerization or free-radical polymerization with termination by disproportionation. This most probable distribution is given by Eq. (34), where p is the conversion of end groups in step growth polymerizations or polycondensation or the probability of propagation in chain growth polymerization. hðiÞ ¼ p i1 ð1 pÞ

(34)

The width of the distributions reﬂects the sharpness of the initiation reaction; with k i > k p all chains start at the same time and the distribution is very narrow, just reﬂecting some statistics of monomer addition. If chain initiation is delayed for some chains (k i < k p ), the distribution is skewed and will become narrower if reac-

0.06 0.05

Poisson

0.04

h(i)

334

0.03

Gold

0.02

Schulz-Flory

0.01 0 0

50

100

150

i Fig. 7.4. Comparison of various distributions for Pn ¼ 51: Schulz–Flory distribution with p ¼ 0:98, Pw ¼ 102; Gold distribution with r ¼ k p =k i ¼ 100, u ¼ 86:513, Pw ¼ 64; and Poisson distribution for n ¼ 50. The corresponding weightaverages are 102, 64, and 52.

200

7.2 Anionic Polymerization 1.4

1 0.8

1.3

xM

D

0.6 1.2 0.4 1.1

0.2

1

0 0

0.2

0.4

0.6

0.8

1

xI

Polydispersity D and monomer conversion xM as a function of initiator conversion xI for r ¼ 10 (full lines) and r ¼ 100 (broken lines) according to Eqs. (29), (32), and (33). ½I0 ¼ 0:101 M, ½M0 ¼ 3 M. Number- and weight-averages at xM ¼ 1 are 31 and 34 for r ¼ 10, and 51 and 64 for r ¼ 100. Fig. 7.5.

tion proceeds and more chains will be initiated and propagate. The broadest distribution is the most probable distribution if, as in step growth polymerization, there is no initiation but only chain propagation. Slow initiation has some important consequences, which are shown in Figure 7.5. The monomer may be consumed before initiator conversion is complete. This may be important when using functionalization techniques or synthesizing block copolymers. Moreover, the polydispersity may be considerably higher than 1, the value which is usually assumed to be typical for living polymerization and used as a criterion for ‘‘livingness’’. Besides the reactivity ratio r, the equilibrium between diﬀerent species – associated and non-associated ones or free ions and ion pairs, active and dormant species – which may have diﬀerent reactivities in propagation reactions, can also have an impact on broadening the molecular weight distribution [61–73]. The MWD discussion up to now has concerned batch reactors, but there are also a number of publications dealing with other reactors. In ideal plug ﬂow reactors under steady-state conditions, polymers with the same characteristics of the MWD as in batch reactors are built – the time axis is transformed to the length axis of the plug ﬂow reactor [74, 75], and for fast initiation the MWD is a Poisson distribution. For laminar plug ﬂow reactors [75, 76] some broadening of the distribution is observed, depending on ½M0 =½I 0 and conversion. The long-term stability of laminar ﬂow tubular reactors is questioned [76], because of the possibility of very long chains growing near the reactor walls, where the residence time approaches inﬁnity, if radial diﬀusion of the monomer occurs from the inner tube to the walls. The MWD resulting from semi-batch operations of a stirred tank reactor with monomer feed under various conditions is treated in Refs. 77–82. In a homogeneous continuous stirred tank reactor (HCSTR), the steady-state concentrations of monomer and initiator can be derived from the monomer and initiator mass bal-

335

336

7 Ionic Polymerization

ance, neglecting volume changes, by Eqs. (35) and (36), which must be solved simultaneously. ½Mf and ½I f are the feed concentrations, V is the reactor volume, and vin; I ; vin; M are the volume feed ﬂows of initiator and monomer. vin; M ½Mf t V ½M ¼ k i ½I þ kP k i ½M 2 ½I t 2 þ ½M

ð35Þ

vin; I ½I f t ½I ¼ V k i ½Mt þ 1

ð36Þ

In a CSTR, the narrow Poisson distribution resulting from the chemistry is superimposed by the broad residence time distribution of the HCSTR – chains can leave the reactor after seconds of growth as very short chains, but there are also chains which reside in the reactor for very long time, growing to very long molecules. This results in a Schulz–Flory distribution [83] [Eqs. (37)–(40)], where t is the mean residence time and [M] is the steady state-monomer concentration. j1 1=t kP ½M hð jÞ ¼ kP ½Mt þ 1=t kP ½M þ 1=t wð jÞ ¼ j

1=t kP ½Mt þ 1=t

2

ð37Þ j1

kP ½M kP ½M þ 1=t

ð38Þ

Pn ¼ 1 þ kP ½Mt

ð39Þ

Pw ¼ 1 þ 2kP ½Mt

ð40Þ

Equation (37) is equivalent to Eq. (34) with the propagation probability given by Eq. (41). p¼

kP ½M kP ½M þ 1=t

ð41Þ

In a segregated stirred tank reactor (SCSTR) [74] the width of the distribution ranges from that of an HCSTR for rather low conversion or mean residence time, up to a value approaching that of a batch reactor for high conversions or long mean residence times. The inﬂuence of termination, chain transfer, and longchain branching in batch and CSTR reactors is described in Refs. 84 and 85. Side Reactions Pure living polymerization of hydrocarbon monomers usually occurs at low temperatures, where side reactions are not important. At higher temperatures, however, two side reactions do become important, elimination of LiH and transfer to compounds with an acidic hydrogen [5, 6, 31, 86, 87]. 7.2.1.3

7.2 Anionic Polymerization

337

There have been some investigations on the stability of alkyllithium derivatives and of growing chains. Generally, linear alkyllithiums are more stable than branched ones, with ﬁrst-order decomposition half-lives of 1–6 h at 87–98 C [88, 89]. From Ref. 89, k tt ¼ 1:0 10 10 expð11798=TÞ s1 for s-BuLi and k tt ¼ 9:5 10 10 expð13407=TÞ s1 for n-BuLi can be evaluated. As with all rate coeﬃcients of ionic polymerization, this elimination also depends on the reaction system. The lithium hydride (LiH) elimination from butyllithium in the presence of polar additives such as lithium butoxide is reported to be several times faster than from pure lithium alkyl [90, 91], and in ethereal solvents proton abstraction or ether cleavage may occur [92]. Growing chains such as polydienyl- and polystyryllithium are less stable. For polystyryllithium, the LiH elimination is followed by proton transfer to another polystyryllithium [93], resulting in an allylic anion, which is unable to propagate (see Scheme 7.1). The same holds for polystyrylsodium [94]. Quantitative data for LiH elimination are given in Refs. 13 (k tt ¼ 295 e5280=T s1 ) and 10 (k tt ¼ 3:92 10 6 eð8700=TÞ ). In Ref. 95 the inﬂuence of THF was investigated, but rather similar values were obtained (k tt ¼ 2:4 10 6 eð8459=TÞ ). It must be noted that these termination reactions are much slower than propagation, that is to say the half-life for the active chains is much higher (hours) than the half-life for propagation (seconds) at temperatures of 80 C and higher.

Bu

Li

Bu

k tt

+

LiH

i

i

+

Li Bu +

+

PsLi

i Scheme 7.1.

-

Bu

i

Termination in anionic polymerization of styrene.

Polybutadienyllithium is somewhat more stable than polystyryllithium (k tt ¼ 6:7 105 s1 versus 1:9 104 s1 at 93 C [13]), but contrary to styrene, the side reactions here may cause coupled and branched polymers [95–98]. The other class of important side reactions is transfer to compounds with acidic hydrogen atoms, such as toluene, ethyl benzene and so on (Scheme 7.2). Some quantitative data have been given for chain transfer from polystyryllithium to aromatic solvents [10, 33, 47, 99–101], and from polydienyllithium to alkenes [102, 103] and toluene, and for the inﬂuence of polar additives [104]. There is usually no eﬀect on the polymerization rate, because the number of active chain ends

PsH

338

7 Ionic Polymerization

Bu

Li

Li

Bu +

k tr

i

+ i Li

Li +

kp

1 Scheme 7.2.

Transfer to ethylbenzene in anionic polymerization of styrene.

is maintained, but the chain length of the macromolecules, that is, the molecular weight, is reduced. The presence of transfer reactions limits the production of block copolymers by sequential addition of monomers to the living chains. However, in Refs. 33 and 99 it is pointed out that the presence of transfer reactions may be advantageous, for example, for homopolymerzation of styrene in a CSTR, because they reduce the amount of initiator necessary to get the desired molecular weight. The eﬀect of side reactions, such as termination by monomer, impurities, or spontaneous termination and transfer to monomer or impurities, on the molecular weight distribution are dealt with in Refs. 66 and 105–119; they generally result in some broadening of the distribution. Copolymerization The most important copolymers are those from styrene and butadiene, either as statistical copolymers or block copolymers. From a kinetic point of view the association behavior discussed above becomes even more complex because of the possible cross-association between the diﬀerent growing chain ends. This issue has seldom been addressed [35, 120]. But in spite of this complex association behavior, in most cases the simple Mayo terminal model [Eq. (42)] with two copolymerization parameters, which was developed originally for free-radical polymerization, is in many cases suﬃcient to describe anionic copolymerization also. 7.2.1.4

F1 ¼

r1 f12 þ f1 f2 þ 2f1 f2 þ r2 f22

r1 f12

ð42Þ

However, there is one important diﬀerence between free-radical and living anionic polymerization, and this is the lifetime of the growing chain. This diﬀerence becomes important when considering the dependence of copolymer composition on conversion. Equation (42) gives the copolymer composition or mole fraction F1 of monomer M1 in the polymer as a function of mole fraction f1 in the monomer

7.2 Anionic Polymerization

feed for incremental conversion. Because of the diﬀerent reactivity of the monomers, usually f1 and consequently F1 change with increasing conversion. In free-radical polymerization, where the lifetime of the growing polymer chain is in the order of seconds or less, compared to hours for the overall polymerization reaction, and where new chains are initiated throughout the overall reaction time by decomposition of radical initiators such as peroxides or azo compounds, those chains which are initiated at diﬀerent times or levels of conversion will diﬀer in their composition, but those which are initiated at the same time will have the same composition. This diﬀerence in composition from chain to chain is called ﬁrst-order chemical heterogeneity. If, however, as in living polymerization, all chains start at the same time and live throughout the polymerization, every chain will see all the changes in monomer feed composition, and consequently the composition will change along the chain and not from chain to chain. This is known as second-order chemical heterogeneity. The diﬀerences between these two kinds of heterogeneities, one between diﬀerent chains, the other within diﬀerent fractions of one chain, are shown in Scheme 7.3.

Increasing conversion

First-order chemical heterogeneity

I-BBBBBSBB

I-BBBBBSBB I-SBSBSSBS

I-BBBBBSBB I-SBSBSSBS I-SSSSBSSS

Second-order chemical heterogeneity R-BBBBBSBB-Li

R-BBBBBSBBSBSBSSBS-Li

R-BBBBBSBBSBSBSSBSSSSSSSSS-Li

Scheme 7.3. Chemical heterogeneity with increasing conversion for free-radical (ﬁrst-order) and living polymerization (second-order).

The severity of the chemical heterogeneity strongly depends on the copolymerization parameters. In free-radical polymerization there is just one pair of parameters, which may depend somewhat on temperature, for one pair of monomers; whereas in ionic polymerization these parameters for every pair of monomers strongly depend on the counter ion and solvent polarity (see Table 7.6). These extreme values have a very drastic eﬀect on the shift not only in composition but also in rates. Taking the absolute rate coeﬃcients given by Ohlinger [35] for Li-initiated copolymerization of butadiene and styrene at 20 C in toluene (kSS ¼ 0:45, kSB ¼ 110, kBB ¼ 0:084, kBS ¼ 0:0066 M1 s1 ), the resulting overall

339

7 Ionic Polymerization Tab. 7.6. Copolymerization parameters for Li-initiated copolymerization of styrene (¼ M1 ) and butadiene (¼ M2 ).[a]

˚

T [ C]

Solvent

r1

r2

25 20 30 50 30 78 0 25 25

bulk toluene benzene cyclohexane heptane THF THF THF diethyl ether

0.04 0.004 0.035 0.025 0.1 11 0.2 0.3 0.4

11.2 12.9 10 15.1 7 0.04 5.3 4 1.7

[a] Examples

taken from Ref. 6, p. 247ﬀ.

conversion versus time curves are given in Figures 7.6 and 7.7. The interesting feature is that even though the homopolymerization of styrene is faster than that of butadiene, during copolymerization nearly all the butadiene is consumed at a lower rate than in homopolymerization before styrene is incorporated to any appreciable extent, and after the butadiene is consumed the overall polymerization rate becomes that of pure styrene.

1 xS = 1 0.8

xS = 0 xS = 0.35

conversion

340

0.6 xS = 0.5 0.4

xS = 0.65

0.2

0 0

5000

10000

15000

20000

t/s

Batch copolymerization of styrene and butadiene with n-butyllithium (1 g) at 20 C in toluene (10 kg) with various styrene feed fractions xS (20 mol monomer overall). Simulation of overall conversion with data from Ref. 35. Fig. 7.6.

25000

30000

7.2 Anionic Polymerization 0.8

x S = 0.65 0.6

xS inpolymer

x S = 0.5

0.4

x S = 0.35

0.2

0 0

5000

10000

15000

20000

25000

30000

t/s

Fig. 7.7. Batch copolymerization of styrene and butadiene with n-butyllithium (1 g) at 20 C in toluene (10 kg) with various styrene feed fractions xS (20 moles monomer overall). Simulation of mole fraction of styrene in copolymer with data from Ref. 35.

The reason is that kSB has a high value and kBS has a low value. This combination is responsible for nearly all chain ends existing as butadienyllithium ends, which predominantly propagate by addition of butadiene (kBB > kBS ). If one of the occasional propagation reactions by addition of styrene gives a styryllithium chain end, there will be a very fast addition of butadiene to regenerate butadienyllithium ends. Thus, in a batch copolymerization, very few styrene units are incorporated into the chains during the ﬁrst part of the reaction and they exist as isolated units. Incorporation of styrene will increase only when nearly all the butadiene is consumed. Overall, therefore, there is a block of nearly pure butadiene with some isolated styrene units, followed by a rather short block where styrene and butadiene are incorporated at almost the same rates – a so-called tapered block, followed by a ﬁnal block of nearly pure styrene (see Scheme 7.3). From Table 7.6 one can see that a similar but reverse situation is valid in polar solvents; here styrene is consumed ﬁrst and then butadiene, but again a copolymer with nearly pure blocks of homopolymer will result. Tailor-made Polymers by Living Polymerization – Optimization Anionic living polymerization oﬀers a unique opportunity for tailor-made polymers because all the chains and their living ends are accessible throughout the course of the reaction. In general, there are two important tasks in tailoring polymers: one concerns their chemical distribution and the other the molecular weight distribution. 7.2.1.5

341

342

7 Ionic Polymerization

The structure of copolymers in terms of composition or sequence length distribution may vary between two extremes – copolymers with a very high level of chemical heterogeneity within each chain, and homogeneous copolymers with the comonomer units randomly distributed along the chain. The various pathways to block copolymers are described in a number of reviews and monographs [6, 121–127]. In principle, block copolymers are accessible by sequential addition of monomers to either monofunctional or bifunctional [128, 129] initiators (see Scheme 7.4, where the route from a monofunctional initiator to a three-block copolymer is shown as an example). The amount of undesired byproducts – homopolymers or di-block copolymers – depends on the reactivity ratios relative to propagation given in the scheme. These are obviously the ratios for transfer and termination at higher temperatures, but also the ratios for initiation and the cross-propagation reactions. From Figure 7.5 and Eq. (29) it can be seen that the monomer for forming one block may be consumed before the initiator or the ends of the previously formed block are completely transferred to the ends of the new block, and so the yield of tri-block copolymers is reduced.

Reactivity ratios i for side reactions

R-Li

ki k p,S

+nS

SSSSSS

R-So-L -Li + R-Li -L

k S Bu k p , Bu

+ m Bu

R-So-Bup-L -L + R-Li -Li + R-Buq-Li

k Bu S k p ,S

k tt k p ,i

SSSSBBBBB + BBBBB

k tr k p ,i

+lS

R-So-Bup -Sr ––Li + R-Buq-Ss-Li + R-S - t-Li Scheme 7.4.

Dead chains

SSSSBBB + SSSSBBBSSSS

Pathway to SBS tri-block copolymer and possible side products.

The optimal production of tri-block copolymers in a series of CSTRs, where the reactors are operated in an isokinetic way, is described in Ref. 130. To produce a copolymer with a more random structure in spite of the extreme values of the copolymerization parameters, there are several possibilities. One is

7.2 Anionic Polymerization

to use small amounts of polar additives such as THF, ethers, alkoxides, and so on as modiﬁers [131, 132], to bring the extreme r-values nearer to unity. Another obvious method is to polymerize in a continuous stirred tank reactor, where there is no shift in composition. For the steady state, the copolymerization equation can be written as a function of feed (index 0) and reactor concentrations [Eq. (43)]. ½M1 0 ½M1 ¼ ½M2 0 ½M2

½M1 þ1 ½M2 ½M2 þ1 r2 ½M1

r1

ð43Þ

A third possibility is model-based feed control [133], where butadiene is fed to a mixture of styrene and butadiene in such a way that a certain monomer ratio is maintained. The other important task is to tailor the molecular weight distribution of the polymer, and especially for this task the ‘‘livingness’’ in anionic polymerization is advantageous [134]. Optimized reactor operations for broadened or bimodal distributions in a tubular reactor are described in Refs. 135 and 136. Semi-batch operation with a programmed initiator feed [137, 138] and oscillating feeds to homogeneous CSTRs oﬀer the possibility of a wide range of MWDs [139–144]. Industrial Aspects – Production of Living Polymers In general, the requirements for purity of the reagents, solvents, and monomers are much higher in anionic polymerization than in free-radical polymerization. Necessary distillations and/or adsorption towers, often with activated aluminum, increase the costs for feed preparation compared to free-radical processes. Anionic polymerization of styrene has attracted much attention during recent years and several publications and patents have been published on this issue. The main reason for this interest is the high rate of polymerization which can be achieved compared to free-radical polymerization, and the low content of residual styrene and oligomers in the ﬁnal polymer [145–147]. There are several concepts for an anionic process, which in many cases are operated in an adiabatic or at least non-isothermal mode. Some are using recirculated loop reactors [148–151] or single-pass tubular reactors [75, 152, 153], which may be segmented [154]. A spray tower [155] has a similar residence time distribution, but heat removal is by a countercurrent nitrogen stream. Boiling CSTRs are described in Refs. 33 and 156. One problem seems to be fouling or gel formation in regions of the reactor where the ﬂow of living chains is limited [76]. Polybutadiene rubbers are produced either with transition metal catalysts to give high-cis-butadiene rubber or with lithium alkyls to give medium-cis-butadiene rubber (see Table 7.7). The latter is mainly used as the rubber component in the production of HIPS (high impact polystyrene). These rubbers are generally produced in aliphatic solvents [6, 157–159] in a CSTR or a series of CSTRs. The vinyl content, which is usually about 10% in aliphatic solvents, can be varied by the addition of polar compounds (see Table 7.3) over a wide range. 7.2.1.6

343

344

7 Ionic Polymerization Tab. 7.7. Microstructure of polybutadienes produced in hydrocarbon solvent with diﬀerent catalyst systems.

Catalyst

cis-1,4 [ %]

trans-1,4 [ %]

1,2 [ %]

Nd Co Ni Ti Li

98 96 96 93 36

1 2 3 3 52

1 2 1 4 12

Copolymers from styrene and butadiene (SBRs) with a more or less random comonomer distribution are produced either in an emulsion polymerization process (cold or hot E-SBR), a free-radical polymerization, which however gives crosslinked polymers, or in a solution process using lithium initiators (S-SBR). There are continuous processes [157, 158, 160–166] as well as (semi-) batch processes (167, 168) with a controlled feed of butadiene [133, 169] to maintain a constant monomer ratio. Randomizers are often used [131, 170] to control the comonomer distribution. Block copolymers and also star-shaped [171] block copolymers from styrene and dienes are generally produced batch-wise with the monomers added in appropriate sequences. These sequences depend on the functionality of the initiator (mono- or bifunctional initiators) [172–175], on the use of coupling agents [176, 177], and on the sharpness of the transition between the various blocks. Another reason for apportioning the monomer feed is the limited heat removal capacity of the reactor. 7.2.2

Anionic Polymerization of Vinyl Monomers Containing Heteroatoms

The possibilities of living polymerization described above have always attracted researchers toward extending this technique to other vinyl monomers such as acrylates (i), methacrylates (ii), cyanoacrylates (iii), nitriles (iv), and vinyl aldehydes or ketones (v) (see Scheme 7.5). However, almost no industrial applications have

H

H

H

CH3

H

CN

H

COOR

H

COOR

H

COOR

i

ii

iii

H

H

H

H

H

CN

H

COR

iv Scheme 7.5.

v Monomers with heteroatoms for anionic polymerization.

7.2 Anionic Polymerization

been developed up to now for this group of monomers. Some overviews can be found in Refs. 5 and 7. An interesting example of the application of these monomers are the cyanoacrylates, which can be polymerized by very weak bases such as water or skin proteins and which are used as superglues. The reason for this deﬁciency in application is the presence of side reactions, which can only be suppressed by proper selection of the structure of the monomer itself and the reaction conditions, especially the polymerization temperature. The propagating center in the anionic polymerization of (meth)acrylates is the ester enolate anion, which is formed according to Eq. (44).

A

+

H

CH3

H

COOMe

A CH2

O

ð44Þ

OMe

H3C

The most important side reaction is the attack at the ester group, which can be inter- or intramolecular [Eqs. (45) and (46)]. H

CH3

H

COOMe

CH3

H

+

R

CH3

CH3 OMe O

H3C

O

O

ð45Þ

CH3

CH2

OMe

O MeO

Me

COR

H

CH3

CH2

+

O

O H3C

+ HC O 3

O OMe

OMe

ð46Þ

Another side reaction is the enolization of acrylate polymer chains [Eq. (47)]. H A

+

CH2 C COOR

CH2 C

+

AH

ð47Þ

COOR

Whether these side reactions are termination or transfer reactions depends on the ability of the anions to re-initiate the polymerization and on the fate of the vinyl ketone formed in Eq. (45). Generally, this vinyl ketone is rapidly incorporated into the chain, and a relatively unreactive dormant end group is produced. The alkoxide from reactions such as Eq. (45) is usually not reactive enough to initiate another chain. These side reactions can be suppressed to some extent at temperatures below 0 C, which make them hardly accessible in industrial applications. The t-butyl group is reported to prevent side reactions. In general, the kinetic formalism for the heteroatom-containing monomers is the same as that described in Section

345

346

7 Ionic Polymerization

7.2.2, but side reactions are more important. Some recent developments in the polymerization of these monomers have been described in Refs. 178 and 179, and these papers also cover the relevant mechanistic literature. However, the anionic polymerization of (meth)acrylate monomers has not yet gained industrial importance. 7.2.3

Anionic Polymerization of Monomers Containing Hetero Double Bonds

Besides monomers containing carbon–carbon double bonds, monomers with heteroatoms at the double bond can also be polymerized via an anionic mechanism. The only monomer with some industrial importance is formaldehyde. It is polymerized [180] via its carbon–oxygen double bond to give polyacetal homopolymer or polyoxymethylene [Eq. (48)]. nCH2 O ! ðaCH2 aOaÞn

ð48Þ

The polymerization is a precipitation polymerization from gaseous formaldehyde in an inert solvent such as cyclohexane at fairly low temperatures to prevent depolymerization reactions. Amines such as tri-n-butyl amine are used as initiators. The polymer precipitates in the dispersing agent as powder. After completion of the polymerization, it is necessary to stabilize the hydroxyl end groups, for example by esteriﬁcation with acetic anhydride to prevent the unzipping reaction of Eq. (49). aðaCH2 aOaÞn CH2 aOH ! aðaCH2 aOaÞn1 CH2 aOH þ CH2 O

ð49Þ

7.2.4

Anionic Polymerization via Ring Opening

There are a number of heterocyclic monomers, for example, epoxides, cyclic sulﬁdes, lactones and lactides, lactams, cyclic carbonates, and cyclosiloxanes, which can be polymerized by ring-opening reactions; many of them can be polymerized by an anionic as well as by a cationic mechanism. They cannot all be covered here, but there are a number of monographs and reviews on this subject [181–185]. In polymerization via double bonds the driving force for chain growth is the energy diﬀerence between one double and two single bonds, which in most cases is high enough to overcome the loss in entropy when going from monomers to polymers. In ring-opening polymerization the number and nature of the bonds remain the same, and there is just the release of ring strain to make chain growth of cyclic monomers an exothermic reaction. The Gibbs free energy of polymerization DGP is given in Eq. (50). DGP ¼ DHP TDSP

(50)

7.2 Anionic Polymerization

The equilibrium constant KP for the chain growth step Pi þ M ! Piþ1 is deﬁned in Eq. (51) and the equilibrium monomer concentration ½Me in Eq. (52). kp ½Piþ1 1 ¼ A kd ½Pi ½Me ½Me

KP ¼

ð51Þ

ln½Me ¼ ðDHP TDSP Þ=RT

ð52Þ

For monomer concentrations below this equilibrium concentration, no polymerization occurs. As for most polymers, the changes for entropy and enthalpy are both negative (see Table 7.8), there exists a limiting temperature TC above which polymerization is thermodynamically forbidden. TC is given by Eq. (53), where the standard state refers to unit concentration. DHP DHP0 ¼ 0 DSP DSP þ R ln½M

TC ¼

ð53Þ

From Table 7.8 it can be seen that for most of the cyclic monomers, the reversibility of the propagation must be taken seriously into account. It has been shown [186] that, even for a living polymerization without any termination or transfer re-

Tab. 7.8. Standard thermodynamic parameters for some cyclic monomers for anionic polymerization [182].

Monomer O

O

Atoms in ring

States [a]

DHp0 [kJ molC1 ]

DSp0 [ J molC1 KC1 ]

[M]e [M]

3

gc

140

174

4 1016

4

lc

82.4

74

3 1011

5

ss

14

13.5

1:6 102

6

lc

7.1

27.6

6

ss

22.9

41.1

7

lc

13.8

4.6

O O P

O

OMe

O

N H

1.58

O O

O

O

N H

1:16 102

O

2:2 103

O

[a] State of monomer (ﬁrst letter) and polymer (second letter): g – gaseous, l – liquid, s – solution, c – condensed.

347

7 Ionic Polymerization

action but where the propagation step is reversible, the limiting molecular weight distribution is not a Poisson distribution but a most probable distribution with D ¼ 2. As the cyclic monomer and the linear polymer chain consist of the same bonds, there are two further side reactions which are inherent to the system – inter- and intramolecular chain transfer to polymer (Scheme 7.6).

X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

X

X

+

X

X

X

X

+

X

348

X

kinter X X

X

X

X

X

kintra +

X

+

X

X

X

X

X

kp X

Scheme 7.6.

X

X

X

X

X

Chain growth, inter- and intramolecular transfer in ring-opening polymerization.

By intermolecular chain transfer there is no change in Pn , because the number of chains remains the same, but a scrambling of the molecular weight distribution will occur, leading to the most probable MWD as a limiting distribution. Intramolecular chain transfer will lead to macrocycles, which also can participate in propagation reactions. The equilibrium concentration according to the Jacobsen– Stockmayer theory [187] of these non-strained macrocycles of size n is given by Eq. (54). ½Mn e ¼

kp ðnÞ A n5=2 k p ðnÞ

ð54Þ

The anionic polymerization of lactams [188, 189] is usually via a so-called activated anionic mechanism with a two-component catalyst system from a strong base and an N-acyllactam. It is interesting in the sense that the active center of the growing chain is an N-acyllactam end group to which a lactamate anion, formed from the monomer, is added (Scheme 7.7). First, there is H-abstraction from the monomer to give the lactamate anion. In principle, this anion could add to another lactam molecule, but this step is rather slow. Instead it will add much faster to the Nacyllactam activator. A subsequent proton transfer gives the active N-acyllactam end group and the lactamate anion is regenerated. The chain propagation consists of the same sequence of elementary reactions. This activated anionic polymerization is living in the sense that the number of active centers is preserved; but the inevitable intermolecular chain transfer to poly-

7.2 Anionic Polymerization

349

initiation H O N

R

R

O

N

O

B

O

N

O

+

+

N

O +

N

N

O R

H O N

R

O

O

N

O

O H O N

N

N

O

O

N

+

O

propagation O H O N *

N

O +

N

O *

O H O N

N

n

O H O N

N n

Scheme 7.7.

O

O

N

O

n

N

O +

H O N

O H O N

O NH

N

O +

N

n

Activated anionic mechanism for ring-opening polymerization of lactams.

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

O

7 Ionic Polymerization

350

lecular weights and applications [22–24]. The most important ones are rather low molecular weight polymers, which are used either as nonionic surfactants [197] or as raw materials for polyurethanes [23, 24]. Scheme 7.8 is a general reaction scheme for the copolymerization of ethylene oxide and propylene oxide with multifunctional alcohols as initiators. OH

OH OH + KOH

HO OH

O K

HO

OK

+

+

+

OK

OK

+

+

+

+

OK

O K

+

+

+

+ H2O initiatior formation

+

OH

HO

OK

O

O

O

OK

O

O

+

OK

O

O

+

OK

O

copolymerization EO / PO

+

+

R

R OK

+

H2O

+

OH

R O

O K

HO

+

R

R nO K

+

+

O

Termination

H2 O

mOH

O

R n OH +

O

mO K

+

chain transfer Scheme 7.8.

Anionic ring-opening copolymerization of ethylene and propylene oxide.

Usually potassium alkoxides, often from multifunctional alcohols, are used as initiators. They are formed in a separate step starting from the respective alcohol and potassium hydroxide. The resulting water must be removed to ensure maximum conversion to the potassium alkoxide. Remaining water will terminate growing chains, forming hydroxyl end groups. Chains with alkoxide end groups can undergo chain transfer reaction with hydroxyl-terminated chains. This proton exchange, which may also happen with multifunctional initiator molecules, is the fastest reaction in this scheme. The acidity of the participating species is rather similar; the diﬀerences are generally those between primary and secondary hydroxyl groups, so the equilibrium constant for the proton exchange is around unity.

7.3 Cationic Polymerization

This equilibrium between active and non-active chains is very fast, so in general a Poisson distribution will result. For the case where the ﬁrst addition is diﬀerent from the subsequent ones, either the Gold distribution [58] in Eq. (30) or a distribution given by Weibull [198] can be used. A recursive formula for a more complex distribution assuming diﬀerent reactivities for each addition of ethylene oxide is given in Ref. 199. However, it has been shown that normally the chain length dependence of the propagation rate coeﬃcient is not important, with the exception of the ﬁrst unit [200, 201]. Both distributions can be used to describe ethoxylation reactions [202]. There are a number of recent publications on the kinetics and copolymerization kinetics of ethylene and propylene oxide initiated with several alcohols and KOH [203–209]. The copolymerization parameters are generally in the ranges of rE ¼ 2–4, rP ¼ 0.15–0.3, showing the higher reactivity of ethylene oxide. One important side reaction, which may occur at higher temperatures, is the isomerization of propylene oxide to allyl alcohol [210, 211], leading to an unsaturated end group (1) which may aﬀect polyurethane production. R O CH2CH CH2

1

The industrial process [23, 212, 213] of ethoxylation and propoxylation is usually a semi-batch process. The starter alcohol and KOH are mixed and water is removed by distillation. In a second step, monomers are fed into the reactor, where the feed rate is chosen so as to be able to remove the heat of polymerization and to keep the latent heat of polymerization of unreacted monomers in a safe state. By this process, homopolymers and random copolymers are accessible. Block copolymers are produced by successive feeds of the respective monomers. Catalyst is removed by addition of acids and subsequent crystallization and ﬁltration of precipitated salts. An optional fourth step is the removal of volatile compounds by distillation.

7.3

Cationic Polymerization

There are many monomers that can be polymerized via a cationic mechanism (see Table 7.1), but the most important polymers from an industrial point of view are homo- and copolymers from isobutene and from some heterocyclic monomers such as trioxane, tetrahydrofuran, and epoxides. There is a detailed discussion on the mechanistic features in Refs. 1–5, 7, and 181–183. 7.3.1

Cationic Polymerization of Vinyl Monomers

Cationic polymerization of vinyl monomers is by electrophilic addition of the monomer to a growing carbenium ion. The reactivity of vinyl monomers in cati-

351

352

7 Ionic Polymerization

onic polymerization follows the electron-donating ability of the substituents at the double bond. Electron-donating substituents increase the nucleophilicity of the double bond in the addition reaction and stabilize the resulting carbenium ion. The general order of reactivity in cationic polymerization is given in Eq. (55). Me H2C

CH N

CH

H2C

CH

H2C

Me H2C

CH

H2C Me

OR

ð55Þ

OMe

Initiation is by either strong protonic acids (perchloric acid, triﬂic acid, and so on) or Lewis acids together with a co-initiator such as BF3/H2 O, AlX3/RX. As discussed for anionic polymerization, for cationic polymerization also several diﬀerent species may be involved in propagation, but it has been pointed out (Ref. 2, pp. 190, 205) that the reactivity diﬀerences between the various species – carbenium ions and ion pairs – are not very great. Covalent species are not active in propagation. In general, propagation rate coeﬃcients in cationic polymerization are very large. A very comprehensive review is given in Ref. 214. For isobutene, values of 1:5 10 8 M1 s1 are reported at 0 and 78 C [215]. For such high rate coeﬃcients the polymerization may become diﬀusion-controlled, which has been addressed [216] for a tubular reactor. In cationic polymerization of alkenes, there is an equilibrium between active ions or ion pairs and inactive covalent species, where the ionization constant is rather low (105 M1 in the isobutene/BCl3 system). The dynamics of this equilibrium strongly aﬀects the width of the molecular weight distribution. For the sequence of reactions [Eq. (56)] the polydispersity of the distribution is given by Eq. (57) [217], if there are no other side reactions such as transfer or irreversible termination. k ass þM H3 CaCHA ! H3 CaCHþ A ! þM

R

k ion

D ¼ Pw =Pn ¼ 1 þ ½I 0 k p =kass

a

a

kp

R

H2C

ð56Þ

ð57Þ

Bimodal distributions may result, if free ions and ion pairs have the same reactivity but diﬀerent lifetimes [218]. However, transfer reactions are nearly unavoidable side reactions in cationic polymerization and may involve b-proton transfer even to rather weak bases like the monomer itself, to transfer agents and solvent as well as Friedel–Crafts alkylation of aromatic rings, if monomers like styrene are polymerized. Proton transfer yields unsaturated chains [Eq. (58)], with the double bond in either the endo or exo position. These may lead to branched polymers, if the double bonds are accessible to homo- or copolymerization.

7.3 Cationic Polymerization

* +

C

*

A

+

+

C

A

+

+

n

ð58Þ

n

* n

As proton transfer usually has the higher activation energy, the molecular weight of the polymer decreases with temperature. Tabulated transfer coeﬃcients may be taken from Refs. 219 and 220. The problem of living or non-living cationic polymerization of alkenes is addressed in Refs. 220 and 221. It is shown that, depending on the molecular weight, even for transfer constants up to 5 104 the system exhibits the behavior of a living system, such as a linear increase of molecular weight with conversion, rather narrow distributions, and so on. Copolymerization parameters for cationic polymerization can be found in Refs. 3, 222, and 223. Note that the scattering of data in higher and so the reliability of copolymerization parameters for cationic polymerization are much less than for radical polymerization. Commercial polymers [2, p. 683ﬀ, 157, 224] from isobutene involve isobutene homopolymers and butene rubbers. Polyisobutene homopolymers comprise a wide range of polymers with molecular weights from a few thousand grams per mole up to 10 6 g mol1 , and butene rubbers are copolymers of isobutene with 1–3 mol% of isoprene. High molecular weight homopolymers are produced in the continuous BASF belt process at 100 C in boiling ethylene to remove the heat of polymerization. The ethylene evaporates during polymerization and the polymer is scraped from the belt. Volatiles are removed in twin-screw extruders. Homopolymers and rubbers are also produced in the Exxon slurry process in methyl chloride at 95 C. The polymerization rate is controlled by initiator feed and cooling is with liquid ethylene and/or propylene. The polymer slurry is withdrawn from the top of the reactor and worked up downstream. Medium and low molecular weight polymers are produced in solution processes in light hydrocarbons in CSTR reactors or in recirculating tubular (loop) reactors. The molecular weight is controlled by monomer purity, initiator, water content, and temperature. 7.3.2

Cationic Ring-opening Polymerization

Monographs and reviews of this subject can be found in Refs. 181–185 and 225– 227. The most important polymers from cationic ring-opening polymerization are polyoxyalkylenes with one or four CH2 groups produced by the polymerization of trioxane and tetrahydrofuran. The smaller epoxides, ethylene and propylene oxide, though able to polymerize by a cationic and anionic mechanism, are not polymerized cationically, because of the existence of side reactions leading to dioxane or

353

354

7 Ionic Polymerization

dioxolane [228]. In contrast to these smaller homologs, THF and trioxane can only be polymerized by a cationic mechanism [23, 229]. Initiation can be by strong protonic acids such as perchloric acid and ﬂuorosulfonic acid, oxonium salts like triethyloxonium (Et3 Oþ , A ¼ BF4 or SbCl6 ), and Lewis acids, for example BX3 , PF5 , SBF5 , and so on. Lewis acids usually need a co-initiator, which in many cases is water, which is present to some extent in the monomer even after puriﬁcation. Hetero polyacids of the general formula Hn Xm Mtz Oy with X ¼ P or Si and Mt ¼ Mo; W or V, have gained more and more importance, especially for THF. The active chain end is usually an oxiranium ion (2). +

R O

2

Ions and ion pairs are both active in propagation, but their respective contribution usually cannot be distinguished because of the high exchange rates between the two. Covalent species may reversibly result from association with counter ions such as CF3 SO 3 or ClO4 . They may participate in propagation reactions, but their reactivity is much less than that of the ionic species (see Table 7.9). A detailed discussion on the kinetics of propagation for THF is given in Ref. 182. Because of the low energy gain from ring strain, the equilibrium monomer concentration for the ﬁve- and six-membered rings such as THF and trioxane during polymerization is rather high (see Figure 7.8). As already discussed (see Section 7.2.4), inter- and intramolecular transfer reactions to polymer are of great importance with regard to the molecular weight distribution (Scheme 7.6). In complete thermodynamic equilibrium, the conversion should be given by the equilibrium monomer concentration and the molecular weight distribution should be the most probable one with D ¼ 2, because of the intermolecular chain transfer, which is responsible for scrambling of the molecular

Tab. 7.9.

Propagation rate coeﬃcients for some monomers for ionic and covalent species [2].

˚

Monomer

Counter ion

Solvent

T [ C]

Propagating species

k p [MC1 sC1 ]

THF

CF3 SO 3

CCl4

25

CH2 Cl2

25

CH3 NO2

25

ionic covalent ionic covalent ionic covalent covalent ionic covalent ionic

4 102 6 105 3:1 102 1:7 104 2:4 102 5 104 1:4 104 4 104 1:9 103 1:8 105

Oxepane

CF3 SO 3

H3 NO2

25

Oxazoline

CH3 C6 H4 O

CD3 CN

40

7.3 Cationic Polymerization

2.5

ln[M]

2 1.5 1 0.5 0 0.0028

0.003

0.0032

0.0034

0.0036

1/T / 1/K Fig. 7.8.

Equilibrium THF concentration during polymerization according to Ref. 252.

weight distribution. Intramolecular chain transfer will give macrocycles, the concentration of which should be proportional to n 5=2 [Eq. (54)], if ring formation is reversible and all equilibrium conditions are fulﬁlled. Examples of this ideal equilibrium between linear and cyclic polymers are found for 1,3-dioxolane and siloxanes [230, 231]. However, for THF it has been shown [232] that, even after monomer conversion has reached the equilibrium concentration, there is still a buildup of cyclic monomers [Eq. (59)]: that is, the system is not yet in equilibrium with respect to macrocycles and transfer reactions may still proceed.

+O +

O

+

O

O

+

O

+

O

ð59Þ

So, in spite of these transfer reactions, linear polymers with a narrow molecular weight distribution can be formed by kinetic control, if, as for THF, the transfer rate coeﬃcient to polymer is rather small. Chain growth, and consequently conversion and kinetic chain length, may have reached their equilibrium faster than the other equilibrium reactions. Depending on reactivity ratios in propagation and transfer, the polymerization can be stopped at low conversions, yielding the desired linear and narrow product [233]. In contrast, transfer rates for trioxane to polymer are extremely fast. This can be seen from the observation that during copolymerization of trioxane with 1,3dioxolane, ethylene oxide, and similar comonomers, the comonomer is nearly completely consumed at an early stage of reaction [229, 234–236]. One would therefore expect longer sequences of these comonomers. However, it has been shown, from

355

356

7 Ionic Polymerization

NMR and stability studies, that the comonomers are randomly distributed along the chains [237, 238]. This is attributed to the fact that intermolecular chain transfer (transacetalization), contrary to THF polymerization, proceeds on a time scale similar to propagation, and the same holds for the intramolecular chain transfer leading to cyclic polymer [239, 240]. Cationic polymerization of trioxane is usually a precipitation polymerization leading to crystalline polymers, and the size of the macrocycles is determined by the size of the crystalline lamellae [240]. The polymerization–depolymerization equilibrium is not between the growing chain and the originally propagating monomer, trioxane, but to formaldehyde [Eq. (60)] [241, 242]. Thus formaldehyde should be considered as a comonomer [236, 243], which will be accumulated until its equilibrium concentration and pressure is reached. þ RaOCH2 aOCH2 aOCH2 aOaCHþ 2 T RaOCH2 aOCH2 aOaCH2 þ CH2 O

ð60Þ

This unzipping depolymerization occurs during polymerization, but it may also occur under thermal stress with the neutralized polymer; unzipping then starts from neutral but unstable end groups such as aOH or aCHO or from statistical chain scission. Unzipping will stop at comonomer units such as those from ethylene oxide, dioxepane, and similar monomers, which are not able to depolymerize, and a then stable copolymer will result. Homopolymers, which are usually polymerized anionically from formaldehyde (see Section 7.2.3) will not be stable unless unstable end groups are transformed to stable ones. There are several industrial processes for poly-THF and polyoxymethylene. Cationic polymerization of poly-THF may be batch-wise or continuous [23, 244] using strong protonic acids in a temperature range of 30 to 60 C, depending on the desired molecular weight. Newer processes use solid catalysts like zirconium oxide together with acetic anhydride to form poly-THF with two acetate end groups, which later are hydrolyzed to give the dihydroxy polymer. Other acidic catalysts such as zeolites or acid montmorillonite clay are also described [245–247]. The cationic copolymerization of trioxane with ethylene oxide, 1,3-dioxolane, and suchlike is initiated either with strong protonic acids or Lewis acids, for example BF3 . Molecular weight is controlled by the catalyst concentration and monomer purity, and also by chain transfer agents such as methylal [248], which may lead to more stable end groups. Most processes are run below the melting temperature of the polymer (164–167 C) in precipitating agents or in bulk, and are carried out in kneaders or double-screw reactors [249, 250], but there are also some descriptions of melt processes [251].

7.4

Conclusion

Ionic polymerization oﬀers several advantages. In many cases, it is much faster than free-radical polymerization or polycondensation. This is either because of

Notation

very high rate coeﬃcients for propagation, especially in the case of cationic polymerization, or because of the high stationary concentration of active centers, for example in living anionic polymerization compared to free-radical polymerization. Furthermore, because of its site-based nature, it oﬀers more control over the structure of the resulting polymer by changing the reactivity of the active center when using diﬀerent counter ions, solvents, complexing agents, or salts as additives. So a wide range of structural isomers can be obtained, and copolymerization parameters may be varied over a wide range, resulting in structures ranging from block copolymers to random copolymers from the same pair of monomers, which is not possible by free-radical polymerization. However, from a kinetic and modeling point of view, this site-based nature of ionic polymerization also has some disadvantages. Because the reactivity of the active center strongly depends on the nature of the initiator and on all the other factors in the polymerizing system, the kinetic scheme and parameters of every system are diﬀerent, and so in general they must be determined again for every change in the system. The high sensitivity to impurities necessitates much more eﬀort to purify monomers and solvents. This may prevent the wider employment of ionic polymerization. Generally, ionic polymerization is used for monomers which cannot be polymerized by free radicals, or for the production of polymer structures which are otherwise not accessible.

Notation

D fi Fi hðiÞ i ½I f kt k tt kAB kass k ion kp kd ki K K ass [M] ½Meq

polydispersity index mole fraction of monomer i in the monomer mixture mole fraction of monomer i in polymer chains being formed frequency distribution of chain length degree of polymerization initiator feed concentration rate coeﬃcient for termination [M1 s1 ] rate coeﬃcient for thermal termination, for example, LiH elimination [s1 ] rate coeﬃcient for chain propagation [M1 s1 ] for monomer B added to chain ending with A rate coeﬃcient of association rate coeﬃcient of ionization, dissociation rate coeﬃcient for chain propagation [M1 s1 ] rate coeﬃcient for decomposition [s1 ] rate coeﬃcient for chain initiation [M1 s1 ] equilibrium constant, general equilibrium constant of association monomer concentration [M] equilibrium monomer concentration

357

358

7 Ionic Polymerization

½Mf Mn Mw ½Ni n Pn Pw r1 ; r2 r Rp R T TC v_ in; I v_ in; M wðiÞ xM xI

monomer feed concentration number-average molecular weight [g mol1 ] weight-average molecular weight [g mol1 ] concentration of polymer chains with degree of polymerization i association number or ring size number average degree of polymerization weight average degree of polymerization copolymerization parameters according to the terminal model ratio of k p =k i for living polymerization rate of propagation [M s1 ] gas constant temperature ceiling temperature initiator feed ﬂow [L s1 ] monomer feed ﬂow [Ls1 ] mass distribution of chain length monomer conversion initiator conversion

Greek DGp DHp DSp t n

Gibbs free energy of polymerization enthalpy of polymerization entropy of polymerization residence time average kinetic chain length

Acronyms s- (t-, n-)Bu CSTR HCSTR M, M T, T THF PaLi PsH MWD Pi HIPS NMR RIM SBR SCSTR

sec.- (tert.-, n-)butyl continuous stirred tank reactor homogeneous continuous stirred tank reactor monomer molecule ( , with active center) transfer agent ( , with active center) tetrahydrofuran growing polymer chain with lithium counter ion polystyrene molecule ending with hydrogen molecular weight distribution polymer chain with degree of polymerization i high-impact polystyrene nuclear magnetic resonance reaction injection molding styrene–butadiene rubber segregated continuous stirred tank reactor

References

References 1 B. L. Erusalimski, Mechanisms of

2

3

4

5

6

7

8 9

10

11 12 13

14

15

Ionic Polymerization, Consultants Bureau, New York, 1986. K. Matyjaszewski (ed.), Cationic Polymerization, Marcel Dekker, New York, 1996. J. P. Kennedy, E. Marechal, Carbocationic Polymerization, WileyInterscience, New York, 1982. J. P. Kennedy, B. Ivan, Designed Polymers by Carbocationic Macromolecular Engineering – Theory and Practice, Hanser Publishers, Mu¨nich, 1992. M. Szwarc, M. Beylen, Ionic Polymerization and Living Polymers, Chapman & Hall, New York, 1993. H. L. Hsieh, R. P. Quirk, Anionic Polymerization, Marcel Dekker, New York, 1996. M. Szwarc, Ionic Polymerization Fundamentals, Hanser Publishers, Mu¨nich, 1996. T. Fukuda, K. Kubo, Y.-D. Ma, Prog. Polym. Sci., 1982, 17, 875. B. R. Morrison, M. C. Piton, M. A. Winnik, R. G. Gilbert, D. H. Napper, Macromolecules, 1993, 26, 4368; M. D. Zammit, T. P. Davis, G. D. Willet, K. F. O’Driscoll, J. Polym. Sci., Part A: Polym. Chem., 1997, 35, 2311. K.-D. Hungenberg, F. Bandermann, L. Janko, K. Knoll, in K. H. Reichert, H. U. Moritz (eds.), DECHEMA Monographs, Vol. 131 (5th Int. Workshop on Polymer Reaction Engineering, 1995), p. 387, VCH, Weinheim, 1995. G. Spach, J. Chem. Soc. (Lond.), 1962, 355. R. A. Finnegan, H. W. Kutta, J. Org. Chem., 1965, 30, 4138. W. J. Kern, J. N. Anderson, H. E. Adams, T. C. Bouton, T. W. Bethea, J. Appl. Polym. Sci., 1972, 16, 3123. W. Cooper, in: W. M. Saltman (ed.), The Stereo Rubbers, Wiley-Interscience, New York, 1977. J. Lal, J. E. Mark, Advances in Elastomers and Rubber Elasticity, Plenum Press, New York, 1986.

16 F. R. Eirich, Science and Technology of

17

18

19

20 21

22

23

24

25

25a

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Coordination Polymerization1 Joa˜o B. P. Soares and Leonardo C. Simon 8.1

Polyoleﬁn Properties and Applications 8.1.1

Introduction

Polyoleﬁns are among the most important commodity plastics today due to their low production costs, reduced environmental impact, and a very wide range of applications. They are found in products as diverse as prosthetic implants, gas pipelines, automobile parts and accessories, synthetic fabrics, ﬁlms, containers, and toys. It may seem surprising that polymers composed only of carbon and hydrogen atoms can be so ﬂexible, but the versatility of polyoleﬁns can be easily explained by the way the monomer molecules – ethylene, propylene, and higher a-oleﬁns – are connected to form the polymer chains. It is true for any polymer, but dramatically so for polyoleﬁns, that chain microstructure is the key to understanding their physical properties. Polyoleﬁns are produced in practically all types of reactor conﬁgurations – autoclaves, tubular reactors, loop reactors, ﬂuidized-bed reactors – making them a prime choice for polymer reaction engineering studies. Polymerization may take place in either gas or liquid phases. For liquid-phase reactors, the monomers can be either liquid (as in the case of propylene and higher a-oleﬁns) or dissolved in an inert diluent. Industrial catalysts for oleﬁn polymerization are mainly heterogeneous, but some processes also use soluble catalysts. There are many diﬀerent types of catalysts for oleﬁn polymerization and they can be used to synthesize polymer chains with very diﬀerent microstructures and properties. Despite the fact that polyoleﬁns use comparatively simple monomers and have been around for many years, it can be said with conﬁdence that the degree of sophistication of polyoleﬁn manufacturing and characterization techniques has 1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

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no equal among other synthetic polymers. In this chapter, we will introduce the reader to this fascinating ﬁeld of polymer reaction engineering by giving an overview of the diﬀerent scientiﬁc and technological aspects related to oleﬁn polymerization with coordination catalysts. 8.1.2

Types of Polyoleﬁns and Their Properties

Polyethylenes are the commercial polyoleﬁns with the highest tonnage. Figure 8.1 compares the molecular structures of some polyethylene resins made by coordination and free-radical polymerization. (Free-radical polyoleﬁns are not the main topic of this chapter and will be described only in contrast with polyoleﬁns made with coordination catalysts.) The oldest type of commercial polyoleﬁn is low-density polyethylene (LDPE) made by free-radical polymerization. Low-density polyethylene is made using supercritical ethylene under severe polymerization conditions in autoclaves (1500–2000 atm, 180–290 C) or tubular reactors (1500–3500 atm, 140–180 C). Because of backbiting and chain transfer to polymer reactions, LDPE has both shortand long-chain branches. These branches decrease the crystallinity and density of the polymer – typically in the range 0.915–0.935 g cm3 – and aﬀect several other mechanical and rheological properties. Low-density polyethylene is used predominantly for making ﬁlms because of its limp feel, transparency, and toughness. Additionally, the long-chain branches give excellent processability and high melt tension to LDPE. High-density polyethylene (HDPE) was ﬁrst made with Ziegler–Natta catalysts. When produced as a homopolymer, HDPE has no short-chain branches, but a

H

H

H

H

H

H

H

H

H

H

H

H

H

H

C

C

C

C

C

C

C

C

C

C

C

C

C

C

H

H

H

H

H

H

H

H

H

H

H

H

H

H

LDPE

LLDPE / VLDPE

0.915-0.935 g/cm3

0.915-0.94 g/cm3 / 0.88 - 0.912 g/cm3

Fig. 8.1.

Polyethylene microstructures.

HDPE

0.96-0.97 g/cm3

8.1 Polyoleﬁn Properties and Applications

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

atactic

Cp2ZrCl2

isotactic

Et(Ind)2ZrCl2

syndiotactic

iPr(Flu)(Cp)HfCl2

isotacticstereoblock

(NMCp)2ZrCl2

isotacticatacticstereoblock

Et(Me4Cp)(Ind)TiCl2

hemi-isotactic

iPr(Cp)(Ind)ZrCl2

Fig. 8.2.

Polypropylene microstructures and some metallocene catalysts used in their synthesis.

367

368

8 Coordination Polymerization

rubber domains

propene + ethene PC

PC

propene catalyst + diluent

product TC

TC

c.w.

c.w. homopolymer matrix

Fig. 8.3.

Process for the production of high-impact polypropylene.

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

8.1 Polyoleﬁn Properties and Applications

5-ethylidene-2-norbornene

1,4-hexadiene

CH - CH3

CH2 = CH - CH2 - CH = CH - CH3

dicyclopentadiene

Fig. 8.4.

Diﬀerent dienes used for the production of EPDM.

8.1.3

The Importance of Proper Microstructural Determination and Control in Polyoleﬁns

Since polyoleﬁns are composed of only carbon and hydrogen atoms, the way these atoms are connected – in other words, the microstructure of a polyoleﬁn – determines their properties. At the most elementary level, the microstructure of a polyoleﬁn is deﬁned by its distributions of molecular weight, chemical composition, long-chain branching, and stereoregularity. Therefore, determining the microstructure of polyoleﬁns can be considered the ﬁrst and most important step toward the understanding of their properties. High-temperature gel permeation chromatography (GPC) is the standard technique for measuring the molecular weight distribution of polyoleﬁns. Since gel permeation chromatography is a very well-established technique it will not be discussed here. Several publications provide additional discussions on the GPC technique [7–9]. The chemical composition distribution of polyoleﬁns is measured (indirectly) by either temperature rising elution fractionation (Tref ) or crystallization analysis fractionation (Crystaf ). These two techniques provide similar information on the chemical composition distribution of polyoleﬁns and can be used interchangeably in the vast majority of cases. Both methods are based on the fact that the crystallizability of HDPE and LLDPE depends strongly on the fraction of a-oleﬁn comonomer incorporated into the polymer chains, that is, chains with an increased aoleﬁn fraction have a decreased crystallizability. A similar statement can be made for polypropylene and other polyoleﬁn resins that are made with prochiral monomers: resins with high stereoregularity and regioregularity have higher crystallizabilities than atactic resins. In Crystaf, polyoleﬁn chains are crystallized from a dilute solution by slowly lowering the solution temperature. Chains with fewer a-oleﬁn units crystallize at higher temperatures, while chains with a higher a-oleﬁn fraction crystallize at lower temperatures. This information is used to generate a Crystaf proﬁle relating crystallization temperatures to the fraction of polymer that crystallizes at those

369

8 Coordination Polymerization

Differential Crystaf Profile

Cumulative Crystaf Profile

Crystaf vessel

dw/dTc

370

w Tc

Tc

Tc Chemical Composition Distribution

w

Calibration Curve

Tc

α-olefin fraction

α-olefin fraction

Estimation of the chemical composition distribution of a polyoleﬁn using a Crystaf proﬁle and a calibration curve.

Fig. 8.5.

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

8.1 Polyoleﬁn Properties and Applications

Weight fraction

Easier to crystallize

More difficult to crystallize

Mol % α-olefin comonomer Crystallization temperature Chemical composition distribution of a polyoleﬁn made with a heterogeneous Ziegler–Natta catalyst. Fig. 8.6.

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

371

8 Coordination Polymerization

0.06

2x105

0.04 105

Mw

Normalized response (a.u.)

372

8x104 0.02 6x104

0.00

4x104 0

2

4

6

8

10

x, mole % butene Average weight-average molecular weight of ethylene– 1-butene copolymers made with a heterogeneous Ziegler–Natta catalyst as a function of average 1-butene content [11a].

Fig. 8.7.

very often subject to signiﬁcant experimental error. Nonetheless, very good evidence of the presence of long-chain branches can be found by studying the rheological response of these resins. Long-chain branch formation with polyoleﬁns will be discussed in detail below in Section 8.3.4.

8.2

Catalysts for Oleﬁn Polymerization

The class of catalysts we currently call Ziegler–Natta was ﬁrst used by Ziegler in 1953 to polymerize ethylene at low pressure, and further developed by Natta in 1954 to produce isotactic polypropylene. Both Ziegler and Natta were awarded the Nobel Prize in chemistry in 1963; since then, this ﬁeld has grown incessantly, with the development of improved catalysts and new industrial processes. Many other catalysts capable of polymerizing oleﬁns have become available since the original Ziegler–Natta catalyst based on crystalline titanium chloride was introduced. More recently, the discovery of soluble metallocene/aluminoxane catalysts opened the doors to a new revolution in the production of polyoleﬁns. These catalyst systems are able to make polyoleﬁns in very high yields and with a degree of microstructural control not possible to achieve using conventional Ziegler–Natta catalysts. Most industrial processes today still use heterogeneous Ziegler–Natta catalysts, although the market share of metallocene resins is increasing due to the enhanced

8.2 Catalysts for Oleﬁn Polymerization

properties of polyoleﬁns made with these catalysts and the fact that polymerization processes that were originally designed to use Ziegler–Natta catalysts can be converted, with minimal changes, to operate with metallocenes – the so called ‘‘dropin’’ technology. Although traditional heterogeneous and homogeneous Ziegler–Natta catalysts are commonly used as the standard example of coordination polymerization, coordination catalysts include any complex of transition metals and organic ligands: Phillips catalysts are heterogeneous, chromium-based complexes that are not classiﬁed as Ziegler–Natta catalysts; metallocenes are complexes of a transition metal – in most cases an early transition metal – and cyclopentadienyl or cyclopentadienylderivative ligands; late transition metal catalysts may have a variety of ligands containing heteroatoms, such as phosphorus, nitrogen, or oxygen, directly bonded to the transition metal. The active site in coordination catalysts for oleﬁn polymerization is, therefore, a transition metal surrounded by ligands. The catalytic properties depend on the ﬁne tuning between the transition metal and the ligands in terms of geometry and electronic character. In most cases the active site is produced by the activation of a complex called a pre-catalyst, or catalyst precursor. The pre-catalyst is commonly stable and easy to handle; sometimes it is stable even to moisture and oxygen. The creation of the active site by reaction of the pre-catalyst with an activator or cocatalyst is made just prior to its injection in the polymerization reactor or inside the polymerization reactor itself. The activator alkylates the pre-catalyst complex to form the active sites and stabilizes the resulting cationic active site. Common activators are based on organoaluminum or organoborane compounds [16]. Because the activator works as a Lewis acid (electron acceptor), it is also used to scavenge polar impurities from the reactor. These impurities are electron donors, such as oxygen, sulfur and nitrogen compounds, and moisture, that poison the cationic active site. Figure 8.8 shows a simpliﬁed chemical equation for the activation mechanism and the corresponding equation used for modeling in the Section 8.3.

L X A X L

L R A L

AlR3

C

Al

AlR2X2

C*

A = transition metal center (Ti, Zr, Ni, …) L = ligands X = halogen (Cl, Br) AlR3 = alkylaluminum cocatalyst R = alkyl group (methyl, ethyl) Fig. 8.8.

Catalyst activation by reaction of a pre-catalyst and a cocatalyst.

373

374

8 Coordination Polymerization

L R A L

L R A L

L A L R

C*

P*1

M

L A L

( L A L

n

)

Pol

n

R P*1 Fig. 8.9.

nM

P*1+n

Monomer coordination and insertion.

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

8.2 Catalysts for Oleﬁn Polymerization

Regioselectivity 1,2-insertion Pol L A L

L A L

2

1

2

Pol

1

2,1-insertion Pol L A L

L A L

2 1

1 2

Pol

Stereoselectivity

Pol L A L

L A L Pol

Pol L A L

L A L Pol

Fig. 8.10.

Regio- and stereoselectivity in coordination polymerization.

trons (generally substances containing oxygen, nitrogen, or sulfur) can coordinate irreversibly with the active site, causing irreversible catalyst deactivation. Bimolecular catalyst deactivation happens when two active sites form a stable complex that is inactive for monomer polymerization. This type of bimolecular intermediate is favored at high catalyst concentrations and is reversible [18]. The term ‘‘latent state’’ can be used to describe certain catalytic intermediates that reduce the catalyst activity. The formation of latent states is diﬃcult to probe and the mechanisms are generally not well understood. Figure 8.12 shows chemical equations for this catalyst deactivation mechanism.

375

376

8 Coordination Polymerization

chain transfer by β-hydride elimination L A L

L H A L

L H A L

Pol

Pol

Pol P*H

P*r

Dr

chain transfer by β-methyl elimination L A L

CH3 L A L

Pol

P*

P*r

L CH3 A L

Pol

Pol

Dr

chain transfer to hydrogen L A L

Pol P*r

L H A 2 L

H2

P*H

H2

H L A L

Pol

Pol

Dr

chain transfer to monomer L A L

L A L

Pol

P*r

P *r

M

L A L

Pol

Dr

chain transfer to cocatalyst L A L

Pol

P*r Fig. 8.11.

P*A

Al

Pol

D Al

Chain termination mechanisms.

L R A L 2 C* Fig. 8.12.

L A L R

AlR3

L L R A A L R L 2 Cd

Catalyst deactivation by bimolecular reactions.

AlR2

Pol

8.2 Catalysts for Oleﬁn Polymerization

Dr

H2

C

Dr

M

AlR3 C*

Dr

P*r

P*H

P*H

M

P*Me

M M M

Fig. 8.13.

Catalytic cycle for polymerization.

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

377

8 Coordination Polymerization

polymerization rate

378

Decay type Build-up type

polymerization time Polymerization kinetics with coordination catalysts showing diﬀerent deactivation rates. Fig. 8.14.

8.2.1

Ziegler–Natta, Phillips, and Vanadium Catalysts

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

8.2 Catalysts for Oleﬁn Polymerization

has a very broad molecular weight distribution, with polydispersities ranging from 12 to 24. Interestingly, hydrogen is not an eﬀective chain-transfer agent and generally decreases catalyst activity. Vanadium catalysts are used to manufacture ethylene–propylene–diene rubbers. The catalyst is produced by activating a mixture of VCl3 and VCl4 with AlR3 (alkylaluminum) compounds. Vanadium catalysts are very sensitive to reaction conditions [23]. Changes in the composition of the solvent, polymerization time, and temperature can aﬀect catalyst activity and selectivity. 8.2.2

Metallocene Catalysts

Metallocene catalysts have structures similar to ferrocene (Cp2 Fe), which is a widely known complex with a ‘‘sandwich’’ structure, where the transition metal lies between two cyclopentadienyl (Cp) rings. Cp2 ZrCl2 (Figure 8.15) is a well studied example of a metallocene catalyst for oleﬁn polymerization. Metallocenes are soluble and therefore can be used as homogeneous catalysts, but they can also be supported on inert carriers such as silica, alumina, and magnesium chloride, among other inorganic and organic supports, and used as heterogeneous catalysts [24]. The cocatalysts commonly used with Ziegler–Natta catalysts (AlEt2 Cl, AlEt3 ) are not able to activate metallocene catalysts very well. Even though metallocenes can be activated with these cocatalysts, they generally deactivate very fast, and consequently have no commercial interest. The use of bulky activators, such as methylaluminoxane (MAO), is necessary to produce highly active metallocene catalysts. In fact, it can be said that it was the discovery of MAO as a good cocatalyst for metallocenes that made possible the metallocene revolution that we are currently experiencing in polyoleﬁn manufacture [25]. Other bulky Lewis acids such as trispentaﬂuorophenylborane (B(C6 F5 )3 ) are also very good cocatalysts for metallocenes. A large variety of metallocene catalysts can be obtained by altering the simple structure of Cp2 ZrCl2 . The nature of the transition metal and the structure of the ligand have a large eﬀect on catalyst behavior. The shape, geometry, and chemical structure of the ligand can aﬀect the activity and selectivity of the catalyst. The symmetry imposed by ligands around the active site determines the geometry for monomer coordination and insertion, and consequently the relative orientation of catalyst and growing polymer chain. The possibility of variations in the chemical

Cl

Zr Fig. 8.15.

Cl Structure of a typical metallocene catalyst, Cp2 ZrCl2 .

379

380

8 Coordination Polymerization

X

X

M

X

M

X

X

M

X

C2v

Cs (meso)

C2 (racemic)

(atactic PP)

(atactic PP)

(isotactic PP)

Ex: Cp2ZrCl2

Et(Ind)2ZrCl2

X

M

X

Ex: Et(Ind)2ZrCl2 X

M

X

Cs

C1

(syndiotactic PP)

(no unequivocal prediction)

Ex: iPr(Flu)(Cp)HfCl2

Ex: (Flu)(MeCp)ZrCl2

Symmetry requirements for the production of polypropylenes with diﬀerent stereoregularities. Fig. 8.16.

substitutions and/or presence of bridging groups on the cyclopentadienyl rings creates an endless set of possible structures, especially if substitutions of ligands with heteroatoms are considered. The size and the type of the bridge connecting the cyclopentadienyl rings aﬀect the opening on the opposite side of the transition metal, and consequently the relative rates of the elementary steps in the polymerization mechanism. This eﬀect is called the ‘‘biting angle’’. The literature on metallocene catalysts is very rich – one may even say opulent – and thousands of metallocene catalyst structures have been documented since the early 1980s [1, 2, 26]. The appropriate selection of ligand structure has a major eﬀect on the selectivity of the polymerization mechanism. A practical and very interesting example is the control of tacticity in polypropylene. There are two mechanisms to explain control of stereoregularity: chain-end or enantiomorphic site control. In the former, the orientation of the growing polymer chain determines the orientation of the monomer during insertion. In the latter, this orientation is determined by the active site, as determined by the arrangements between the transition metal and its ligands [27]. Since the stereochemical control of most metallocenes is of the enantiomorphic type, it is possible to classify them according to the type of polypropylene chain they will produce based merely on the symmetry of the ligands around the active site, as elegantly illustrated in Figure 8.16. Furthermore, certain ligands can change the orientation during the growth of a polymer chain, for instance oscillating between conﬁgurations that produce atactic and isotactic polypropylene blocks. By the proper choice of polymerization conditions it is possible to produce polypropylenes that have chains varying from mainly atactic through block atactic–isotactic to mainly isotactic conﬁgurations [28].

8.2 Catalysts for Oleﬁn Polymerization

CH3

H3C

CH3 H3C H3C

Si

H3C

CH3 N

H3C Fig. 8.17.

CH3

Ti

C

CH3 CH3

A half-sandwich metallocene.

Another important class of metallocene catalysts, the ‘‘half-sandwich’’ metallocenes (Figure 8.17) have a more open structure because only one cyclopentadienyl ring is coordinated to the transition metal. Catalysts with open structures have very high reactivity ratios toward the incorporation of longer a-oleﬁns and are used for the production of ethylene/1-octene copolymers [29]. More importantly, these catalysts can form polyoleﬁns containing long-chain branches via a mechanism that is analogous to the copolymerization of long a-oleﬁns, as will be described in Section 8.3.4. 8.2.3

Late Transition Metal Catalysts

This family of coordination catalysts is characterized by having a transition metal from groups 8, 9, or 10. A great advantage of some polymerization catalysts based on late transition metals is that the active sites are more tolerant to polar comonomers (and impurities) than the early transition metals used with metallocenes. The tolerance toward polar comonomers is attributed to the electronic conﬁguration of the central metal and its interaction with its ligands. This feature is very attractive for modifying the chemical composition of polyoleﬁns by copolymerization with vinyl alcohols, acrylates, or other vinyl polar comonomers. A few polar functional groups can signiﬁcantly increase the hydrophilicity of polyoleﬁns and enhance several of their properties, such as dyeability, adhesion, and compatibility with other polar polymers. Late transition metal catalysts are very active and produce high molecular weight polyethylene. The catalyst can be used in homogeneous solution or supported in inert carriers such as silica. The active site can be generated by activation of a metal halide complex such as (diimine)NiCl2 with organoaluminum compounds such as (CH3 )3 Al or MAO. As for metallocenes, the active species are cationic. These catalysts have a well deﬁned structure and in homogeneous systems can behave as single-site catalysts.

381

382

8 Coordination Polymerization Forward Chain Walking P

Forward Chain Walking P

A

Chain Walking followed by Insertion

Monomer Insertion P

P

Backward Chain Walking

A

A

A

A

P Ethyl Branch Insertion

A = metal active center P = polymer chain

Fig. 8.18.

Chain walking mechanism.

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

8 Coordination Polymerization

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

8 Coordination Polymerization

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

Pr þ M ! P1 þ D¼ r

ð8Þ

Finally, the cocatalyst may also act as a chain-transfer agent [Eq. (9)]. If the 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

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts

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

8 Coordination Polymerization

In Eq. (15), wðrÞ is the weight distribution of chains of length r and rn is the number-average chain length of the polymer population. Therefore, from a 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

8 Coordination Polymerization

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

8 Coordination Polymerization

tribution becomes broader as the value of rn =k decreases, reﬂecting the fact that polymers with shorter number-average chain lengths and a tendency to form blocks (see Eq. (17): k increases when rA rB increases) have broader chemical 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)

8.3 Polymerization Kinetics and Mechanism with Coordination Catalysts

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

8 Coordination Polymerization

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

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

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

8 Coordination Polymerization

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

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

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

8 Coordination Polymerization

state concentration at t ¼ 0 to obtain the initial condition for Eq. (41). Unless one is interested in the monomer proﬁles for the very ﬁrst seconds of polymerization, this approximation generally leads to a system of partial diﬀerential equations that 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

8 Coordination Polymerization

taking place on the surface of the catalyst. The method of moments is generally the preferred technique used in conjunction with the multigrain model, but its 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

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

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

8 Coordination Polymerization

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.

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

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.

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

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

8 Coordination Polymerization

We will ﬁrst show how to derive such population balances for living polymer chains, but instead of applying the same approach to all other species we will 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

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

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

8 Coordination Polymerization

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

8.6 Types of Industrial Reactors

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

8 Coordination Polymerization

Fig. 8.38.

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

These polymerization processes were originally designed to polymerize oleﬁns with heterogeneous Ziegler–Natta catalysts, with the exception of some solution processes that were designed to work with homogeneous Ziegler–Natta catalysts for the production of EPDM rubbers or some types of polyethylene resins. 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.

8.6 Types of Industrial Reactors

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

8 Coordination Polymerization

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

For the particular case of ﬂuidized-bed reactors, ﬂuidization is not a trivial 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-

8.6 Types of Industrial Reactors

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

8 Coordination Polymerization

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

8 Coordination Polymerization

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

8 Coordination Polymerization

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

References 1 G. W. Coates, Chem. Rev., 2000, 100,

1223. 2 L. Resconi, L. Cavallo, A. Fait, F. Piemontesi, Chem. Rev., 2000, 100, 1253. 3 K. Angermund, G. Fink, V. R. Jensen, R. Kleinschmidt, Chem. Rev., 2000, 100, 1457. 4 J. B. P. Soares, R. F. Abbott, J. D.

Kim, J. Polym. Sci.: Part B: Polym. Phys., 2000, 38, 1267. ¨ hm, J. C. Boot, 5 J. Scheirs, L. L. Bo P. S. Leevers, Trends Polym. Sci., 1996, 4, 408. 6 G. Ver Strate, in Encyclopedia of Polymer Science and Engineering, Vol. 6, J. I. Kroschwitz (Ed.), John Wiley & Sons, New York, 1986, p. 522.

References 7 J. B. P. Soares. Fractionation, in

8

9

10

11

12 13 14 15 16 17 18

19 20 21

22

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.

23 S. Gambarotta, Coord. Chem. Rev.,

2003, 237, 229. 24 G. G. Hlatky, Chem. Rev., 2000, 100,

1347. 25 H. Sinn, I. Schimmel, M. Ott, N.

26 27

28 29 30 31

32 33

34 35 36 37 38 39

40 41 42 43

von Thienen, A. Harder, W. Hagendorf, B. Heitmann, E. Haupt, in Metalorganic Catalsyts for Synthesis and Polymerization, W. Kaminsky (Ed.), Springer, Berlin, 1999, pp. 105– 122. H. Butenscho¨n, Chem. Rev., 2000, 100, 1527. S. A. Miller, R. Waymouth, in Ziegler Catalysts, G. Fink, R. Mulhaupt, H. H. Brintzinger (Eds.), Springer, Berlin, 1995, pp. 441–454. H. G. Alt, A. Ko¨ppl, Chem. Rev., 2000, 100, 1205. S. Chum, W. J. Kruper, M. J. Guest, Adv. Mater., 2000, 12, 1759. S. D. Ittel, L. K. Johnson, Chem. Rev., 2000, 100, 1169–1203. L. C. Simon, C. P. Williams, J. B. P. Soares, R. F. de Souza, Chem. Eng. Sci., 2001, 56, 4181. J. B. P. Soares, Chem. Eng. Sci., 2001, 56, 4131. P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953. W. H. Stockmayer, J. Chem. Phys., 1945, 13, 199. P. C. Hiemenz, Polymer Chemistry, Marcel Dekker, New York, 1984. T. F. McKenna, J. B. P. Soares, Chem. Eng. Sci., 2001, 56, 3931. J. B. P. Soares, A. E. Hamielec, Polym. React. Eng., 1995, 3, 261. J. B. P. Soares, A. E. Hamielec, Polymer, 1995, 36, 2257. J. B. P. Soares, A. E. Hamielec, Macromol. Theory. Simul., 1995, 4, 305. J. B. P. Soares, Polym. React. Eng., 1998, 6, 225. T. Keii, Macromol. Theory Simul., 1995, 4, 947. J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul., 1996, 5, 547. M. R. Spiegel, J. Liu, Mathematical Handbook of Formulas and Tables, McGraw-Hill, New York, 1999.

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45 46 47

48 49 50 51 52

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54

Methods for Scientists and Engineers, University Science Books, Sausalito, California, 2003. J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul., 1997, 6, 591. J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul., 1995, 1085. A. Prasetya, L. Liu, J. Litster, F. Watanabe, K. Mitsutani, G. H. Ko, Chem. Eng. Sci., 1999, 3263. J. J. Zacca, J. A. Debling, W. H. Ray, Chem. Eng. Sci., 1996, 51, 4859. J. J. Zacca, J. A. Debling, W. H. Ray, Chem. Eng. Sci., 1997, 52, 1941. J. A. Debling, J. J. Zacca, W. H. Ray, Chem. Eng. Sci., 1997, 52, 1969. J. A. Debling, W. H. Ray, Ind. Eng. Chem. Res., 1995, 34, 3466. S. Floyd, K. Y. Choi, T. W. Taylor, W. H. Ray, J. Appl. Polym. Sci., 1986, 32, 2935. S. Floyd, K. Y. Choi, T. W. Taylor, W. H. Ray, J. Appl. Polym. Sci., 1986, 31, 2231. S. Floyd, R. A. Hutchinson, W. H.

55

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57 58 59 60 61 62 63 64 65

Ray, J. Appl. Polym. Sci., 1986, 32, 5451. S. Floyd, T. Heiskanen, T. W. Taylor, G. E. Mann, W. H. Ray, J. Appl. Polym. Sci., 1987, 33, 1021. R. A. Hutchinson, C. M. Chen, W. H. Ray, J. Appl. Polym. Sci., 1992, 44, 1389. R. A. Hutchinson, W. H. Ray, J. Appl. Polym. Sci., 1990, 41, 51. R. A. Hutchinson, W. H. Ray, J. Appl. Polym. Sci., 1991, 43, 1271. R. A. Hutchinson, W. H. Ray, J. Appl. Polym. Sci., 1991, 43, 1387. R. A. Hutchinson, W. H. Ray, J. Appl. Polym. Sci., 1987, 34, 657. P. Canu, W. H. Ray, Comp. Chem. Eng., 1991, 15, 549. M. Wulkov, Macromol. Theory Simul., 1996, 5, 393. T. Xie, A. E. Hamielec, Makromol. Chem., Theory Simul., 1993, 2, 421. T. Xie, A. E. Hamielec, Makromol. Chem., Theory Simul., 1993, 2, 455. M. Covezzi, G. Mei, Chem. Eng. Sci., 2001, 56, 4059.

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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

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

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

9 Mathematical Methods

(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

9 Mathematical Methods

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

9 Mathematical Methods

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

9 Mathematical Methods

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

9 Mathematical Methods

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

9 Mathematical Methods

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

Termination by recombination

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

Terminal double bond propagation

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

Termination by recombination

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 .

9.6 Pseudo-distribution Approach

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.

9.6 Pseudo-distribution Approach

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

9 Mathematical Methods

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

9.6 Pseudo-distribution Approach

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

9 Mathematical Methods

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

9 Mathematical Methods

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

9.6 Pseudo-distribution Approach

459

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

Propagation Termination by disproportionation

Termination by recombination

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

Terminal double bond propagation

!

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

9.6 Pseudo-distribution Approach

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.

Termination by disproportionation

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

Termination by recombination

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

9 Mathematical Methods

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

9 Mathematical Methods

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.

Termination by disproportionation

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

Termination by recombination

dFn2; 0 þ¼ k tc l 0 Fn2; 0 ; dt

Propagation

Transfer to monomer

Transfer to polymer

Terminal double bond propagation

! 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:

9.6 Pseudo-distribution Approach

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

9 Mathematical Methods

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

9 Mathematical Methods

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

9 Mathematical Methods

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

9.6 Pseudo-distribution Approach

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

9 Mathematical Methods

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

9 Mathematical Methods

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

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9 Mathematical Methods

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.

9.6 Pseudo-distribution Approach

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.

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9 Mathematical Methods

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

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9 Mathematical Methods

(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

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9 Mathematical Methods

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

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9 Mathematical Methods

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-

9.8 Monte Carlo Simulations

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

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9 Mathematical Methods

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:

9.8 Monte Carlo Simulations

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,

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9 Mathematical Methods

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.

9.8 Monte Carlo Simulations

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 (<0.3) pp: birth conversion 0.22, length 180. This concludes generation 1, which generated four pps in total. In the second generation only two of these turn out to have one extra branch point each. After ﬁnding birth conversions and lengths for these two, the algorithm stops. 9.8.4

CSTR with Transfer to Polymer [14]

The algorithm is similar to that for the batch reactor, but now the residence time distribution (RTD) has to be taken into account, while the expressions for the (conditional) probability are also slightly modiﬁed. The RTD in terms of the reduced residence time y ¼ t=t, where t is the average residence time, is given by: FðyÞ ¼ expðyÞ

ð143Þ

In contrast to the batch reactor, here we take the reduced residence time y as the independent variable. The y of the ﬁrst pp is sampled from this distribution [using sampling formula Eq. (133)], while its length is chosen from the weight-fraction distribution, wn , Eq. (132) [using sampling formula Eq. (133) twice and adding]. Note that in a CSTR this distribution is at steady-state. Essentially, the longer a pp stays in the reactor, the higher its probability of receiving branch points. Stated another way, the branching density in a pp is proportional to residence time y: rðyÞ ¼ ry

ð144Þ

This is consistent with Eq. (143) since:

r¼

ðy

FðyÞrðyÞ

ð145Þ

0

It follows from the steady-state variants of the balances of monomer and branching density, Eqs. (135)–(137), that the average branching density r is related to (steadystate) conversion x, according to: r ¼ C tp x=ð1 xÞ

ð146Þ

The actual number of branch points on the ﬁrst pp again follows from the binomial distribution, Eq. (138). The residence time u of each of these pps that has

491

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9 Mathematical Methods

grown on the ﬁrst pp should be less than y, or 0 < u < y. They are sampled from the conditional (given the connection to the pp created at y) probability: CPa ðujyÞ ¼ u=y;

ð147Þ

because of the linear dependence on exposure time, branching density being constant. The sampling procedure is identical to that for CPa and CPi in the batch MC algorithm. The lengths of these pps follow from the (steady-state) number-fraction distribution, nn , according to Eq. (140), for the same reasons as have been explained for the batch reactor. The probability that the ﬁrst pp has itself been initiated at a secondary radical site on a previously created pp is given by Pb as formulated in Eq. (141); in a CSTR Pb is a constant. ‘‘Previous’’ here means a longer residence time than y: y < z < y. Residence time z is sampled (as before) from a conditional probability expression containing the RTD:

CPi ðzjyÞ ¼

1 FðzÞ 1 FðyÞ

ð148Þ

Sampling is easily performed by applying Eq. (143) and adding y to the value found. 9.8.5

Comparison of Galerkin-FEM Classes Model and CSTR with Transfer to Polymer

We implemented the Monte Carlo code in MATLAB and performed simulations for a CSTR using the kinetic data shown in Figure 9.16. For comparison, calculations with the Galerkin-FEM two- and ﬁve-classes multiradical models were made. For a sample of 20,000 molecules good agreement could be observed with the Galerkin-FEM models. The CLD tail is interesting (Figure 9.17). We observe a signiﬁcant contribution of living chains to the overall concentration in both GalerkinFEM models. In both models transfer to polymer of living chains was included, which in the ﬁve-classes model yielded a considerably higher concentration of living chains. It is the overall concentration of the ﬁve-classes model that is most in line with the MC simulations. We conclude that the two-classes model underestimates the tail and there deviates from the MC results. The extremely large molecules of the tail turned out to be very time-consuming in the MC simulations performed with a code written in MATLAB (not speed-optimized). One molecule with CL > 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

9.8 Monte Carlo Simulations

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.

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9 Mathematical Methods

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

9.8 Monte Carlo Simulations

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

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9 Mathematical Methods

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

9.8 Monte Carlo Simulations

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

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9 Mathematical Methods

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-

9.8 Monte Carlo Simulations

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

9 Mathematical Methods

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-

9.8 Monte Carlo Simulations

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

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9 Mathematical Methods

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

9 Mathematical Methods

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

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

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

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

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): <ðN1 Þ j PN ¼

PN1 PN1N1 N 1 X

Pj PN1j

with PN ¼

y X

Pn; N

ð177Þ

n¼1

j¼0

Note that here Pn; N represents the concentration at the branching catalyst CGC– Ti, Pn;b N . In this case in the absence of linear catalyst, Pn; N ¼ Pn;b N . For a chain without a terminal double bond, the branch point probability density function (NPDF) is modiﬁed to: <ðN1 ÞjPN ¼

P N1 PN1N1 N 1 X Pj PN1j

ð178Þ

j¼0

The PDF of Eq. (178) expresses the probability of an insertion of a chain with N1 branch points under the condition that a polymer with N branch points and a terminal double bond is formed. This formula is symmetrical by deﬁnition. It turns out that this PDF (apart from round-oﬀ errors) is identical to the PDF deﬁned in Eq. (177). This is due to the kinetics of the system, which allows the concentration of polymers with a terminal double bond to be written as a fraction of the concentration of polymers without terminal double bonds [35], as formulated in: Pn; N Pn; N

¼c

with c as an arbitrary constant.

ð179Þ

507

508

9 Mathematical Methods

The PDF <ðN1 ÞjPN is a symmetric distribution with peaks at N1 ¼ 0 and N1 ¼ N 1. This implies that the probability of inserting a chain with zero or N 1 branch points is much higher than that of chains with and intermediate number of branch points. Once the number of branch points on the chain inserted is known, the problem is divided into two smaller sub-problems. The architecture of the inserted chain with N1 branch points as well as the architecture of the growing chain with N 1 N1 branch points must be determined. For both the inserted and the growing chain the same line of reasoning can be used as above for a smaller number of branches, N1 and N N1 1, respectively. In our algorithm the procedure described above in a recursive manner continues until no more structures with branch points have to be attached. Then the topology is known. The lengths of the segments are determined in exactly the same manner as for the radical polymerization case. Note, however, that with N branch points we here have 2N þ 1 segments. Mixed-catalyst System The synthesis of architectures in this case has similarities to, but also marked diﬀerences from, that of the single-catalyst case. Notice that although the mixed system algorithm is constructed to deal with those systems, it obviously should be able to describe the single catalyst correctly as a limiting case as well. In the latter case both algorithms should lead to the same results. Again, the procedure starts at the most recent insertion (Figure 9.21) of a branched structure into a growing structure. Now, the main diﬀerence from the single-catalyst case is that separation of topology and segment length distribution is no longer possible. This is due to the fact that in the mixed-catalyst case segments originate from two diﬀerent catalysts and therefore possess diﬀerent statistical properties. This also implies that the interchange of arbitrary pairs of segments is not allowed now. Hence, the assignment of lengths to segments can no longer be performed in an independent way and has to be included in the topology generation procedure. This means that for the structure to be inserted we have to specify both the number of branch points N1 , and the chain length n1 . Note that the inserted structures originate from both linear and branching catalyst and therefore have mixed statistical properties. The insertion of a structure can be realized in many diﬀerent ways. The probabilities of these diﬀerent options follow from the bivariate chain length/number of branch points distribution in a manner to be explained next. The insertion probability of a polymer chain containing a certain combination of number of branch points N1 and chain length n1 is proportional to this structure’s concentration:
9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

the insertion point. The value of k may vary between 2 and n n1 2, while each insertion option has its own probability. The sum of all these probabilities yields the desired growing structure probability,
at each time instant the probability of future growth of a structure is completely independent of its growth history. Hence, under the condition that a polymer with N branch points and n monomer units is formed, the probability of a ﬁrst insertion with N1 branch points and n1 monomer units equals the normalized product of the probabilities
Pn1 ; N1

nn 1 1 X

b

<ðn1 ; N1 ÞjPn; N ¼

b

b

Pk; 0 Pnn1 k; NN1 1

k¼1 n2 X N 1 X

Pi; j

i¼1 j¼1

ni1 X

ð180Þ b b Pk; 0 Pnik; Nj1

k¼1

The n; N-PDF for chains without TDB likewise is:

Pn1 ; N1 <ðn1 ; N1 ÞjPn;b N

¼

nn 1 1 X

b Pk;b 0 Pnn 1 k; NN1 1

k¼1 n2 X N1 X i¼1 j¼1

Pi; j

ni1 X

ð181Þ b Pk;b 0 Pnik; Nj1

k¼1

Again, due to the similarity between the distributions of chains with and without TDBs [Eq. (179)]. The PDFs possess the same shape. Now, it is instructive to derive monovariate N-PDFs from the n; N-PDFs. Doing this for the single-catalyst case should then generate N-PDFs identical to the monovariate N-PDFs as obtained from the single catalyst algorithm. The N-PDF describing the probability that a structure with N1 branch points will be inserted to form a molecule with N branch points in total is obtained from the n; N-PDF by taking the sum over chain lengths n1 :

<ðN1 ÞjPn;b N ¼

n1 X n1 ¼1

<ðn1 ; N1 Þ j Pn;b N

ð182Þ

509

510

9 Mathematical Methods

One may realize that this can be performed for various total chain lengths n. This implies that in principle this N-PDF <ðN1 Þ j Pn;b N is still a function of n. Now, the monovariate N-PDF created by the single-catalyst algorithm, <ðN1 Þ j PNb , does not depend on n. A clearly interesting test for the mixed system algorithm is to check whether it indeed yields an n-independent N-PDF for the single-catalyst system, which turns out to be the case. Equations (181) and (182) give the PDFs describing the probability of having a ﬁrst insertion of a branched structure with n1 monomer units and N1 branch points. Hence, this ﬁrst step in the algorithm should provide n1 and N1 . The sampling from the bivariate distribution proceeds in two sub-steps. First, N1 is sampled from the monovariate N-PDF derived from the bivariate n; N-PDF as described above. Secondly, for the sampled N1 a chain length PDF is extracted from the full bivariate n; N-PDF as the cross-section at N1 . From this n-PDF the n1 is sampled. The ﬁrst step of the algorithm is completed by determining the length of the linear segment, k. This is realized by sampling from the PDF that describes the probability of ﬁnding such a linear segment of length k. This probability was already introduced above as being proportional to both the concentration of these linear segments Pk;b 0 and the concentration of the complementary branched strucb . Hence, sampling proceeds from the PDF being formed by tures Pnn 1 k; NN1 1 the normalized product of the two concentrations: b <ðkÞjPnn ¼ 1 ; NN1 1

b Pk;b 0 Pnn 1 k; NN1 1 nn 1 1 X

ð183Þ

b b Pk; 0 Pnn1 k; NN1 1

k¼2

After the last insertion as the ﬁrst step in the algorithm the problem is split into two sub-problems, the tasks of ﬁnding the architectures of the inserted structure and that of the growing branched structure. These tasks proceed in exactly the same manner as the ﬁrst step and they may each lead to further architectureﬁnding steps. For instance, ﬁnding the architecture of the inserted chain involves the derivation of the n; N-PDF <ðn2 ; N2 ÞjPnb1 ; N1 . The procedure described above in a recursive manner continues until no more structures with branch points have to be attached. Notice that at that instant of time – in contrast to the single-system algorithm – not only the topology is known, but also the segment lengths, and hence the architecture synthesis process is fully complete. 9.9.4

Mathematical Methods for Characterization of Branched Architectures Graph Theoretical Connectivity Matrices Branched topologies as generated by the conditional Monte Carlo methods described in this section are most conveniently represented in matrix forms from graph theory [33, 53]. We name two of them the adjacency matrix A and the incidence matrix C (see Figure 9.22). They both describe connectivity. Note that in 9.9.4.1

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

1 a 2 b 3 c 4

N = 10 5

d a

7

f

6

e 2 1

3 b g

c

d 5 e 6

10

i 4

f 7 g 8 h 9

h 8

9

i 10 Comb

Caylee tree 1 2 3 4 5 6 7 . 1 1 1 . . . 1 . . . 1 1 . 1 . . . . . 1 1 . . 1 A Cayley tree = . 1 . . . . .

. . .

a b –1 –1 1 . . 1

511

8 . . 1

9 10 . . 1 . . 2 . . 3

. . . 1

. . . .

. . . .

. . . .

. . . .

. 1 1 4 . . . 5 . . . 6 . . . 7

1 . .

. 1 1

. . .

. . .

. . .

. . .

c –1 . .

d e f g h . . . . . –1 –1 . . . . . –1 –1 .

1 2 3 4 5 6 . 1 . . . . 1 . 1 . . . . 1 . 1 . . A comb

i . . .

. . .

1 . .

. 1 .

. . 1

. . .

. . .

–1 –1 4 . . 5 . . 6

. . .

. . .

. . .

. . .

. . .

1 . .

. 1 .

. . 1

. . .

.

.

.

.

.

.

.

.

9 . . .

10 . 1 . 2 . 3

. . 1 .

. . . 1

. . . .

. . . .

4 5 6 7

. . . .

1 . . .

. 1 . .

1 . 1 .

. 1 . 1

. . .

. . .

. . .

. . .

. . .

. 1 . 1 . 8 . . 1 . 1 9 . . . 1 . 10

a b c –1 . . 1 –1 . . 1 –1

1 2 3

. . CCayley tree = .

8 . . .

. . = . .

. 8 . 9 . 10

. . .

7 . . .

d . . .

e . . .

f . . .

g . . .

h . . .

i . . .

1 2 3

. . .

. . .

4 5 6

. . Ccomb = .

. . .

1 –1 . . . 1 –1 . . . 1 –1

. . .

7 8 9

. . .

. . .

. . .

. . .

. . .

1 . .

–1 . . 7 1 –1 . 8 . 1 –1 9

1 10

.

.

.

.

.

.

.

Fig. 9.22. Branched topologies of molecules with N ¼ 10 branch points (terminal segments not shown). Extreme cases: Cayley tree and comb. Adjacency and incidence matrices, A and C of the topologies shown, are represented below.

.

1 10

512

9 Mathematical Methods

addition to C, a vector of segment lengths corresponding to the segments in the vector fa; b; c; . . .g completes the architectural information. The mathematical methods for characterization to be discussed here are designed in such a way that they utilize either of these two representations as input. Characterization of Architectures by Radius of Gyration Molecular architectures can be structurally classiﬁed as being more comb-like or Cayley tree-like. Structure has impact on the radius of gyration, which is larger for linear molecules than for branched molecules of the same weight (number of monomer units), since the latter are more compact. The ratio between branched and linear radius is usually described by a ‘‘contraction factor’’. Furthermore, Cayley tree-like structures are more compact than comb-like structures [33, 56]. We will show here how to obtain the contraction factor from the architectural information. The squared radius of gyration hs 2 i is expressed in monomer sizes. According to a statistical-mechanical model [55] it follows from the architecture as represented in graph theoretical terms, the Kirchhoﬀ matrix, K, which is derived from the incidence matrix, C [33]: 9.9.4.2

hs 2 i ¼ n1 TrðL1 n1 Þ

ð184Þ

1 Here, n is the number of monomer units and TrðL1 n1 Þ denotes the trace of Ln1 , being the matrix with n 1 reciprocals of the eigenvalues of the Kirchhoﬀ matrix K. The full n n sized matrix K is calculated from:

K ¼ CgC T

ð185Þ

where C T is the transpose of C (size ðnÞ ðn 1Þ and g is a vector of length ðn 1Þ related to the size of monomer units. For the computation of the radius of gyration, we apply a coarse graining method to save on computational eﬀort to ﬁnd the smallest eigenvalues of K [56]. Thus, the Kirchhoﬀ matrix reduces in size, now based on the number of branch points, ð2N þ 2Þ ð2N þ 2Þ. In this case the vector g of length (2N þ 1) contains the N 1 interbranch segment lengths and the N þ 2 free arm lengths of the molecule. It should be realized that the incidence and Kirchhoﬀ matrices contain purely ‘‘topological’’ information (connectivity between branch points). Thus, we see that the radius of gyration is determined by both topology (K) and segment length distribution (g). We now deﬁne a purely topological structure qualiﬁer ‘‘topological radius’’, based on a topological Kirchhoﬀ matrix: K ¼ CC T

ð186Þ

Note that the comb–Cayley tree ranking of molecules within a certain population in Eq. (186) is diﬀerent in principle from that in Eq. (185), since the former excludes segment length eﬀects.

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

9.9.4.3

Characterization of Architectures by Seniorities and Priorities

Introduction The rheological meaning of the concepts of seniority and priority has been explained previously elsewhere [46, 52]. Here, we will introduce them as merely topological qualiﬁers. The seniority s j of a segment j is deﬁned as the molecular distance to the nearest free arm. The seniority of a free arm is 1; the value for a segment ending on a terminal branch point is 2. Priority is deﬁned as follows. Each inner segment is connected to two trees, each of these trees having a number of terminal segments (‘‘free arms’’). The priority is the smaller of these numbers of terminal segments. Our method of ﬁnding the distribution is based on a graph theoretical representation of branched molecules. Seniority The seniority of a molecular segment between two branch points (‘‘inner’’ segment) is related to the longest chemical path (LCP) of that segment. In terms of graph theory we say that segments ending on a terminal branch point (‘‘outer’’ segments) are attached to one and all other segments are attached to two parts of the ‘‘tree’’ that represents the molecule. The LCP is deﬁned as the highest number of segments in the path to a terminal segment; the inner segments have two such LCPs one on each side. In a comb most segments have much longer LCPs than those in a Cayley tree. The seniority of a segment is deﬁned as the shorter of the two LCPs of that segment. The seniority of an outer segment is 1; the value for a segment ending on a terminal branch point is 2. The algorithm ﬁnds the seniority distribution by checking all the possible end-to-end paths of the molecule while numbering the inner segments according to the order in which they are passed in the path. (Note that to ﬁnd the number of segments with seniority 1, the outer segments, is a trivial problem since any molecule with N branch points possesses N þ 2 outer segments). The following steps are taken in the algorithm:

1. Find the Nt sets of Nt terminal branches Jt by identifying all columns of A, for which the sum equals 1; start with one vertex Jt ð jÞ, j A f1:Nt g, corresponding to column J of A. 2. Find adjacent vertices; from a selection of nj columns of A, for each of these columns Jð jÞ, j A f1:nj g, check the nonzero entries being found in nij rows nj P nij row indices IðiÞ, IðiÞ, i A f1:nij g for each j; this produces in total ni ¼ j¼1

i A f1:ni g; thus from a vertex set Jð jÞ, j A f1:nj g, all the vertices adjacent to the vertices in the set are found and stored in a new vertex set IðiÞ, i A f1:ni g. 3. Assign sequence numbers to edges; for each of the vertices Jð jÞ a set of nij adjacent vertices IðiÞ is found, associated with nij directed edges pointing from vertex Jð jÞ to vertices IðiÞ; for certain Jð jÞ this is the set of edges eJI , where J ¼ Jð jÞ and I ¼ IðiÞ, i A f1:nij g; thus ﬁnding the edges eIJ for this Jð jÞ means that they are spotted on a path and should receive a sequence number ns; for all other Jð jÞ other edge sets are found, but all ni edges found from one vertex set Jð jÞ,

513

514

9 Mathematical Methods

1 ns(e12) = 2 ns(e21) = [3 4]

5

ns(e24) = [3 3] ns(e54) = 2 ns(e45) = [3 4] ns(e42) = 3

S=2

4

S=3

S=2 2

S=2

ns(e32) = 2 ns(e23) = [3 4]

3

Sequence numbers per segment for each of the three possible path sequences through one architectural alternative of a ﬁve-branch molecule and the resulting seniorities. Fig. 9.23.

j A f1:nj g, are assigned the same sequence number; by deﬁnition, ns equals 2 for the (one) edge found from the starting vertex. 4. Update adjacency matrix; for each I; J pair found in step 2: AðI; JÞ ¼ Að J; IÞ ¼ 0. 5. The new vertex set IðiÞ, i A f1:ni g, found in step 2 is the starting point for a new cycle of ﬁnding subsequent adjacent vertex sets, but this should happen in the updated A; thus, the new set of columns follows as J ¼ IðiÞ, where all zero columns in the updated A are left out; the algorithm then repeats steps 2–5 until A has only zeros. 6. Collect all sequence numbers for all 2ðN 1Þ edges in both directions, the vector ns ðeij Þ; for any directed edge ﬁnd maxfns ðeij Þg; ﬁnd the seniority of each edge by taking the minimum of [maxfns ðeij Þg; maxfns ðeji Þg]. For illustration, the algorithm has been applied to the simple N ¼ 5 problem of Figure 9.23. Find terminal branches: Nt ¼ 3, Jt ¼ f1 3 5g; start with Jt ¼ 3. Nonzero entry in column J ¼ 3 is in row 2, hence nij ¼ 1, i ¼ 1, Ið1Þ ¼ 2. The edge found is eJI ¼ e32 and it assigned the sequence number 1: ns ðe32 Þ ¼ 2. 2

0 61 6 6 A is updated with Að2; 3Þ ¼ Að3; 2Þ ¼ 0: A ¼ 6 0 6 40 0

1 0 0 1 0

0 0 0 0 0

0 1 0 0 1

3 0 07 7 7 07 7 15 0

New vertex set from updated A: J ¼ I ¼ 2; repeat from step 2; Non-zero entries in column J ¼ 2 are in rows 1 and 4, hence nij ¼ 2, i ¼ 1; 2; Ið1Þ ¼ 1; Ið2Þ ¼ 4; The edges found are e21 and e24 and they are assigned the sequence number 3: ns ðe21 Þ ¼ ns ðe24 Þ ¼ 2; A is updated with Að2; 1Þ ¼ Að1; 2Þ ¼ 0 and Að2; 4Þ ¼ Að4; 2Þ ¼ 0:

9.9 Prediction of Branched Architectures by Conditional Monte Carlo Sampling

2

0 60 6 6 A ¼6 0 6 40 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 1

3

0 07 7 7 07 7 15 0

New vertex set from J ¼ Ið1Þ ¼ 1 and Ið2Þ ¼ 4; since column 1 has all zeros, the new J ¼ 4; repeat from step 2. Nonzero entry in column J ¼ 4 is in row 5, hence nij ¼ 1, i ¼ 1; Ið1Þ ¼ 5. The edge found is eJI ¼ e45 and it assigned the sequence number 1: ns ðe45 Þ ¼ 4. A is updated with Að4; 5Þ ¼ Að5; 4Þ ¼ 0, so A has all zeros and the collection of sequence numbers stops. This run yields the sequence fe32 e21 e24 e45 g with the associated sequence numbers f2 3 3 4g (see Figure 9.23). Similarly, starting with vertex 1 and vertex 5 yields the sequences fe12 e23 e24 e45 g and fe54 e42 e21 e23 g with sequence numbers f2 3 3 4g and f2 3 4 4g, respectively. Collecting the sequence numbers per edge yields the vectors ns ðeij Þ shown in Figure 9.23 and then taking maxima and minima according to step 6 of the algorithm yields the seniority distribution dS . This seniority algorithm is a transparent one in the sense that it explicitly calculates the segments’ longest chemical paths, on which the deﬁnition of the seniority is based. However, it involves a lengthy procedure since the sequence numbering has to be applied as many times as there are terminal branches. In the case of N ¼ 1534 (perfect Cayley tree with six generations) this number amounts to 768. For this reason a faster and simpler algorithm has been developed, in which no explicit longest chemical path evaluation takes place, but instead a decomposition of the graph is realized. It consists of the following steps (see Figure 9.24): 1. Find the set of Nt terminal branches Jt by identifying all the columns or rows of A for which the sum equals 1 (asterisk positions in Figure 9.24). 2. Assign seniority values to (terminal) edges; each of the vertices Jt is adjacent to a penultimate vertex Ipu , (some diﬀerent Jt to the same Ipu ,) so that the set Ipu may have fewer elements than Nt : Ipu a Nt (in the ﬁgure the penultimate vertices have an asterisk in the second conﬁguration); all vertices Jt to Ipu represent Nt edges; to all of these edges a seniority value is assigned: S ¼ 2 þ iu , where iu is the updated number of the adjacency matrix (0 the ﬁrst time). 3. Update A by putting zeros in rows and columns Jt . The eﬀect is two-fold: the original terminal vertices (asterisks in the ﬁrst conﬁguration of Figure 9.24) vanish, being replaced by new ones (asterisks in the second conﬁguration), but the structure is also decomposed at certain vertices, namely those marked by an asterisk on a non-terminal position. Note that thus all asterisks in fact denote terminal vertices: that is, they have sums of rows and columns equal to 1 in A. 4. Repeat steps 1 through 3 until A is all zeros. We have ensured that this decomposition algorithm reproduces exactly the same seniority distributions as the longest chemical path algorithm.

515

516

9 Mathematical Methods

*2

3

2

*

*2

*

2

*

2

*

*

*3

*

3

3

3

2

*2

2

*2 *

2

2

5 5

*

1

*

*

* *

*4

3 5

3

4

*

5

p24 = 4 p42 = 3 4

p12 = 2 p21 = 5

4

*

*

3

3

*

2

4 4

4

*

4

*

4

*

three on positions where the structure is broken (into four parts); segments adjacent to these vertices get S ¼ 3. By the second and third updates similarly the segments with S ¼ 4 and S ¼ 5 are identiﬁed.

1

2 p32 = 2 p23 = 5

dP = {3 1}

3

3

*

*

Fig. 9.24. Decomposition algorithm to ﬁnd seniority distributions for a molecule with N ¼ 36. The ﬁrst conﬁguration has 11 terminal segments, yielding 11 segments with S ¼ 2. The ﬁrst update produces the second conﬁguration with nine ‘terminal’ vertices, including

p54 = 2 p45 = 5

*

*

3

3

ds = {38 11 12 8 4} (incl. ones)

5

*

3

Fig. 9.25. ‘‘Two-sided’’ and real priorities for one architectural alternative of a ﬁve-branch molecule.

Priority Priority values are found for each segment (again, inner segments only) by constructing balances of terminal segments on each side of the segments. An example has been worked out, based on Figure 9.25. The full set of balance equations reads:

9.10 Computational Fluid Dynamics for Polymerization Reactors

p12 ¼ 2

p24 ¼ p12 þ p32

p45 ¼ p24 þ 1

p32 ¼ 2

p21 ¼ p42 þ p32

p42 ¼ p54 þ 1

p54 ¼ 2

p23 ¼ p42 þ p12

These eight equations with eight unknowns can be solved to yield the results shown in Figure 9.25. Thus the number of free segments on each side of each segment is obtained. The lower value is the priority of that segment. In general the set of equations can be written as:

p ¼ B1 :r

where r ¼

2ðN1Þ X

Bij þ 1:

j¼1

Here, B is a matrix of 2ðN 1Þ 2ðN 1Þ coeﬃcients that is obtained from the adjacency matrix A. This yields the two priorities per edge, one for each direction, of which the smallest value is the priority.

9.10

Computational Fluid Dynamics for Polymerization Reactors 9.10.1

Introduction

Chemically reacting ﬂows in polymerization reactors display complex behavior due to the interactions between physical and chemical processes. These processes occur over a wide range of length and time scales, and often result in tight coupling between ﬂuid dynamics and chemistry. Early numerical simulations of these ﬂows were based on combinations of idealized ﬂow reactor models such as the CSTR, the plug-ﬂow reactor (PFR) and the batch reactor (BR). These models either neglect radial mass and heat transfer or include only eﬀective transfer coeﬃcients, and solve a system of governing diﬀerential equations as an initial value problem. They often assume that the reactants are mixed rapidly and thus the concentrations can be considered to be uniform at small scales. For diverse polymerization processes, however, inadequate mixing of chemical species or interphase mass transfer limitations coupled with fast reactions can signiﬁcantly aﬀect reactor performance (yield, stability, operability) and product quality (selectivity). The state-ofthe-art computational ﬂuid dynamics (CFD) models provide cost-eﬀective tools to better understand complex reacting ﬂows while designing and operating reliable industrial reactors. Modeling Challenges The accurate mathematical formulation of polymerizing ﬂows is a challenging task for several reasons. For example, turbulent ﬂows require complete resolution of all 9.10.1.1

517

518

9 Mathematical Methods

length scales in the turbulence physics, such as large-scale transport due to convection and turbulent diﬀusion (macromixing), mid-scale transport due to dispersion (mesomixing), and small-scale transport due to molecular diﬀusion (micromixing). Polymerization mechanisms, on the other hand, often involve a large number of chemical species representing complex reaction networks. The disparate chemical time scales result in a stiﬀ set of equations, and require enormous computational eﬀort to integrate the species rate expressions. Since reactions occur on the molecular level, the turbulence–chemistry interactions can change the progress of fast, exothermic reactions [60, 61] and may initiate runaway conditions, or overheat polymerizing particles, thus aﬀecting the ﬁnal polymer properties. These challenges are further compounded while modeling multiple phases in heterogeneous polymerization reactors such as bubble column reactors (BCRs) or ﬂuidized bed reactors (FBRs). A range of complexities associated with the dynamics of each phase, the interaction between and within phases, sub-grid-scale heterogeneities (such as size distributions within each phase), and coupling with reactions at the micro-scale need to be addressed. For example, bubble size distribution is important to correctly model interfacial area and local mass-transfer coefﬁcients, which can further aﬀect polymerization reactions. Although phenomenological models describing such physical eﬀects have greatly improved over the years, this area still lacks reliable multiphase turbulence closures, or experimentally validated intraphase and interphase transport models. As with any other emerging technology, CFD capabilities have been understood to a certain degree for reacting ﬂows, and applied at the industrial level to model polymerization reactors. A complete review of the application of CFD for homogeneous and heterogeneous reactors would be quite extensive, and the capabilities for multiphase reacting ﬂows are still evolving. This section, therefore, focuses on the relatively well-developed CFD models for single-phase reacting ﬂows. We present a comprehensive CFD algorithm for reactor modeling and a brief review of multiphase ﬂow capabilities, followed by illustration of single-phase CFD capabilities using representative industrial test cases for the two fundamental classes of polymerization (that is, addition and condensation) processes.

9.10.2

Development and Optimization of Modern Polymerization Reactors

Figure 9.26 shows the traditional commercialization process for new polymerization technology, involving a number of development stages from discovery to commercial production. Each stage is used to gain a scientiﬁc understanding of the new technology and to make real-world assessments of design decisions required for commercialization success. This interactive route progresses through expensive and time-consuming experimental and pilot-plant testing. In recent years, the focus has shifted to employ CFD as an alternative or a complement to this traditional route while reducing the process risks and increasing conﬁdence in the new technology. Experience has shown that CFD, when implemented correctly and eﬃ-

9.10 Computational Fluid Dynamics for Polymerization Reactors

519

Optimize/improve Reacting CFD existing reactors Technology New process • Technology • Kinetics data

Lab Scale Tests

Pilot Plant Tests

Scale-up

Plant Trials

Reacting CFD Design/scale-up new reactors Technology Stage 1

Stage 2

Stage 3

Stage 4

Discovery

Commercial Scale Input Validate Implement Stage 5

Commercial Production

CFD application strategy for design, scaleup and optimization of polymerization reactors. Fig. 9.26.

ciently, can be a cost-eﬀective tool for design and scaleup of new reactors and optimization of existing reactors. Beneﬁts of CFD In the early days of its development, using even simple CFD techniques for reactor design and analysis was impractical due to limited computational power. With today’s high-performance computers as well as eﬃcient numerical and chemstry algorithms, the use of CFD is expanding for detailed reacting ﬂow analysis in the initial stages of design. The sophisticated analysis can be used to investigate reactor conditions that are outside the range of existing experimental facilities, or study dynamics of reactors for extreme conditions that are diﬃcult to reproduce experimentally due to instabilities. In general, CFD can help process engineers make well-informed design and operational decisions while meeting stringent environmental and safety laws as well as higher-quality standards for speciﬁc end-uses. 9.10.2.1

Limitations of CFD The validation of a CFD simulation against key experimental and pilot-plant data remains an integral and essential part of design. For entirely new reactor designs, it is diﬃcult to estimate the accuracy of a simulation without prior knowledge of similar ﬂow regimes. Despite the development of easy-to-use commercial CFD codes, the initial setup for complex polymerization reactors requires signiﬁcant investments of time and expense. This investment increases enormously due to the computing and meshing limitations when modeling complicated geometric shapes with rotating parts (for example, CSTRs), with changes in liquid levels (for example, in semi-BRs), with changes in ﬂow regimes (for example, dilute to dense, turbulent to laminar in BRs), or with compressibility eﬀects. Commercial codes oﬀer no direct way of incorporating the possible dependence of kinetic data on physical properties or the interdependence of physical properties on operating conditions. 9.10.2.2

520

9 Mathematical Methods

Moreover, the convergence and stability of solution techniques are severely hampered while solving stiﬀ chemistry, or changes in physical properties (such as viscosity or density) during ﬂow simulation. The grid size required to resolve all length scales (for example, the turbulence–chemistry interactions) often becomes prohibitive for reactors involving complex chemistry or complicated geometries. 9.10.3

Integration of CFD with Polymerization Kinetics

Figure 9.27 presents a comprehensive CFD algorithm based on the classical CFD approach. This algorithm formulates ﬂow characteristics using Eulerian Reynoldsaveraged Navier–Stokes (RANS) equations for the conservation of mass and momentum, and energy equations for the convective, conductive, and radiative heat eﬀects. Homogeneous or heterogeneous reactors can be modeled by increasing the complexity of physical models according to the ﬂow physics, coupled with the exact treatment of reactions. The governing model equations result in a set of coupled, nonlinear, partial diﬀerential equations, which are diﬃcult to solve analytically in closed form, except for a few special cases. For most engineering ﬂow problems, the equations can be solved numerically with appropriate boundary conditions using discretization techniques such as ﬁnite diﬀerence, ﬁnite volume, ﬁnite element, or boundary element methods. Commercial CFD codes perform a numerical simulation of the reactor geometry using a representative axisymmetric two- or three-dimensional mesh with a large number of computational cells (that is, ﬁnite elements used to solve model equa-

• 2D/3D geometry • Mesh Micro

Boundary Conditions Properties • Physical • Thermodynamic Kinetic data

Fig. 9.27.

CFD Formulation • Continuity equation • Momentum balances • Turbulence model • Mass transfer • Heat transfer Chemistry, MWD Chemistry algorithms ISAT technique Chemical look-up tables Built-in integrators

mixing

Micromixing Model Full PDF MEM model

Multi-phase Multi-phase model physics e.g. Multi-fluid model PSD

Population Balances Moment methods Discrete-sectional methods Monte Carlo methods

Generalized CFD algorithm for modeling polymerization reactors.

Increasing physical details

9.10 Computational Fluid Dynamics for Polymerization Reactors

tions). A typical simulation requires 100,000 to several million cells to accurately resolve all length scales. The use of a ﬁner mesh adds enormous computational expense in terms of memory and CPU time. Cells can be distributed selectively so that denser cells are clustered in a region of interest to resolve stiﬀ concentration gradients or small-scale phenomena. This approach also requires a reasonable approximation of the turbulent ﬂow characteristics using turbulence models that are adapted to the Navier–Stokes equations. Classiﬁcation and Complexity of CFD Models Other CFD approaches, namely direct numerical simulation (DNS) and large-eddy simulation (LES), solve the governing transport equations without any approximation, but require enormous computational expense for even simple reacting ﬂows. DNS and LES are limited in their ability to capture the micromixing eﬀect on slow diﬀusion/fast reaction processes, and are restricted to the fundamental understanding of turbulence or veriﬁcation of mixing closures for use in engineering models [62]. Another sophisticated approach is to solve joint probability density function (PDF) transport equations of species and temperature using a stochastic Monte Carlo algorithm [61, 63]. On the other hand, simpliﬁed Lagrangian micromixing models [64] (for example, the interaction-by-exchange-with-the-mean model [65, 66], or engulfment models using experimentally measured mixing characteristics [67], etc.) have been proposed for investigating the inﬂuence of micromixing on reactions using residence time distributions. Micromixing models are particularly attractive to the polymerization industry because of their simplicity. Inherent assumptions regarding the turbulent ﬂow ﬁeld, however, make them inadequate to describe the strong coupling between ﬂuid dynamics and complex reaction networks or to provide generalized scaleup rules. The complementary capabilities of the Lagrangian micromixing models and the Eulerian CFD framework can be integrated to describe turbulent micromixing in detail [68]. For heterogeneous polymerization reactors, the ﬂow properties of multiple phases or mixtures of several components must be included in the CFD simulation. The motion of bubbles (or droplets or catalyst particles) in a ﬂow is modeled by transport equations for the additional phases as well as interphase and intraphase heat and mass transfer. Recent reviews of BCRs and FBRs will provide detailed understanding of various phenomena in multiphase ﬂows and the governing transport equations [69–74]. Commercial CFD codes provide formulations based on the volume of ﬂuid model with multiple ﬂuids or phases, the mixture model with a single ﬂuid, or the Eulerian model. Micromixing models are often necessary to account for the sub-grid-scale interfacial and concentration heterogeneities, whereas the distribution of particle sizes is described by solving population balances within the CFD framework. Various closure methods are used to model population balances depending upon the particle dynamics. For example, a recently proposed direct quadrature method of moments (DQMOM) provides an accurate and cost-eﬀective representation of particulate processes (that is, nucleation, growth, aggregation, and breakage) and predicts the complete particle size distribution using a small number of transport equations [75]. The technology for multi9.10.3.1

521

522

9 Mathematical Methods

phase reacting ﬂows is still evolving and has yet to realize its full potential for important applications in the polymerization industry. Treatment of Polymerization Kinetics The CFD algorithm treats polymerization reactions exactly by integrating the species rate expressions, which form a set of highly stiﬀ ordinary diﬀerential equations (ODEs). Although there is no theoretical limitation preventing the use of detailed polymerization chemistry (with an arbitrary number of species), there is a practical limit dictated by the high computational expense of direct integration of the rate expressions involving the exponential functions. Thus, appropriate ODE solvers along with other simpliﬁcations must be used. For example, the total number of species can be reduced by assuming the quasi-steady-state approximation for free radicals, or by including only reactions that are sensitive to the transport processes. Inherent time scale diﬀerences of reaction and transport processes can be used to handle them separately by employing the fractional time-stepping formulation [63]. Even with these simpliﬁcations, repeated integrations for all grid points at each time step present a computationally formidable problem, particularly for three-dimensional simulations. Alternative eﬃcient chemistry algorithms such as multilinear interpolation using a chemical look-up table [76, 77] or the in-situ adaptive tabulation (ISAT) [78] are often explored to reduce computational time. For example, the ISAT improves the computational eﬃciency by a factor of up to 100 by tabulating the look-up tables only in the accessed composition region for a given ﬂow simulation. More details of the CFD models and the chemistry algorithms for polymerizing ﬂows (advantages, disadvantages, and implementation in commercial codes) can be found in Ref. 79. 9.10.3.2

Illustration of Homogeneous Reactor Model Formulation The application of the CFD algorithm for single-phase turbulent reacting ﬂows is illustrated in the following case. The multienvironment micromixing (MEM) model describes concentration ﬂuctuations that arise from two poorly mixed reactants in a tubular reactor. The theoretical framework and the parameter selection for the MEM model are discussed in greater detail by Fox [68]. The formulation of a three-environment model (Figure 9.28) to quantify the turbulent mixing between environment 1 (E-1) and environment 2 (E-2) follows. Formation of a reacting environment (E-3) is governed by the transport equations for volume fraction of environments ( pn ): 9.10.3.3

E-1, p1 (Pure species I) r1 = -γ (1-p1) p1

r1

E-3, p3, (3) (Reacting) r 3 = r1 + r2

r2

E-2, p2 (Pure species M) r2 = -γ (1-p2) p2

Mixing rate γ = Cφ (ε/k) Mixing parameter C = 1 for the fully-developed turbulent flow. Fig. 9.28.

Mixing parameter Cf ¼ 1 for the fully-developed turbulent ﬂow.

9.10 Computational Fluid Dynamics for Polymerization Reactors

qpn qpn q qpn þ hUi i þ Gn ð pÞ ¼ GT qt qx i qx i qx i

n ¼ 1; 2; 3

ð187Þ

The species mixing, heat transfer and reactions in E-3 are governed by the transport equations for the volume-weighted species concentration and temperature in ð3Þ E-3 ðsð3Þ a ¼ fa p3 Þ: qsð3Þ qsð3Þ q qsð3Þ a GT a þ Mað3Þ ðp; sð3Þ Þ þ p3 Sa ðfð3Þ Þ þ hUi i a ¼ qx i qt qx i qx i

ð188Þ

Repeated roman indices imply summation, and a denotes the species or temperature in E-3. The Navier–Stokes equations, the standard k–e equations, describe the turbulence physics, and compute the mean velocity hUi i along with the turbulent parameters (kinetic energy k, dissipation rate e, diﬀusivity GT ). The micromixing functions Gn ðpÞ and Mað3Þ ðp; sð3Þ Þ are deﬁned in terms of the probability ﬂuxes rn for environment n, and the chemical source terms Sa ðfð3Þ Þ are estimated on the basis of the species rate expressions. This formulation is used to investigate the turbulence–chemistry interactions in the example in Section 9.10.4.1. 9.10.4

Target Applications

Many aspects of reacting ﬂows have been studied for developing CFD models of polymerization reactors. Table 9.20 documents representative examples of these studies for various polymerization processes by summarizing reactor type, number of phases, number of species, CFD techniques, and engineering challenges that are addressed using CFD. Table 9.21 reports potential CFD applications for other polymerization processes. Speciﬁc examples of single-phase turbulent reacting ﬂows in a tubular jet reactor are discussed in Section 9.10.4.1. We select representative industrial problems of engineering interest for the two fundamental classes of polymerization reactions, namely addition and condensation polymerization. For each example, we present a general overview of the problem and detailed reacting ﬂow analyses, followed by useful process design and operational information. 9.10.4.1

Illustrative Case Studies

Free-radical polymerization [79, 93] Low-density polyethylene (LDPE) accounts for almost two-ﬁfths of the global polyethylene production capacity, reaching about 45 million tons per year in 1996. LDPE is exclusively produced by free-radical polymerization in a tubular or autoclave reactor, each of which accounts for about 50% of the total capacity. In the tubular reactor, a small amount of initiator is injected into a turbulent monomer ﬂow for initiating the exothermic free-radical chemistry. Under extreme operating conditions (T @ 140–300 C, p @ 1000–3500 atm), these

523

524 Tab. 9.20.

9 Mathematical Methods

Target applications in the polymerization (PP) industry.

PP product, PP process

Reactor, phases,[a] species

CFD techniques

Design, process engineering challenges

References

Polyethylene, Free-radical PP

PFR, S, 6

joint PDF method, simple chemistry joint PDF method, detailed chemistry

local hot-spots, MWD

80

initiator loss, MWD, local/global decomp. (reactor stability) Initiator consumption, conversion

81

conversion, MWD, temp. distribution, design changes laminar striation thinning, MWD

84–86

PFR, S, 16

autoclave, S, 3

Eulerian CFD, eddy-dissipation,[b] simple chemistry Eulerian CFD, simple chemistry

autoclave, S, 3 to 6 Polypropylene, Free-radical PP

static mixer reactor, S, 0

Polyoleﬁn, Catalyzed PP

FBR, M, 6

Polyoleﬁn, Catalyzed PP

FBR, S, 0

Polystyrene, PVC, etc., Suspension PP

BR, M, 0

cis-Polyisoprene, Anionic PP

PFR, S, 6

Polyamide, etc., Condensation PP

Multijet reactor, S, 6

[a] S:

single-phase; M: multiphase.

Tab. 9.21.

[b] Ref.

Eulerian CFD, micromixing theory,[c] no chemistry multi-ﬂuid model, DQMOM, simple chemistry Eulerian CFD, heat transfer, no chemistry Eulerian CFD, compartment-mixing, population balance, no chemistry Eulerian CFD, representative simple chemistry Eulerian CFD, eddy-dissipation,[b] simple chemistry

83

57

polymer particle overheating

75

heat removal, polymer particle overheating PSD control

88

MWD prediction

91

reactor fouling

92

89, 90

82. [c] Ref. 87.

Potential applications in the polymerization industry.

PP product, PP process

Reactor type

Design, process engineering challenges

a-Oleﬁns, Metallocene PP Polysulfone, Linear condensation PP Ethylene–vinyl acetate, Freeradical PP Polyamide-6, Hydrolytic PP Polyisobutylene, Cationic PP Polyester, Polyesteriﬁcation Latex, Emulsion PP

PFR CSTR

reactor fouling sensitivity of chain-length distribution

PFR

initiator loss, MWD, phase separation

CSTR/PFR PFR semi-BR CSTR

yield, selectivity reactor stability polymer molecular properties polymer properties, yield

9.10 Computational Fluid Dynamics for Polymerization Reactors

reactors are sensitive to local mixing conditions and serve as an excellent test case to study the performance and stability of commercial reactors involving singlephase turbulent reacting ﬂows. In this example, the mixing between initiator (E-1) and monomer (E-2) is quantiﬁed using the MEM model introduced in Section 9.10.3.3. The ISAT algorithm is customized for a total of 16 species in the LDPE chemistry, including chain initiation, propagation, termination, chain transfer, and other branching reactions as well as simpliﬁed ethylene decomposition reactions (reported in Table 9.22). More details of the kinetic scheme, the rate expressions, and the rate constants as well as the customized ISAT algorithm can be found in Ref. 81. The poorly mixed region is modeled using a two-dimensional axisymmetric computational domain of length 10 m and radius 1.9 cm. The standard k–e model describes the turbulence, as viscosity eﬀects are negligible due to the fully-developed turbulent ﬂow. The MEM model and the ISAT algorithm (about ten-fold faster than the direct integration) are solved using an interactive interface via external routines in the Fluent2 v5.3

Tab. 9.22.

Comprehensive free-radical polymerization reactions for modeling chemical source

terms [81]. Detailed low-density polyethylene polymerization chemistry: Organic peroxide initiation

fni ; kdI

ni

Ini ! 2Ani

(ni ¼ 1; 2)

kIni

Ani þ M ! R1 Propagation

kp

Ri þ M ! Riþ1 k tc

Termination by combination

Ri þ Rj ! Piþj

Termination by disproportionation

Ri þ Rj ! P¼ i þ Pj

Chain transfer to monomer

Ri þ M ! P¼ i þ R1

k td

k trm

Ri þ CTA ! Pi þ RCTA

Chain transfer to polymer

Ri þ Pj ! Pi þ Rj0

k trp

kSCB

Ri ! Ri

(SCB)

b-Scission

kLCB Ri0 þ M ! Riþ1 kb 0 Riþj ! Ri þ P¼ j

Simpliﬁed ethylene decomposition chemistry: Initiation

2M ! C2 H3 þ C2 H5

Propagation

C2 H5

LCB formation

k1

C2 H5

k2 ; k20

! M þ H k3

þ M ! C2 H6 þ C2 H3 k4

H þ M ! H2 þ C2 H3

Termination

(DB)

kCTA

Chain transfer to transfer-agent Backbiting (SCB formation)

(DB)

k5 C2 H3 ! C þ CH3 k6 CH3 þ M ! CH4 þ C2 H3 kt CH3 þ CH3 ! C2 H6 kt C2 H3 þ CH3 ! C2 H2 þ CH4 kt C2 H3 þ C2 H3 ! C2 H2 þ M

(LCB) (DB)

525

526

9 Mathematical Methods

software. An unsteady coupled implicit solver is used to limit the eﬀects of truncation errors on the solution. The interface also includes the interdependence of the kinetic, physical, and thermodynamic properties of the reaction mixture. Figures 9.29(b)–(f ) present the detailed reacting ﬂow analysis and the performance evaluation using a typical CFD simulation for the feed temperature (Tfeed ) of 250 C. As seen in Figure 9.29(b), center-mode initiator injection causes sharp radial concentration gradients near the injection point. This phenomenon leads to the higher termination of initiator free radicals and loss of @64% of the initiator compared to the plug-ﬂow conditions. Contours of the mean monomer massfraction (Figure 9.29(c)) and the mean temperature (Figure 9.29(d)) show a conversion of @5% and a steady-state temperature rise of @57 C, respectively. The poor micromixing also aﬀects other reaction rates. For example, the localized release of the exothermic energy due to higher propagation rates leads to local temperature ﬂuctuations in E-3 (Figure 9.29(e)), whereas non-uniform chain-transfer rates cause a broader MWD, as indicated by higher polydispersity (Figure 9.29(f )). Thus, the CFD algorithm is able to capture the eﬀect of the turbulence–chemistry interactions on the reactor performance, which cannot be predicted by the traditional reactor models. Detailed CFD simulations are used to validate and increase the accuracy of a simpliﬁed one-dimensional MEM model. The high computational eﬃciency of the simpliﬁed approach is extremely useful to evaluate the performance for a range of operating and mixing conditions. For example, proﬁles in Figure 9.29(g) summarize high-temperature initiator eﬃciency for a range of feed temperatures and mixing conditions, whereas the reactor stability map in Figure 9.29(h) identiﬁes unsafe regions of reactor operation. Thus, in this example, CFD oﬀered a low-cost alternative to replace pilot-plant tests and explore a variety of options for optimizing initiator consumption while developing oﬄine strategies to control the reactor safety and product quality in industrial reactors. Condensation polymerization [92] Several industrially signiﬁcant resins – polyamides (for example, nylon), polyester, polyurethanes, polyacetals, phenol-aldehydes, polyurea, and silicones – are produced by condensation polymerization. In this example, a multi-jet tubular reactor is used for the polycondensation between monomers A and B, each having two reactive groups. Multiple turbulent jets of B are impinged onto radial ﬂow of A at the inlet of the reactor to achieve the rapid mixing required for starting the step-growth polymerization. The reaction forms polymer linkages L and condenses out a small molecule by-product C forming a by-product complex SC (Table 9.23). Solvent S acts as a catalyst and dissolves all chemical species, thereby maintaining single-phase ﬂow. Subsequent secondary and branching reactions control the structure and MWD of the ﬁnal polymer. The inlet region (length L @ 3:5D and residence time t) is sensitive to various ﬂuid dynamic processes and can aﬀect the reactor operability due to deposition of products. The continuity and Navier–Stokes equations, the standard k–e turbulence model, and the transport equations for species concentration and enthalpy are solved using a three-dimensional computational domain for the inlet region (Figure 9.30(a)).

9.10 Computational Fluid Dynamics for Polymerization Reactors Co-axial jet, djet = 0.2 mm Monomer, QM (m3/s) 〈Uz 〉 = 13.16 m/s p1 = 0 p2 = 1

527

L = 10 m Grid size: 400 x 19

D = 3.8 cm

Outlet

For velocity profiles: ρ = 560 kg/m3; µ = 0.0016 kg/m.s Tfeed = 250 °C QI p1,0 = QI + QM

Initiator, QI (m3/s) 〈Uz 〉 = 4.52 m/s p1 = 0.1 p2 = 0.9

(a) Two-dimensional axisymmetric computational domain

Poorly micromixed region (0 to 0.2 m) (Scale-up factor: D ~ 1:85, L ~ 1:40)

(b) Initiator mass fraction (blue to red: (0–1.15) ×10 –3)

(c) Monomer mass fraction (blue to red: 0.95–1)

(g) Optimization of high-temperature initiator efficiency

(d) Mean temperature (blue to red: 250–307°C)

(e) Local temperature in the reacting environment (blue to red: 250–329°C)

(f ) Polydispersity (blue to red: 0–7.15)

(h) Reactor stability map for commercial-scale operation

Fig. 9.29. Typical reacting CFD simulation results and performance evaluation for the coaxial jet tubular LDPE reactor (scaleup factor for diameter is 1:50).

528

9 Mathematical Methods Tab. 9.23. Primary condensation polymerization reactions for modeling chemical source terms [92].

4 4 k1 ¼ 1:2810 exp t

8:9410 RgT

Condensation reaction:

A þ B þ S ! L þ SC DHr =RTA0 ¼51:3

6 4

Solvent regeneration:

A þ SC !

AC þ S 6 5

Transfer reaction:

AC þ S ! A þ SC

k2 ¼ 4:6910 exp t

8:3310 RgT

k3 ¼ 3:8610 exp t

1:5010 RgT

The chemistry is simpliﬁed to include only condensation, solvent regeneration, and transfer reactions (reported in Table 9.23) that are important in the region of interest. Finite-rate/eddy-dissipation formulation [82] in the Fluent2 v6.0 software is used to model the turbulence–chemistry interactions. This formulation uses the minimum of the Arrhenius and eddy-dissipation rates to account for mixing limitations. CFD simulations are performed to visualize the velocity, pressure, concentration (with and without chemical reactions), and temperature distributions for a range of operating rates. For example, Figure 9.30(b) shows the non-uniform radial distribution of the reacting species A at the turndown ratio of 0.6 compared to the maximum operating rate. The presence of a recirculation zone near the inlet results in a higher mass fraction of the polymer linkage (gL ). This phenomenon is highlighted in the inset of Figure 9.30(c) by clipping the contours to the maximum gL in the recirculation zone (gLmax; rc ). Visual observations of internals in a fouled reactor conﬁrm the deposition of products in regions of higher polymer linkage mass fraction (that is, contours at the reactor entrance and the bottom wall of the reactor) indicated by CFD simulations. Although nonreacting CFD simulations indicate improved mixing at lower operating rates (Figure 9.30(d)), the reactor operation is limited due to high values of gLmax; rc . The lower operating limit is predicted to be a @0.42 turndown ratio corresponding to the experientially estimated gLmax; rc of 7%. The operation at higher rates (turndown ratio > 0:75 in Figure 9.30(d)) is limited by the maximum available premix pressure for stream B. Thus, the analysis is helpful in predicting the feasible operating range, and can be used to optimize other process conditions as well as design parameters for enhancing the operating ﬂexibility. 9.10.5

Concluding Remarks

CFD is emerging as a powerful tool for the design and optimization of polymerization reactors with the rapid growth in high-performance computing resources. Experimentally validated CFD models can now be used to complement the traditional design procedure for cost-eﬀective development of new reactors or for the productivity improvement of existing reactors. While limitations in handling detailed chemistry in real time still exist, the ability to visualize and quantify the reactor

9.10 Computational Fluid Dynamics for Polymerization Reactors

529

Cross-section locations for post-processing 0.1 0.25 0.35 0.5 0.75 1.0 djet

0

x/L

Stream B, UB D

gB = 1 gT = 1.33 pr = 1.22 Stream A, UA

Injection slot

gA = 0.1; gB = 0.9 gT = 1 pr = 0.85

Outlet

L / D ~ 3.5 gT = T/TA0 pr = p/(∆pB)max ∆pr = ∆p/(∆ pB)max

(a) Schematic diagram of the multi-jet tubular reactor

Reactor cross-sectional map

Symmetry plane (b) Mass fraction of reactant A

Clipped between 0 & 0.045

3D contours inside the reactor

(c) Mass fraction of polymer linkage L; inset contours highlight formation of high polymer mass at the inlet inle

(d) Reactor performance as a function of the turndown ratio

Fig. 9.30. Typical reacting CFD simulation results and performance evaluation for the multijet tubular condensation polymerization reactor.

530

9 Mathematical Methods

performance without expensive experimentation is accelerating use of these models in the polymerization industry. As with other emerging technologies, CFD’s capabilities have been understood to a certain degree for reacting ﬂow modeling. It can predict single-phase turbulent reacting ﬂows in industrial reactors with suﬃcient detail and increased accuracy. Multiphase reacting ﬂow modeling, however, lacks experimentally veriﬁable physical models to describe the complex multiphase ﬂuid dynamics (for example, multiphase turbulence) and its interaction with heterogeneous reactions. Thus, advanced physical models and eﬃcient numerical algorithms are needed to realize CFD’s full potential for applications involving heterogeneous polymerizations. Finally, the emerging high-performance supercomputing technology is likely to have a major impact in polymer reaction engineering for modeling plant-scale reactors with an increased level of physical, mechanical, and chemical details. Steady-state or dynamic CFD calculations can be integrated with a general-purpose process simulator, using an interface that allows the automatic exchange of critical variables. This approach will open up new opportunities for real-time design modiﬁcations to develop safe, reliable, and eﬃcient polymerization processes and to control polymer quality by carrying out sophisticated design and analysis at a substantially reduced cost.

Acknowledgments

N.K. thanks Professor R. O. Fox and his research group at the Iowa State University, Dr. C. LaMarca and colleagues in the DuPont Engineering Research and Technology department for the review of Sec. 9.10, and DuPont management for the support.

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Theory Simul., 10, 144–157. Eichinger, B. E. (1980). Macromolecules, 13, 1. Iedema, P. D., Hoefsloot, H. C. J. (2001). Macromol. Theory Simul., 10, 870–880. Fourcade, E., Wadley, E. R., Hoefsloot, H. C. J., Green, A., Iedema, P. D. (2001). Chem. Eng. Sci., 56, 6729–6741. Iedema, P. D., Willems, C., Van Vliet, G., Bunge, W., Mutsers, S. M. P., Hoefsloot, H. C. J. (2001). Chem. Eng. Sci., 56, 3659–3669. Fourcade, E., Hoefsloot, H. C. J., van Vliet, G., Bunge, W., Mutsers, S. M. P., Iedema, P. D. (2001). Chem. Eng. Sci., 56, 6589–6603. Baldyga, J., Bourne, J. R. (1999). Turbulent Mixing and Chemical Reactions, John Wiley and Sons, UK. Fox, R. O. (2003). Computational Models for Turbulent Reacting Flows, Cambridge University Press, Cambridge. Fox, R. O. (1996). Rev. Inst. Franc¸. Pe´trole, 51, 215. Pope, S. B. (2000). Turbulent Flows, Cambridge University Press, New York. Villermaux, J., Falk, L. (1994). Chem. Eng. Sci., 49, 5127. Harada, M. (1962). Mem. Fac. Eng. – Kyoto University, 24, 431. Villermaux, J., Devillon, J. C. (1972). In Proc. 2nd Int. Symp. Chem. React. Eng., Amersterdam, B, p. 1. Baldyga, J., Bourne, J. R. (1988). Chem. Eng. Res. Design, 66, 33. Fox, R. O. (1998). Chem. Eng. Proc., 37(6), 521. Enwald, H., Peirano, Almstedt, A.-E. (1996). Int. J. Multiphase Flows, 22(S1), 21. Mathiesen, V., Solberg, T., Arastoopour, H., Hjertager, B. H. (1999). AIChE J., 45, 2503. Hatzantonis, H., Yiannoulakis, H., Yiagopoulos, A., Kiparissides, C. (2000). Chem. Eng. Sci., 55, 3237. Joshi, J. B. (2001). Chem. Eng. Sci., 56, 5893. Li, J., Kuipers, J. A. M. (2003). Chem. Eng. Sci., 58, 711.

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10

Scaleup of Polymerization Processes1 E. Bruce Nauman 10.1

Historic and Economic Perspective

Since the mid-1960s the world has seen remarkable increases in production rates for commercial polymers. Single-train line rates have increased by factors of 20 to 40 for polymers such as polyethylene, polypropylene, and polystyrene. A rate of 1 t h1 that would have been economical in 1964 has become 20–40 t h1 . Part of this improvement is attributable to improved catalysis and innovative processes, but the greatest increase in productivity is due to classical scaleup. Larger plants produce more polymer using substantially the same manpower and with reduced capital investment per ton of annual capacity. The selling prices of the major commodity polymers have increased by factors of 3 to 4 since the 1960s, but the US consumer price index has increased by a factor of 6. The result has been an ongoing decrease in the real cost of polymers that leads to their greater use and a continuing incentive to build still larger plants. The success of the petrochemical industry in the 20th century is attributable in large part to the scalability of typical chemical processes. This chapter discusses the technical basis for achieving capacity increases without major changes in technology. These techniques are broadly applicable to the development of new polymers and new polymerization processes. They continue to be applied to existing processes, although no process is inﬁnitely scalable. Eventually, a limit must be reached.

10.2

The Limits of Scale

Consider the scaleup of a stirred tank reactor. It is common practice to scale using geometric similarity so that the larger reactor will have the same shape as the small reactor. Suppose the volume scaleup factor is given by Eq. (1). 1) The symbols used in this chapter are listed at

the end of the text, under ‘‘Notation’’. Handbook of Polymer Reaction Engineering. Edited by T. Meyer, J. Keurentjes Copyright 8 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31014-2

534

10 Scaleup of Polymerization Processes

S¼

V large Vsmall

ð1Þ

All linear dimensions such as reactor diameter, impeller diameter, and liquid height will scale as S 1/3 . Surface areas will scale as S 2/3 while volume itself scales as S 1 . The total amount of heat transferred to a jacket will increase as S increases, but heat transfer per unit volume of reactor contents will decrease. Sooner or later, the reactor will become adiabatic. Similarly, any mass transfer to an external surface will become limiting. These limitations are well known and can be avoided, for example, by employing processes without external mass transfer and by using enough inerts so that the adiabatic temperature rise is acceptable. However, there is still a mixing limit that will ultimately emerge. In very large vessels, it will become impossible to mix the reactive components in a time that is reasonable compared to the reaction half-life. Consider a pilot-scale vessel that has a mixing time of 30 s using a small electric motor. Suppose it is desired to maintain this mixing time when the reactor is scaled, using geometric similarity, by a factor of 10 in linear dimensions and by a factor of 1000 in volume. Standard correlations show that it is necessary to maintain the same rotational speed of the agitator, say 100 rpm, in both large and small vessels. The agitator power increases by a factor of S 5/3 ¼ 100 000, so that a 0.5 kW motor scales to 50 000 kW. The lesson from this is that even a simple phenomenon – such blending of two ﬂuids – requires longer and longer times as the size of the process increases. Eventually, the operation becomes unfeasible.

10.3

Scaleup Goals

The goals of a process scaleup are to obtain acceptable product quality with good manufacturing economics. Product quality is usually best at the laboratory scale because mixing and heat transfer are excellent and because the academic training of chemists puts a high premium on yield and selectivity. The same product quality can be achieved upon scaleup by employing many bench chemists working in parallel, but the economics will be poor. The scaleup engineer seeks to design a single-train process that meets anticipated market demands or else is the largest capacity unit for which adequate quality can be achieved. A ‘‘single-train’’ process may include some equipment in parallel. Shell-and-tube heat exchangers are a ubiquitous example. They are relatively inexpensive and are sold as integral units. Another example in the polymer industry is pelletization. The maximum economic size for extrusion and pelleting has lagged behind the maximum size for polymerization reactors. Thus it is common for a polymerization reactor to feed two or three pelleting lines. These lines are expensive, so the economics of scaleup are compromised. However, the major cost of a polymerization line is in the feed preparation, reaction, and recovery sections, and these are generally single-train. They have one control system, one operating crew and –

10.4 General Approaches

with the possible exception of additives added in the extrusion or pelleting steps – make one product at a time. They are single-train by the practical, working deﬁnition of single train. Since the ability to mix and transfer heat generally decreases upon scaleup, the uniformity of a product usually decreases as well. This typically means an increase in polydispersity of a polymer. It is easy to make polystyrene having a polydispersity of 2 by a laboratory-scale, free-radical process. Industrial processes have diﬃculty going below 2.2 and operate best when making products with a polydispersity of 2.5 and higher. Such products are sellable and are even preferred for injection molding. However, had the early polystyrene markets required the low polydispersity achievable in the laboratory, the commercial processes would have failed. Residual volatiles are a similar example. Coagulation and vacuum drying in the laboratory have always been able to produce the low volatilities that are now possible on a commercial scale after 50 years or so of process development. For a new polymer, product quality goals and qualiﬁcation tests must be based on what is possible to achieve in commercial-scale equipment, not on what is possible in the laboratory. This may require a change in the laboratory-scale operating conditions so that the product is economically achievable in a commercial process. The intentional lessening of product goals in the laboratory or pilot plant to enable economic scaleup is called ‘‘diplomatic scaleup’’ because it may require diplomacy to convince the bench chemist and management.

10.4

General Approaches

There are many ways of increasing the capacity of a chemical reactor. The simplest methods are scaling in parallel or scaling in series. When applied to a tubular reactor, the tube diameter is held constant, and capacity is increased by adding tubes in parallel or in series. There are several scaleup possibilities:

Add identical reactors in parallel. The shell-and-tube design used for heat exchangers is a common and inexpensive way of increasing capacity. Gas-phase, heterogeneously catalyzed reactions are usually scaled in this way since temperatures and pressures are identical in the small and large scales. The ultimate scaleup limit is uniform distribution of feed across a large tube bank, but the limit has not been reached with 10 000 tubes. In the polymer industry, 1000tube heat exchanges are used as post-reactors and devolatilization preheaters for polystyrene. Shell-and-tube designs are also used in solution process loop reactors, but instability or fouling problems arise if the discharge stream has a much higher viscosity than the feed stream. Make the reactor longer or add reactors in series. Capacity can be increased in the high-pressure process for LDPE by adding an extra mile to the tubular reactor. Adding stirred-tank or stirred-tube reactors in series is sometimes used to scale up polystyrene processes. For incompressible ﬂuids, scaling in series is lim-

535

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10 Scaleup of Polymerization Processes

ited by pressure drop. For compressible ﬂuids, lengthening a reactor can lead to a dramatic increase in density so that the ratio of heat transfer area to volume decreases dramatically. Simultaneously increase two or more characteristic dimensions of the reactor. Scaling with geometric similarity, so that all linear dimensions increase by a factor of S 1/3 , is common for stirred-tank reactors but uncommon for tubes. Geometric similarity for a tube means keeping the same length-to-diameter ratio, L/d t , upon scaleup. It is also possible to scale while maintaining a constant pressure drop across the reactor. Turbulent reactors require a reduction in length-todiameter ratio to scale at constant pressure drop. Scaling with geometric similarity gives a constant pressure drop in laminar-ﬂow reactors. Scaling in parallel or series is preferred when heat transfer is a dominant consideration. The third method, scaling with geometric similarity, is cheaper for reactions that permit adiabatic operation. Two other methods of scaleup are:

Lower product quality expectations to what is readily achievable in a commercial reactor, ideally an adiabatic reactor with premixed feed. This is the diplomatic form of scaleup mentioned in Section 10.3. Go from batch to continuous operation. Scaleup is easier, especially for postreactor processing techniques that are typically continuous. For a stirred-tank reactor, the reaction rate will decline because the entire reaction is conducted at the highest conversion and thus, typically, at the lowest reaction rate. However, the decline in rate is more than compensated for by the full time utilization of the reactor compared to occasional utilization in a batch process. In what follows, we will draw heavily on Ref. 1, in which scaleup is treated in the general context of reactor design. Another general reference work is by Bisos and Kabel [2] while McGreavy [3] brieﬂy discusses scaleup of polymerization processes. Many references purport to discuss scaleup but only give modeling equations that contain scale-dependent parameters, the values of which are left as an exercise to the reader. Another common approach is to enumerate the various dimensionless numbers that govern a reaction system. One quickly concludes that scaleup of a pilot-scale reactor is impossible because it is not feasible to maintain the various dimensionless numbers at the same values as they had in a pilot plant, except by scaling in parallel. The secret, of course, is to know when constancy of a dimensionless number can be sacriﬁced without compromising the scaleup. As an obvious example, the Reynolds number will usually increase upon scaleup, but laminar ﬂow patterns are largely indiﬀerent to Reynolds number until some critical value is reached. Similarly, the Froude number is unimportant in a baﬄed stirred tank. In contrast, dimensionless groups involving temperature (for example, T/Tact where Tact is the Arrhenius activation temperature) must be kept nearly constant upon scaleup of a chemical reaction.

10.6 Stirred-tank Reactors

10.5

Scaleup Factors

Equation (1) deﬁned S as the ratio of large and small volumes. A more rigorous deﬁnition bases S on the ratio of holdup (inventory) in the large and small reactors. This accounts for possible density variations. However, the goal in almost all scaleups is to achieve the same reaction environment in the large and small scales and thus the same density. For this case, S is deﬁned in Eq. (2). Holdup Scaling Factor ¼

Holdup in Large System ½ r^VLarge VLarge ¼ A ¼ S ð2Þ Holdup in Small System ½ r^VSmall VSmall

The throughput scaling factor, ST , is the ratio of production rates for the large and small reactors [Eq. (3)]. Throughput Scaling Factor ¼ ¼

Throughput for Large System Throughput for Small System ½rin Q in Large ½rin Q in Small

A

Q Large ¼ ST Q Small

ð3Þ

The mean residence time, t, is the ratio of holdup to throughput. Reaction systems are almost always scaled with a constant value for t. This means that ST ¼ S is the almost universal way in which ﬂow reactors are scaled. Batch reactors are usually scaled using the same batch reaction time.

10.6

Stirred-tank Reactors

The conventional stirred-tank reactor is an agitated vessel, typically a jacketed pressure vessel, and often with provisions for reﬂux of a solvent or monomer. The continuous-feed version is the CSTR. Continuous operation is typical of highvolume polymers but large batch and fed-batch stirred-tank reactors are occasionally used. Reactors other than stirred tanks may be functionally equivalent to stirred tanks. Loop reactors are widely used in the polymer industry, especially for solution and slurry oleﬁn polymerizations; the agitator in the stirred tank is replaced with a circulation pump. The loop many consist of jacketed pipe or there may be heat exchangers and even ﬂash vessels in the loop. The loop may consist of many ‘‘legs’’ for space considerations, but the legs are connected in series and there is only one circulation pump. Fluidized-bed reactors can also be treated as stirred tanks for many purposes; so can the stirred-tube reactors used for styrenic polymerizations. A reactor will behave like a stirred tank when the contents are approximately uniform in tempera-

537

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10 Scaleup of Polymerization Processes

ture and composition and when the contents have a nearly exponential distribution of residence times. If the reaction is ﬁrst order or pseudo-ﬁrst order, or if the conversion is low, the yield behavior will be similar to that of an ideal, perfectly mixed CSTR provided only that the contents are approximately isothermal and have an exponential distribution of residence times. It is not necessary for the contents to be well mixed on the molecular level. Mixing on the molecular level, that is, micromixing, is of great academic interest but has so far not been identiﬁed as important in commercial polymerization vessels. Instead, circulation times (mixing times) are limiting. Typical mixing times in large vessels are a few minutes. In turbulent vessels, the distance scale over which diﬀusive mixing is limiting is on the order of the Kolmogorov scale, typically a few tens of microns, and diﬀusive mixing occurs in a fraction of a second. The Kolmogorov scale varies with power per unit volume and thus remains constant for a conventional scaleup. Although the underlying mechanisms are diﬀerent, circulation times are typically much greater than diﬀusion times in laminar ﬂow vessels as well. The scaling of stirred-tank reactors is usually based on geometric similarity so that the large vessel looks like the small reactor, and all linear dimensions scale as S 1/3 . Table 10.1 gives a variety of scaleup factors for this form of scaleup. In the general case, the agitator rotational speed, NI , is independently adjustable. The throughput scaleup factor, ST , is also independently adjustable but has no appreciable eﬀect on the scaleup factors in Table 10.1 because the internal circulation due to the agitator is normally much higher than the throughput of a CSTR. Thus, batch stirred tanks and normally designed CSTRs behave in much the same way with respect to the parameters listed in Table 10.1. The scaleup factors

Tab. 10.1.

Scaleup factors for turbulent stirred tanks.

Vessel diameter Impeller diameter Vessel cross-section Vessel volume Reynolds number Froude number Agitator speed Power in turbulent ﬂow Power in laminar ﬂow Power per volume Mixing time Circulation rate Heat transfer area, A ext Inside coeﬃcient, h Coeﬃcient area, hA ext Driving force, DT

General scaling factor

Scaling factor for constant power per unit volume

Numerical scaling factor for S F 512

S 1/3 S 1/3 S 2/3 S NI S 2/3 NI2 S 1/3 NI NI3 S 5/3 NI2 S NI3 S 2/3 NI1 NI S S 2/3 2/3 NI S 1/9 2/3 NI S 7/9 2/3 2/9 NI S

S 1/3 S 1/3 S 2/3 S S 4/9 S1/9 S2/9 S S 1.0 S 2/9 S 7/9 S 2/3 S1/27 S 17/27 S 10/27

8 8 64 512 8 0.5 0.25 512 512 1 4 128 64 0.79 50.8 10.1

10.6 Stirred-tank Reactors

for power and heat transfer coeﬃcients shown in Table 10.1 are based on generally accepted correlations for turbulent stirred tanks. Nauman [1] gives details. As shown in Table 10.1, power in the turbulent regime varies as NI3 S 5/3 . As a practical matter, NI must be lowered upon scaleup. Agitation experts have many criteria for setting NI , but scaling with constant power per unit volume is a good general rule. For fully turbulent vessels, this means that NI will vary as S2/9 . Table 10.1 gives speciﬁc scaling factors for scaleups with constant power per unit volume, and also gives numerical examples for such scaleups with S ¼ 512. In these examples, the power calculation assumes turbulent ﬂow. Note that for turbineagitated vessels, ﬂow is substantially turbulent ﬂow when the impeller Reynolds number, rNI D 2 /m, is above about 100. Power in laminar ﬂow scales as NI2 S. The specialized literature, and especially Paul et al. [4], provide details on closeclearance impellors (for example, anchors and helical ribbons) and correlations suitable for deep laminar ﬂow. As a generalization, heat and mass transfer to the external environment is very low in such devices. However, it is usually possible to achieve a suﬃciently high circulation rate in the vessel for the tank to be reasonably well mixed. The volumetric scaleup by a factor of 512 anticipates an increase in throughput by the same factor. Is this possible while maintaining reasonable product quality? The results in Table 10.1 suggest three possible problems. The ﬁrst is mass transfer to the external environment. For geometric similarity, surface areas increase by a factor of only S 2/3, or 64, while throughput and volume increase by a factor of 512. A limitation can arise if a condensation product has to be removed across the free surface at the top of the vessel. A practical example is the removal of byproduct ethylene glycol in the glycolysis route to PET. Similarly, very high boiling rates in a reﬂux reactor can cause slugging and liquid entrainment. A potential solution to such mass transfer problems is the use of an external recycle loop, as shown in Figure 10.1. The recycled material is ﬂashed into the dome of the stirred tank, creating extra surface area by foaming (that is, by ﬂash devolatilization). A second possible limitation is heat transfer. If the vessel contents are cooled through a jacket or by internal coils, the cooling demand increases by a factor of 512 but the surface area increases by a factor of only 64. Additionally, the heat transfer coeﬃcient decreases when scaling at constant power per unit volume. The result is that the driving force at the wall must increase by a factor of about 10 when scaling with S ¼ 512 and constant power per unit volume. This may be acceptable when the pilot unit operates with a DT of 2 C but becomes problematic when the pilot plant operates with a 20 C DT. There are many solutions. In a CSTR, use cold feed. Some processes for PMMA use a 40 C feed to control the reaction exotherm. Diluents and low per-pass conversions can also be used: this approach is typical of solution polyoleﬁn processes. Reﬂux boiling can be used: it is common in styrenic polymerizations where the reﬂux solvent is normally returned as a liquid. In some polypropylene processes, the returning propylene is ﬂashed into the ﬁrst reaction vessel. Finally, the external loop shown in Figure 10.1 can be used. Indeed, a loop reactor with the elimination of the stirred tank performs identically to a CSTR provided that the circulation rate is high compared

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10 Scaleup of Polymerization Processes

Vapor

Condensate

• Lowers distance scales for heat transfer and mixing.

Shell-andTube Heat Exchanger

Stirred Tank

Circulation Pump

• Allows a controlled dilution rate for incoming feed. • Can be detuned at the small scale to simulate achievable results at the large scale.

Reactor Outlet

Static Mixer Fig. 10.1.

CSTR Inlet and Fed Batch Additives

A Generalized Loop Reactor.

to the throughput. Heat transfer can be better managed using a shell-and-tube exchanger in the loop than by the jacket or internal coils in a stirred tank. A ﬂash can be built into the loop for mass transfer or to cool the reaction mass. At large scales, loop reactors are economically superior to stirred vessels and are preferred by sophisticated companies for high-volume processes. The third possible limitation is mixing time. As the scale increases, it becomes increasingly diﬃcult to mix fresh feed (whether in a CSTR or a fed-batch reactor) with the contents in the vessel. In the example of S ¼ 512, constant power per unit volume causes the mixing time to increase by a factor of 4. Does this hurt the product? Test for this by decreasing the agitator speed in the pilot plant to give a mixing time that can reasonably be achieved upon scaleup. Diplomatic scaleup may be possible. If fast mixing times are really required, multiple injection points can be used. Loop reactors do not solve the mixing time problem as they are subject to exactly the same limitations of impeller speed, power, and mixing time. However, they do allow the dilution of feed at a controlled rate. Also, faster mixing times achieved through the use of multiple injection points are easier to implement in a loop reactor.

10.7 Design Considerations for Stirred Tanks

The real question in the scaleup of a stirred-tank reactor is simply whether the contents of the vessel will remain reasonably well mixed upon scaleup. If there is good compositional homogeneity, then there will be thermal homogeneity as well, since thermal diﬀusivities are typically two orders of magnitude higher than molecular diﬀusivities. The absolute value of the temperature in the larger vessel can be regulated by scalable techniques such as boiling, cold feed, or external heat transfer if wall cooling is inadequate.

10.7

Design Considerations for Stirred Tanks

Removal of the heat of reaction is the critical factor in vinyl polymerizations. It is less important for ring-opening polymerizations, such as for Nylon 6, and is unimportant for condensation polymerizations, such as for PET and polycarbonate. The removal of heat through the walls of a stirred tank or by internal cooling coils is diﬃcult to scale and not recommended. Scaleup of even a moderately exothermic reaction can lead to a vessel half-full of cooling coils for which the normal correlations for heat transfer are overly optimistic. Scalable techniques include cold feed, reﬂux (autorefrigeration) of monomer or solvent, use of diluents, low per pass conversions, and external heat exchange in shell-and-tube reactors. Reﬂux reactors remove the heat of polymerization through the heat of vaporization of monomer or solvent. Boiling by homogeneous nucleation scales directly with volume. Vapor bubbles break the free surface of the reactor. The vapor is condensed and returned directly to the reactor or is recycled to the feed. Concerns about boiling reactors include polymerization on the dome of the reactor and fouling of the condensate heat exchanger. The issue is simple. If the polymer is soluble in the monomer or solvent mixture as it condenses, there is little problem. This fact seems self-evident, although it took 30-odd years for a company otherwise noted for its engineering skills to be shamed into recognizing it by a European subsidiary. If the polymer in insoluble in the condensate, fouling can be expected. PET ﬁnishing reactors (which are more like tubes than stirred tanks) exemplify this problem, and 30-plus years of research have failed to yield a good solution. Perhaps the best solution is a hot spare that can be switched into duty as the other condenser is cleaned. Another limitation of a boiling reactor is the eﬀect of pressure on boiling point. A very tall reactor will have higher temperatures at the bottom that will typically give lower molecular weight polymer and higher polydispersity in the product. Field erection of stirred tanks is much more expensive than shop erection, and the maximum size of a stirred-tank reactor may be limited by the size that can be shipped. Thus plants immediately adjacent to water may be able to have larger reactors than plants accessed only by rail or road. Rail and road shipments limit reactor sizes to about 100 m 3 . Much larger vessels are ﬁeld-erected, but these designs are usually more suited as low-pressure storage tanks or fermentation reactors than as polymerization reactors that are typically rated for 10–20 atm pressure.

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10.8

Multiphase Stirred Tanks

Polymerization reactors are occasionally multiphase. The boiling reactor is an example where scaleup has been easy because homogeneous nucleation depends on temperature and pressure but is otherwise independent of scale. As discussed above, potential limits exist for very large vessels, but these have not yet been encountered. The precipitation of rubber particles in a HIPS process is another example where the phase separation mechanism, that is, spinodal decomposition, is homogeneous and thus potentially independent of scale. The local concentrations that drive spinodal decomposition are due to a combination of chemical reaction and mass transfer between the entering feed and the vessel contents. A scaleup limitation could occur, for example if the mixing time in the CSTR became too large, but some compensation is possible by slightly altering conversion in the reactor. As a practical matter, vessels diﬀering by more than 25 000-fold in volume have achieved substantially the same results in HIPS polymerizations. Sparged reactor vessels present a greater problem for scaleup, although sparging is not very common in continuous polymerization reactors. The European targeted research activity on polymer materials (TRA-PM) has addressed the scaleup of multiphase reactions with the comment: ‘‘A newer integrated, but still semi-empirical approach uses experience-based equipment selection, laboratory experiments generating starting parameters for CFD calculation, numerical calculations using advanced CFD tools, pilot scale experiments to verify CFD calculations and ﬁnal design and scale-up with CFD.’’ This approach may become possible as CFD codes become better able to solve the convective diﬀusion equations for heat and mass transfer in the presence of a chemical reaction and in addition to their more established but not completely perfect role in the prediction of the ﬂuid mechanics. At the moment, this procedure lacks validation even for single-phase stirred-tank reactors.

10.9

Stirred Tanks in Series

One way to double the output of a stirred tank process is to increase each linear dimension by 2 1/3 ¼ 1:26, but this may lead to heat transfer or mixing limitations. Another way is to add a stirred tank in parallel, but better results and a more ﬂexible process are usually achieved by adding a stirred tank in series. Some polystyrene processes use ﬁve stirred-tank and stirred-tube reactors in series. Some polypropylene processes use two or three loop and stirred-tank reactors in series. Modern ﬂuidized-bed processes for polypropylene use two beds in series. One advantage of using tanks in series is the improvement in residence time distribution from the exponential distribution of a single stirred tank toward that of piston (plug) ﬂow. The series arrangement leads to higher conversion and thus eases the burden of downstream recovery and recycling. The usual case is for two

10.10 Tubular Reactors

reactors in series to produce more than two reactors in parallel, although operation at very high conversions (for example, 70% for styrene) may be an exception because of the gel eﬀect. A major advantage of using tanks in series is the ability to have a diﬀerent reaction environment in the various tanks. This advantage is especially useful in polypropylene processes where the last reactor adds ethylene for impact modiﬁcation.

10.10

Tubular Reactors

Scaling tubes in parallel is the standard method for low-viscosity ﬂuids when the reaction exotherm is signiﬁcant, and particularly for gas-phase reactions where the pressure drop is important. Shell-and-tube designs allow tubular polymerization reactors to be scaled in parallel, subject to the limitation of distributing the feed across the tube bundle. Distribution becomes a problem when the viscosity of the feed is much less than the viscosity of the product. This viscous ﬁngering problem that arises in secondary oil recovery is also a problem in multitubular polymerization reactors. The relatively inviscid material entering cannot displace polymerized or devolatilized material that may form in the tubes. The result can be many tubes that are completely plugged, with a few tubes gushing unreacted feed. Static mixers cannot eliminate this problem since they too oﬀer parallel paths for ﬂow, some of which can plug. The known, practical solution is to limit per pass conversion so that the outlet viscosity is within about an order of magnitude of the inlet viscosity. Polystyrene post-reactors and devolatilization preheaters function satisfactorily when the entering feed is about 70% polymer. Solution HDPE processes function satisfactorily when the per pass conversion is limited to about 15%. Multitubular devolatilization preheaters can function with entering polymer concentrations below 15% by using a motor-controlled, backpressure valve that prevents ﬂashing in the tube bundle. When scaling in parallel, the number of tubes scales directly with the inventory scaleup factor, S. Factors for other forms of scaleup for incompressible ﬂuids are given in Table 10.2. This table includes three geometric scaleup factors. They are for volume, S, radius SR , and length, SL . They are related by Eq. (4) so that only two are arbitrarily adjustable. S ¼ SL SR2

ð4Þ

Thus the general scaling equation for pressure drop in a single tube in laminar ﬂow can be given in several equivalent forms [Eqs. (5)]. 2 6 SDP ¼ ST S4 R SL ¼ ST S SL ¼ ST SSR

ð5Þ

The results in Table 10.2 emphasize the constraint of Eq. (4) by eliminating SL from the scaleup factors in the column that speciﬁes ST ¼ S.

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10 Scaleup of Polymerization Processes Tab. 10.2.

Scaleup factors for tubes.

Flow regime

Laminar diameter scaling factor Length scaling factor Reynolds number Length-to-diameter ratio Pressure scaling factor, DP Laminar with Gz < 75 Inside coeﬃcient, h Coeﬃcient area, hA ext Driving force, DT Laminar with Gz > 75 Inside coeﬃcient, h Coeﬃcient area, hA ext Driving force, DT Turbulent Diameter scaling factor Length scaling factor Length-to-diameter ratio Reynolds number Pressure scaling factor, DP Heat transfer area, A ext Inside coeﬃcient, h Coeﬃcient area, hA ext Driving force, DT

General scaleup factors

General factor for ST F S

Series scaleup

Geometric similarity

Constant pressure scaleup

SR SL ST S1 R SL S1 R ST S4 R SL

SR SS2 R SS1 R SS3 R S 2 S6 R

1 S SL S S2

S 1/3 S 1/3 S 2/3 1 1

S 1/3 S 1/3 S 2/3 1 1

S1 R SL ST S1 L

S1 R SS2 R SR2

1 S 1

S1/3 S 1/3 S 2/3

S1/3 S 1/3 S 2/3

1/3

1/3

1/3

ST S1 R SL 1/3 2/3 ST SL 2/3 2/3 ST SL

SR 4/3 SSR 4/3 SR

1 S 1

S1/9 S 5/9 S 4/9

S1/9 S 5/9 S 4/9

SR SL SL S1 R ST S1 R 1:75 4:75 ST SR SL SS1 R ST0:8 SR1:8 ST0:8 SR0:8 SL ST0:2 SR0:8 S1 L

SR SS2 R SS3 R SS1 R 2:75 6:75 S SR SS1 R S 0:8 S1:8 R 1:8 2:8 19 S SR S0:8 SR2:8

1 S S SL S 2:75 S S 0:8 S 1:8 S0:8

S 1/3 S 1/3 1 S 2/3 S 1/2 S 2/3 S 0:2 S 0:87 S 0:13

S 11/27 S 5/27 S2/9 S 0:59 1 S 0:59 S 0:07 S 0:66 S 0:34

The general scaleup factor for the Reynolds number is given by Eq. (6). ðd t uÞLarge ReLarge ¼ ¼ ST S1 R ReSmall ðd t uÞSmall

ð6Þ

Scaleup at constant Reynolds number, with ST ¼ S, would require SS1 R ¼ 1, so SR ¼ S and SL ¼ S1 . The scaled-up reactor would have to be fatter but shorter, which is a strange form of scaleup. For scaleups other than in parallel, the Reynolds number will increase, although typically not by enough to cause turbulence or even transitional ﬂow in polymer systems. Scaling a tube in series is conservative with respect to heat transfer. The necessary wall driving force remains constant in laminar ﬂow and decreases in turbulent ﬂow. The major diﬃculty is the pressure drop that scales as S 2 for laminar ﬂow and as S 2:75 in turbulent ﬂow. Polymer solutions are mildly shear thinning, so these factors somewhat overestimate the pressure drop. Still, the pressure drop in the pilot-scale reactor had better be small.

10.11 Static Mixers

Scaling a laminar-ﬂow, tubular reactor with geometric similarity has the interesting consequence of constant pressure drop. Thus, scaling with geometric similarity is great for adiabatic reactors. A constant pressure drop is obtained for gases as well as liquids. However, even though temperature eﬀects may be negligible, the reactor will have radial concentration gradients due to the laminar velocity proﬁle. Molecular diﬀusion will alleviate this proﬁle and usually helps the performance of a laminar-ﬂow reactor. The beneﬁcial eﬀects of diﬀusion will diminish upon scaleup. Geometrically similar scaleup of laminar ﬂow in tubes cannot be recommended unless radial diﬀusion was negligible in the pilot-scale reactor. Diﬀusion depends on the dimensionless group Dt/dt2 where D is the molecular diﬀusivity. If this group is less than about 0.001, then diﬀusion has little eﬀect except possibly for subtle reactions such as crosslinking. A geometrically similar scaleup in turbulent ﬂow gives a pressure drop that increases as S 1/2 . More energy goes into pumping. The pumping energy increases as ST S 1/2 and the pumping energy per unit throughput increases as S 1/2 . The extra energy causes more eddy diﬀusion and gives a better environment for reaction. Unlike molecular diﬀusion in a laminar-ﬂow system, the beneﬁt from eddy diﬀusion increases with size, but turbulence is unlikely when the continuous phase contains a high molecular weight polymer. For laminar ﬂow of a Newtonian ﬂuid, scaleup at constant pressure drop is identical to scaleup with geometric similarity. For constant pressure drop in turbulent ﬂow, the reactor diameter increases somewhat faster, S 11/27 , than for scaleup with geometric similarity. The reactor length increases somewhat more slowly, S 5/27 , so that the reactor becomes shorter and fatter and heat transfer becomes substantially more diﬃcult. Scaleup with constant heat transfer is theoretically possible with any form of scaleup, provided the wall driving force can be adjusted as shown in Table 10.2. As a practical matter, the required DT soon becomes excessive if signiﬁcant heating or cooling was needed in the pilot unit.

10.11

Static Mixers

Rather diﬀerent considerations apply when a tube containing a static mixer is scaled up. If the tube diameter increases upon scaleup, then the number of mixing elements should also increase, but the increase is minor. For a scaleup based on geometric similarity, the extra number of elements is given by Eq. (7). Nextra ¼

logðSÞ 3 logð2Þ

ð7Þ

This result assumes the static mixer is of the Kenics type that repeatedly divides the ﬂuid into two portions (that is, one that follows the 2 N model for striation thickness). See Ref. 5 for details and results for other types of static mixers. Equa-

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10 Scaleup of Polymerization Processes

tion (7) is a slowly increasing function of S; for example, an increase by a factor of 512 in throughput requires only three extra elements.

10.12

Design Considerations for Tubular Reactors

As discussed above, shell-and-tube reactors and heat exchangers can be employed in polymerization and recovery trains, provided the inlet and outlet streams have similar viscosities. Subject to this caveat, scaleup is simple and inexpensive. Static mixers are sometimes used to enhance heat exchange, but their beneﬁt is marginal compared to the alternative of using open tubes with a somewhat smaller diameter and greater length to give the same pressure drop. In this connection, there is no minimum tube diameter. Some companies have used 0.5-inch (12.5-mm) tubes in a variety of polymer processes since the 1960s. An insistence on large-diameter tubes can lead product property problems. Tubular reactors are rarely scaled in series because of pressure limitations. The great exception to this statement is the high-pressure process for LDPE, where reactor lengths are measured in kilometers. The diameter of the tube is strictly limited to about 5 cm to avoid thermal runaway. A few tubes in parallel, possibly with separate ethylene compressors, can be used to achieve greater capacity, but conventional shell-and-tube designs present a serious problem of tube-to-tube instability. Scaleup to large-diameter tubes is generally possible only when the adiabatic temperature rise is acceptable. A practical example of such scaleup is the adiabatic post-reactor that is sometimes added to polystyrene reaction trains as an inexpensive way of boosting capacity. Tubular pilot reactors should be run adiabatically; or if some removal of heat is needed, they should be run at the same value of the thermal diﬀusion group, aT t/dt2 where aT is the thermal diﬀusivity. This means running the pilot unit at the same value of t and d t but with a lower ﬂow rate and a shorter tube than expected in the scaled-up reactor. It is equivalent to scaling in series. This approach will also give the same value of the molecular diﬀusion group, Dt/dt2 , where D is the molecular diﬀusivity. Typical vinyl polymerizations, for example of ethylene, propylene, vinyl chloride, styrene, and methyl methacrylate, have adiabatic temperature rises between 200 and 1600 C. This leads to the possibility of parametric sensitivity and thermal runaway. Figure 10.2 demonstrates parametric sensitivity for styrene polymerization in a tube that is assumed to be well mixed in the radial direction, as might result from the use of static mixers (that is, the model consists of simultaneous, ordinary diﬀerential equations for composition and temperature). The dramatic transition from a controlled reaction to a thermal runaway corresponds to a mere 1 mm diﬀerence in tube diameter. Although inlet and wall temperatures can be varied, reaction in a large-diameter tube is not feasible for this and similarly energetic reactions absent an excellent control system [6]. A detailed model of this polymerization in an open tube (the model consists of simultaneous, partial diﬀerential equations for composition, temperature, and velocity) shows that hydrody-

10.12 Design Considerations for Tubular Reactors

500.0 0.062

Temperature, °C

400.0

300.0

0.061

200.0

100.0

0.060

0.0 0.00

0.60

1.20

1.80

2.40

3.00

3.60

Axial Position, m Temperature proﬁles for the polymerization of undiluted styrene in a tubular reactor with good radial mixing. Tin ¼ 135 C and Twall ¼ 20 C. The parameter in the plot is the tube diameter in meters. Fig. 10.2.

namic instabilities can arise in the absence of static mixers due to large viscosity diﬀerences between the wall and centerline. For the illustrated case of Tin ¼ 135 C and Twall ¼ 20 C, a hydrodynamic instability occurs at a smaller tube diameter than for a thermal runaway. Installation of static mixers merely postpones the problem until the marginally larger diameters shown in Figure 10.2 are reached. The overall conclusion is that single-tube polymerization from undiluted styrene is not feasible if the tube diameter exceeds about 0.06 m or if the conversion exceeds about 20%. Figure 10.3 illustrates the conversion and stability limits for open tubes (tubes without a static mixer). Scaling in series using a long tube appears to be feasible for styrene polymerization, as it is for ethylene polymerization. However, high pressures are not required for the styrene polymerization, and stirred-tube reactors or boiling stirred tanks have been preferred. Scaleup of polycondensation reactions involving multifunctional monomers (for example, phenol/formaldehyde) in tubular reactors has proven especially diﬃcult. Even though the overall stoichiometry (for example, formaldehyde at 75 mol% of the entering phenol concentration) is set to avoid crosslinking, locally crosslinked regions near the tube walls can result when the diﬀusion group, Dt/dt2 , is too large. The ultimate result is a plugged tube. This empirical observation has been conﬁrmed in modeling studies. A similar problem may exist in anionically catalyzed polymerizations although modeling studies and some experimental studies indicate that such polymerizations should be possible in tubular reactors.

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10 Scaleup of Polymerization Processes

Fig. 10.3. Polymerization of undiluted styrene in an open tube with radial mixing only by molecular diﬀusion. The mean residence time used for conversion calculations is 1 hour.

10.14 Casting Systems

10.13

Extruder and Extruder-like Reactors

Extruders are sometimes used for polymer modiﬁcation reactions such as grafting. As a general rule, scaleup is diﬃcult and expensive. The diﬃculty is primarily due to the very limited ability of extruders to exchange heat with an external source. Most production-scale extruders operate almost adiabatically, with energy input from the motor appearing as enthalpy of the product. This is in addition to any reaction exotherm. The expense is due to the very high cost of reaction volume in an extruder. As a practical matter, residence times are limited to a few minutes. Like stirred tank reactors, extruders are generally scaled using geometric similarity so that the ability to transfer heat decreases upon scaleup. Similarly, the screw speed must generally decrease, or at least not decrease, upon scaleup. Extruders operate in the laminar-ﬂow regime, and the scaleup factor for power, as a function of screw diameter, is less dramatic than in stirred tanks, particularly for shearthinning polymer melts. As a rough approximation, power varies as conveying capacity, NI D 3 , so that the enthalpy input to the polymer is approximately constant. However, extruders with a screw diameter of less than about 50 mm are generally considered unscalable because the ratio of external area to volume is too large to approximate to the nearly adiabatic operation expected upon scaleup. The ﬁnishing reactors used to drive polycondensation reactions (for example, for PET) to high molecular weights are extruder-like devices that generate surface area for the removal of the condensation byproduct (such as ethylene glycol). Conven1/2 tional devolatilization theory suggests they scale as NI D 2 .

10.14

Casting Systems

We consider here two kinds of casting systems, mold casting and casting onto a moving web. Reliable scaleup of mold casting requires an absence of polymerization byproducts, especially gaseous byproducts, and a manageable heat of reaction. The polyurethane formulations used for RIM (reactive injection molding) have no byproduct and a negligible heat of polymerization so that the molding process can be scaled to quite large parts. Similarly, the ring-opening polymerization of caprolactam has no byproducts and an adiabatic temperature rise of about 60 C that can be tolerated in the molding process. Again, quite large parts can be molded. Methyl methacrylate has a substantial heat of reaction so that a large casting of pure monomer would boil and ruin the part. Addition of the polymer, PMMA, to the casting mixture lowers the possible temperature rise. It also increases the viscosity so that the gel eﬀect is enhanced. In PMMA, the gel eﬀect is pronounced but the polymerization stops well short of complete conversion because the mixture solidiﬁes. The result is that an adiabatic casting of a PMMA/methyl methacrylate mixture never reaches the boiling point of the monomer and a defect-free part is possible, even a large one.

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Web casting can be used for polymerizations that do have reaction byproducts and large reaction exotherms. Examples of web-cast products include photographic ﬁlms and coated abrasives. Here, the desired product is obtained directly as a ﬁlm or sheet. Web casting can be used in belt-plus-ﬂaker operations where the web acts a temporary polymerization reactor. Web casting is scaled up at constant web thickness and at constant residence time on the web. Production increases are achieved by using wider webs or longer, higher-speed webs. The major issues for scaleup are primarily the mechanical and control systems need to achieve a desired and uniform ﬁlm thickness.

10.15

Concluding Remarks

The scaleup of polymerization processes is conceptually identical to the scaleup of ordinary chemical reactions. The principal diﬀerence is that polymer systems are more likely to be highly viscous and in laminar ﬂow. Although polymer melts can be markedly non-Newtonian, this is rarely a critical factor in the scaleup of polymer reaction and recovery systems. Vinyl polymerizations have strong exotherms so that parametric sensitivity and thermal runaways can be a problem, but so do many other chemical reactions. Condensation polymerizations tend to be equilibrium-limited, but so do many other chemical reactions. Laminar ﬂow decreases heat and mass transfer both to the environment and within the ﬂuid as molecular and thermal diﬀusion replaces eddy diﬀusion as the primary mechanism, and their beneﬁts diminish upon scaleup. The large viscosity diﬀerences possible during polymerizations can lead to hydrodynamic instabilities within a tube or between tubes. These problems are seldom found in ordinary chemical reactions and prevent the easy application of shell-and-tube designs. The net result is that scaling in series, either by increasing reactor length or by adding reactors in series, is more common in the polymer industry than in the ordinary chemical industry. The complications of laminar ﬂow can also be avoided by suspending the polymer phase in a continuous, low-viscosity phase. Examples include the gas-phase and slurry processes for polyoleﬁns as well as suspension and emulsion polymerizations. Despite the complications that sometimes result from laminar ﬂow, history has shown that polymer processes can be scaled up with the same technical certainty and the same good economics as for ordinary chemicals. If anything, the gains in single-train production rates have been higher in polymers than in their associated monomers.

Notation

A ext dt

external area available for heat transfer tube diameter

References

D D Gz H L N NI Q Re S SL SR ST t T DT u V

impeller or extruder screw diameter molecular diﬀusivity Graetz number, dt2 u/ðaT LÞ inside heat transfer coeﬃcient tube length number of elements in a static mixer rotational velocity of impeller (rotations per unit time) volumetric ﬂow rate Reynolds number, rd t u/m or rNI D 2 /m volume or inventory scaling factor length scaling factor radius scaling factor throughput scaling factor mean residence time temperature temperature driving force mean velocity volume

Greek m r r^

viscosity density volume-average density

References 1 Nauman, E. B., Chemical Reactor

Design, Optimization and Scaleup, McGraw-Hill, NewYork, 2002. 2 Bisio, A., Kabel, R., Editors, Scaleup in the Chemical Process Industries, Wiley, New York, 1985. 3 McGreavy, C., Editor, Polymer Reactor Engineering, Blackie, London, 1994. 4 Paul, E. L., Atiemo-Obeng, V. A., Kresta, S. M., Editors, Handbook of

Industrial Mixing, Wiley, New York, 2004. 5 Thakur, R. K., Vial, Ch., Nigam, K. D. P., Nauman, E. B., Djelveh, G., ‘‘Static mixers in the process industries’’, Chem. Eng. Res. Des., 81, 787–826 (2003). 6 Mallikarjun, R., Nauman, E. B., ‘‘Optimal processes for crystal polystyrene’’, Polym. Plast. Tech. Eng., 28, 137–149 (1989).

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Safety of Polymerization Processes1 Francis Stoessel 11.1

Introduction

Polymerization – that is, polycondensation and polyaddition – performed on an industrial scale presents a number of speciﬁc reaction and process engineering aspects which diﬀerentiates these processes from reactions of low molecular weight molecules. This is also true for aspects of the dynamic control of polymerization reactors. Therefore the concepts developed for low-molecular-weight chemistry must be adapted to the speciﬁc problems of polymerization reactions. The scope of the present chapter is the assessment of risks linked with the performance of polymerization reactions on an industrial scale. Moreover, the focus is on the control of the course of reaction by means of chemical reaction engineering [1]. Risks linked with handling of raw material or products are not treated in this chapter. Part of the chapter is based on a comprehensive text by Moritz [2], who is acknowledged for his authorization to use it here. Barton and Nolan [3] and later Maddison and Rogers [4] analyzed the causes of chemical runaway incidents based on statistics from the British Health and Safety Executive. A classiﬁcation by reaction type (Figure 11.1) reveals how prominent polymerizations are. From 134 cases where the causes could be identiﬁed, 64 were due to polymerization reactions, including polycondensations and polyadditions. From these 64 polymerization incidents, 13 (20%) involved phenol–formaldehyde resin production. This led the British Plastics Federation (BPF) to publish a booklet, Guidance for the Safe Production of Phenolic Resins [5]. A further investigation among more recent incidents involving polymer reactions showed that most of the incidents occur in vinyl chloride, vinyl acetate, and polyester resins. Since polymerization is the type of reaction where most runaway situations occur, it is worth devoting special attention to process safety. Understanding how a reaction may leave its ‘‘normal’’ course and enter a runaway course is essential for the design of safe and economic processes. This behavior has its roots in the thermodynamics 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

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11 Safety of Polymerization Processes

Esterification Oxidation Diazotization Amination Alklation Halogenation Salt formation Hydrolysis Sulpfonation Nitration Polymerization 0 Fig. 11.1.

10

20

30

40

50

60

70

Number of runaway incidents, classiﬁed by reaction type.

and chemical engineering aspects of the reactions. This will be explained in the present chapter. In Section 11.2, general principles of reactor safety and heat balance of reactors are presented, with an emphasis on speciﬁc aspects of polymerizations. Section 11.3 is devoted to safety-related thermodynamics and reaction engineering aspects of polymerization reactions. In Section 11.4, cooling of polymerization reactors is reviewed. The chapter is concluded by a section describing safety aspects of industrial processes, together with technical risk-reducing solutions.

11.2

Principles of Chemical Reactor Safety Applied to Polymerization

Accidents happening in polymerization reactors are practically always due to a lack of control of the course of reaction caused by a disturbance of the heat balance, which results in a temperature increase leading to loss of control of the reactor and a runaway reaction. In this section a systematic procedure based on a failure scenario with six key questions, allowing assessment of the criticality of a process, is presented. Since the heat balance is at the center of our concerns in matters of thermal control of reactors, the diﬀerent terms of the heat balance will be examined. Finally, aspects of the dynamic stability of reactors and of the thermal stability of reaction masses are analyzed. 11.2.1

Cooling Failure Scenario

A common practice in the assessment of risks due to runaway reactions is to use the cooling failure scenario as developed by Gygax [1, 6]. This is a worst-case as-

11.2 Principles of Chemical Reactor Safety Applied to Polymerization

Temperature Tend Tad

Decomposition on Reaction

3

MTSR

Main i Reaction n

Reacti

TMRad

∆T ad 2

6

5

Tp 1 Normal Process

4

Cooling r l n Failure Time

Fig. 11.2. Cooling failure scenario, presenting the consequences to the desired reaction of loss of cooling and triggering of a secondary decomposition reaction. The numbers represent the key questions used in the assessment of thermal risks (see text).

sumption that is useful for the risk assessment. It is assumed that, while the reactor is at reaction temperature, a cooling failure occurs. If at this instant unconverted monomer is still present in the reactor, the temperature will increase due to the completion of the reaction. This temperature increase is proportional to the amount of nonreacted material. At the temperature reached at the end of this period, a secondary decomposition reaction may be triggered and the heat produced by this reaction may lead to a further increase in temperature. The following questions help to develop the runaway scenario and to determine the data required for the risk assessment. This scenario (Figure 11.2) was developed for chemical reactions in general and is well suited to polymerization reactions. 1. Can the process temperature be controlled by the cooling system? This is a typical question which should be answered during process development. To ensure thermal control of the reaction, the power of the cooling system must be suﬃcient to dissipate the heat produced in the reactor at any time. Special attention must be devoted to the strong change in the viscosity of the reaction mass during polymerization and to possible fouling at the reactor wall (see Section 11.4). An additional condition, which must be fulﬁlled, is that the reactor is operated in the dynamic stability region as described in Section 11.2.4. 2. What temperature can be attained after runaway of the desired reaction? The

555

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11 Safety of Polymerization Processes

3.

4.

5.

6.

answer to this question necessitates a study of the kinetics of the reaction, in order to determine the degree of accumulation of monomer in the reactor as a function of time. The concept of MTSR (maximum temperature of the synthesis reaction) was developed for this purpose. In the case of polymerization reactions, the accumulation of monomer is not the only factor which must be known; the number of living chains must also be considered. What temperature can be attained after runaway by decomposition? The thermal data of the secondary decomposition reactions allow calculation of the adiabatic temperature rise and determination of the ﬁnal temperature, starting from the level of the MTSR. This temperature gives a direct indication of the possible consequences of a runaway. In polymerization reactions, the temperature reached after loss of control of the reaction itself will often determine the consequences. Polymerization reaction masses are often thermally stable, so this question is often not relevant. Exceptions are polycondensation reactions, especially hot melts or reactive resins. At which moment does the cooling failure have the worst consequences? Since the amount of unconverted reactants and the thermal stability of the reaction mass may vary with time, it is important to know at which instant the accumulation, and therefore the thermal potential, is highest. This will be the worst case, and obviously the safety measures have to account for it. How fast is the runaway of the desired reaction? Generally, industrial reactors are operated at temperatures where the desired reaction is fast. Hence, a temperature increase above the normal process temperature will cause a signiﬁcant acceleration of the reaction: therefore, in most cases, this period of time is short. For polymerization reactions, where decomposition of the reaction mass is not critical, this time will determine the choice of technical risk reduction measures. The concept of time to maximum rate under adiabatic conditions (TMR ad ) as used for decomposition reactions can be applied to the polymerization itself, starting from the process temperature. It allows estimation of the probability of entering a runaway situation, as explained below for decomposition reactions. How fast is the runaway of the decomposition starting at the MTSR? The dynamics of the decomposition reaction play an important role in the determination of the probability of an incident. The TMR ad concept was developed for that purpose [7, 8]. The principles of using the time to explosion (TMR ad ) is that the longer the time to explosion, the greater is the chance of recovering the situation. This means that for a longer time to explosion, the probability of triggering the runaway is lower and, conversely, for a shorter TMR ad this probability is higher.

The questions mentioned in this scenario can be answered using the results of calorimetric experiments which can be directly used in the determination of the characteristic temperature levels: after a cooling failure they will give us ﬁrst the temperature due to the runaway of the desired reaction, and then the temperature reached after the runaway of the decomposition reaction (Tend ).

11.2 Principles of Chemical Reactor Safety Applied to Polymerization

11.2.2

Criticality Classes Applied to Polymerization Reactors

For reactions presenting a thermal potential we can consider the relative position of four temperature levels [9]:

The process temperature (TProcess ) is the initial temperature in the cooling failure scenario. In the case of a non-isothermal process, the initial temperature will be taken at the instant when the cooling failure has the heaviest consequences (worst case) according to Question 4 in the scenario, above. The maximum temperature of synthesis reaction (MTSR) depends essentially on the degree of accumulation of unconverted monomer and catalytic activity. Therefore it is strongly dependent on the process design. In polymerization processes, this temperature is often high enough to become critical. The maximum temperature for technical reasons (MTT) represents the technical limit of the equipment. In an open system, as for solution and emulsion polymerizations, it can be the boiling point. For a closed system, it is the temperature at which the pressure reaches the maximum permissible, that is, the set pressure of a safety valve or bursting disk. This could be the case for mass polymerization. This temperature (MTT) is often the critical factor for polymerization reactions. The temperature at which the TMR ad is 24 h (TD24 ): this temperature is deﬁned by the thermal stability of the reaction mixture. It is the highest temperature at which the thermal stability of the reaction mass is unproblematic. This is often a secondary factor since polymerization masses are usually thermally stable. These four temperature levels allow division of the scenarios into ﬁve diﬀerent classes, going from the least critical (1) to the most critical (Figure 11.3).

Temperature MTT

MTT MTSR

MTSR

TD24 MTT MTSR

MTT MTSR

MTSR MTT

TP Class

1

2

3

4

5 Criticality lt

Fig. 11.3.

Criticality classes of failure scenarios.

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11 Safety of Polymerization Processes

11.2.2.1

Description of the Criticality Classes

Class 1: After loss of control of the synthesis reaction, the MTT cannot be reached and the decomposition reaction cannot be triggered. Only if the reaction mass was maintained for a long time under heat accumulation conditions, could the MTT be reached. Then the evaporative cooling may serve as an additional safety barrier. The process is thermally safe. Class 2: After loss of control of the synthesis reaction, the MTT cannot be reached and the decomposition reaction cannot be triggered. The situation is very similar to Class 1, but if the reaction mass is maintained for a longer time under heat accumulation conditions, the decomposition reaction could be triggered and reach the MTT. In this case, reaching this temperature could be a hazard if the heat release rate at the MTT is too high with respect to vaporization or pressure rise. If the reaction mass is not kept for a longer time under heat accumulation conditions, the process is thermally safe. Class 3: After loss of control of the synthesis reaction, the MTT will be reached, but the decomposition reaction cannot be triggered. In this situation, the safety of the process depends on the heat release rate of the synthesis reaction and on the pressure, respectively, at the MTT. Class 4: After loss of control of the synthesis reaction, the MTT will be reached and the decomposition reaction could theoretically be triggered. In this situation, the safety of the process depends on the heat release rate of both the synthesis reaction and the decomposition reaction at the MTT. Evaporative cooling or the emergency pressure relief may serve as a safety barrier. Class 5: After loss of control of the synthesis reaction, the decomposition reaction will be triggered and the MTT will be reached during the runaway of the decomposition reaction. It is very unlikely that the evaporative cooling or the emergency pressure relief can serve as a safety barrier in this case. The heat release rate of the decomposition at the MTT determines the thermal safety of the process. This is the most critical of all the scenarios.

For polymerization reactions the relative positions of the MTSR and the MTT are dominant: decomposition reactions play a role only in speciﬁc cases, such as resins. Thus the most important classes are 1 to 3, whereas the consequences of exceeding the MTT level may be dramatic. These criticality classes are very useful in the decision making process of choosing the protection strategy for a reactor. Depending on the criticality class of the scenario, diﬀerent measures can be applied to avoid, to control, or to stop the runaway:

Class 1: No special measure is required for this class of scenario. The evaporative cooling or the emergency pressure relief could serve as a barrier. Class 2: No special measure is required. The reaction mass should not be held for longer time under heat accumulation conditions. The evaporative cooling or the emergency pressure relief could eventually serve as a barrier.

11.2 Principles of Chemical Reactor Safety Applied to Polymerization

Class 3: The reaction medium temperature exceeds the technical possibilities of the reactor. Thus technical measures are required. In a ﬁrst approach one should try to avoid conditions in which a runaway situation develops. If this cannot be ensured, then emergency measures are required. Class 4: Similar to Class 3. The same measures apply here but the additional heat release rate due to the secondary reaction has also to be taken into account. Class 5: In this class the MTT is very unlikely to serve as a safety barrier. Therefore only emergency measures can be used. Since in most cases the decomposition reactions release very high energies, particular attention has to be paid to the design of safety measures. It is worthwhile to consider an alternative design of the process in order to reduce the severity, or at least the probability, of triggering a runaway reaction. 11.2.3

Heat Balance of Reactors

The heat balance is important as well for the design of reactors, for their scaleup, for the risk assessment, and especially for the assessment of the reactor stability. The heat balance is also at the center of the evaluation of calorimetric experiments as used for safety studies. Thus understanding the heat balance of a reactor is essential for the design of safe processes. Hereafter the diﬀerent contributions to the heat balance, such as the heat release rate of the reaction, the heat exchange at the wall of the reactor, the heat dissipated by the stirrer, the heat accumulation in the reactor, the eﬀects of the sensible heat of the feed, and the heat losses, will be discussed in detail. The diﬀerent terms of the heat balance are expressed as heat release rates or thermal power. Heat Production The heat production corresponds to the rate of heat release by the reaction. Therefore it is proportional to the reaction enthalpy and to the reaction rate [Eq. (1)]. 11.2.3.1

qRX ¼ rA ðDHR Þ V

ð1Þ

This term will be of primary importance with respect to reactor safety: mastering the heat release by the reaction is the key to reactor safety. For a single reaction of order n, the reaction rate can be expressed as Equation (2).

rA ¼ k 0 exp

EA CAn RT

ð2Þ

Two features of these expressions are important for safety purposes: the heat release rate of a reaction is an exponential function of temperature. Secondly the heat release rate is proportional to the volume of the reaction mass. Therefore it will vary with the cube of the linear dimension of the vessel (L 3 ) containing the

559

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11 Safety of Polymerization Processes

reacting mass. It can also be expressed in a form allowing enhancement of the conversion rate [Eq. (3)].

qRX ¼ NA0 ðDHR Þ

dX dt

ð3Þ

Consequently, the heat of reaction is obtained by integration, which allows one to express the conversion as thermal conversion in Equation (4). ðt 0 Xth ¼ ð y

ðt

qRX dt ¼

qRX dt

0

qRX dt

Q RX

ð4Þ

0

The thermal conversion is often used in the kinetic evaluation of calorimetric experiments. It is also very useful for safety assessments, since it gives direct information on the accumulation of reactants that could react even after a cooling failure. The thermal conversion is used for the calculation of the MTSR. Heat Exchange There are several mechanisms for heat exchange between a reacting medium and a heat carrier: radiation, conduction, and forced or natural convection. Here we shall consider convection only. The heat exchanged with a heat carrier across the reactor wall by forced convection is proportional to the heat exchange area and to the driving force, that is, the temperature diﬀerence between the reaction medium and the heat carrier. The proportionality coeﬃcient is the overall heat transfer coeﬃcient [Eq. (5)] 11.2.3.2

qEX ¼ U A ðT TC Þ

ð5Þ

In the case of signiﬁcant change in the physical chemical properties of the reaction mixture, which is quite common with polymerization reactions, the overall heat exchange coeﬃcient U will also become a function of time. This is essentially due to the viscosity change during polymerization and to possible fouling at the reactor wall, which may sometimes become important. With respect to safety, two important features must be considered here: the heat removal is a linear function of temperature and since it is proportional to the heat exchange area, it varies as the square of the linear dimension of the equipment (L 2 ). This means that when the dimensions of a reactor have to be changed, as for scaleup, the heat removal capacity increases more slowly than the heat production rate. Therefore the heat balance becomes more critical for larger reactors. The contribution of heat exchange to the heat balance is examined in more detail in Section 11.4.1.

11.2 Principles of Chemical Reactor Safety Applied to Polymerization

Heat Accumulation Heat accumulation represents the variation of the energy content of a system with temperature [Eq. (6)]. 11.2.3.3

qAC ¼ MR cp

dT dt

ð6Þ

This includes all system compounds, the reaction mass as well as the reactor itself, or at least the parts directly in contact with the reacting system. However, in safety assessments, and as a worst case assumption, the heat capacity of the reactor itself is often neglected. Since heat accumulation is the consequence of a diﬀerence between heat production rate and cooling rate, it results in a variation of the temperature of the reactor contents. Convective Heat Transport due to Feed If a feed stream to a reactor is at a diﬀerent temperature from the contents of the reactor, there is convective heat transport to the reactor. The thermal eﬀect of the feed stream must be accounted for in the heat balance [Eq. (7)]. 11.2.3.4

qFeed ¼ m_ Feed cp; Feed ðTFeed TÞ

ð7Þ

This eﬀect is also called ‘‘sensible heat’’. It can also be used as an additional means of cooling in semi-batch and in continuous reactors. When the temperature diﬀerence between reactor and feed is important and/or the feed rate is high, this term may play an important role. In such cases, when the feed is stopped an abrupt increase in the reactor temperature may result. This term is also important in calorimetric measurements, where the appropriate correction must be performed. Stirrer The mechanical energy dissipated by the agitator is converted into viscous friction energy and ﬁnally altered into thermal energy. In most cases this term may be neglected when compared to the heat released by a chemical reaction. But with viscous reaction masses, as for example with polymerization reactions, this term must be integrated in the heat balance. It can be estimated from Equation (8). 11.2.3.5

qS ¼ Ne r n 3 dS5

ð8Þ

Computation of the thermal energy dissipated by a stirrer requires knowledge of the Newton number, also called the power number, which depends on the stirrer type and the ﬂow regime characterized by the Reynolds number. The contribution of the stirrer power to the heat balance is examined in more detail in Section 4.3.

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11 Safety of Polymerization Processes

Heat Losses Industrial reactors are thermally insulated for safety reasons (hot surfaces) and for economic reasons (heat losses). Nevertheless, at higher temperatures heat losses may become important. Their calculation may become tedious, since heat losses are often due to a combination of losses by radiation and by natural convection. If estimation is required, a simpliﬁed expression using a global overall heat transfer coeﬃcient may be useful [Eq. (9)]. 11.2.3.6

qLoss ¼ h ðT TAmb Þ

ð9Þ

The simplest way of estimating the overall heat transfer coeﬃcient (h) is by direct measurement at plant scale. Simpliﬁed Expression of the Heat Balance An overall heat balance taking all the terms explained above into account can be established [Eq. (10)]. 11.2.3.7

qAC ¼ qRX þ qFeed þ qS qEX qLoss

ð10Þ

But in most cases a simpliﬁed heat balance, which comprises only the most important terms on the right-hand side of Equation (10), is suﬃcient for safety purposes. Such an expression will be used in Section 11.2.4. The problems linked with the cooling of polymerization reactors are presented in detail in Section 11.4. 11.2.4

Dynamic Control of Reactors

In fact a chemical reactor is governed simultaneously by its heat balance and its mass balance. If we consider a single reaction of order n, we obtain a set of equations [Eqs. (11)]. 8 dT dXA U A ðT TC Þ > > ¼ DTad > < dt M R cp dt > > dX EA > n : A ¼ k 0 exp CA0 ð1 XA Þ n dt RT

ð11Þ

In this system of coupled diﬀerential equations, the mass balance corresponds to the reaction rate and the heat balance is a simpliﬁed version showing only the heat production by the reaction and the heat removal by the cooling system, both terms resulting in heat accumulation. This system presents the property of parametric sensitivity, meaning that a small change in one of the parameters may lead to dramatic changes in the solution of the system of equations, that is, in the behavior of the reactor. This is an old [10–12], but always real, problem [13, 14]. This behavior may be observed for batch reactors and for tubular reactors (plug ﬂow reactors) and also for bed reactors [15, 16]. Calorimetric methods make it possible to

11.2 Principles of Chemical Reactor Safety Applied to Polymerization

1

X

0.8

C

0.6 B

0.4 0.2

A

T(°C)

0 0

50

100

150

200

Multiplicity of solutions for an exothermal reaction performed in an adiabatic CSTR. The straight line represents the heat balance and the S-shaped line the mass balance. A: cold branch operating point. B: instable operating point. C: hot branch operating point. Fig. 11.4.

determine the data used to assess the dynamic stability of a reactor as a function of the production scale [17–19]. When performing this type of assessment, it is important to consider also variations in the overall heat transfer coeﬃcient and in the temperature of the cooling system, which may vary with time or even with season. A further speciﬁc aspect of the reactor heat balance is the multiplicity of solutions to the system of equations. This situation may arise with a CSTR in which an exothermal reaction is performed. The mass balance [Eq. (12)] is coupled with the heat balance [Eq. (13)], which gives a system of equations [Eq. (14)] that is represented graphically in Figure 11.4. FA0 XA ¼ rA V

ð12Þ

qRX ¼ rA V ðDHR Þ ¼ FA0 cp ðTR T0 Þ

ð13Þ

XA ¼

cp 1 ðTR T0 Þ ¼ ðTR T0 Þ DTad DHR

ð14Þ

In such cases, the temperature of the reactor may jump from the low-temperature solution to the high-temperature solution, which can be catastrophic if the reactor is not designed to be operated under these conditions. This problem was extensively studied by Ray and co-workers [20]. 11.2.5

Thermal Stability of Polymerization Reaction Masses

Since polymerizations are very exothermal, the potential temperature increase in the case of loss of control of the reaction itself is often great enough to cause evaporation of some components of the reaction mixture or even a partial cracking of

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11 Safety of Polymerization Processes

the polymer chains. Hence the pressure of the system may increase, and cause the rupture of the vessel and consequently heavy damage to the plant and/or its environment. In most cases the thermal stability of polymerization reaction masses is not critical by itself: the main risk stems from the intended polymerization reaction. In polyaddition reactions as used for the production of resins, and especially of hot melts, the reaction mass remains reactive even after the end of the polymerization itself. Often curing is followed by a decomposition of the products that is accompanied by large gas releases, which means that decomposition may easily result in the explosion of the vessel containing the resin. This kind of product may present critical situations in the case of storage or transportation in large amounts. This problem goes beyond the scope of this handbook, however.

11.3

Speciﬁc Safety Aspects of Polymerization Reactions

Polymerization reactions form a well deﬁned and speciﬁc class of reactions. They present some special features linked to the reaction kinetics and also to the thermodynamic aspects. These topics are presented in this section together with the factors that may aﬀect the reaction rate. 11.3.1

Kinetic Aspects

The reactions taking place during the synthesis of a polymer are rather complex in nature. The description of the chemistry of a polymerization reaction often involves over 20 diﬀerent elementary reactions. This means that control of the overall reaction rate that governs the process safety may be rather complicated. Nevertheless the kinetically determining step in polymerization reactions is the chain growth reaction. With respect to chain growth, two main types of reaction may be distinguished (Table 11.1). One type is the addition of one monomer molecule at a time at the

Tab. 11.1.

Polymerization mechanisms.

Type of chain growth [a]

Reaction steps

Terminology for reaction

Monomer addition Pn þ M ! Pnþ1

initiation growth termination transfer growth

polymerization (radical, ionic, coordination)

Polymer addition Pn þ Pm T Pmþn þ X [a] P :

n active monomer molecule with chain length n; M: monomer; Pn : bifunctional polymer molecule with length n; X: low molecular condensation product.

polycondensation polyaddition

11.3 Speciﬁc Safety Aspects of Polymerization Reactions

end of a growing chain until the chain is terminated. This is called a chain growth polymerization. The active center may be a free radical, an ion, or an available coordination bond on a transition metal complex of the catalyst. Once the active center is created, the growth of the chain is very fast, which results in a very short living time for the active chain – in the order of seconds – until it is deactivated by termination. Thus the concentration of active chains is extremely low, at approximately 108 mol L1 . This renders these polymerizations very sensitive to impurities. After termination a dead polymer chain remains, but in the case of free-radical polymerization the dead chain may react again with further radicals. The other possible growth mechanism involves two molecules carrying reactive groups, which react together leading to the addition of monomers, oligomers, or polymers. In this case the growth is achieved by the addition of longer elements than in chain growth. This type of reaction is called step growth polymerization, polyaddition, or polycondensation. With this mechanism, the growth of the chain proceeds more slowly, so the lifetime of the active chain and therefore the polymerization time are longer. These are often equilibrated reactions, so that the low molecular by-products must be removed from the reaction mass in order to shift the equilibrium toward the products. These diﬀerent chain growth mechanisms result also in diﬀerent activation energies. These are in the order of magnitude of only 20 kJ mol1 for chain growth polymerizations. But polycondensations and polyaddition present high activation energies in the order of 100 kJ mol1 . Consequently the reaction temperatures are very diﬀerent: whereas polymerizations are often performed at modest temperature levels from 50 to 100 C, polycondensations require higher temperatures between 150 and 250 C. Since the activation energy dictates also the variation of the reaction rate with temperature, chain growth polymerizations are less sensitive to temperature excursions than step growth polymerizations are. 11.3.2

Thermochemical Aspects

Polymerizations are generally exothermal reactions with speciﬁc energies up to 3600 kJ kg1 , corresponding to an adiabatic temperature rise of up to 1800 K. Some typical reaction enthalpies are presented in Table 11.2, together with the speciﬁc heat of reaction and adiabatic temperature rise obtained for mass polymerization. Most free-radical and ionic polymerizations have negative standard enthalpies and standard entropies; thus at higher temperatures these reactions must be considered reversible [Eq. (15)]. Pn þ M Ð Pnþ1

ð15Þ

As for every reversible exothermal reaction, there is an equilibrium temperature at which the chain growth and depolymerization rates are equal. This temperature is called the ceiling temperature and is related to the equilibrium concentration of the monomer according to Eq. (16).

565

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11 Safety of Polymerization Processes Typical reaction enthalpies and corresponding adiabatic temperature rises.

Tab. 11.2.

Monomer

Ethylene Propene Isobutene Butadiene Isoprene Chloroprene Acrylamide[b] Acrylonitrile Acrylic acid Methyl acrylate Methyl methacrylate Vinyl acetate Vinyl propionate Vinyl chloride Vinylidene chloride Styrene a-Methylstyrene Ethylene oxide Propylene oxide Trioxane[c] e-Caprolactam 2-Pyrrolidone

˚

Physical state [a]

Temperature [ C]

CDHR [kJ molC1 ]

CDHR [kJ kgC1 ]

CDT ad [ C]

gc lc lc lc lc lc s l c0 lc lc lc lc lc lc l c0 lc lc l c0 g s c0 lc lc

25 25 25 25 25 61.3 74.5 74.5 74.5 76.8 74.5 74.5 74.5 25 25 25 25 25 25 30 200 75

101.5 84 48 73 75 68 81.5 76.5 67 78 55.5 88 86 71 75.5 70 35 94.5 75.5 19.5 15.5 4.5

3620 2000 855 1350 1100 768 1147 1423 930 1100 550 1022 860 1135 780 672 296 2145 1300 216 137 53

1810 1000 428 676 559 384 574 721 465 435 277 511 430 542 390 336 148 1073 650 108 68 26

˚

[a] Physical state: l: liquid; g: gas; s: in solution, c: condensed amorphous, c 0 : crystalline or part crystalline. [b] Solvent is water. [c] Solvent is methylene chloride.

TC ¼

DHR DS 0 þ R ln CM; C

ð16Þ

It is important to realize that the ceiling temperature is not a constant, but it is a function of the monomer concentration. For most monomers, this equilibrium concentration is lower than the detection limit of common analytical methods. An exception is a-methylstyrene, with a ceiling temperature of 61 C for 100% monomer and a concentration of 2.2 mol L1 at 25 C. In the runaway of a polymerization, if a temperature in the range of the ceiling temperature may be reached, or in other words the MTSR may be close to the ceiling temperature (MTSR A TC ), then the safety analysis must account for the contribution of the depolymerization reaction, which produces low molecular species and may result in a pressure increase. This can easily be realized with the criticality classes presented above, by choosing the temperature TD24 on the basis of the depolymerization reaction.

11.3 Speciﬁc Safety Aspects of Polymerization Reactions

q kW

X

800

0.8

600

0.6

400

0.4

200

0.2

0

0 0

2

4

t (h)

Heat release rate (solid line) and monomer conversion (broken line) during the polymerization of 5000 kg vinyl chloride and 5000 kg water as a suspension. (after Hamielec [21]). Fig. 11.5.

The high enthalpy of reaction and the high concentrations often used in industrial polymerization processes lead to very high heat release rates. As an example [21], the polymerization of vinyl chloride as a 50% (by weight) suspension in water with a charge of 5000 kg vinyl chloride and 5000 kg water releases an energy of 5.7 GJ and the heat release rate is about 780 kW (Figure 11.5). This high heat ﬂow presents an engineering challenge in order to ensure a suﬃcient heat transfer coeﬃcient. The high heat release rates observed in polymerization reactions do not represent by themselves the only safety engineering problem. A further issue is the fact that the reaction dynamics leads to fast changes of the heat release rate during the process. This in turn requires a fast-acting temperature control system, which may be diﬃcult to achieve for large reactor volumes. In the example given in Figure 11.5, after about 40 to 50 minutes of reaction time, corresponding to 5 to 10% conversion, the heat release rate increases from 100 to over 700 kW. This is even more diﬃcult, in the sense that the reaction temperature should be maintained constant in order to obtain the required properties of the ﬁnal product. The accumulated heat at this stage represents approximately 90%, or 5.4 GJ. This energy could lead to an adiabatic temperature increase of over 1900 K if not controlled by the temperature control system. The thermodynamic characteristics of the reaction and the heat transfer capability of the reactor may lead to a parameter combination giving rise to the parametric sensitivity as explained in Section 11.2.4 and Eq. (11).

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11 Safety of Polymerization Processes

7.7 % MAA

q (W) 30

6.3 % MAA 20

1.7 % MAA

10

0 0

2

1

Fig. 11.6. Heat release rate as a function of time for the copolymerization of methacrylic acid (MAA) with styrene in emulsion with diﬀerent mass fractions of methacrylic acid. The self-accelerating behavior corresponding

t (h)

to the gel eﬀect is evident after 20 min. Composition of the reaction mass: 418 g water, 120 g styrene, 10/8/2 g methacrylic acid, 0.505 g K2 S2 O8 , 7 g Na dodecylbenzenesulfonate.

11.3.3

Factors Leading to Changing Heat Release Rates

Some polymerization processes present a particular kinetic behavior: after a certain time the reaction accelerates itself, leading to an increase in the heat release rate. This can be exempliﬁed with the copolymerization of styrene and methacrylic acid in emulsion [22]. At the beginning of the polymerization, a fast increase of the heat release rate is observed (Figure 11.6). This is due to the formation of latex particles that are essentially nucleated during the initial reaction phase: the reaction rate is often proportional to the number of latex particles. For higher concentrations of methacrylic acid, a second acceleration of the reaction can be observed: this self-acceleration is also called gel eﬀect or the Tromsdorﬀ eﬀect. It is due to the fact that the viscosity of the reaction mixture increases and the termination reactions of radical chains become diﬀusion controlled. Thus the concentration of living chains increases and so does the heat release of the reaction. This phenomenon appears as a self-acceleration, and like other types of self-accelerated reactions, this is also sensitive to small changes in the process conditions. This is shown in Figure 11.6 by the changes in the initial concentration of methacrylic acid, which cause ampliﬁed changes in the maximum heat release rate and also in the time point where this maximum occurs. The maximum heat release rate increases from 14 W kg1 with 1.7% methacrylic acid to 30 or 40 W kg1 when the concentration is increased to 6.3 or 7.7%.

11.3 Speciﬁc Safety Aspects of Polymerization Reactions

q (W)

20

10

0 0

2

Heat release rate as a function of time for the copolymerization of methacrylic acid (MAA) with styrene in emulsion following a seeding process. The mass fraction of methacrylic acid was varied: solid line, 10%; Fig. 11.7.

4

t (h)

broken line, 4.2%; dotted line, 1.7%. Composition of the reaction mass: 451 g water, 9 g styrene as seed (dP ¼ 112 nm), 108/115/ 118 g styrene, 12/5/2 g methacrylic acid, 0.505 g K2 S2 O8 , 0.26 g Na dodecylbenzenesulfonate.

The same copolymerization performed as a seeding process shows a totally different behavior that implies dramatic changes in the heat release rate as a function of time. In a seeding polymerization, the time required for the formation of the latex particles disappears because a given amount of latex particles are added at the beginning of the process [22]. Therefore the particles swell in the presence of the monomer and only the phases of particle growth and monomer depletion can be observed (Figure 11.7). This leads to an initial heat release rate, which remains practically constant during particle growth. The reaction seems to follow a zeroorder rate law. Then after a period of time depending on the initial concentration of methacrylic acid, the gel eﬀect leads to an increasing reaction rate. Consequently the heat release rate increases practically exponentially, even under isothermal conditions. The time interval after which this increase occurs and the maximum heat release rate depend on the concentration of methacrylic acid. Such a behavior may easily lead to a runaway situation if not anticipated. Additionally the particle size of the seed aﬀects the dynamic behavior of the reaction. Therefore it is essential to study the thermal behavior of a polymerization reaction during process development. A safe process may only be designed if these phenomena are thoroughly understood and if the corresponding engineering means are used for scaleup. A well suited tool for this kind of study is reaction calorimetry [23–26].

569

570

11 Safety of Polymerization Processes

11.4

Cooling of Polymerization Reactors

Diﬀerent technical means may be used for cooling industrial reactors. Among them we consider three methods: direct cooling using the sensible heat of a feed stream, which was treated in Section 11.2.3; indirect cooling using heat exchange across the reactor wall or with internal cooling systems; and hot cooling using the latent heat of evaporation. These latter two cooling techniques are discussed in Sections 11.4.1 and 11.4.2. In Section 11.4.3, the problem of viscosity change is examined. 11.4.1

Indirect Cooling: Heat Exchange Across the Reactor Wall

The heat transfer across a wall can be expressed by the general equation, Eq. (5). In practice, the temperature diﬀerence between the reaction medium and the cooling system must remain within reasonable limits. One limitation is given by the dynamic stability in the case of a batch reaction. But another limitation is due to the fact that too low a wall temperature may cause the buildup of a polymer ﬁlm, or fouling at the inner side of the wall. In a homogeneous reaction mixture, even without fouling, the resistance of the laminar heat transfer ﬁlm increases as the wall temperature decreases. The result is that too low a cooling medium temperature may have adverse results, since the reduction of the heat transfer coeﬃcient may overcome the beneﬁts of the higher temperature gradient. Increasing the heat transfer area by using inserts such as cooling coils or other shapes of built-in heat exchangers is often problematic. These inserts may enhance the formation of a polymer ﬁlm at the wall and they also may have a negative impact on the agitation of the reaction mass. Further, they render the cleaning operations of the reactor more diﬃcult. Another method, which is often used to increase the heat exchange area in low molecular reactions, is loop circulation through an external heat exchanger. This technique may be inadequate if the polymers are sensitive to the shear forces that will arise in a circulation pump. In certain cases reactors with an increased height to diameter ratio are used in order to increase the speciﬁc cooling area. The problem of the determination of the overall heat transfer coeﬃcient can be simpliﬁed if one considers that the overall resistance to heat transfer consists of several resistances in series [Eq. (17)]. 1 1 dp dw df 1 1 dp 1 ¼ þ þ þ þ ¼ þ þ U hR lp lw lf hC hR lp j |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} depends on reaction mass

ð17Þ

depends on reactor

The ﬁrst two terms, representing the liquid ﬁlm at the inner wall of the reactor and any polymer deposit, only depend on the reactor contents, that is, on the agitation

11.4 Cooling of Polymerization Reactors

and on the physical properties of the reaction mass. Here the resistance of the internal laminar ﬁlm and of the polymer deposit at the wall play a key role. Therefore the reactor should be cleaned regularly with high-pressure cleaner. The last three terms depend on the reactor itself, especially on its heat exchange system, that is, on the reactor wall, on fouling in the jacket and on external liquid ﬁlm. They are often grouped into one term: the equipment heat transfer coeﬃcient (j). For the description of the heat transfer coeﬃcient of the internal ﬁlm, there are several correlations available, the most popular of which is presented in Eq. (18) [27]. Nu ¼ C te Re 2/3 Pr 1/3

m mW

0:14 ð18Þ

This expression is valid for Newtonian ﬂuids; therefore with polymers its validity must be veriﬁed. In this correlation, the dimensionless numbers are deﬁned in Eq. (19). Here the Reynolds number (Re) is expressed for a stirred tank where the ﬂow rate corresponds to the tip speed of the agitator. Nu ¼

hR dR l

Re ¼

n dS2 r m

Pr ¼

m cp l

ð19Þ

The last term in Eq. (18) represents the ratio of the viscosity of the reaction mass at reaction (bulk) temperature to its viscosity at wall temperature. It accounts for the changes of the heat transfer coeﬃcient, when switching from heating to cooling. This produces an inversion of the temperature gradient and therefore aﬀects the viscosity of the product close to the reactor wall. With reactions performed in solvents it can generally be neglected, but may become important in the case of polymers: the viscosity of the reaction mass is often important and its temperature dependence may give this term a value that cannot be neglected. Equation (18) can be solved for hR, the internal ﬁlm heat transfer coeﬃcient. By grouping the terms in an appropriate way, the heat transfer coeﬃcient of the reaction mass can be split into two parts, one (z) representing the technical data of the reactor and the other (g) grouping the physical properties of the reaction mass [Eq. (20)]. The gravity constant g is introduced to rend the ﬁrst term dimensionless and to give the second the dimensions of a heat transfer coeﬃcient. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2/3 4/3 2 2 3 r l cp g dS te n ¼ z:g ð20Þ hR ¼ C 1/3 dR g m |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} technical data physical--chemical data of the reactor

of the reaction mass

Thus, for a given reaction mass, the heat transfer coeﬃcient of the internal ﬁlm can be inﬂuenced by the revolution speed of the agitator and its diameter. The value of z, characterizing the internal part of the equipment factor, can be calculated using the geometric characteristics of the reactor. Some typical values of the agitator constant are given in Table 11.3. The value of g can either be calculated

571

572

11 Safety of Polymerization Processes Tab. 11.3.

Typical agitator constants.

Agitator

Constant

Plate stirrer Rushton turbine Rushton turbine with pitched blades Propeller Anchor Impeller Intermig (Ekato)

0.36 0.54 0.53 0.54 0.36 0.33 0.54

0.01

1/U

0.009

slope = z..γ =0.00192

0.008

1/φ(T) = 0.00802

0.007 0.006

/ (n/n0)-2/3

0

0.1

0.2 2

0.3

0.4 .

Wilson plot measured with toluene in a reaction calorimeter. The reciprocal heat transfer coeﬃcient is plotted against the agitator revolution speed to the power 2/3. Fig. 11.8.

0.5

0.6 .

0.7

0.8 .

0.9

1

The intercept with the ordinate represents the reciprocal heat transfer coeﬃcient of the equipment and either z or g can be determined from the slope of the straight line.

from the physical properties of the reactor contents – as far as they are known – or measured in a reaction calorimeter by the Wilson plot method [28, 29]. This parameter is independent of the geometry or size of the reactor. Thus it can be determined at laboratory scale and used at industrial scale. The Wilson plot consists of the determination of the overall heat transfer coeﬃcient as a function of the agitator revolution speed in a reaction calorimeter. The Wilson plot (Figure 11.8) makes it possible to verify that the correlation in Eq. (18) is valid: if the measures ﬁt on a straight line, a validation is built into the method. The intercept with the ordinate represents the reciprocal heat transfer coeﬃcient of the equipment, that is, the wall and the external cooling system of the calorimeter. The slope is the product of z and g, which allows the determination of either one of these parameters. In a ﬁrst stage, z is determined by a calibration performed using a solvent with known physical properties. In a second stage g is determined during the actual measurement with the reaction mixture. Moreover, the contribution of polymer deposits to the overall resistance to heat transfer may be important. Since the thermal conductivity of polymers is low, they

11.4 Cooling of Polymerization Reactors Tab. 11.4. Factors inﬂuencing the heat transfer with some typical values of heat transfer coeﬃcients in an agitated reactor.

Type

Inﬂuencing factors

Typical values [ W mC2 KC1 ]

Internal ﬁlm hR forced convection

stirrer: speed and type reaction mass cp ; l; r; h physical data, especially r ¼ f ðTÞ

water toluene glycerol

1000 300 50

water gases With d ¼ 1 mm, PE PVC, PS With d ¼ 10 mm, iron stainless steel glass glass lined With d ¼ 0:1 mm, Gallert Kesselstein water with ﬂow no ﬂow evaporation

100 10

1000 100 3000

water with ﬂow no ﬂow

2000 200

hR natural convection (stirrer failure) Polymer deposit

Reactor wall l/d

Fouling at external wall

External ﬁlm hc

thermal conductivity ðlÞ thickness of deposit construction wall thickness ðdÞ construction material coating thermal conductivity ðlÞ thickness of deposit jacket construction, ﬂow rate heat carrier, physical properties, phase change welded half coil construction, ﬂow rate physical properties

300 170 4800 1600 100 800 3000 5000

act as insulators, and thus even thin deposits may aﬀect strongly the heat transfer coeﬃcient, as shown in Table 11.4. The resistance of the reactor wall and external ﬁlm (hc ) can be determined in a cooling experiment realized directly with the production reactor ﬁlled with a known amount of a substance (M) with known physical chemical properties. The temperature TR of the contents of the reactor and the average temperature TC of the cooling system are recorded during this experiment. A heat balance can be calculated between two instants t1 and t2 : the heat removed from the contents of the reactor is given by Eq. (21), and this is realized with an average cooling power given by Eq. (22) with the average temperature diﬀerence as expressed in Eq. (23). Q ¼ M cp ðTR1 TR2 Þ

ð21Þ

qEX ¼ U A DT

ð22Þ

DT ¼

ðTR1 TC2 Þ ðTR2 TC2 Þ lnðTR1 TC1 Þ lnðTR2 TC2 Þ

ð23Þ

573

574

11 Safety of Polymerization Processes

The overall heat transfer coeﬃcient can then be obtained from the heat balance equation [Eq. (24)]. U¼

M cP ðTR1 TR2 Þ A DT ðt2 t1 Þ

ð24Þ

This is a simpliﬁed method using only two points. A more accurate method is to use more points by applying the diﬀerential equation of Newtonian cooling, which expresses the variation with time of the temperature diﬀerence between reactor contents and cooling system [Eq. (25)]. M cP

dðDTÞ ¼ U A dt DT

ð25Þ

By integration, with the initial condition: t ¼ 0, DT ¼ DT0 ¼ T0 TC , one obtains Eq. (26) with the thermal time constant tc of the reactor given by Eq. (27). DT t ¼ ln DT0 tC tC ¼

or

DT DT0

t ¼ exp tC

M cP UA

ð26Þ ð27Þ

Thus a plot of the logarithm of the ratio of the instantaneous temperature diﬀerence to the initial temperature diﬀerence, as a function of time, should give a straight line, as far as cooling is Newtonian. The slope of this line is the thermal time constant of the reactor, from which the overall heat transfer coeﬃcient U can be calculated. Then the equipment heat transfer coeﬃcient j can be calculated from Eq. (28) written for a clean reactor. If there was some polymer deposit at the reactor wall during the cooling experiment, its resistance must be taken into account. 1 1 1 ¼ j U zg

ð28Þ

For the calculation of the heat transfer coeﬃcient of the external ﬁlm some models are also available. These models describe the hydraulics of the ﬂow in the jacket or in the half-welded coils. The results depend strongly on the technical design of these parts of the equipment. Direct measurement is mostly preferred. 11.4.2

Hot Cooling: Cooling by Evaporation

Hot cooling, which uses the latent heat of evaporation of a solvent, is a very eﬃcient technique: on one hand it is independent of the heat transfer at the reactor

11.4 Cooling of Polymerization Reactors

wall, and on the other hand the condenser can be designed independently of the reactor’s geometry. This allows relatively high speciﬁc cooling powers to be reached. In case a reaction cannot be performed at boiling temperature, it is possible to work under partial vacuum in order to decrease the boiling point and to work at reﬂux. Hot cooling can be used as the main cooling system for a reactor working under normal operating conditions; it can also be used as an emergency cooling system, in cases when the boiling point is reached during the temperature increase following a failure of the main cooling system. Obviously, this is only possible provided the condenser is equipped with an independent cooling system and the equipment has been designed for this purpose. The boiling rate of the solvent, depending on the instantaneous heat release rate of the reaction, governs the whole design of the reﬂux system. Some technical aspects and limitations must be considered in the design of such cooling systems. Too high a boiling rate could lead to ﬂooding of the vapor tube, when the condensate ﬂows down in countercurrent to the rising vapor. Further, the presence of vapor bubbles in the reaction mass increases its apparent volume. The reaction mass swells, and if its level is high enough to enter the vapor tube, again ﬂooding will occur. A method was developed for predicting these limitations [30, 31]. If the boiling point is reached during runaway, like in scenario of criticality classes 3 or 4, a possible secondary eﬀect of the evaporation is the formation of an explosive vapor cloud, which in turn can lead to a severe room explosion if ignited. In some cases, there is enough solvent present in the reaction mixture to compensate for the energy release, allowing the temperature to be stabilized at the boiling point. This is only possible if the solvent can be distilled oﬀ in a safe way, to a catch pot or a scrubber. The thermal stability of the concentrated reaction mixture must also be ensured. In most cases, however, the condensed solvent is reﬂuxed to the reactor. If the capacity of the reﬂux system is suﬃcient, all of the vapor produced by an exothermal reaction can be conducted from the reactor to the condenser, where it is entirely condensed. In such a case, the boiling point may act as a safety barrier. In the opposite situation, if the ﬂow rate is too high with respect to the capacity of the system, a pressure increase will result. This may be due to ﬂooding of the vapor tube or to swelling of the reaction mass. This pressure increase may in turn lead to the rupture of the weakest element in the distillation system, followed by the release of a vapor cloud, which could be ignited and cause a vapor cloud explosion. Further, if the condenser capacity is too short, uncondensed vapor will pass to the ventilation system, where it could cause a secondary incident, if it is not compatible with the design. For these reasons, it is essential to know the vapor ﬂow rate due to an exothermic reaction, which depends on the latent heat of evaporation of the solvent. The capacity of the reﬂux system will be assessed using the vapor velocity in the diﬀerent parts of the equipment [Eq. (29)]. u¼

qRX DHV rG S

ð29Þ

If the diameter of the vapor tube is insuﬃcient for a given vapor release rate, the

575

576 Tab. 11.5.

11 Safety of Polymerization Processes

Limiting vapour velocity for diﬀerent solvents. Solvent

DHv [kJ kg1 ] Tb [ C] Mw [g mol1 ] P [mbar] rg [kg m3 ] umax [m s1 ]

water

methanol

ethanol

acetone

dichloromethane

chlorobenzene

toluene

m-xylene

2260 100 18 1013 0.59 10.2

1100 65 32 1013 1.15 6.6

846 78 46 1013 1.60 5.3

523 56 58 1013 2.15 5.1

329 40 85 1013 3.31 4.5

325 132 112 1013 3.37 4.4

356 111 92 1013 2.92 4.8

343 139 106 1013 3.13 4.6

high vapor velocity results in a pressure increase in the reactor leading to a temperature increase and a further acceleration of the reaction. The consequence will be a thermal explosion until the rupture of weak parts of the equipment allows pressure relief. In order to avoid this type of reaction course, it is important to know the maximum vapor velocity admissible in a given tube and consequently the maximum admissible heat release rate for the reaction. To predict whether ﬂooding will occur in existing equipment, an empirical correlation was established experimentally [31]. The experimental study was performed in the laboratory, in the pilot plant and on an industrial scale with various organic solvents and water for tubes with an inside diameter between 6 and 141 mm. The maximum allowable heat release rate is obviously a function of the latent heat of evaporation and of the tube cross-section. It can be calculated by Eq. (30). q max ¼ ð4:52DHV þ 3:37 10 6 ÞS

ð30Þ

Calculations performed for diﬀerent common solvents show that the limiting velocity remains relatively constant for diﬀerent classes of solvents. Some values are shown in Table 11.5. This allows the required vapor tube diameter d for a given heat release rate for diﬀerent solvents to be calculated (Figure 11.9). The relative volume increase due to swelling of the reaction mass can be estimated using the Wilson correlation (Eqs. (31), (32) with the conditions: if u~ < 2 then K ¼ 0:68, a ¼ 0:62; or if u~ b 2 then K ¼ 0:88, a ¼ 0:40) [32, 33]. This correlation was ﬁrst established for air in water; it is easy to use and was found to describe with enough accuracy the swelling of a liquid by bubbles of its vapor. a¼K a¼

rV rL rV

HB H0 HB

0:17

~0:1 u~ a D H

DH ~H ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D s g ðrL rV Þ

ð31Þ u u~ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s g g ðrL rV Þ

ð32Þ

11.4 Cooling of Polymerization Reactors

5000

q (kW)

Water

4000 3000 Methanol Ethanol i-Propanol Acetone Other solvents

2000 1000 d (m)

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Heat release rate at ﬂooding as a function of the vapor tube (riser) diameter for diﬀerent solvents. Fig. 11.9.

5 ug (m/s)

Water

4 3

Ethanol

2

Toluene

1 α

0 0

0.1

0.2 .

0.3

0.4 .

0.5

0.6 .

Maximum allowed vapor velocity ug across the surface of the reaction mass as a function of its degree of swelling a. Fig. 11.10.

The correlation allows the maximum admissible heat release rate for given plant equipment to be calculated, or equipment for the requirements of a given process to be designed (Figure 11.10). They are both based on easily accessible physicalchemical properties of the boiling solvent and on geometric data of the reactor. Such considerations make it possible to adapt the equipment or the process, that is, the degree of ﬁlling of the reactor, to the safety requirements. This kind of measure often allows processes to be run under safe conditions, whereas a classical assessment would consider them to be critical.

577

578

11 Safety of Polymerization Processes

Nu

mPas

Fig. 11.11. Nusselt number as a function of the apparent viscosity during reticulation of poly(vinyl alcohol) in aqueous solution in a laboratory reactor with an anchor stirrer.

Geometry VR ¼ 1:3 L, D ¼ 0:1 m, dR /D ¼ 0:95, H/D ¼ 1:9, r ¼ 1000 kg m3 , l ¼ 0:65 W m1 K1 . After Schmidt and Reichert [38].

11.4.3

Importance of the Viscosity

In polymerizations the ﬂuid often does not behave in a Newtonian manner. In such cases Eqs. (18)–(20) do not apply as such, and modiﬁed equations must be used. It could be shown experimentally [34–36] that the shear forces are proportional to the agitator revolution speed, meaning that an average shear force in the reactor can be used and the dynamic viscosity can be replaced by an apparent viscosity [37]. Such an example is illustrated in Figure 11.11, where the variation of the Nusselt number Nu during a polymerization shows the decrease in the overall heat transfer by a factor of three when the viscosity increases by over one order of magnitude [38]. In mass polymerization an increase of the viscosity by six or seven orders of magnitude is common. This means that the viscosity is a key factor for the control of a polymerization reactor, that is, for its safety. In fact the viscosity inﬂuences both the heat balance and the mass balance. It has been shown how the heat transfer coeﬃcient is aﬀected by the viscosity. But the energy dissipation by the stirrer is also strongly dependent on viscosity (see Section 11.4.4). Furthermore, viscosity aﬀects the molecular diﬀusion, the mass transport, the mixing time, or the residence time distribution, and therefore the reaction rate. Since the reaction rates inﬂuence the chain length and particle sizes, they have a direct eﬀect on the polymer properties. In turn they aﬀect the viscosity and the shear forces – there is a feedback eﬀect. Such complex interactions cannot be described by analytical equations, so empirical models must be used. Often

11.5 Chemical Engineering for the Safety of Polymerization Processes

these models are very speciﬁc for a given reaction system and even for a reactor geometry. Their use aﬀords deep chemical and engineering knowledge and experience. To give an example, increasing the stirrer speed in order to improve the heat transfer may result in an adverse situation: one reason is that the heat dissipation by the stirrer is increased. But a further eﬀect may arise at the same time: the mixing time is reduced, enhancing the mass transfer and resulting in an increase in the reaction rate that leads to an eﬀect opposite to the initial intention.

11.5

Chemical Engineering for the Safety of Polymerization Processes

Chemical reaction engineering traditionally addresses the design of processes operated under nominal operating conditions. But if the same principles are used to design the process to remain safe even under conditions deviating from the intended conditions, it becomes a strong tool for improving process safety [1]. For this reason, in this section the diﬀerent reactor types are analyzed with regard to their safe operation and especially to the design in order to prevent runaway. Some further risk-reducing techniques for stopping a commencing runaway or mitigating its consequences are also presented. 11.5.1

Batch Processes

Batch processes are characterized by a closed mass balance. This means that during the reaction nothing is added to or withdrawn from the reactor. In fact we consider as batch processes those where the monomer and the solvent, if there is one, are added at the beginning of the reaction. The initiator or catalyst is also added at the beginning of an operation. We extend this deﬁnition to processes where a volatile compound is evaporated in order to shift the equilibrium to the products, as is often the case with polyadditions or polycondensations. In these processes the control of the temperature is the unique way of controlling the reaction course. Diﬀerent strategies of temperature control may be used in this situation. An isothermal operation requires the cooling capacity to be suﬃcient at any time to compensate for the heat release rate of the reaction. Moreover, the dynamics of the cooling system must allow sharp changes in the reaction rate, which are often observed in polymerization reactions (see Section 3.3), to be followed. Due to the high energy of reaction, this kind of process is only possible in diluted systems as emulsion polymerization or polymerization in solution. In most cases non-isothermal processes are used. A part of the energy of the reaction is used to heat the reactor to the desired process temperature in the so-called polytropic reaction mode. A typical temperature control strategy is to use an adiabatic phase, during which there is no cooling, so that the correct initiation of the reaction can be checked. Then, in a second stage, after reaching a speciﬁed temperature, the cooling system is switched. The correct choice of the initial temperature

579

580

11 Safety of Polymerization Processes

and of the cooling system switching temperature is critical for the process safety. This type of strategy is often used in polyaddition or polycondensation reactions, where high reaction temperatures are desired. Advanced temperature control strategies allowing the optimization of productivity, but taking technical limitations of the plant equipment into account, are proposed [39]. Great care must be used to avoid working in the parametrically sensitive ﬁeld. Thus a thorough study of the process by reaction calorimetry is essential for the safe design of such processes. The aim of process design should be to develop processes that are tolerant to a failure of the cooling system. This is often a utopian dream, due to the high energy of polymerization reactions. In batch operation, the correct charge of the reactor is essential, and requires great care by the operators, because this is often a manual operation. In every case a batch process requires emergency measures to be taken in order to recover control of the reactor when the cooling system fails, or at least to mitigate the consequences of a runaway. The scenario presented in Section 11.2 is a great help for this purpose. 11.5.2

Semi-batch Processes

In semi-batch operation only a part of the monomer, or no monomer at all, is initially charged. The main part of the monomer is added to the reactor during the process, allowing additional control of the reaction course and of the polymer properties by adjusting the feed rate. In such processes the accumulation of monomer and therefore of heat is limited to a fraction of the overall heat. Such processes are often used for emulsion polymerizations. The continuous feed also reduces the instantaneous heat release by the reaction and requires less cooling capacity. A correct choice of the concentrations and of the temperature even makes it possible to achieve a so-called feed-controlled process. This means that the monomer entering the reactor is immediately converted; that is, there is no monomer accumulation. In such a situation the reaction rate is equal to the feed rate, which can easily be adapted to the cooling capacity of the reactor (Figure 11.12). In the case of any failure, stopping the feed immediately stops the reaction and the heat production. Thus the temperature can be stabilized, making it possible to design fail safe processes. The design of a safe semi-batch process requires a strict control of the accumulation of unconverted monomer. In fact the accumulation results from a balance between monomer addition by the feed, and monomer conversion by the reaction. Thus it can be inﬂuenced by the feed rate, but also by the reaction rate, which in turn can be governed by the temperature and the initiator or catalyst concentration. These parameters must be optimized during process development. Here again, reaction calorimetry is of great help for the determination of the accumulation [40]. It can be calculated from thermal conversion by means of Eq. (4) obtained directly from the calorimetric measurement and the actual amount fed. Just by varying the

11.5 Chemical Engineering for the Safety of Polymerization Processes q RX (W/kg)

60 50

X acc

0.3 0.2

40 30 20

0.1 t (h))

t (h)

10 0

0 0

2

4

6

8

10

2

0

2

4

6

8

10

12

Fig. 11.12. Heat release rate and accumulation obtained in a semi-batch reaction for diﬀerent feed rates: 2, 4 and 6 h.

feed rate and the process temperature, one can verify that the heat release rate of the reaction never exceeds the cooling capacity of the reactor. A second constraint is the maximum allowed temperature in the case of a cooling failure, that is, the accumulation, and therefore the MTSR should remain below a given level (Figure 11.12) [41, 42]. In practice, emulsion polymerizations are often performed according to the monomer feed process with an initial charge of the monomer, the remaining being fed over time. Figure 11.13 represents a schematic example of an emulsion polymerization according to the monomer feed process [2]. In such processes three phases may be distinguished: the initiation, the feed and the end of reaction. In this example, the delay between initiator addition and continuous feed was varied. Initiation In a ﬁrst stage, an initial amount of monomer is added to the reactor. This addition is immediately followed by the addition of the initiator. During the delay between initiator addition and monomer feed, inhibition of the polymerization may occur. This depends on the purity of the reactants and on the oxygen concentration remaining in the reactor. Then a fast increase in the thermal power follows: it is due to formation of the particles. On an industrial scale, such a fast increase of heat release rates may generally not be compensated by the comparatively slow cooling system, and results in an accumulation of heat, that is, an increase in the reactor temperature. But at this stage, the monomer concentration is still low. Often the total amount of water was charged initially, conferring a high heat capacity on the system. Thus the resulting temperature increase during this stage remains low. In fact the initial charge must be optimized in such a way that the initiation remains safe. In the case of emulsion feed polymerizations, the initial contents of the reactor represent an aliquot of the total batch. The situation is similar to the monomer feed process, since the initial charge may be designed in such a way that the temperature increase remains within critical limits. The cooling capacity, and sometimes also the heat capacity, of the reactor are high enough to limit the initial temperature excursion. 11.5.2.1

581

11 Safety of Polymerization Processes

Initial i l monomer charge

dN/dt

Initiation

Optimal feed

Feed d

t

Initial i monomer charge

dN/dt

Initiation

r too early Feed start

t

Feed

dN/dt Initial i monomer charge

582

Initiation

Feed start too late

Feed

Fig. 11.13. Schematic representations of the molar ﬂow rates of monomer in a semi-batch emulsion polymerization. Top graph: the feed is optimal; it is immediately converted. Middle graph: the feed was started too early; the

t reaction course is no longer controlled by the feed. Bottom graph: the feed was started too late; the accumulation results in a quasi runaway situation.

Feed After the particle growth stage of the emulsion polymerization is terminated, the monomer concentration within the latex particles decreases and so does the heat production rate. In the meantime, the temperature control system is able to reduce the temperature of the jacket. When the molar ﬂow rate (mol s1 ) of the monomer 11.5.2.2

11.5 Chemical Engineering for the Safety of Polymerization Processes

depletion reaches a value corresponding approximately to the molar ﬂow rate of the intended feed, the monomer feed should be started (Figure 11.13, top graph). This allows compensation in order to maintain the concentration constant; that is, a quasi steady state can be achieved. The reactor is cooled by two methods: heat exchange across the wall, and via the sensible heat of the cold feed. Here, the emulsion feed processes obviously show an advantage due to the higher heat capacity of the emulsion compared to the monomer. The right time point for starting the feed is an important safety-relevant parameter. If the feed is started too early during the particle formation period (Figure 11.13, central graph), an accumulation of monomer may result. This will cause an increase in the reaction rate and result in an uncontrollable temperature increase – even a runaway reaction. Matching of feed rate and reaction rate, allowing control of the course of the polymerization, is only achieved in the very late stages of the feed time, if ever. If the feed is started too late, the situation may become even worse (Figure 11.13, bottom graph): the monomer initially present is already converted and the heat release rate is low. But the jacket may still be at a low temperature due to the previous exotherm, and undercool the reactor. Since the feed is increasing the monomer concentration at too low a reaction rate, an accumulation will occur and may lead to a sudden ‘‘wake up’’ of the reaction, which in turn could result in a runaway situation. Final Stage After the end of the feed period, the heat release rate decreases as a consequence of the monomer depletion. This third stage is very similar to a batch reaction, because the reaction rate can no longer be inﬂuenced by the feed. 11.5.2.3

Practical Aspects The diﬃculty concerning the initiation on an industrial scale is that the actual initiation of the polymerization must be recognized before a great amount of monomer is added to the reactor. This can be realized either by chemical in-line analysis, or by a heat balance on the industrial reactor [43]. If the initiation fails and the feed is continued, it may lead to an accumulation of monomer that could react spontaneously in an uncontrolled way. Thus the process becomes a batch reaction, for which the reactor is not designed. In a semi-batch reaction, overcooling, or too low a temperature, may be as critical as too high a temperature, because the reaction rate is lower and the resulting accumulation may be important. If the temperature is adjusted to its nominal value, the accumulated monomer may also react in an uncontrolled way, leading to a runaway reaction. This may be due to degradation of the initiator which slows down the reaction. Such eﬀects can be detected by a heat balance on industrial reactors. For polymerization or copolymerization processes performed in semi-batch reactors, the feed rate can also be adjusted to meet quality criteria [44–48]. Advanced techniques are proposed to control the feed rate by taking into account simultaneously productivity and safety criteria [49, 50]. 11.5.2.4

583

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11 Safety of Polymerization Processes

Thus not only does a safe process result from sound development, working out the right process parameters, but on an industrial scale it must be checked that the polymerization remains on the ‘‘right path’’. 11.5.3

Continuous Processes

In continuous polymerizations, the main problem is the dynamic stability of the reactor. The stability problems have various diﬀerent aspects: the thermal stability as introduced above (see Section 11.2.4), the concentration stability, the particle number stability, and the viscosity stability. Even under isothermal conditions these problems may lead to multiplicity or oscillatory behavior. It is worth emphasizing the fact that stability and safety are in no case synonymous: a reactor may be unsafe even if working at a stable working point, or conversely it may be run safely at an instable working point. But knowledge of stability limits of the reactor is essential for the design of a safe process. Concentration Stability In a continuous stirred tank reactor (CSTR) where a homogeneous polymerization is being performed, the mass balance or the performance equation can be written in a very simple way [Eq. (33)]. 11.5.3.1

dX X ¼ dt t

ð33Þ

If the reaction follows an autocatalytic kinetic equation, like polymerizations exhibiting the Tromsdorﬀ eﬀect, then the conversion rate as a function of conversion, dX/dt ¼ f ðXÞ, shows a typical maximum curve [51]. The operating point is found at the intercept of the kinetics line and a straight line with the space time reciprocal as the slope. This conﬁguration leads to a multiplicity of solutions as represented in Figure 11.14 for several values of the space time. In a range of space time between t1 and t3 , there are three solutions. The upper and lower solutions are stable, whereas the intermediate one is instable. In such a situation, the CSTR should be designed to be operated at the high conversion point (hot branch, or ignited reactor). This allows for a higher productivity and a safer behavior if the high cooling capacity required can be achieved. If the reactor is operated at the lower conversion point (cold branch), a small perturbation, for example of a feed pump, may ‘‘ignite’’ the reactor, which suddenly jumps to the hot branch, with high conversion. This could lead to a runaway. Particle Number Stability The particle size stability is a special form of the dynamic concentration stability [52]. If the emulsiﬁer is present at a concentration above the critical concentration, new particles will be created. Thus the area growth rate, and therefore the emulsiﬁer consumption, are increased. Then the emulsiﬁer supply by the feed may be in11.5.3.2

11.5 Chemical Engineering for the Safety of Polymerization Processes

dX/dtt

1 2

3

X Fig. 11.14. Schematic representation of the conversion rate as a function of conversion (solid line) in an isothermal CSTR with a polymerization presenting a gel eﬀect. Three

diﬀerent space times are represented by the three operating lines (broken lines). Filled circles represent stable operating points and the white circle an unstable operating point.

suﬃcient to ensure coverage of the particle’s surface. This stops the nucleation process of new particles. Since the particles are washed out by the reactor outlet, only fewer, but larger, particles remain in the reactor. The speciﬁc interfacial area decreases until the emulsiﬁer consumption becomes lower than the supply. Then the concentration increases, and a new cycle starts. These oscillations may occur especially during start-up of the reactor. They result in oscillations of the conversion that impinge on the quality of the product. They can be avoided by using a adequate seeding policy for the reactor. 11.5.4

Design Measures for Safety

Safety measures or risk-reducing measures must be considered under the aspect of inherent safety. Kletz [53], who was the promoter of these ideas for a long time, formulated some principles for reduction of risks:

intensiﬁcation, using so little hazardous material that it will not matter if it all leaks out; substitution, using a safer material instead; attenuation, using a hazardous material in a safer form; limitation of the eﬀects of failures, not by adding on protective equipment but by equipment design or changing the conditions of use; simpliﬁcation, as complex plants provide more opportunities for human error and contain more equipment that can go wrong;

585

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11 Safety of Polymerization Processes

avoiding knock-on or domino eﬀects; making incorrect assembly diﬃcult or impossible; making the state of equipment, such as open or shut, clear; designing equipment that is able to withstand incorrect installation or operation; making equipment easy to control.

Following these principles in a more speciﬁc way for polymerization reactions, three levels of priority can be deﬁned: in decreasing order, the ﬁrst priority is the reduction of severity by design. As a second priority, technical measures for control of the reaction to avoid runaway should be considered. The aim is to obtain a fail safe process by reduction of the probability of occurrence of an incident. As last resort only, emergency measures should be taken in order to mitigate the consequences of runaway. In any case, the basic principle remains: ‘‘Avoid runaway rather than mitigate its consequences.’’ With reference to this principle, diﬀerent types of risk-reducing measures can be considered for polymerization reactions (Sections 11.5.4.1–11.5.4.10). Process Design Semi-batch processes are preferred to batch processes where possible. This point was made sbove (see Section 11.4). Continuous processes allow the inventory of reaction mass to be reduced, and represent an elegant way of reducing the severity of a process. 11.5.4.1

Reactor Design Many design elements are in fact preventive measures. Some of them will be mentioned here. One may consider building a reactor resistant to the maximum pressure even in the case of loss of control of the polymerization. Often the ﬁnal pressure obtained with polymerization reactions is not so high. Another important element is the agitation. The agitator ensures uniform mixing over a large range of viscosities, avoids local concentration of reactants that could lead to hot spots, and ensures heat transfer. The agitation is also required for temperature measurement, that in turn ensures control of the reactor. In this context it is important to keep heat exchange surfaces (and also the thermometer) clean. Temperature is often the key parameter for process safety; thus redundant systems should be used, since failure rates in the order of magnitude of one per year are reported for them. The temperature probe must be correctly located and the temperature recorded so that trends can be followed. This is essential for safety as well as for product quality. The rate of temperature change can also represent very important information for process control. Utility-independent thermometers are also very useful. The temperature of the cooling system (if possible at the inlet and outlet of the jacket) should be monitored. The process should be designed in order to provide adequate heating proﬁles, especially to avoid inadvertent fast heating. An interlock may stop heating in the case of malfunction. When reﬂux cooling is used, one must avoid deposits inside the condenser 11.5.4.2

11.5 Chemical Engineering for the Safety of Polymerization Processes

tubes. To do so, the condenser can be pitched to assist drainage, and must be sized to prevent ﬂooding. The process must be designed to avoid foam-up of the reaction mixture (swelling; see Section 11.4.2). Services are very important for the control of the reaction. Redundant systems should be provided for electrical power, as well as compressed air or inert gases. The electricity supply must be examined in detail during risk analysis: motors – that is, feeding and cooling pumps, agitators, vacuum, and feed – will be stopped if the electrical power fails. Moreover, instruments dependent on compressed air become inoperative. For this reason, the fail safe position of valves (open for cooling, closed for heating) should be considered. Failure of vacuum at reﬂux may lead to the development of a hazardous situation: the boiling point, and thus the reaction temperature, increase and may end up in a runaway situation. Control of Feed In semi-batch or continuous operation, the feed rate allows control of the reaction course. Hence it plays an important role concerning the safety of the process. With an exothermal reaction, it is important to be able to limit the feed rate by technical means. One possibility is feed by portions, a method that is only applicable for semi-batch reactors. This mode of addition is the traditional way of limiting accumulation. In this case, the addition must be controlled by the conversion: that is, the next portion is added only if the previous portion has been consumed by the reaction. Diﬀerent criteria can be used to follow the reaction: the temperature, the appearance of the reaction mass, chemical analysis, and so on. For a well designed process, the additions can also be performed on a time basis. Another possibility is to use a feed tank with a control valve and gravimetric ﬂow. The valve can be controlled by the weight of the reactor, by the weight of the tank, by the level in the tank, or by a ﬂow meter. The maximal feed rate can be limited by the clearance of the valve or by a calibrated oriﬁce. If a centrifugal pump is used, it is necessary to provide also a control valve to limit the ﬂow rate. The ﬂow control strategies are the same as those described above. With a metering pump, the through ﬂow rate can be controlled by a stroke adjusting mechanism or a variable speed drive acting on the stoke frequency. The control can be realized by a ﬁxed adjustment or by through a ﬂow meter. 11.5.4.3

Emergency Cooling If the cooling system fails, water can be supplied from a hydrant to the cooling jacket or coil, or an independent emergency cooling system can be provided. With such a system, it is critical that the temperature does not fall below the solidiﬁcation point of the reaction mass, or otherwise a crust would form, resulting in reduced heat transfer, which again may favor the runaway situation. The agitation of the reaction mass is also critical in such a situation: large reaction masses behave adiabatically in practice, even if cooled on the outside. Here the injection of nitrogen into the bottom of the reaction mass has proven to be helpful for an emergency mixing. Such equipment must be tested under practical conditions. 11.5.4.4

587

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11 Safety of Polymerization Processes

Inhibition This technique may be applied to catalytic reactions where an inhibitor can be added in small amounts. Mixing is an especially critical factor here, since a small amount of inhibitor has to be added and mixed into a large volume of reaction mixture [54, 55]. 11.5.4.5

Quenching Some reactions can be stopped by the addition of a suitable component. Dilution by an inert and cold material may lower the temperature to slow down the reaction. For this type of measure, the critical factors are the amount and rate of addition and the temperature of the quenching material. The required empty volume must be also be available in the reactor. Calorimetric methods are of great help in the design of such measures, because they allow measurement of the heat of mixing, which is often important, and the thermal stability of the resulting mixture. 11.5.4.6

Dumping This measure is similar to quenching, with the diﬀerence that the reaction mass is not kept within the reactor, but transferred into a vessel containing the inhibitor or the diluting compound. This vessel must be prepared to receive the reaction mass at any instant during the process. The transfer line is critical for the success of this measure. It must be designed to allow an emergency transfer even in the case of breakdown of the utilities. This measure is particularly suitable in cases where the reaction mass must be transferred for workup after a normal operation. 11.5.4.7

Controlled Depressurization If a runaway is detected in an early stage, where the temperature and the pressure increase is slow, a controlled depressurization of the reactor can be considered. This is only suitable for cases where a volatile solvent is present; the reactor is slowly depressurized until evaporative cooling occurs. Obviously, a scrubber and/or a reﬂux condenser must be installed and designed to work with independent utilities. 11.5.4.8

Pressure Relief This measure will not be described in detail in this chapter. The design of venting lines for reactions with thermal potential is a complex matter. There are examples where pressure release was able to protect reactors from an explosion, but also cases where a reactor exploded, even with an open manhole. This measure only applies with reaction systems where the pressure still increases signiﬁcantly for small temperature increases above the normal operating level. In addition, the discharge line must end in a catch tank or in a scrubber to avoid spillage with possibly toxic or ﬂammable material. The behavior of the reaction mass – that is, the temperature and pressure increase – under runaway conditions must be known. Critical factors are foaming and two-phase ﬂow, which require speciﬁc design methods [56–59]. Special care must be taken to deﬁne the runaway scenario against which the reactor has to be protected. Emergency vent systems may also be designed for liquid discharge (dumping). 11.5.4.9

11.6 Conclusion

T Time to effect Time to action Alarm level

Effect

Time to discovery

t Time to Maximum Rate Fig. 11.15.

Chronological development of contingency measures.

Time Factor Time plays a primary role in the eﬃciency of a measure. The following steps must be taken from the instant a failure occurs up to recovery of the control of the process (see Figure 11.15): First, when a failure or a malfunction occurs, it must be detected. The detection time can be inﬂuenced by the choice of appropriate alarm settings and instrumentation. But the most important is the choice of an adequate parameter which must be monitored to detect a malfunction. The design of alarms, interlocks, and control strategies is an important part of process design. Once the alarm is switched on, some time is required for the measure to be applied. Quenching or dumping requires some time for the transfer of the mass, an emergency cooling system must be switched on, and the cooling medium must ﬂow at the required temperature with the required ﬂow rate. Finally, the measure must become eﬀective: Some time also elapses from the instant when the measure was applied until its eﬀects become sensitive to the process. This is especially true with large reactors such as those used in polymerization processes. This time factor must be estimated for an eﬀective design of safety measures, and compared with the dynamics of the runaway of the desired reaction and of decomposition reactions. Here the TMR ad concept (Section 11.2.1) is of great help. 11.5.4.10

11.6

Conclusion

Even if polymerization reactions are critical from the point of view of thermal process safety, there are means to systematically identify the risks and to design the

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11 Safety of Polymerization Processes

reactors in order to minimize the risks. The systematic evaluation scheme presented in this text has proven to be well suited for risk identiﬁcation. A method based on the criticality classes and on calorimetric tools for process development has been presented. Despite the fact that more advanced methods are available, these ﬁnd only few applications in the industrial practice. This is probably not due to their inherent quality, but may be an eﬀect of long-term traditions in the polymer industry, which render innovation more diﬃcult. References 1 R. Gygax, Chemical Engineering 2

3

4 5

6

7 8 9 10 11 12

13

14 15 16 17

Science, 1988, 43(8), 1759. H. U. Moritz, Sichere Handhabung chemischer Reaktionen, Dechema, Frankfurt a.M., 1995, Vol. 3, p. 115. J. A. Barton, P. F. Nolan, Safety of chemical batch reactors and storage tanks, ECSC, EEC, EAEC, Brussels, 1991. N. Maddison, R. L. Rogers, Chemical Technology Europe, 1994, 11–12, 28. P. A. Dunn, Guidance for the safe production of phenolic resins, British Plastics Federation, London, 1979. R. Gygax, ESCIS (Expert Commission for Safety in the Swiss Chemical Industry), Lucerne, 1993, Vol. 8. R. Gygax, Chemical Engineering Progress, 1990, 53. D. I. Townsend, J. C. Tou, Thermochimica Acta, 1980, 37, 1. F. Stoessel, Chemical Engineering Progress, 1993, 10, 68. C. Van-Heerden, Industrial Engineering Chemistry, 1953, 45(6), 1242. N. R. Amundson, O. Bilous, AIChE Journal, 1955, 1(4), 513. R. Aris, Introduction to the analysis of chemical reactors, Prentice-Hall, Englewood Cliﬀs, 1965. A. Varma, M. Morbidelli, H. Wu, Cambridge University Press, Cambridge, UK, 1999. P. Hugo, Chemie Ingenieur Technik, 1980, 52(9), 712. K. Y. Choi, W. H. Ray, Chemical Engineering Science, 1988, 43(10), 2587. K. Y. Choi, W. H. Ray, Chemical Engineering Science, 1985, 40(12), 2261. B. Hentschel, Chemie Ingenieur Technik, 1979, 51(8), 823.

18 H. U. Moritz, Third Berlin Inter-

19

20

21

22 23 24

25

26 27 28

29 30 31 32

33

national Workshop on Polymer Reaction Engineering, VCH, Weinheim, 1989. H. G. Poersch-Parke, A. Avela, K. H. Reichert, Chemie Ingenieur Technik, 1989, 61(10), 808. A. Uppal, W. H. Ray, A. B. Poore, Chemical Engineering Science, 1974, 29(4), 967. A. E. Hamielec, Course Notes, Course on ‘‘Polymer Reaction Engineering’’, McMaster University, Hamilton, Ontario, 1980. M. Klocke, Thesis, Universita¨t-GH, Paderborn, 1995. F. Stoessel, Journal of Thermal Analysis, 1997, 49, 1677. R. N. Landau, L. R. Williams, Chemical Engineering Progress, 1991, 12, 65. W. Regenass, 8th RC1 User Forum Hilton Head SC, USA, Mettler-Toledo, 1996. W. Regenass, Chimia, 1997, 51, 189. VDI-Wa¨rmeatlas, VDI-Verlag, Du¨sseldorf, 1974. J. R. Bourne, M. Buerli, W. Regenass, Chemical Engineering Science, 1981, 36, 347. S. Choudhury, L. Utiger, Chemie Ingenieur Technik, 1990, 62(2), 154. J. Wiss, F. Stoessel, G. Kille´, Chimia, 1990, 44(12), 401. J. Wiss, F. Stoessel, G. Kille´, Chimia, 1993, 47(11), 417. J. F. Wilson, R. J. Grenda, J. F. Patterson, Trans. Am. Nucl. Soc., 1961, 4(37), 356. J. F. Wilson, R. J. Grenda, J. F. Patterson, Trans. Am. Nucl. Soc., 1962, 5(25), 151.

Notation 34 M. F. Edwards, W. L. Wilkinson,

35 36 37

38

39 40 41

42

43 44

45 46

47

Chemical Engineering (London), 1972, 265, 328. A. B. Metzner, R. E. Otto, AIChE Journal, 1957, 3, 3. H. U. Moritz, Chemical Engineering Technology, 1989, 71. A. Steiff, R. Poggemann, P. M. Weinspach, Chemie Ingenieur Technik, 1980, 52(6), 492. C. U. Schmidt, K. H. Reichert, Chemie Ingenieur Technik, 1987, 59, 739. C. Gentric et al., Chemical Engineering Journal, 1999, 75, 31. F. Stoessel, Chemical Engineering Progress, 1995, 9, 46. P. Hugo, J. Steinbach, F. Stoessel, Chemical Engineering Science, 1988, 43(8), 2147. F. Stoessel, O. Ubrich, Journal of Thermal Analysis and Calorimetry, 2001, 64, 61. J. Schmidt, H. Giesbrecht, Chemie Ingenieur Technik, 1996, 68(9), 1144. I. S. d. Buruaga, M. Arotc¸arena, P. D. Armitage, L. M. Gugliotta, Chemical Engineering Science, 1996, 51(11), 2781. G. Arzamendi, J. Asua, Ind. Eng. Chem. Res., 1991, 30, 1342. A. Urretabizkaia, J. R. Leiza, J. M. Asua, AIChE Journal, 1994, 40(11), 1850. I. S. d. Buruaga, A. Echevarria, P. D.

48

49 50 51

52 53 54

55

56

57

58 59

Armitage, J. C. D. L. Cal, R. Leiza, J. M. Asua, AIChE Journal, 1997, 43(4), 1069. L. M. Gugliotta, J. R. Leiza, M. Arotc¸arena, P. D. Armitage, J. M. Asua, Ind. Eng. Chem. Res., 1995, 34, 3899. T. J. Crowley, K. Y. Choi, Journal of Process Control, 1996, 6(2–3), 119. O. Abel et al., Journal of Process Control, 2000, 10(4), 351. R. S. Knorr, K. F. O’Driscoll, Journal of Applied Polymer Science, 1970, 14, 2683. G. Ley, H. Gerrens, Makromolekulare Chemie, 1974, 75, 563. T. A. Kletz, Process Safety Progress, 1996, 15(1), 5. ¨ ter, A. Steiff, S. Kammel, U. Schlu P. M. Weinspach, Chemical Engineering Science, 1996, 51(10), 2253. H. Niemann, S. Lighart, S. Duffield, A. Schipper, PPSS96, Houston, TX, 1996. J. Etchells, J. Wilday, Workbook for chemical reactor relief system sizing, HSE, Norwich, UK, 1998. Guidelines for pressure relief and eﬄuent handling systems, CCPS-AICHE, New York, 1998. J. Schmidt, F. Westphal, Chemie Ingenieur Technik, 1997, 69(6), 776. J. Schmidt, F. Westphal, Chemie Ingenieur Technik, 1997, 69(8), 1074.

Notation

Symbol

Name

Practical unit

SI unit

A A; P

heat exchange area chemical compounds concentration speciﬁc heat capacity diameter or thickness diameter activation energy

m2

m2

mol L1 J kg1 K1

mol m3 J g1 K1

m

m

m J mol1

m J mol1

C cP d D Ea

591

592

11 Safety of Polymerization Processes

F g h H DHR DHV k k0 m_ M MR n n N q Q r R S S t T u U V X z

molar ﬂow rate acceleration due to gravity heat transfer coeﬃcient height molar enthalpy of reaction speciﬁc enthalpy of vaporization rate constant frequency factor mass ﬂow rate monomer mass of reaction mixture order of reaction revolution frequency number of moles heat release rate thermal energy (heat) rate of reaction universal gas constant cross-section entropy time temperature linear velocity overall heat transfer coeﬃcient volume conversion equipment constant (stirred tank)

mol h1 m s2

mol s1 m s2

W m2 K1

W m2 K1

m kJ mol1

m J mol1

kJ kg1

J g1

function of rate law function of rate law kg h1

function of rate law function of rate law g s1

kg

g

– rpm – W kJ

– s1 – W J

mol m3 h1 J mol1 K1 L mbar mol1 K1 2 m J K1 h C m s1 W m2 K1

mol m3 s1 J mol1 K1

s K m s1 W m2 K1

m3 – –

m3 – –

m2

Subscripts

Subscript

Meaning

Example

0 A; B; P; R; S

initial value chemical compounds

T0 CA

initial temperature concentration of A

Notation

AC ad C ex f Feed G Loss M P P R RX S TAmb th V W

accumulation adiabatic coolant exchange (heat exchange) fouling feed gas loss monomer process polymer reactor, reaction mass reaction stirrer ambient thermal vapor wall

Xac DTad TAmb TC qex df TFeed rG qLoss CM TP lP dR qRX dS Xth mV dW

degree of accumulation adiabatic temperature rise ambient temperature temperature of coolant heat dissipation rate thickness of fouling ﬁlm speciﬁc weight of gas Monomer concentration process temperature thermal conductivity of polymer diameter of reactor heat release rate of reaction stirrer diameter thermal conversion mass ﬂow rate of vapor wall thickness

Greek

Symbol

Name

Practical unit

SI unit

a g

relative volume increase material constant for heat transfer diﬀerence (used as preﬁx) heat transfer coeﬃcient of equipment thermal conductivity dynamic viscosity speciﬁc weight surface tension time constant space time in continuous reactor

– W m2 K1

– W m2 K1

– W m2 K1

– W m2 K1

W m1 K1 cP ¼ mPa s kg m3 N m1 ¼ 10 3 dyn cm1 h h

W m1 K1 Pa s g m3 N m1 (¼ kg s2 ) s s

D j l m r s t t

Acronyms MTSR MTT TMR ad

maximum temperature of synthesis reaction maximum temperature for technical reasons time to maximum rate under adiabatic conditions (time to explosion)

593

594

11 Safety of Polymerization Processes

Dimensionless groups

Symbol

Name

Expression

Ne

Newton (power) number

Ne ¼

P r nS3 dS5

Nu

Nusselt number

Nu ¼

hd l

Pr

Prandtl number

Pr ¼

m Cp l

Re

Reynolds number (stirred tank)

Re ¼

n d2 r m

Parameters P ¼ power of stirrer r ¼ speciﬁc weight of ﬂuid nS ¼ revolution speed dS ¼ stirrer diameter h ¼ ﬁlm heat transfer coeﬃcient d ¼ characteristic length l ¼ thermal conductivity m ¼ dynamic viscosity Cp ¼ speciﬁc heat capacity l ¼ thermal conductivity n ¼ stirrer frequency d ¼ diameter of agitator r ¼ speciﬁc weight m ¼ dynamic viscosity

595

12

Measurement and Control of Polymerization Reactors1 John R. R. Richards and John P. C. Congalidis 12.1

Introduction

Consistent polymer properties are of paramount importance to end-user manufacturers who must produce the polymer in its ﬁnal form and shape for the intended application. These properties are the result of complex polymer architecture and composition formed in reaction and perhaps further inﬂuenced in isolation and extrusion processes. Producing consistent, uniform, and in-speciﬁcation polymer for the end-user are the tasks of the polymer process measurement and control systems. Polymer processes, whether batch or continuous, rarely run under stable equilibrium conditions. However, in order to operate such processes safely and in order to set the characteristics of the products optimally, a set of process manipulated variables must be kept constant or systematically modiﬁed over the duration of the reaction or in the course of the various reaction steps. The main process variables of this type are temperature, pressure, concentration, amount, ﬂow, and level. Speeds (agitator, gear pumps, extruders), power input and viscosity can be of substantial importance also. This chapter will discuss various measurement techniques of importance to engineers and scientists designing and operating polymer reactors and associated equipment. We will then discuss basic control concepts and conclude with more advanced control strategies. We have attempted to expand and update the ﬁne chapter from the previous edition of this book [1]. 12.1.1

Deﬁnitions

The ﬁrst step in the objective of obtaining constant or systematically adjusted operating conditions is the proper measurement of process variables. There are vari1) 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

596

12 Measurement and Control of Polymerization Reactors

ous organizations developing standards pertaining to measurement and control, including the US National Institute of Standards and Technology (NIST), the Deutsches Institut fu¨r Normung (DIN), the International Organization for Standardization (ISO), and the International Electrotechnical Commission (IEC). In measurement technology there are a number of standard measurement concepts [2–8]:

A measurement installation translates the measurement quantity to a measured value. The measured value consists of a numerical value and a unit of measurement. The result of measurement consists of several measured values, which can be reproduced as a value or a curve of values. The measurement follows certain measurement principles whose application results in measurement methods. The measuring method is executed with a measuring installation, which consists of diﬀerent measuring instruments and auxiliary devices. Measuring instruments are situated in the signal ﬂow and are essential to the measuring installation. Auxiliary devices usually serve to provide the auxiliary energy which is needed to provide measuring signals. If the measuring signals pass through several measuring devices connected in series, then a measurement chain is present.

According to Figure 12.1 a measurement installation translates the measurement quantity to a measured value. Here we follow the connections of the sensor, transmitter, and transducer as described in Ref. 7:

The chain consists of a sensing element (sensor), a measuring instrument whose input consists of the measured quantity and its output consists of the corresponding measurement signal. At the end of the chain lies the output unit, which provides the desired measurement value.

Measured quantity (process variable) Sensing element

Transmitter (signal generator/ line driver)

Transducer

A typical process transducer [7]. D. E. Seborg, T. F. Edgar, D. A. Mellichamp, Process Dynamics and Control, Copyright 8 2003 John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. Fig. 12.1.

Transmitted signal (to controller)

12.1 Introduction

Between the measured quantity and the measured variable to the output unit lie transducers (see Figure 12.1), which process (transform and amplify) the signal from the measurement instruments ahead of them and feed their signal to the output measurement device. Measurement transmitters are those devices which convert an analog input signal to an unambiguous standardized analog output signal [4]. The transmitter is usually required to convert the sensor element output to the standardized form compatible with, for example, a controller input. Measurement transducers are devices which have input and output signals in the same or diﬀerent structure (analog/digital or digital/analog). Sometimes the terms ‘‘transmitter’’ and ‘‘transducer’’ are used interchangeably. One calls the measuring device, with its case and accessories, a measuring instrument. The instrument gauge or indicator is the part of the measurement device whose movement allows the reading of the measurement value. To the measurement instrument also belongs a scale. Between the end points of the scale lies the range, or span, of the measurement device within which individual values of the measured quantity are obtained. In order to calibrate and adjust the scale standards are used, which associate a precisely deﬁned output measurement value to a measured quantity.

12.1.2

Measurement Error

Accuracy of a measurement is the degree of conformity of an indicated value to an accepted standard or ideal value [4]. Repeatability is the closeness of agreement among a number of consecutive measurements of the output for the same value of the input. Each individual measurement x i deviates from the correct or ideal measured value m [2, 8]. This diﬀerence is called the measuring error or deviation d i [Eq. (1)]. di ¼ xi m

ð1Þ

Measurement errors can be systematic errors, which are found in the measuring method or in the equipment; with knowledge of the correct measured value these errors can be compensated by a correction term or a correction factor. Random errors, which oscillate in sign and amount, cannot be corrected. Averaging eliminates these errors. The simplest mean is the arithmetic average value x, which is an estimate of the true mean value m [Eq. (2)].

x¼

n 1X xi n i¼1

ð2Þ

597

598

12 Measurement and Control of Polymerization Reactors

The smaller the random errors of the individual measurements and the larger the number of measurements, n, the smaller the deviation of the arithmetic average from the true mean. The empirical sample standard deviation, s, is a measure of the measurement error [Eq. (3)].

s¼

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX n u u ðx i xÞ 2 t i¼1

n1

ð3Þ

With very large n, the discrete measurements x i can be thought of as continuous measurements x and the sample standard deviation s becomes the true standard deviation s of a Gauss normal diﬀerential distribution function [Eq. (4)]. " # 1 1 xm 2 f ðxÞ ¼ pﬃﬃﬃﬃﬃ exp 2 s 2ps

ð4Þ

A tabulation or histogram of how often the various values of x i occur in replication is approximated by this distribution. From the Gaussian distribution the statistical conﬁdence of the individual values is obtained. From the cumulative values of this distribution, 68.3% of all measurement values are within the interval x G s, 95.4% are within x G 2s, and 99.7% are within x G 3s. Further statistical concepts can be found in Refs. 5 and 8.

12.2

Measurement Techniques

The measurement technique to be chosen is principally determined by the measured quantity and by the accuracy with which the variable must be measured. The measuring instrument produces a signal, which must be transformed in such a way that it can be registered by an indicator or recorder and further processed. This requirement is fulﬁlled directly by some measuring methods; however, in most cases a measurement transmitter is operated between the sensor and the measurement device. Pneumatic pressures between 3 and 15 psig, or between 0.2 and 1.0 bar, or electrical currents between 4 and 20 mA are used [7] as standard signals. Pneumatic signals are transmitted in plastic tubing and a distance of approximately 300 m between the sensor and the transducer can be accommodated. Electrical direct current signals can be transferred over a distance of a few kilometers. Electrical signals are much more commonly used today than pneumatic signals. To accommodate larger distances the analog signals must be converted to digital signals; in that case the distance between measurement transducer and receiver can be expanded practically at will. In the Sections 12.2.1– 12.2.11 some particularly common measurement methods from operational practice will be discussed.

12.2 Measurement Techniques

12.2.1

Temperature

In polymer technology temperature is one of the very important variables of state, probably the most important. Depending on the measurement range, it is measured with electrical or mechanical contact thermometers or with optical methods. With contact thermometers one attempts to keep the thermal capacity of the measuring device and heat transfer as small as possible so that actual temperature changes can be obtained as soon as possible. Often the sensitive measurement sensors are protectively shielded. Here compromises must be made which are appropriate for individual cases. The most widely used thermometer in polymer technology is the resistance temperature detector (RTD). For higher temperatures or with point measurements, the thermocouple is often preferred. Mechanical contact thermometers (expansion thermometers, rod thermometers, or bimetallic thermometers) are nearly always used only for local measurements. For optical thermometers, radiation pyrometers are used. Resistance Thermometers In resistance thermometers (or RTDs) one uses the dependency of electrical resistance on the temperature [2]. The resistance of metals almost always rises with temperature. In practice one uses platinum thermometers. A light platinum wire on a ceramic support is covered with a protective layer of glass, enamel, or magnesium oxide. This ‘‘measurement resistance’’ is the end of one so-called measurement element, which ends in connecting terminals. The whole assembly is in a sleeve, which is solidly connected to the measuring point in the apparatus with ﬂanges or screws (Figure 12.2). For pure metals, resistance thermometry is governed by Equation (5), where R t is the resistance at temperature t, R 0 is the resistance at the reference temperature, usually 0 C, and a; b; c are coeﬃcients based on calibration points [3]. 12.2.1.1

R t ¼ R 0 ð1 þ at þ bt 2 þ ct 3 þ Þ

3

2

5

Resistance thermometer thermowell assembly. 1: Measuring resistor; 2: terminal block; 3: connection head; 4: protective pipe; 5: screw tap [1]. Fig. 12.2.

ð5Þ

4

1

599

600

12 Measurement and Control of Polymerization Reactors

For example, platinum has a temperature coeﬃcient of resistance a ¼ 0:00385 W/ (W C) in the range 0 to 100 C. The measuring range of commercial-purity platinum thermometers is situated between about 200 and 850 C and the accuracy of these thermometers is G0.15% of span over the full range [4]. Resistance versus temperature can usually be described by just a quadratic function, which can be linearized in measurement ampliﬁers. If large temperature ranges are not present, the linearization can often be omitted. The resistance is usually measured in a bridge circuit [2]. Thermocouples A thermocouple consists in principle of two wires made of dissimilar metals soldered together in two junctions, which form a closed circuit (Figure 12.3). At the contact junctions a voltage gradient is produced and a current, which depends on the temperature, ﬂows around the circuit. So long as both junctions have the same temperature the voltages compensate and no current ﬂows; if the junctions have diﬀerent temperatures, then a voltage gradient is produced, which can be measured in an open circuit and which is approximately proportional to the temperature (although the relationship is nonlinear over large temperature scales). If one of the two soldered connections is held at a constant reference temperature, then the other connection can serve as a temperature probe. Suitable metal pairs, which supply suﬃciently large thermal voltages to deliver 12.2.1.2

Constantan Copper

Copper

Measuring Junction Reference Junction Thermocouple wiring diagram [1].

Fig. 12.3.

12.2 Measurement Techniques

reliable measured values, are copper/constantan (type T), iron/constantan (type J), chromel/alumel (type K) and platinum (70%)–rhodium/platinum (94%)–rhodium (type B). Their thermal voltages permit measurements up to 371 C, 760 C, 1149 C, and 1700 C respectively [3]. The standard-grade error limits for the ﬁrst three types are G0.8 C, G2.2 C, and G2.2 C respectively. Thermocouples oﬀer the beneﬁt that one can obtain point measurements and they are also suitable for measurements of high temperature. The outside shape resembles a resistance thermometer. The usually inevitable shield tube makes the beneﬁt of the point measurement often invalid. A possibility to prevent this is to solder together the head of the shield tube with a soldered joint; at the very least, it is then possible that the components are less sluggish (with time constants of a few seconds). If that is not possible, one must accept time constants of some minutes. Expansion Thermometers The thermal expansion of gases, liquids, and solids is still used to build simple thermometers, which indicate the temperature in the direct proximity of the measurement location. The most usual measuring media are alcohol and mercury. However, mercury poses a health hazard and should only be used when necessary. The measuring range with liquid-ﬁlled thermometers (mercury with a glass stem) is situated between 200 and þ370 C, with an accuracy of G1% of full scale, usually less than G0.5 C [4]. The principle of the thermal expansion of solids is used in bimetallic thermometers, which consist of two metal strips of diﬀerent materials with diﬀerent coeﬃcients of expansion ﬁrmly connected together. Temperature diﬀerences bend these bimetallic springs and they are usually wound in spirals or coils. The useful measuring range is from 73 to þ537 C with an accuracy of G1% of full scale [4]. Bimetallic strips are used for temperature compensation in mechanical instrument movements. They are also used in switching relays to indicate whether a limiting value is exceeded. 12.2.1.3

Radiation Pyrometers High temperatures are measured with radiation pyrometers, which capture the radiation emitted by the object [4, 9]. They are important mostly for hightemperature, non-contact measurements. According to the Stefan–Boltzmann law the intensity of radiation emitted by a black body increases with the fourth power of the absolute temperature. According to Wien’s law the wavelength at maximum radiation is inversely proportional to the absolute temperature. One can build thermometers based on these two laws, by measuring either the radiation at all wavelengths or the radiation at speciﬁc wavelengths [4]. In the thermocouple pyrometer (Figure 12.4) one collects the radiation emitted from the object on a blackened area whose temperature rise is measured with a thermocouple. In portable devices an eyepiece lens is behind the measuring area so that one can aim at the object. Measuring errors result from space losses between object and measuring instrument; for well-known distances compensation can be made for the errors. Addi12.2.1.4

601

602

12 Measurement and Control of Polymerization Reactors

4

1

2

3

7

5

8

6

Principle of the radiation pyrometer. 1: Radiation source; 2: radiant heat; 3: collecting lens; 4: black surface; 5: thermocouple; 6: temperature indication; 7: eyepiece; 8: measuring instrument [1]. Fig. 12.4.

tionally most objects are not black-body radiation emitters; this can be corrected by using the Planck radiation law to compare the radiation intensities for diﬀerent wavelengths. Optical pyrometers can be used for temperature ranges of 760–3500 C [4]. In special cases (narrow-band and total radiation pyrometers) they can be used for much lower temperatures (between 40 and þ4000 C) if a non-contact measurement is desired [4]. Pyrometers are important also for qualitative observations, for example as automatic ﬂame guards, and for protection in the combustion chambers of thermal installations so that, after the ﬂame is extinguished, fuel oil or heating gas does not enter the combustion chamber to form an explosive mixture. 12.2.2

Pressure Measurement

Not only is the pressure a signiﬁcant thermodynamic process variable, but pressure measurements are also fundamentally important for safety reasons. Besides the absolute pressure in an apparatus, one is often interested in a pressure diﬀerence, for instance in cases of ﬂow measurements with oriﬁces, and level measurements. Process equipment pressures are usually measured by the elastic deformation of bellows, a Bourdon tube, a diaphragm, or a capsule [3]. Bellows pressure gauges (Figure 12.5) are very compact and permit linear pressure/path relationships. They are particularly useful for the measurement of

12.2 Measurement Techniques

p1

Fig. 12.5.

p2

Bellows diﬀerential pressure gauge [1].

pneumatic pressure signals and diﬀerential pressure measurement, and have been used in pneumatic controller feedback elements. The gauges are available for pressures of 0–12 mbar (minimum range) and 0– 140 bar (maximum range) [3]. For vacuum applications the range is 0–12 mbar vacuum. Accuracy is G0.1% to G2% of span [4]. The Bourdon pressure gauge (Figure 12.6) is a versatile instrument. In this instrument a tube with an oval cross-section (Bourdon tube) is bent or coiled inside the instrument. If the internal pressure rises, then the oval cross-section expands and the tube increases its radius of curvature. At its end a pointer is attached, which is carried by the movement of the free end. Pressure gauges are available for pressures of 0–0.3 bar (minimum range) and 0–7000 bar (maximum range) [3]. Above 100 bar the tubes are arranged in several windings. For vacuum applications the range is 0–1 bar vacuum. In a diaphragm pressure gauge the elastic deﬂection of a circularly clamped metal disk is transferred to a pointer. Such devices are suitable for measuring ranges of 0–0.5 mbar and 0–70 bar [3]. For vacuum applications the range is 0– 0.5 mbar and 0–1 bar vacuum. They are also suitable for diﬀerential pressure measurement if both sides of the diaphragm are connected with the respective measuring points. Bourdon tube and diaphragm pressure gauge designs are easily calibrated and have standard designs. A capsule is formed by joining two or more diaphragms together, with the total deﬂection of the assembly equal to the sum of the deﬂections of each capsule.

603

604

12 Measurement and Control of Polymerization Reactors

Fig. 12.6.

P Principle of the Bourdon pressure gauge [1].

Pressure sensor selection guidelines can be found in Ref. 3. Pressure sensors have various transducers connected to them to convert the measurement into a usable signal. Strain gauges and piezoresistive and piezoelectric pressure transducers are commonly used to produce the measured values. Currently, smart transducers are used; they are characterized by inclusion of microprocessors and electronics to store ranges, calibrations and diagnostics, and for other tasks of data handling from the ﬁeld instruments to the control room. 12.2.3

Weight

Balances compare unknown weights with one standard weight, or alternatively with a well-known force. Balances for weight measurement are very reliable and accurate instruments; their measuring accuracy is often better than G0.1% [4]. It must be kept in mind that the gravitational attraction on the Earth’s surface can vary by as much as 0.5% but this can be compensated by calibration standards [4]. The hydraulic and electrical strain gauge load cells and the electrical strain

12.2 Measurement Techniques

gauge are now used as primary sensors. The strain gauge type of load cell is capable of a sensitivity of 1 part in 20 000 [6]. Other more accurate sensors are available, such as the variable capacitance transducer capable of a sensitivity of 1 part in 1 000 000 [6]. Either bulk weighing of vessels or weighing platforms are generally used in the polymer industry. Types of balances used include weighing platforms, portable platform scales, and truck scales. Common methods of weighing include mechanical lever scales, spring-balance scales, and load cell weighing systems. Portable bench scales have capacities from 10 to 1000 kg. Beam-type weighing platforms have capacities from 200 to 5000 kg [4]. In electronic balances the force exercised by the load on the base is measured with a force measurement cell (for example, an electrical strain gauge) and transformed into an electrical signal. This signal is processed within the balance in a digital or analog fashion and is then displayed. These systems can be calibrated and used in fully automatic installations. As force measurement cells one can use pressure sensors with lower requirements for accuracy. These are installations in which the weight is transformed into pressure pneumatically or hydraulically; one can determine levels in tanks or reaction vessels by this method, which can also be used to meter automatically the main ingredients of a batch process, for example the water and monomer phase in an emulsion or suspension polymerization. Strain gauges are used widely in force measurement cells. When a strip of conductive metal is stretched or compressed by a mechanical load, without reaching its elastic limit, its electrical resistance changes. In order to use this principle, one sticks strain gauges onto elastic cylinders or bending rods, whose deformation is thus transferred to the strain gauges. An elastic cylinder carries four to eight resistance strain gauges, which are arranged in a bridge circuit. The load on the cell then detunes the bridge and creates a measuring voltage proportional to the detuning. This conﬁguration also provides compensation for resistance variations due to temperature. Modern units contain almost exclusively digital balances for signal indication and analysis. With these units one can meter in weights of ingredients in a given order. Belt balances (Figure 12.7) allow simultaneous conveying and weighing, so they are used frequently as metering devices. In these balances, an endless conveyor belt is designed to be the load part of a balance [6]. Weight changes are measured by load cells, and then integrated over short intervals to give the rate of ﬂow. Totalizers can give the total weight over a given time. The belt is covered continuously with the material to be weighed. The feed requires a variable metering device, for example a vibrating hopper or a screw-type feeder. 12.2.4

Liquid Level

Level measurements are used either to control level in vessels or apparatus, or to measure throughput. Level measurements are important in order to estimate inventories and their movement, or to assure the smooth operation of processes. Fur-

605

606

12 Measurement and Control of Polymerization Reactors

slidegate valve

conveyor

M

motor balance bridge force receiver Fig. 12.7.

data processing

Conveyor belt balance [1].

thermore, in batch processes, level measurements are important to limit ﬁlling and emptying steps as well as for alarming. For continuous processes, they serve to maintain a constant level in the vessels where the ﬂow occurs (for example, the sump in a rectiﬁcation column or the contents of a chemical reactor). Flow measurements are indispensable to metering and feeding functions. A great variety of level measurement techniques are available. These involve point-contact, visual, buoyancy, ﬂoat, and hydrostatic methods, and radio-frequency, ultrasonic, microwave, nuclear radiation, resistance tape, and thermal level systems [3]. Point-contact measurements of liquid level are basically length measurements. For example, one observes the wetting of a dipstick through a sight glass in the vessel, or the movement of a ﬂoat that follows the surface of the liquid in the reservoir. Similarly the point-contact level in a solid hopper can be measured directly by a mechanical plumb line [4], which is attached at the end of a measuring tape that is unwound from a barrel so that the plumb line hits the solid surface; markings on it permit the measurement of the length digitally. Likewise, the length of a hanging string can monitor the position of a ﬂoat on a liquid surface. In the ﬂoat method, a ﬂoating gauge is not introduced into the vessel itself but into a diﬀerent container, such as a side gauge glass, which is arranged in parallel and communicates with the main vessel. In this way disturbances arising from the movement of the ﬂuid (boiling or circulation) are avoided. Floating gauges can also be used in order to indicate the position of the separation surface between two immiscible liquids. In the buoyancy method, displacement bodies measure levels in liquid containers. By measuring the diﬀerence in weight of a partially submerged body at various degrees of submergence, one may determine the level of the liquid in which it is

12.2 Measurement Techniques

submerged [3]. The device fails if the weight or the volume of the displacement body changes during the process, for example because of scaling or if the density of the medium changes. In the hydrostatic method, the pressure at the base of the liquid, which is related to the height of the liquid above the base, is measured. The measured pressure depends directly on the liquid height according to the Bernoulli equation or mechanical energy balance [10–12] [Eq. (6), where z2 is the level at the surface of the liquid, z1 is the level at the base, p2 is the pressure at the surface, p1 is the pressure at the base, r is the liquid density, and g is the acceleration due to gravity]. In pressurized reservoirs one measures the pressure diﬀerence p1 p2 between the base and the gas space. z2 ¼ z1 þ

p1 p2 rg

ð6Þ

In a variant of this method, an inert gas is bubbled through the liquid with a dip tube. If the density of the liquid is known, one can also determine the level through weighing. It should be noted that in equations such as Eq. (6) a conversion factor may be necessary in unit systems that are termed inconsistent. In the US Customary System (USCS) the force unit is the lb-force (lb f ) and in SI units the force unit is the s2 is used in USCS Newton (N). The conversion factor g c ¼ 32:174 lb m ft lb1 f units to resolve expressions involving forces in lb f and masses in lb m so that Newton’s law [Eq. (7), where F is force, m is the mass, and a is the acceleration] is satisﬁed. F¼

ma gc

ð7Þ

In the SI unit system, which is used in this book, g c ¼ 1 kg m s2 N1 and so this conversion factor is not necessary. In polymer reactors one has to deal often with very viscous liquids or melts. Here ﬂoating gauges or displacement methods fail. An alternative is level measurement with radioactive gamma-ray beams (Figure 12.8) that can penetrate metal walls [4]. The entire reactor is traversed by gamma-radiation; its intensity is given according to the Beer–Lambert law [Eq. (8), where I; I0 are the intensities behind and in front of the object, e is the absorption coeﬃcient, r is the density of the medium and l is the path length. I ¼ I0 expðerlÞ

ð8Þ

Radiation sources include cobalt-60 and cesium-137. Receivers are scintillation counters. The specimen is usually arranged in a rod-like shape and installed on a point-shaped detector, particularly if a continuous measurement is concerned. The beneﬁt of the method is that it operates in a non-contact fashion. However, it re-

607

608

12 Measurement and Control of Polymerization Reactors

Melt entrance

Radioactive preparation

Counter

Screen

M

Melt exit Fig. 12.8.

Level measurement using radiation [1].

quires extensive safety measures [4]. The specimen must be shielded with lead against the environment, so that only the direction toward the detector is free. Mounting, dismantling, and maintenance may be undertaken by trained personnel only, and the radiation limiting values are to be checked constantly. For this reason this method should be used only when other methods fail. The accuracy is 3 mm to 1% of height span [4]. Other level-measuring methods are also possible [3]. Capacitance measurements use the change in capacitance of a capacitor, which is built in a probe through the vessel wall (Figure 12.9). Level can also be measured by the change in resistance of a resistance tape on the vessel wall. One can also obtain levels by introducing vibrating sensors into the container at diﬀerent levels until the vibrations are damped by the liquid (Figure 12.10). Microwave radar level systems are also possible, which are not aﬀected by density changes in the beam path as are ultrasonic beam techniques. 12.2.5

Flow

Flow measurements establish the volume or mass of a ﬂuid per unit of time through the measurement device. The methods are similar for both liquids and gases; nevertheless it must be considered that liquids have a substantially higher

12.2 Measurement Techniques

b a

a

c

c

b

A

B

Electrical measurement of liquid level. (A) Capacitive measurement; (B) conductivity measurement; a: electrode; b: measuring instrument; c: material [1]. Fig. 12.9.

density than gases. Also, the density dependence on pressure and temperature must be particularly considered with gases. Because of these considerations, different forms of implementation result [2–4]. Types of measurement systems include diﬀerential pressure, magnetic, turbine, oscillatory, mass ﬂow, ultrasonic, and positive-displacement meters [3]. Direct ﬂowmeters are generally counters. They couple the number of selfrepeating periodic processes (for example, circulation) with time measurement. The most widely used device of this type is the oval gear ﬂowmeter (see Figure 12.11). For cold water the ﬂow ranges are 0.8–5 L min1 to 416–2669 L min1

A Mechanical measurement of liquid level. (A) Vibration probes; (B) rotary wing probe [1]. Fig. 12.10.

B

609

610

12 Measurement and Control of Polymerization Reactors

V1

V2 A Fig. 12.11.

B

C

Oval-shaped gear ﬂowmeter [1].

and for heavy oil (5–300 cP) the ﬂow ranges are 0.2–7.2 L min1 to 167–3826 L min1 [3]. In the oval wheel counter, two oval wheels run in gears moving in opposite directions; the pressure drop along the device drives them. The wheels lock successively and the volumes V1 and V2 convey them and release them again (Figure 12.11). With each full rotation of the pair of wheels, the volume V ¼ 2ðV1 þ V2 Þ is conveyed through the counter [1]. The devices can be calibrated and high accuracies can be obtained (0.1% [4]); with the smallest types, inaccuracy is highest. The devices are very susceptible to contamination and must therefore be protected by ﬁne ﬁlters of solid particles. Gases can be measured with the rotating lobe ﬂowmeter, which is similar to the oval gear ﬂowmeter (see Figure 12.12). The diﬀerence is that the rotating lemniscate-like lobed impellers do not run directly on one another and are not geared together. They are coupled through a gearbox. The devices are calibrated and can be constructed for ﬂow stream capacities between 2 and 4800 m 3 h1 [4]. They have good repeatability (0.015%) at high ﬂow rates [4]. They can be used for temperatures up to 205 C and pressures up to 83 bar [4]. The temperature and pressure of the gas are captured and processed so that the volume is indicated in standard conditions of temperature and pressure (STP). The measured variable is deduced from the speed of the rotors, which is obtained through a magnetic clutch from the pressure chamber.

A Fig. 12.12.

B

Lobed-impeller ﬂowmeter [1].

C

D

12.2 Measurement Techniques

Among the indirect volume ﬂowmeters, turbine ﬂowmeters are the most important. They measure the speed of the ﬂowing medium from the speed of an impeller, or rotor. There are devices arranged in an axial or radial direction to the ﬂow. Turbine ﬂowmeters can be used for liquids, gases, and vapors. The rotating vane meter, which is arranged in the radial direction to the ﬂow, is the most well known and is used for ﬂows of 1–4800 m 3 h1 [4]. The measuring range is 1:10 with an accuracy of about G0.1% [4]. The turbine ﬂowmeter with electrical impulse pick-up, arranged in the axial direction of the ﬂow, is of importance for gas ﬂows. It contains an impeller with magnetic wings which, with each passage, induce a voltage surge in a solenoid coil situated outside. The frequency of this alternating current is proportional to the ﬂow. Reproducible measurements require turbulent ﬂow, because only then does the slip not depend on the ﬂow velocity. The measuring range also depends on the viscosity of the ﬂowing medium. There exist gas turbine ﬂowmeters for ﬂows up to 4500 m 3 h1 [4]. Linearity is G1% over a ﬂow range of 20:1 [4]. Rangeabilities can be as much as 100:1. If one wants total volumes using ﬂowmeters, for example for accounting purposes that require high accuracy, then the instantaneous measurement must be integrated with time. This can be accomplished with pneumatic or electrical counters. Measuring methods which deduce the ﬂow directly from ﬂow characteristics are very common. There are versatile methods being used, which all depend on the velocity of the ﬂowing medium. The most important are measurements with oriﬁce devices, with suspended bodies, and according to a magnetic-inductive method. Beyond that there are ﬂowmeters which use the pressure drop of a ﬂowing liquid in a capillary, and magnetic-inductive, ultrasonic, and Coriolis ﬂowmeters. Restrictor devices such as venturi tubes, oriﬁce plates, and nozzles operate on the basis of the Bernoulli equation [10, 11]. A venturi tube is depicted in Figure 12.13. For incompressible ﬂuids without friction or pump work at points upstream (1) and downstream (2) on a streamline in the ﬂow, the Bernoulli equation becomes Eq. (9), where z1 ; z2 are the levels at the two points, p1 ; p2 are the pressures at the

Fig. 12.13.

d1

d2

p1

p2

Venturi nozzle. d1 : Diameter of the pipe; d2 : diameter of the throat.

611

612

12 Measurement and Control of Polymerization Reactors

two points, v1 ; v2 are the average velocities at the two points, a1 ; a2 are the kinetic energy correction factors at the two points, r is the liquid density, and g is the acceleration due to gravity. p1 a1 v12 p2 a2 v22 þ gz1 þ ¼ þ gz2 þ r 2 r 2

ð9Þ

The deﬁnition of the kinetic energy correction factor is the area average of the cubed velocity v 3 over the average velocity cubed, v 3 [Eq. (10)] [10]. ð 1 3 v dA v3 A a¼ ð 3 ¼ 3 v 1 v dA A

ð10Þ

Note that a ¼ 2 for laminar ﬂow and a ¼ 1 for plug ﬂow. For equal heights (horizontal restrictors) the Bernoulli equation becomes Eq. (11). p1 a1 v12 p2 a2 v22 þ ¼ þ r 2 r 2

ð11Þ

The continuity equation can be written at two points on a streamline in the ﬂow [Eq. (12), where A1 ; A2 are the ﬂow areas at the two points, r1 ; r2 are the liquid densities at the two points] [10]. r1 v1 A1 ¼ r2 v2 A2

ð12Þ

Since the ﬂuid is incompressible, Eq. (13) holds. r1 ¼ r2 ¼ r

ð13Þ

Combining Eqs. (11)–(13) and eliminating v1 gives Eq. (14), where b is the ratio of diameters d2 =d1 . 1 v2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 a1 b 4

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ð p1 p2 Þ r

ð14Þ

To compensate for the friction loss and kinetic energy factor assumptions, the velocity equation is modiﬁed to Eq. (15), [10] where Cd is an empirical discharge coeﬃcient. Cd v2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ4 1b

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ð p1 p2 Þ r

ð15Þ

12.2 Measurement Techniques

Therefore, for restrictors, the volumetric ﬂow qv relationship is Eq. (16) [10]. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cd A2 2ð p1 p2 Þ ð16Þ qv ¼ v2 A2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ4 r 1b The mass ﬂow qm relationship is Eq. (17) [11]. Cd A2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qm ¼ rv2 A2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ð p1 p2 Þr 1 b4

ð17Þ

For compressible ﬂow, similar equations may be used. The mass ﬂow equation is modiﬁed to be Eq. (18) [6, 10, 12]. Y is a dimensionless expansion factor that is a function of p2 = p1 ; b, and k, where k ¼ Cp =Cv is the ratio of speciﬁc heats of the gas [6, 10]. Cd YA2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qm ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ð p1 p2 Þr1 1 b4

ð18Þ

Note that Y ¼ 1 for an incompressible ﬂuid. Both factors Cd and Y can be determined for diﬀerent tube nominal sizes and ﬂow velocities with air or water, and can also be obtained from charts in Refs. 6 and 12, or tables in Ref. 2. Typically the data can also be used for other media, if the Reynolds numbers are kept constant. The most important restrictors are the venturi tube (Figure 12.13) and the oriﬁce plate (Figure 12.14). The discharge coefﬁcient Cd is about 0.94–0.99 for a venturi tube and about 0.6 for an oriﬁce plate [6, 11]. The simplest implementation of a restrictor is the standard oriﬁce plate, which also supplies the most accurate values (Figure 12.14). The oriﬁce is a sharp-edged disk with diameter d2 in a pipe with diameter d1. The pressure tap positions relative to the plate vary depending on the desired design [6]. For so-called radius taps, the pressure-measuring drillings are situated inside the ﬂanges in each case at a distance of one pipe diameter upstream and half a pipe diameter downstream of the oriﬁce [6]. The accuracy of the oriﬁce plates is G0.25% to G0.5% of actual ﬂow [4]. The rangeability with a measurement error of less than G1% of the actual ﬂow is 3:1, which can be improved with smart transmitters [4]. The oriﬁce plate is the restrictor most frequently used in practice. For the measuring accuracy of all standard oriﬁces it is essential that the edge sharpness be maintained in continuous operation. If this is not the case, then the standard nozzles have advantages, and their permanent pressure loss is somewhat lower than in standard oriﬁces. They also have advantages for very small throughputs and are used for very dirty liquids. In a venturi tube the so-called diﬀuser is attached to the normal nozzle piece and serves for backpressure gain. This has the smallest permanent pressure loss among the various restrictors. The accuracy of venturi tubes is G0.75% of the uncalibrated rate and G0.25% of the rate calibrated in a laboratory.

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12 Measurement and Control of Polymerization Reactors

1

2

d2

d1

3

3

p1

p2

Fig. 12.14. Oriﬁce plate and ﬂanges. 1: Oriﬁce plate; 2: ﬂange supports; 3: pressure measurement drill taps [1]. Note that the pressure tap positions vary, depending on the desired design [6].

All measurements with restrictors require turbulent, evenly distributed ﬂow. In order to smooth out disturbances in the ﬂow stream, oriﬁces and nozzles require a straight and smooth upstream piping length of between six and 40 pipe diameters [6] depending on the nature of the upstream ﬁtting and between two and four pipe diameters downstream to the nearest ﬁtting. Sources of error are clusters of gas bubbles or dirt particles (in ﬂuids) or condensing liquids (for gases) in the diﬀerential pressure tubes. For gases the pressure ratio p2 = p1 at the restriction must not be less than about 0.53; otherwise sonic velocity is achieved in the restriction [10]. Measuring methods with ﬂoating bodies work with vertically arranged, inverted cone-shaped or slightly tapered tubes, where the ﬂuid ﬂows through from bottom to top. In the rotameter, the ﬂow holds thereby a ﬂoating body in equilibrium (see Figure 12.15), which depends on the ﬂow velocity and the lift of the body against its weight. The force balance is written as Eq. (19) [10], where FW is the (downward) gravity force, FB is the (upward) buoyancy force, and FD is the (upward) drag force.

12.2 Measurement Techniques

FB, FD

a

b

FW

Fig. 12.15. Flow measurement with ﬂoat (rotameter). a: Conical ﬂow pipe; b: ﬂoat; FW : gravity force; FB : buoyancy force; FD : drag force [1].

FW ¼ F B þ FD

ð19Þ

If Vf is the volume of the ﬂoat, rf is the density of the ﬂoat, and r is the density of the ﬂuid, the force balance becomes Eq. (20). FD ¼ Vf rf g Vf rg

ð20Þ

The drag force is given by Eq. (21), where CD is the drag coeﬃcient, A f is the projected area of the ﬂoat, and vmax is the maximum velocity past the ﬂoat. FD ¼ CD A f

2 rvmax 2

ð21Þ

615

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12 Measurement and Control of Polymerization Reactors

Substituting the drag force into the force balance we get Eq. (22). CD A f

2 rvmax ¼ Vf gðrf rÞ 2

ð22Þ

The volumetric ﬂow rate qv can be related to the maximum velocity past the ﬂoat by Eq. (23), where A t is the tube area at the point of constriction and A f is the ﬂoat area. qv ¼ vmax ðA t A f Þ

ð23Þ

To size the rotameter, vmax can be eliminated between these two equations and the expression simpliﬁed so that the volumetric ﬂow rate qv is obtained as Eq. (24), pﬃﬃﬃﬃﬃﬃ where Cd ¼ 1= CD is a discharge coeﬃcient with values between about 0.6 and 0.8 [9]. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2gVf ðrf rÞ ð24Þ qv ¼ Cd ðA t A f Þ rA f For a linearly tapered tube with the bottom diameter the same as the ﬂoat diameter, the area for ﬂow is a linear function of the height of the ﬂoat h [Eq. (25)] [10]. dt2 d 2f ¼ ðd f þ ahÞ 2 d 2f ¼ 2d f ah þ ðahÞ 2 A 2d f ah

ð25Þ

Substituting this into the mass ﬂow equation gives Eq. (26), from which the ﬂow rate depends almost linearly on the height of the ﬂoating element h and the viscosity of the medium; in each case it is calculated with the help of calibration curves, which depend on the instrument and the calibration medium (air or water). p qm ¼ rqv ¼ rvmax ðdt2 d 2f Þ ¼ 4

2p rvmax d f a h 4

ð26Þ

The ﬂoating elements are available in diﬀerent forms; for simple gas measurements the ﬂoating element is a sphere; for ﬂuids one mostly ﬁnds a cylindrical ﬂoating element (Figure 12.15), which is shaped downward like a cone and at the upper edge like a ﬂat truncated cone-like disk. This disk is sharply grooved in the side, in order to induce a rotation in the equilibrium position. This movement stabilizes its position in the center of the conical pipe. The position of the ﬂoating element can be observed directly, if the conical guide tube consists of glass or quartz. If that is not possible, then its position can also be measured magnetically, if it contains a magnet and the pipe consists of a nonmagnetic material. Rotameters are useful for ﬂows from 0.01 cm 3 min1 to 920 m 3 h1 of liquid, or 0.3 cm 3 min1 to 2210 m 3 h1 of gas [4]. The measurement accuracy for industrial rotameters is approximately G1% to G2% of full scale over a 10:1 range [4].

12.2 Measurement Techniques

Magnetic-inductive ﬂowmeters consist of one nonmagnetic, insulated pipe, which is traversed by a magnetic ﬁeld. If a conductive ﬂuid moves through the pipe, the ions contained in it are deﬂected crosswise to the ﬂow direction and to the magnetic ﬁeld. Two electrodes, which are attached at the same height inside the pipe, then indicate a voltage diﬀerence, which is proportional to the ﬂow velocity. The method is independent of the ﬂow proﬁle, density, viscosity, and contamination. Moreover it does not build up an additional pressure drop in the line. The measuring accuracy is within G1% [1]. For the time being, there are still diﬃculties for nominal pressures over 16 bar. When a ﬂuid moves in a tube that is rotated, it experiences a Coriolis force proportional to its mass and velocity and to the angular velocity of the tube. In the Coriolis ﬂowmeter, the tube is not rotated but vibrated. Vibrations in the tube are set up and a small elastic deformation in the tube results from the Coriolis force. The deformation magnitude is related to the mass ﬂow rate. These ﬂowmeters have the advantage that the measurement is independent of the temperature, density, pressure, viscosity, or ﬂow proﬁle [3]. The accuracy is high, G0.15% within the range of 10:1 of full scale rate [4], but the initial cost is high. The ﬂow range is 0– 28300 kg min1 [4]. The high accuracy can be very useful for ingredients that are precisely metered to polymerizers, despite the high cost. Because of their operating principle, Corilis ﬂowmeters can simultaneously obtain the density of the ﬂuid. 12.2.6

Densitometry, Dilatometery, and Gravimetry

Density is the mass per unit volume of a substance. Speciﬁc gravity is the density of the substance relative to a standard such as liquid water at 4 C. The density of liquids is monitored by hydrometers, weighing a ﬁxed volume (density balance), the Coriolis method (described in Section 12.2.5), and vibrating methods. Hydrostatic, displacement ﬂoat, sonic, and radiometric methods are among others also used [4]. A hydrometer consists of a tube closed at both ends, with one end enlarged into a bulb that contains ﬁne lead shot or mercury to cause the instrument to ﬂoat upright in a liquid. In the glass tube is a scale calibrated so that the reading on it, level with the surface of the liquid in which the hydrometer is ﬂoating, indicates the number of times the liquid is heavier or lighter than water, that is, the speciﬁc gravity of the liquid. Hydrometers have been connected photometrically or mechanically to produce a useable electrical signal [4]. The density balance operates on the principle of weighing a ﬁxed volume. In it the liquid ﬂows through a U-tube, which forms the load part of a balance (Figure 12.16). The deﬂection of the balance can be indicated directly, and usually one uses electrical or pneumatic transducers so that the measured values can be transferred and processed further. Density balances must be installed so as to avoid vibrations. In the tuning fork densitometer one utilizes the dependency of the frequency of a tuning fork or a vibrating plate, which is excited to produce an under-damped oscillation, on the density of the surrounding medium. The resonant frequency of

617

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12 Measurement and Control of Polymerization Reactors Balance system Indicator dial

Flow

U-tube Fig. 12.16.

Flexible tube

Density balance [1].

an oscillating system is approximately proportional to the square root of the oscillating mass [4]. A vibrating ﬂow U-tube may also be used for density measurement [8]. Vibrating and density balance densitometers can also be used for compressed gases (r > 2 kg m3 ). Gases at standard pressure are measured with gas density balances, with which one determines the buoyancy of a closed hollow sphere ﬁlled with a comparison gas. A sonic densitometer consists of a device for measuring the speed of sound in the liquid [4, 8]. The speed of sound c is given by Eq. (27), where E is the bulk modulus and r is the mass density [4]. Unfortunately the liquid must be clear of particles that might scatter the sound waves and limit the received signal. sﬃﬃﬃ E c¼ r

ð27Þ

The density of liquids can also be measured radiometrically, by the absorption of gamma-rays in the medium. The intensity of the exiting radiation depends on the density of the medium and is given according to the Beer–Lambert law [Eq. (28), where I; I0 are the intensities behind and in front of the object, e the absorption coeﬃcient, r the density of the medium and l the path length].

12.2 Measurement Techniques

I ¼ I0 expðerlÞ

ð28Þ

Unlike level measurements the radiometric measurement of the density requires high outlet intensity and thus higher safety precautions. However, it has a broad range of application and is particularly interesting for measurements in polymer technology; one can measure very viscous melts, emulsions, and suspensions. Dilatometers measure the volume shrinkage during the course of liquid polymerization reactions and are mainly used for laboratory measurement of monomer conversion. They are based on the principle that polymers are denser than their monomers. Volume changes are monitored as monomer is converted to polymer by following the change in height of the solution inside a graduated capillary tube. Conversion is monitored with a computer-linked photodetector that tracks the meniscus in the capillary and records the height changes [13]. The percentage of total solids in a polymer sample can be determined by the gravimetric method through moisture weight loss. The sample is loaded onto a pan and the weight determined. Then it is put into an oven at high temperature for a time to remove all volatiles. It is then reweighed and the percentage of solids is determined. 12.2.7

Viscosity

Viscosities are of interest in polymer technology in order to follow the course of a polymerization reaction or to monitor continuously the quality of a product. Viscosity m is the constant of proportionality between applied shear stress t and the resulting shear rate g_ according to Newton’s Law of viscosity [Eq. (29)] [11, 13]. t ¼ mg_

ð29Þ

Viscosity may be constant (Newtonian), shear thickening (dilatant), or shear thinning (pseudoplastic) with shear rate. For polymer systems, solution or melt, the viscosity can be related to the molecular weight of the polymer, as discussed in Refs. 4, 8, and 13. In most cases viscosity is measured by capillary viscometers or rotating viscometers. In a capillary viscometer one measures the pressure drop by means of constant laminar ﬂow in a capillary; the constant ﬂow can be achieved by a pump and the pressure drop is obtained by a diﬀerential pressure transmitter whose ‘‘plus’’ and ‘‘minus’’ sides are connected to the capillary. The pressure drop is then directly proportional to the viscosity according to the Hagen–Poiseuille law [4, 11] [Eq. (30), where m is the viscosity, r is the capillary radius, l is the capillary length, Dp is the pressure drop, and qm is the mass ﬂow rate]. The capillary viscometer may also be employed in-line for monitoring of molecular weight in polymerizations, as described in Ref. 14.

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12 Measurement and Control of Polymerization Reactors

m¼

pr 4 Dpr 8qm l

ð30Þ

A method to obtain a measure of molecular size that is quick and cheap is the melt indexer [13]. The melt index is deﬁned as the number of grams of polymer extruded in 10 min through a capillary 2.1 mm in diameter and 8 mm long at a certain temperature and pressure (ASTM D1238). Among the diﬀerent possible ways to measure viscosities in rotating viscometers, the coaxial cylinder apparatus is the most commonly used in practice. The measured liquid intersperses the annular gap between the stationary inner cylinder (bob) and the rotating outer cylinder (cup). Therefore a velocity gradient builds between the inner and outer cylinders (Couette ﬂow). The momentum, which is transferred by this downward gradient to the inner cylinder, is directly proportional to the viscosity. Deﬂection is compensated by a torsion bar and the equilibrium deﬂection is measured electrically. The measurement of the angular velocity of the cup and the angular deﬂection of the bob makes it possible to determine the viscosity [4, 11]. Besides the coaxial device, the cone-and-plate viscometer is also used. In this device an inverted cone faces a solid plate and the apex of the cone just touches the plate. The measured liquid is in the free gap. The viscosity of the measured ﬂuid is computed from the torque on the cylinder-driving shaft [4, 11]. The Mooney viscometer, used particularly in the rubber industry, is a variant of the cone-and-plate viscometer; it restricts the sample to a disk-shaped cavity (ASTM D1646) [4]. Vibrating-reed viscometers can also be used for continuous in-line polymer viscosity [4]. The amplitude of the probe vibration depends on the viscosity of the ﬂuid. As the viscosity of the ﬂuid increases, the resistance to probe vibration increases. 12.2.8

Composition

The composition of raw materials, ﬁnished products, and samples of the various steps of a reaction is normally measured at the laboratory using the appropriate physical and chemical analytical methods. However, sampling and analysis are time-consuming and in many cases the result of the analysis is only of current interest and too late for control decisions to be made. In order to monitor compositions continuously, one needs automatically functioning analytical instruments that can continuously obtain and show the composition of a mixture. Some devices are fast and precise enough to be able to generate signals for control loops. The controllers would then adjust the desired values of other input variables such as ﬂow, temperature, or pressure in a cascade control scheme. Optical methods are common [4, 8, 13]. Infrared (IR) spectrographic analysis makes it possible in many cases to follow the appearance or the disappearance of one or more characteristic absorption frequency bands. These frequency bands cor-

12.2 Measurement Techniques

respond to frequencies of vibrations of the bonds in the molecules. One must ﬁrst analyze the spectrum of the IR radiation and then measure the corresponding frequencies. More recently the Fourier transform infrared technique (FTIR) has been used for faster data acquisition and handling than traditional IR spectrographic analysis. IR and FTIR can be applied to polymer solutions or solid ﬁlms for composition analysis and are particularly useful for copolymer composition determination. Optical analytical devices are also built for measuring radiation in the ultraviolet (UV) and the visible spectral region, but the spectral absorption bands obtained here are usually so broad that these devices are used only for special tasks. The refractive index (RI) of a mixture is a function of the composition of the mixture and the respective refractive indices of the constituents [8]. The mixture refractive index follows mixture laws such as the Lorentz–Lorenz law. Operational measuring instruments are usually diﬀerential refractometers or critical-angle refractometers [4]. A large disadvantage in the method is that it only provides meaningful results when a two-component system is considered. However, a diﬀerential refractometer is commonly used as a concentration detector in the eﬄuent of a gel permeation chromatography (GPC) column for molecular weight determination. Raman spectroscopy is dependent on the collision of incident light quanta with the molecule, inducing the molecule to undergo a change [13]. It is now being used to provide a means of studying pure rotational and vibrational transitions in molecules. Raman scattering of light by molecules may be used to provide their chemical composition and molecular structure and is currently being applied to polymers, as shown in Refs. 8, 15, and 16. Apart from optical methods, one uses magnetic (for paramagnetic materials, for example oxygen in diamagnetic gases) and electrical methods. Examples of the latter are conductivity measurements (of ionic liquids, for example purity of boiler feeding water), ionization methods (for example, the ﬂame ionization detector in gas chromatographs or the photo-ionization of gases with UV light to measure tracking of hydrocarbons in air), electro-chemical potential methods (for example, pH measurements), and occasionally polarographic methods. Nuclear magnetic resonance (NMR) is based on the principle that when a hydrogen-containing compound is in a strong magnetic ﬁeld and exposed to radio-frequency signals the compound absorbs energy at discrete frequencies [13]. This technique can be used to measure chain molecular structure, copolymer composition, and copolymer sequence lengths. It can also deduce isotactic/atactic ratios and other structure variations, as shown, for example, in Ref. 17. Mass spectrometry and NMR are currently not in routine on-line process use, but can be used to calibrate other on-line methods. Many methods depend on the separation of a ﬂuid mixture. Among these, gas chromatography (GC) stands out [8]. Suitable devices for on-line control were developed from laboratory gas chromatographs and operate very reliably. However, they can be expensive because of the associated program controls. The principle of gas chromatography is that a carrier gas (helium) is passed over a tubular column of a ﬁne solid. A sample is injected into the carrier gas stream and the gas

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12 Measurement and Control of Polymerization Reactors

eﬄuent from the column is run past a detector such as a ﬂame ionization detector. Calibration is based on the fact that, all conditions being equal, a given hydrocarbon will require the same length of time to pass through the column to the detector (elution time) [4]. A mass spectrometer source produces ions, and information about a sample may be obtained by analyzing the dispersion of ions when they interact with the sample using the mass-to-charge ratio. Sometimes mass spectrometers are used after a separation step such as gas chromatography or liquid chromatography for fraction identiﬁcation. 12.2.9

Surface Tension

In emulsion polymerizations particularly, it may be of some interest to measure the surface tension of the polymerization. The surface tension can give an indication of whether or not micelles are present, which is important in particle nucleation above the critical micelle concentration (CMC) [18, 19]. The on-line method used is usually the bubble pressure method. A dip tube is inserted below the liquid surface and bubbles are formed by compressed gas. Bubbles formed within a liquid are compressed by surface tension. The resulting pressure rises with decreasing bubble radius. This increased pressure, in comparison to the outside of the bubble, is used to measure surface tension. During the process of bubble formation and breakage, the pressure can be measured in the bubble. From the pressure oscillation the surface tension can be calculated. 12.2.10

Molecular Weight Distribution (MWD)

It is widely recognized that a reliable method of monitoring molecular weight distribution (MWD), and the various molecular weight averages (Mn ; Mw , and Mz ) during the polymerization process is of importance to ﬁnal polymer quality. The polydispersity index is deﬁned as the ratio of the weight to the number average molecular weight [Eq. (31)], which is a measure of the spread of the MWD. PD ¼

Mw Mn

ð31Þ

Monitoring the molecular weight distribution or its averages from a batch or continuous polymer reactor in real time would be desirable. Monitoring and feedback control of polymerizations can provide fundamental beneﬁts for improved quality. Traditionally, gel permeation chromatography (GPC) or size-exclusion chromatography (SEC) has been used to determine MWD [8, 13]. In GPC/SEC, polymer solutions are injected into one or more columns in series, packed with porous particles. The packing has small pores and during elution the polymer molecules may

12.2 Measurement Techniques

or may not, depending on their size, penetrate into the pores. Therefore, smaller molecules have access to a larger fraction of pores than the larger ones, and the chains elute in a decreasing order of molecular weights. For each type of polymer an empirical correlation exists between molecular weights and elution volumes. This can be used to calibrate the GPC/SEC, which allows the evaluation of average molecular weights and molecular weight distributions. Direct column calibration for a given polymer requires the use of narrow MWD samples of that polymer. The chromatograms of such standards give narrow peaks and with each standard is associated the retention volume of the peak maximum. There are a number of polymers for which narrow MWD standards are commercially available. More recently triple-detector instruments have been designed which include a diﬀerential viscometer, a light-scattering instrument, and a diﬀerential refractometer that monitors the column eﬄuent. A calibration curve can be obtained from this arrangement as long as all signals are calibrated [13]. For on-line purposes, the viscosity measures previously mentioned (see Section 12.2.7) have been used as a proxy for molecular weight averages in on-line control. Some vendors are commercializing more rapid GPC/SEC instruments for on-line control, with certain instruments already available. 12.2.11

Particle Size Distribution (PSD)

The particle size distribution (PSD) can have a fundamental eﬀect on the physical properties of dispersions that are common polymer products. The measurement of just the average particle size may not be suﬃcient. For example, the presence of populations of diﬀerent sizes resulting in a multimodal distribution could have a strong inﬂuence on ﬁnal properties and may need to be controlled. There are several particle size measurement techniques in use, such as optical imaging, electron imaging, optical diﬀraction and scattering, electrical resistance changes, sieving, sedimentation, and ultrasonic attenuation [4]. Optical (larger than 1 mm) and scanning electron microscopy (SEM) techniques literally give the clearest picture of a PSD. However, analyzing the images may be tedious without image analyzers. Nevertheless, this method can be used as a check or calibration on the indirect methods. The Coulter-counter particle size analyzer (larger than 0.5 mm) is used for measuring volumes of individual particles. Particles are suspended in a conductive ﬂuid, into which electrodes are placed. As a particle passes through an aperture between the electrodes, it displaces its own volume of electrolyte, and there is a measurable change in the electrical resistance of the system. The change becomes a precise measure of particle volume. These volumes can then be put into size bins and the PSD can be constructed. Today there are two principal light scattering technologies that are commercially available: light scattering intensity measurement (also known as static, or Rayleigh, scattering) and dynamic light scattering measurement [also known as quasi-elastic

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12 Measurement and Control of Polymerization Reactors

light scattering (QELS) or photon correlation spectroscopy (PCS)]. It should be noted that light scattering techniques are applied not only to particle size measurement but also to macromolecule size measurement. In static light scattering measurements, the light intensity scattered in solution by a particle, which is small compared with the wavelength of the incident laser beam, is proportional to the concentration multiplied by the molecular weight. When the particles are very small compared with the wavelength of the light, the intensity of the scattered light is uniform in all directions (Rayleigh scattering); for larger particles (above approximately 250 nm diameter), the intensity is angledependent (Mie scattering). If the concentration of the particles in solution is known or is measured during the analysis process, the particle size averages and distributions can be determined. Dynamic light scattering provides a relatively fast and simple method for submicron particle sizing [8]. When a beam of light passes through a colloidal dispersion, the particles or droplets scatter some of the light in all directions. Random intensity ﬂuctuations in scattered laser light arising from the Brownian motion of colloidal particles are analyzed to give either a simple mean size and polydispersity index or complete distribution data, even for multimodal distributions. Turbidimetry has been used traditionally in industry to obtain a measure of average particle size and even the entire PSD, and is a measure of the attenuation of a beam of light passing through a suspended particle sample [8]. Acoustic attenuation spectroscopy measurements can be made without the need for sample dilution and can be used in the particle size range of 10 nm to 100 mm. As sound travels through a slurry or colloid, it is attenuated. The level of attenuation is related to the particle size distribution as discussed in Refs. 20 and 21. Acoustic attenuation measurements can be made on high-concentration and/or opaque samples [8]. Packed column hydrodynamic chromatography (HDC), a technique for separating particles based on their size by eluting in the order largest to smallest, operates on a principle similar to that of GPC/SEC [8]. The sample under investigation and a small-molecule marker solution are introduced so that the eﬄuent ﬂow is not interrupted. HDC has a dynamic operating range from about 20 nm to 1.2 mm. An ultraviolet (UV) detector response is used to calculate the concentration of particles of diﬀerent sizes present in the sample. The subsequent computation of particle size distribution requires a calibration procedure employing a particle size standard. Capillary hydrodynamic fractionation (CHDF) is a hydrodynamic method for measurement of nanometer-sized particles. In this method, slurry containing the particles is forced through a capillary. Flow rate through the capillary is highest in the center of the capillary due to the laminar ﬂow velocity proﬁle. Larger particles extend into the high-ﬂow region while smaller particles travel closer to the wall of the capillary. Larger particles are swept through the capillary ahead of the smaller particles, producing particle size fractionation. The particles are detected at the end of the capillary by a UV detector. The measurable particle size range for CHDF is about 15 nm to 2 mm.

12.3 Sensor Signal Processing

For on-line control these techniques must be evaluated for speed, reliability, and sample dead time [4]. As in the MWD techniques, manufacturers are moving more toward on-line implementation of the more recent methods, but many of these techniques still remain in the control laboratory.

12.3

Sensor Signal Processing 12.3.1

Sensors and Transmitters

Sensor signal processing involves operations such as ampliﬁcation, compensation, ﬁltering, and normalization. As discussed (see Section 12.1), transmitters transform the signals leaving the sensor into a standardized signal, and the sensor plus the transmitter are considered to be the transducer or measurement instrument [7]. Transmitters should be attached operationally as closely as possible to the sensor location in order to minimize transmission error. To protect instruments against outside eﬀects from contamination, damage, and corrosion, they should be placed in sturdy housings. A measurement ampliﬁer is connected to the signal source of an electrical sensor and converts the signal coming from the sensor into the standardized direct current signal. The pneumatic or electrical signals depict only the analog measured variable. The direct measured variable is not always the desired result, however. Thus, for example, one obtains from ﬂow measurements with diaphragms a diﬀerential pressure, not the mass ﬂow, qm , which for compressible ﬂow in an oriﬁce is given by Eq. (32), where Dp ¼ p1 p2 is the pressure drop between the taps upstream and downstream respectively, r1 is the upstream density, Cd is an empirical discharge coeﬃcient, A2 is the area of the oriﬁce, Y is a dimensionless expansion factor, and b ¼ d2 =d1 is the ratio of the oriﬁce to the upstream pipe diameter (see Section 12.2.5) [10]. Cd YA2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qm ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2r1 Dp 1 b4

ð32Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ As can be seen from Eq. (32), the mass ﬂow is proportional to the product r1 Dp. As density and ﬂow can be subject to variations during the process, the downstream pressure will change and so will the density, and the two measured values are coupled together. Additionally, density is also a function of temperature so this must also be measured and a correction added. A smart transducer provides functions beyond those necessary for generating a correct representation of a sensed or controlled quantity. A smart transducer can be used in this situation to correct the ﬂow for pressure and temperature.

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12 Measurement and Control of Polymerization Reactors

12.3.2

Converters

If one wants to use the analog signals for more complicated calculations or in a process control system, one must digitize them and after processing transform them back to analog signals, for example to drive the ﬁnal control elements or actuators. Converters are used for this purpose. Measurement converters transform the analog signal into a digital one (A/D converter) or a digital signal into an analog one (D/A converter). Any decimal number can be converted into the binary system by summing the appropriate multiples of the diﬀerent powers of 2. Equation (33) demonstrates the conversion of binary into decimal numbers as well as that of decimal into binary numbers; thus 10101101 (binary) is equivalent to 173 (decimal). ð1 2 7 Þ þ ð0 2 6 Þ þ ð1 2 5 Þ þ ð0 2 4 Þ þ ð1 2 3 Þ þ ð1 2 2 Þ þ ð0 2 1 Þ þ ð1 2 0 Þ ¼ 173

ð33Þ

The raw binary sensor data can be encoded according to several binary formats, such as: pure binary, two’s complement binary, signed binary, binary coded decimal (BCD), ASCII, or Gray codes. In order to convert the digits of the decimal system to binary digits, which are required for digital technology, one can directly code these digits. Very frequently BCD is used. Every decimal digit is assigned to the corresponding binary digit. In the original binary code the number 35 would be shown as 100011, whereas in the BCD code it would be shown as 0011;0101. It must be cautioned that during this conversion a quantization error results, because of round-oﬀ errors in the decimal places; for example, 1.354 V (analog) might become 1:35 ¼ 0001;0011;0101 (BCD). Many process control applications use A/D and D/A conversions of at least 12-bit unsigned binary representation to obtain a resolution of 1 part in 2 12 ¼ 0.024% [7]. 12.3.3

Indicators

The numerous measurement signals of a process-engineering unit must be indicated so that they can be used and also recorded in some fashion. They are therefore combined in a central control room; there they provide a current overview of the state of the entire plant. Indicators show the measured value in an analog or digital way. The analog representation consists of a pointer before a calibrated scale, and recently also in LED displays and analog representations on monitors. Digital outputs are indicated as number displays or on counters. Apart from the measured value, indicator instrument panels also contain alarm functionality (optical and/or acoustical), which indicates if a critical measured value is exceeded in order to automatically shut down the installation. Once the signal is processed it is sent to the controller, the output of which, the controlled variable, is then sent to the actuator.

12.4 Regulatory Control Engineering

In the past these functions were performed by panel boards consisting of indicators, alarms, strip-chart recorders, and single-loop controllers. Today, distributed control systems (DCS) and data historians perform these functions (see Section 12.4.6). 12.3.4

Filtering Techniques

As mentioned above (see Section 12.1.2), during a measurement stochastic errors can be introduced. The eﬀects of process and measurement noise can be minimized by signal conditioning or ﬁltering [7]. Analog ﬁlters have been commonly used for many years. A high-frequency noise (low-pass) ﬁlter can be represented by Eq. (34), where xðtÞ is the ﬁlter input, yðtÞ is the ﬁlter output, and tF is the ﬁlter time constant. This equation can be discretized to produce a digital ﬁlter with sampling interval Dt [Eq. (35)]. tF

dyðtÞ þ yðtÞ ¼ xðtÞ dt

ð34Þ

tF

yn yn1 þ yn ¼ x n Dt

ð35Þ

When rearranged this gives the exponential smoothing ﬁlter in Eq. (36), where the ﬁlter factor, 0 < a a 1, is deﬁned by Eq. (37). yn ¼ ax n þ ð1 aÞy n1 1

a1

1þ

tF Dt

ð36Þ ð37Þ

This ﬁlter can be implemented in a digital computer. Other ﬁlters, such as a moving-average ﬁlter, are also possible [7].

12.4

Regulatory Control Engineering 12.4.1

General

Measurement instruments supply information on the current operating conditions of a plant. These form the basis to control the process and to keep conditions constant so that the optimal-quality results are obtained. For this purpose one uses controllers which determine automatically the sequence of control events. For example, a process could be that of the tank shown in Figure 12.17. The tank has a

627

628

12 Measurement and Control of Polymerization Reactors

qv0 FT

h

LT qv1 FT

R Tank with liquid level height h, inlet valve with volumetric ﬂow qv0, and outlet volumetric ﬂow qv1 through a restrictor with resistance R. Fig. 12.17.

stream ﬂowing into it through a valve and a stream leaving it through a restriction. A more comprehensive example of a polymer process as shown in Figure 12.33 will be discussed later (see Section 12.5). In order to be able to keep process or measured variables constant, or to change them systematically, they must ﬁrst be measured continuously and recorded. Flow rates into and out of the tank (denoted by the ﬂow transducer, or FT) and the level (denoted by the level transducer, or LT) are measured variables. One therefore obtains information about the instantaneous state of the desired measured variable to be controlled, the controlled variable, and about its change with time. In the case of the tank, this variable would be the level in the tank as measured by the level transducer. The controlled variable value must then be compared with the desired value, called the ‘‘set point’’. Any deviations from the set point, called the ‘‘error’’, can then be corrected through suitable actions. Devices that execute this comparison and provide output signals, which can be used for the correction sent to the manipulated variable, are called feedback controllers or regulators. In the case of the tank, the manipulated variable could be the ﬂow into the tank measured by the ﬂow transducer. If one wants to have a process follow certain time-varying conditions (as is the case often with batch processes), then one can use controllers to send the suitable programmed sequence signals. Often measured or unmeasured variables that aﬀect the process and cause deviations from the set point are called ‘‘disturbance variables’’. In the case of the tank, this variable could be an uncontrolled leak from the tank or a secondary ﬂow into the tank.

12.4 Regulatory Control Engineering

LC

qv0

I/P

FT

h

LT qv1 FT

R Fig. 12.18.

Feedback level control on a tank.

The signals leaving the controller are used to aﬀect temperatures, pressures, and ﬂow rates by means of actuators. Actuators are usually valves whose ﬂow area changes with the signal coming from the controller. Metering machines, or positive displacement pumps, can also be used as actuators. Measurement, comparison, and adjustment constitute an entity called a control loop. Very schematically one may describe the control loop in terms of sensors, controllers, actuators, and the process to be controlled. Shown in Figure 12.18 is a level control system for a tank. Broken lines denote control signals. The controller (denoted by LC) receives the level signal from the level transducer (LT) that sends a signal to the current-to-pressure transducer (I/P) that then applies a pneumatic pressure to the valve diaphragm. Shown in Figure 12.19 is the control loop pictured as a block diagram. In the control loop the controlled variable y is continuously measured and compared with the set point ysp ; it is desirable to have the deviation or error e ¼ ysp y made as small as possible. The controller then decides what control action to make on the manipulated variable u. Disturbances in the process denoted by d inﬂuence the controlled variable y and are compensated for by the manipulated variable u. In contrast to the controlled system, the tank in Figure 12.17 is not under feedback control. It could be operated in a manual semi-batch mode, where the inlet

629

630

12 Measurement and Control of Polymerization Reactors

Disturbance Variable Setpoint ysp

Error e +-

Manipulated Variable u Controller

Valve

d

Controlled Variable y

Tank

Level Transducer Fig. 12.19.

Block diagram of the tank level control loop.

valve is used to ﬁll the tank to a certain level. The control consists of a sequence of operating conditions, which must be carried out under a well-deﬁned plan. It is used around well-deﬁned processes, for example to automate the loading step of a batch emulsion. By analogy with the control loop hierarchy, one can split the control chain into the process to be controlled and the control system, as shown (for example) in Figure 12.18. In the given example, the process consists of the tank, which can be ﬁlled, the control valve with the appropriate piping, the restrictor, the level and ﬂow transducers and perhaps a recorder; the control system covers the programmable controller, the signal transmission, and the I/P converter. Expenditure for measurement, control, and actuator systems in the process industries is substantial. The proportional cost is higher the smaller the process is, because the absolute expenditure depends only on the number of measurement points and control loops, not on the size of the installation. 12.4.2

Process Dynamics

It is the task of the control system of a plant to drive the controlled variable (temperature, pressure, ﬂow, and so on) to the desired values, by either constant or prescribed trajectories. In order to accomplish this, the controlled variable must be continuously measured and the dynamics of the process must be known. In control technology one distinguishes processes with self-regulation and without self-regulation. In the ﬁrst case the output variable y assumes a new equilibrium state after the input variable u is changed; an example is the ﬂow through a pipe after a change at the control valve. Another example is the tank mentioned above: when the inlet ﬂow is changed a new level is reached, as long as the outlet ﬂow has a restrictor whose ﬂow depends on the height of liquid in the tank. One calls such systems ‘‘self-regulating’’, because the output reaches a new steady state when sub-

12.4 Regulatory Control Engineering

jected to step-changes in the input. In the second case a change in the input does not produce a new steady state. For instance, in a tank that is continuously fed and drained by manipulation of an outlet valve, the level remains constant only if the diﬀerence between output ﬂow and input ﬂow is zero. One calls such systems ‘‘non-self-regulating’’, because the output does not reach a new steady state when subjected to step-changes in the input. First-order System In many cases the relationships between plant inputs and outputs may be derived in the form of a mathematical model, which can then be used to understand and control the plant. For the tank example in Figure 12.17 the dynamic mass balance becomes Eq. (38), where r is the liquid density, A is the cross-sectional area of the tank, h is the height of liquid in the tank, and qv0 ; qv1 are the inlet and outlet volumetric ﬂow rates [7, 22]. If we assume constant density this becomes Eq. (39). 12.4.2.1

d ðrAhÞ ¼ rqv0 rqv1 dt A

dh ¼ qv0 qv1 dt

ð38Þ ð39Þ

The ﬂow restrictor may be modeled, to a ﬁrst approximation, as a linear relationship between ﬂow and height of liquid in the tank [Eq. (40), where R is the valve resistance]. The material balance becomes Eq. (41). qv1 ¼ RA

h R

dh ¼ Rqv0 h dt

ð40Þ ð41Þ

Since this is a self-regulating process, a steady state exists, governed by Eq. (42), where the subscript s denotes the steady-state value. Subtracting Eq. (42) from Eq. (41) yields Eq. (43). 0 ¼ Rqv0s hs RA

dh ¼ Rðqv0 qv0s Þ ðh hs Þ dt

ð42Þ ð43Þ

Since we are mainly only interested in deviations from the steady state, in this case we may deﬁne deviation variables by Eqs. (43), where u is the manipulated ﬂow deviation variable and y is the controlled level deviation variable. u ¼ qv0 qv0s y ¼ h hs

ð44Þ

631

632

12 Measurement and Control of Polymerization Reactors

In terms of the deviation variables, our mass balance becomes Eqs. (45). This is in the form of a ﬁrst-order system [Eq. (46), where the time constant is given by Eq. (47) and the process gain by Eq. (48). RA t

dy ¼ Ru y dt

dy ¼ Kpu y dt

ð45Þ ð46Þ

t ¼ RA

ð47Þ

Kp ¼ R

ð48Þ

The dynamic behavior of a dynamic system can be well represented through the so-called ‘‘step response’’. The dynamic evolution of the output variable can be monitored in response to a step-change of the input. We may ask how this system responds to a step-change of magnitude M in the input ﬂow rate. The ﬁrst-order system becomes that described by Eqs. (49), where HðtÞ is the Heaviside step function deﬁned by Eq. (50). t

dy ¼ K p MHðtÞ y dt

yð0Þ ¼ 0 0 HðtÞ ¼ 1

if t < 0 if t b 0

ð49Þ

ð50Þ

The solution to this diﬀerential equation, which may be found with the assistance of Laplace transforms [7, 22] is Eq. (51). yðtÞ ¼ K p Mð1 et=t Þ

ð51Þ

This result is plotted in Figure 12.20 (denoted by one tank) for the case of M ¼ 1 m 3 min1 , K p ¼ 1 min m2 , t ¼ 1 min. Note that the output is over-damped and approaches a steady value after long times. Second-order System In a similar fashion one may analyze the system of two non-interacting tanks in series, as shown in Figure 12.21. The system mass balances are Eqs. (52), where the subscripts 1 and 2 denote tanks 1 and 2 respectively [22]. 12.4.2.2

A1

dh1 ¼ qv0 qv1 dt

dh2 ¼ qv1 qv2 A2 dt

ð52Þ

12.4 Regulatory Control Engineering

1.2

1.2 u

1

1

0.8

y, m

1 tank (first order) 0.6

0.6

2 tanks (second order) 5 tanks

0.4

u, m 3 min -1

0.8

0.4

10 tanks Infinite tanks

0.2

0.2

0

0 -1

0

1

2

3

4

5

t, min Fig. 12.20.

Step responses of n ¼ 1; 2; 5; 10, and an inﬁnite number of tanks in series.

The outlet restrictors may again be considered linear as a ﬁrst approximation [Eqs. (53)].

qv1 ¼

h1 R1 ð53Þ

h2 qv2 ¼ R2 The material balances become those given by Eqs. (54), and at steady state they are given by Eqs. (55).

R1 A 1

dh1 ¼ R1 qv0 h1 dt

dh2 R2 ¼ R2 A 2 h1 h2 dt R1

ð54Þ

633

634

12 Measurement and Control of Polymerization Reactors

qv0 FT

LT

h1

R1 FT

h2

qv1

LT qv2 FT

R2 Fig. 12.21.

Two tanks in series without feedback control.

0 ¼ R1 qv0s h1s R2 h1s h2s 0¼ R1

ð55Þ

Subtracting Eqs. (55) from (54) yields Eqs. (56). dh1 ¼ R1 ðqv0 qv0s Þ ðh1 h1s Þ dt dh2 R2 R2 A2 ðh1 h1s Þ ðh2 h2s Þ ¼ dt R1

R1 A1

ð56Þ

12.4 Regulatory Control Engineering

Introducing deviation variables given by Eqs. (57), the mass balances become Eqs. (58), which are in the form of two ﬁrst-order systems [Eqs. (59)] where the time constants and process gains are given by Eqs. (60). u 0 ¼ qv0 qv0s y1 ¼ h1 h1s

ð57Þ

y2 ¼ h2 h2s dy1 ¼ R1 u 0 y1 dt dy2 R2 R2 A 2 y1 y2 ¼ dt R1 R1 A 1

t1

ð58Þ

dy1 ¼ K p1 u 0 y1 dt

ð59Þ

dy2 ¼ K p2 y1 y2 t2 dt t1 ¼ R1 A1 ;

K p1 ¼ R1

t2 ¼ R2 A2 ;

K p2 ¼

ð60Þ

R2 R1

These mass balances [Eqs. (58)] may be combined into one equation in y2 by differentiating the second and then eliminating y1 by successive substitutions [22] to aﬀord Eq. (61). t1 t2

d 2 y2 dy2 þ y2 ¼ K p1 K p2 u 0 þ ðt1 þ t2 Þ dt 2 dt

ð61Þ

This is the equation of a second-order system. Again we may ask how this system responds to a step-change of magnitude M in the input ﬂow rate. The second-order system becomes that described by Eqs. (62). t1 t2

d 2 y2 dy2 þ ðt1 þ t2 Þ þ y2 ¼ K p1 K p2 MHðtÞ dt 2 dt

ð62Þ

y2 ð0Þ ¼ 0 The solution to this diﬀerential equation, which again may be found with the assistance of Laplace transforms, is Eq. (63) [22].

t1 t2 y2 ðtÞ ¼ K p1 K p2 M 1 et=t1 et=t2 t1 t 2 t2 t1

ð63Þ

635

636

12 Measurement and Control of Polymerization Reactors

For the special case of equal time constants (t1 ¼ t2 ¼ t=2), the solution is Eq. (64). 2t y2 ðtÞ ¼ K p1 K p2 M 1 e2t=t te2t=t t

ð64Þ

This result is plotted in Figure 12.20 (denoted by two tanks) for the case of M ¼ 1 m 3 min1 , K p1 K p2 ¼ 1 min m2 , t ¼ 1 min. Note that the response of the two-tank process is over-damped and slightly lagged compared with the single tank, producing an S-shaped curve. High-order and Dead Time Systems One may proceed in a similar fashion to obtain solutions for more and more tanks in series with equal time constants, with the constraint that the total time constant is equal to the sum of the identical individual time constants so that t1 ¼ t2 ¼ ¼ tn ¼ t=n, creating systems of higher and higher order. These results are plotted in Figure 12.20 for the cases with the numbers of tanks in series n ¼ 1; 2; 5; 10; y with M ¼ 1 m 3 min1 , K p1 K p2 . . . K pn ¼ 1 min m2 , t ¼ 1 min. As the number of tanks increases, the response approaches that of a pure delay or dead time, y, equal to the total time constant of the inﬁnite tanks, y ¼ t. 12.4.2.3

First-order Plus Dead Time System Since dead time is a frequently occurring phenomenon, many systems can be represented by a combination of a ﬁrst-order system plus dead time (FOPDT) system. In the time domain the FOPDT system equation is Eq. (65). 12.4.2.4

t

dyðtÞ ¼ K p uðt yÞ yðtÞ dt

ð65Þ

For a step-change in input ﬂow rate of magnitude M the system becomes Eqs. (66), the solution to which is Eq. (67). t

dyðtÞ ¼ K p MHðt yÞ yðtÞ dt

ð66Þ

yð0Þ ¼ 0 yðtÞ ¼ K p ð1 eðtyÞ=t ÞMHðt yÞ

ð67Þ

This result is plotted in Figure 12.22 in original height h ¼ y þ hs variable for the case of the FOPDT tank with parameters M ¼ 1 m 3 min1 , K p ¼ 10 min m2 , t ¼ 20 min, y ¼ 2 min, hs ¼ 4 m and t 0 ¼ 10 min is the initial time of the stepchange in ﬂow rate. If at time t 0 in a system the input variable qv0 is increased by the amount Dqv0 and the output variable instantaneously changes by Dh, then we are dealing with a self-regulating system without time delay. If the output variable follows with de-

12.4 Regulatory Control Engineering

16

16

14

14 h

12 Kp = ∆h / ∆qv0

h, m

10

hs

10

∆h 8

8

0.63 ∆h

6

6

4

4

qv0, m 3 min-1

12

qv0 2

θ

2

τ ∆qv0

0 0

20

40

60

80

0 100

t, min Fig. 12.22.

Step response of a ﬁrst-order plus dead time process.

lays, then delay elements are present such as storage/accumulation or ﬂow lags. If the response of the output variable begins also after a certain time y, then there exists a system with dead time. Figure 12.22 suggests a method to determine these parameters from a step test experiment [6, 7, 22]. The parameters can be geometrically obtained from Figure 12.22 and this forms the basis for model identiﬁcation. At time t 0 the steady-state input to the process is stepped by an amount Dqv0 and the steady-state output change Dh measured. Then the process gain is given by Eq. (68). Kp ¼

Dh Dqv0

ð68Þ

The dead time y is the delay in the response after the input step at time t 0 . The time constant t is the time after the output has risen 0.63Dh above its original steady value hs after the input step at time t 0 . Dead time is always to be expected if there are transport processes in the system. A pure dead time is observed during the metering of compounds with a weigh belt,

637

638

12 Measurement and Control of Polymerization Reactors

because for every adjustment at the entrance slide, the new quantity is only observed at the output after the residence time on the belt (Figure 12.7). Dead time and time lags are to be expected in mixing control in a pipe, because in addition to the transport time longitudinal mixing also plays a role. Higher-order lags are often hard to distinguish from dead time, as shown in Section 12.4.2.3. Integrating System Suppose the situation of the tank in Figure 12.17 is that the eﬄuent ﬂow restrictor is replaced by a ﬁxed ﬂow rate. This is a non-self-regulating or integrating process. The mass balance is Eq. (69), as before [22]. 12.4.2.5

A

dh ¼ qv0 qv1 dt

ð69Þ

However, this time the eﬄuent ﬂow is ﬂow controlled by a valve or pump, not the height of liquid in the tank. At steady state for a ﬁxed value of qv1 Eq. (70) applies; that is, at steady state, the inlet and outlet ﬂows are exactly matched. 0 ¼ qv0s qv1

ð70Þ

Subtracting Eq. (70) from Eq. (69) gives Eq. (71). A

dh ¼ qv0 qv0s dt

ð71Þ

Introducing deviation variables [Eqs. (72)], the mass balance becomes Eq. (73), where the process gain is given by Eq. (74). u ¼ qv0 qv0s y ¼ h hs dy ¼ Kpu dt Kp ¼

1 A

ð72Þ

ð73Þ ð74Þ

For a step-change in input ﬂow rate of magnitude M the system becomes that described by Eqs. (75), the solution to which is Eq. (76), which is a ramp function with slope K p . dy ¼ K p MHðtÞ dt yð0Þ ¼ 0

ð75Þ

12.4 Regulatory Control Engineering

yðtÞ ¼ K p Mt

ð76Þ

Thus the tank will overﬂow or run dry if the inlet and outlet ﬂows are not exactly matched. Integrator plus Dead Time System With dynamic systems that are non-self-regulating one ﬁnds, likewise, step responses with and without dead time. An integrator plus dead time process would be modeled like Eq. (77). 12.4.2.6

dyðtÞ ¼ K p uðt yÞ dt

ð77Þ

For a step-change in input ﬂow rate of magnitude M the system becomes as in Eqs. (78), the solution to which is Eq. (79). dyðtÞ ¼ K p MHðt yÞ dt

ð78Þ

yð0Þ ¼ 0 yðtÞ ¼ K p Mðt yÞHðt yÞ

ð79Þ

This result is plotted in Figure 12.23 in original height variable h ¼ y þ hs with parameters M ¼ 1 m 3 min1 , K p ¼ 2 m2 , y ¼ 2 min, hs ¼ 4 m and t 0 ¼ 10 min. The parameters for this process may be obtained from a step test. At time t 0 the steady-state input to the process is stepped by an amount Dqv0 and the gain K p is found from the measured slope [Eq. (80)]. Slope ¼

Dh ¼ K p Dqv0 Dt

ð80Þ

The dead time y is the delay in the response after the input step at time t 0 . 12.4.3

Controllers

Controllers are devices that are meant to keep a speciﬁc controlled variable constant despite outside disturbances. One diﬀerentiates between continuous, digital, and On–Oﬀ controllers, depending on whether the output is continuous, discrete, or on–oﬀ. The great majority of controllers have been continuous, but the digital controller is now commonplace due to the widespread use of computers and distributed control systems. Referring again to Figure 12.19, the controller compares the measured value of the controlled variable with the desired value, the set point, computes the manipulated variable, and inﬂuences the actuator so that the set point and the controlled

639

12 Measurement and Control of Polymerization Reactors

25

25

20

20

3

15

qv0, m min

15

-1

h

h, m

640

Kp ∆q v0 = ∆h / ∆t 10

10

∆h hs

5

5

∆qv0

θ

∆t

qv0

0

0

0 Fig. 12.23.

5

10 t, min

15

20

Step response of an integrator plus dead time process.

variable can be as close to each other as possible. Independently of design and application, one ﬁnds three basic functions, which can be simply represented mathematically: namely the proportional, integral, and derivative functions. Proportional Control The proportional controller (P controller) assigns to each value of the deviation a speciﬁc value of the manipulated variable. For our tank example, the P controller mode takes the form of Eq. (81), where qv0 ; qvs are the new output and the steadystate values of the ﬂow rate respectively, K c is the controller gain, and hsp ; h are the current level input height set point and measured value respectively. 12.4.3.1

qv0 ðtÞ ¼ qvs þ K c ½hsp hðtÞ

ð81Þ

The steady-state value of the ﬂow rate qvs , or bias, is the ﬂow rate when the deviation from the set point is zero. The sign on the gain term, or controller action, is adjusted to be either positive or negative to create negative feedback control in which the controlled variable approaches a stable steady state. We may write this controller in terms of deviation variables, according to Eqs. (82), where e is called the error and u is the controller output, to obtain the deviation form [Eq. (83)].

12.4 Regulatory Control Engineering

e ¼ hsp hðtÞ uðtÞ ¼ qv0 ðtÞ qvs uðtÞ ¼ K c eðtÞ

ð82Þ ð83Þ

Pure proportional controllers have the disadvantage that in order to change the actuator, the controller needs a certain deviation; therefore this deviation can never be entirely eliminated and is termed ‘‘oﬀset’’. As given here the controller gain has units of ﬂow/height or m 2 min1 . In commercial controllers the manipulated and controlled variables are often made dimensionless with the range and standard signal of the transmitter and actuator so that the units of the gain would be, for example, %/% or mA/mA. Also, in older controllers the term ‘‘proportional band’’ (PB) is used instead of ‘‘controller gain’’ and is deﬁned by Eq. (84).

PB ¼

100% Kc

ð84Þ

Integral Control The integral controller (I controller) assigns the integral over time of the error to a speciﬁc change in the output variable. In the case of large errors over time, the action is large; in the case of small errors over time, the action is small. The I controller obeys Eq. (85), where tI is the reset time, in deviation variables. 12.4.3.2

1 uðtÞ ¼ tI

ðt

eðtÞ dt

ð85Þ

0

The output changes until the error has disappeared. Pure integral-only controllers have the disadvantage that they are sluggish for large tI and that they may be prone to oscillation for small tI . A disadvantage of integral control is reset windup. This results from the situation where, if the controller output saturates and the error remains large for an extended period of time, the integral gets large and does not recover immediately even if the error then goes to zero or reverses sign. The integral has to be manually reset in this situation in actual implementation. Derivative Control Derivative controllers (D controllers) are practical only in combination with other controller modes. They provide a fast engagement at the beginning of the control action. Here the derivative controller output is proportional to the rate of change of the error [Eq. (86), where tD is the derivative time]. 12.4.3.3

uðtÞ ¼ tD

deðtÞ dt

ð86Þ

641

642

12 Measurement and Control of Polymerization Reactors

The derivative mode of a D controller causes theoretically a large change in output of extremely short duration for a step-change in error. In practice, the eﬀect of a derivative controller is to cause a short increase in the output variable, which then returns to the previous value. PI, PD, and PID Control These basic controllers are used in combinations; P, PI, PID, and also PD controllers are common. The appropriate equations are linear combinations of the selected components. The PI controller obeys the relationship of Eq. (87). It reacts to a deviation with fast correcting action, and corrects the rest of the deviation slowly. It kicks in immediately and operates very accurately in the long term. 12.4.3.4

ð 1 t eðtÞ dt uðtÞ ¼ K c e þ tI 0

ð87Þ

The PD controller obeys Eq. (88). It kicks in ﬁrst very strongly; the output then reverts back to a value that corresponds to the output of a P controller. deðtÞ uðtÞ ¼ K c e þ tD dt

ð88Þ

If one combines all three controller actions, then a PID controller results. It obeys Eq. (89). This is the universal type of controller, which acts fast and avoids lasting oﬀset. ð 1 t deðtÞ ð89Þ eðtÞ dt þ tD uðtÞ ¼ K c e þ tI 0 dt The PID form implemented usually includes a derivative mode ﬁlter such as a ﬁrst-order ﬁlter to eliminate noise, which would be written in the time domain as Eqs. (90), where eF ðtÞ is the ﬁltered error and 0:05 < a < 0:2 is the dimensionless ﬁlter constant [7]. deF ðtÞ ¼ eF ðtÞ þ eðtÞ dt ð 1 t deF ðtÞ eðtÞ dt þ tD uðtÞ ¼ K c e þ tI 0 dt

atD

ð90Þ

The PID controller may also use the rate of change of the measured variable (for example, hðtÞ) instead of the error eðtÞ to eliminate set point change derivative kick. Derivative kick is mitigated if a derivative mode ﬁlter is used. Digital Controllers Modern implementations of controllers are not analog but digital due to the widespread use of computers and digital control systems. Digital control algorithms can 12.4.3.5

12.4 Regulatory Control Engineering

be derived from the continuous versions using standard numerical approximations of the analog controllers. Digital signals are discrete in nature and arise from sampling continuous measurements at equal time intervals of width Dt, or they may arise from naturally discrete signals such as, for example, from analyzers. We may illustrate the nature of digital PID algorithms by starting with the ideal PID controller [7, 22, 23] according to Eq. (91), where qv ðtÞ; qvs are the new output and the steady-state bias values of the ﬂow rate respectively, and eðtÞ ¼ hsp hðtÞ is the current error in the level height set point and measured value. ð 1 t deðtÞ qv ðtÞ ¼ qvs þ K c eðtÞ þ eðtÞ dt þ tD tI 0 dt

ð91Þ

A simple digital form of this equation may be written for the nth time interval using the rectangular rule of integration approximation to the integral and a ﬁrstorder backward ﬁnite diﬀerence approximation of the derivative to yield the positional form, Eq. (92), where e n ; qv; n are the error and controller output respectively at the nth sampling instant, and Dt is the sampling period. "

qv; n

# n 1X e n e n1 ¼ qvs þ K c e n þ e i Dt þ tD tI i¼1 Dt

ð92Þ

We may write this equation for the (n 1)st sampling instant [Eq. (93)]. "

qv; n1

# n1 1X e n1 e n2 ¼ qvs þ K c e n1 þ e i Dt þ tD tI i¼1 Dt

ð93Þ

If we subtract this Eq. (93) from Eq. (92), we obtain the velocity form [Eq. (94)], since it calculates the change in output. 1 e n 2e n1 þ e n2 qv; n qv; n1 ¼ K c ðe n e n1 Þ þ e n Dt þ tD tI Dt

ð94Þ

Solving for the output at the nth instant, we have a form [Eq. (95)] that calculates the output directly. 1 e n 2e n1 þ e n2 qv; n ¼ qv; n1 þ K c ðe n e n1 Þ þ e n Dt þ tD tI Dt

ð95Þ

One major advantage of the velocity form Eq. (95) over the positional form Eq. (92) is that, as the summation is lacking, it has inherent anti-reset windup. Also, an initial value of the output bias is not required [7]. If the sampling period is small compared with the process time constant

643

644

12 Measurement and Control of Polymerization Reactors

(Dt f t) these integral and derivative approximations are accurate and the conventional tuning techniques used for continuous controllers discussed in Section 12.4.3.6 may also be used for digital controllers [7]. Controller Tuning Controller parameters are tuned to provide both performance and stability and there are many diﬀerent rules for the tuning parameters of a control loop [6, 7, 22]. After deciding on the controller structure, one decides on the desired closedloop response criteria. Then one must distinguish between processes where there is at least an approximate model with known parameters and the case when the process model is unknown. 12.4.3.6

Process model is known An approximate model of the process may be obtained by the step response test noted earlier (see Section 12.4.2.4) (or from ﬁrst principles). When an approximate model of the process is known we may obtain the tuning parameters directly. We use here the example of the ﬁrst-order plus dead time process, since its dynamics are so representative of the polymer equipment dynamics. Here we have chosen for the tuning criteria to minimize the integral of the timeweighted absolute error (ITAE) [Eq. (96)] [7].

ITAE ¼

ðy

ð96Þ

tjeðtÞj dt

0

Tab. 12.1.

Minimum ITAE model controller tuning rules based on a FOPDT process [22].[a]

Controller type

Type of response

P

disturbance

PI

set point

PI

disturbance

PID

set point

PID

disturbance

[a] Tunings

Kc 0:49 t 1:084 Kp y 0:586 t 0:916 Kp y 0:859 t 0:977 Kp y 0:855 0:965 t Kp y 1:357 t 0:947 Kp y

tI

tD

–

–

t y 1:03 0:165 t 0:680 t y 0:674 t t y 0:796 0:147 t 0:738 t y 0:842 t

should only be applied in the range 0:1 < y=t < 1:0. B. A. Ogunnaike, W. H. Ray, Process Dynamics, Modeling, and Control, Copyright ( 1994 Oxford University Press. This material is used by permission of Oxford University Press.

–

– 0:929 y 0.308t t 0:995 y 0:381t t

12.4 Regulatory Control Engineering

1.2

ysp 1

Offset

y, m

0.8

P

PID 0.6

PI 0.4

0.2

0 0

5

10

15

20

25

t, min Responses for step-change in set point with ITAE tuned P, PI, and PID controllers on a ﬁrst-order plus dead time process. Fig. 12.24.

The minimum ITAE tuning rules based on a FOPDT process are given in Table 12.1 [22]. Note that the tunings in Table 12.1 should only be applied in the range y 0:1 < < 1:0. t Figure 12.24 shows the dynamic response of P, PI, and PID controller types to a step-change in the input of the ﬁrst-order plus dead time (FOPDT) process of Figure 12.22 with parameters K p ¼ 10 min m2 , t ¼ 20 min, y ¼ 2 min. For the FOPDT example the tuning for the P controller is K c ¼ 0:595 min m2 , for the PI controller it is K c ¼ 10 min m2 , tI ¼ 19:7 min, and for the PID controller it is K p ¼ 0:691 min m2 , tI ¼ 25:6 min, tD ¼ 0:725 min. The derivative mode ﬁlter was used for the PID controller with a ﬁlter constant of a ¼ 0:1. The control loop was simulated numerically for Figure 12.24. It can be seen that the P controller produces a long-term oﬀset, which the PI controller eliminates, but with some overshoot of the set point. The addition of the derivative action for the PID controller eliminates the overshoot and produces the best controller performance. Process model is unknown If the process model is unknown, the continuous cycling method may be used [7]. The rules of Ziegler and Nichols [24] have proven

645

646

12 Measurement and Control of Polymerization Reactors Tab. 12.2. Zeigler–Nichols controller tuning parameters based on the continuous cycling method [7, 22, 24].

Controller type

Kc

tI

tD

P

0.5K cu

–

PI

0.45K cu

PID

0.6K cu

– Pu 1:2 Pu 2

– Pu 8

themselves useful in practice. For this test the controller is ﬁrst brought into automatic as a pure P controller with small gain, the reset time tI is set to the largest value, and the derivative time tD is set to the lowest value. Then the gain K c is slowly increased. When the controlled variable has an oscillatory response, the ultimate gain K cu has been reached and the oscillation has the ultimate period Pu . The rules of Ziegler and Nichols then give the controller settings in Table 12.2. From these starting values one can then ﬁnd the optimal tunings very quickly by small systematic modiﬁcations. Figure 12.25 shows responses for a step-change in set point with P and PID controllers on the ﬁrst-order plus dead time process with parameters. The ultimate gain for the P-only controller, found by trial and error, is K cu ¼ 1:68 m 2 min1 , and the ultimate period from Figure 12.25 is found to be Pu ¼ 7:4 min. The PID controller is tuned with Ziegler–Nichols parameters from Table 12.2 of K c ¼ 1:008 m 2 min1 , tI ¼ 3:7 min, tD ¼ 0:925 min. Note that the PID response in Figure 12.25 has a much greater overshoot and much longer settling time than the ITAE response given in Figure 12.24. Other alternative tuning rules have been developed since the Ziegler–Nichols rules [7]. On–Oﬀ Controllers On–Oﬀ controllers output the manipulated variable in discrete values (on/oﬀ, in/ out) at discrete time intervals. The ideal On–Oﬀ controller is given by Eqs. (97), where u max and u min are the on and oﬀ values of the output and eðtÞ is the error [7]. 12.4.3.7

uðtÞ ¼

u max

if eðtÞ b 0

u min

if eðtÞ < 0

ð97Þ

Figure 12.26 illustrates an On–Oﬀ controller applied to the FOPDT process with parameters K p ¼ 10 min m2 , t ¼ 20 min, y ¼ 2 min. Controller outputs Oﬀ and On are u min ¼ 0 m 3 min1 and u max ¼ 0:2 m 3 min1 respectively. Although the controller is very simple and there are no tuning parameters other than the on and oﬀ values, it can be seen that continuous cycling results, which may or may not be desirable.

12.4 Regulatory Control Engineering

2

ysp

1.5

PID control

y, m

1 P control K c = K cu

ysp

Pu

0.5

0

-0.5 0

5

10

15

20

25

t, min Responses for a step-change in set point with P and PID controllers on a ﬁrst-order plus dead time process. The gain for the P controller is the ultimate gain K cu and cycles at the ultimate period Pu . The PID controller is tuned with Ziegler–Nichols tunings. Fig. 12.25.

A modiﬁcation of the ideal On–Oﬀ controller is the two-point controller, which switches the output when an upper or lower limit value is exceeded, the diﬀerence between these two points being the deadband [6]. Limit signal transducers can be used as two-point controllers (such as contact thermometers). The beneﬁts of the two-point controller as compared with the continuous controller are that it is simple, durable, reliable, and cheap, requires no auxiliary power, and does not lead to instability in the control loop. The disadvantages are the wear on the actuator working parts, the cyclical oﬀset, and the fact that large power circuits must be switched. Two-point controllers are therefore hardly ever used in an industrial environment. Self-operated Regulators Controllers which acquire suﬃcient force from the measurement to move the actuator do not require auxiliary power. Such controllers have a simple and durable construction, and many combine measuring instrument, controller, and actuator 12.4.3.8

647

12 Measurement and Control of Polymerization Reactors

1.2

1.2 ysp 1

1

0.8

y

0.6

0.6

0.4

u, m3 min-1

0.8

y, m

648

0.4 u

0.2

0.2

0

0 0

10

20

30

40

50

t, min Responses for step-change in set point with an On–Oﬀ controller on a ﬁrst order plus dead time process. Fig. 12.26.

functions. They have the advantage that they keep functioning if the auxiliary power fails. Therefore simple pressure control valves of this type are suitable as safety valves, in which a spring holds the plug of a valve in the seat, until an adjustable overpressure overcomes the force of the spring (Figure 12.27). Here the exit pressure aﬀects a diaphragm and holds a valve in an equilibrium position against the force of a spring. If the pressure rises, this closes the crosssection of the plug against the seat, and if the pressure lowers it opens again. This is also the method of operation of the pressure controllers in pneumatic transmission lines. Supply pressure-reducing valves are controllers without auxiliary power. Simple ﬂow controllers use the diﬀerential pressure at an adjustable restrictor: the plus and minus sides are on the two sides of a spring-supported diaphragm. These controllers are used as backﬂow (check) valves or as isolation valves, which shut oﬀ if the ﬂow is reversed or too high. Level controllers with ﬂoats are in condensate separators often coupled with simple temperature controllers as in Figure 12.28. These devices contain an electrical jump switch and operate therefore as two-point On–Oﬀ controllers. Controllers without auxiliary energy are used, where it is not essential to have high control accuracy and when one must do without the auxiliary power for safety reasons.

12.4 Regulatory Control Engineering

spring (reference)

diaphragm

exit pressure (controlled variable)

intake valve (control member) inlet pressure Fig. 12.27.

Pressure controller with no auxiliary power [1].

Single-loop controllers with auxiliary power use compressed air or, more commonly now, electrical power to receive and transmit signals. These devices are much more accurate than those without the auxiliary power; also they are not integrated in the measurement apparatus or the actuator. Distributed control systems are rapidly replacing the single-loop controller panel boards unless only a small number of control loops are involved.

Setpoint adjuster

lever jump switch

Fig. 12.28.

Electrical discrete temperature controller [1].

feeler (staff of stretchthermometer)

649

650

12 Measurement and Control of Polymerization Reactors

12.4.4

Valve Position Controllers

The output signal of a controller is sent electrically or pneumatically to a valve positioner that modulates the supply pressure to the control valve actuator and must make the stem position proportional to the input signal from the controller despite loads on the valve [6, 25]. There the electrical signal must be converted to a pneumatic positioning pressure or the pneumatic signal must be brought to a higher pressure in order to move the stem of the actuator reliably and to adjust the position of the valve accurately. In the pneumatic position controller, which is attached directly to the valve actuator, the pneumatic adjusting signal output from the controller works on a membrane and summing beam [6]. If the pressure signal rises, then a connection to the plant compressed-air network is opened to a pneumatic spool valve then to the actuator that moves the valve stem downward until a compensating spring on the summing beam holds the signal line pressure in equilibrium. Then the valve stem lifts until a new equilibrium position that is sensed by a cam lever is achieved. Thus a complete feedback loop exists where in this case the actuator is the process, the stem position is the controlled variable, the input signal is the set point, and the pneumatic ampliﬁer network is the controller. Electro-pneumatic valve positioners are used with diaphragm-actuated, slidingstem control valves. The electro-pneumatic valve positioner receives an electronic input signal from a control device and modulates the supply pressure to the control valve actuator, providing an accurate valve stem position [25]. 12.4.5

Single-loop Controllers

Single-loop controllers, which used to be more prevalent, are still used in some applications, such as laboratory or pilot plant. They are a rectangular package that has a faceplate for displaying the set point of a controller, to compare the set point with the actual value, and to enable the switch from manual control to automatic control or to cascade control. Single-loop controllers can be combined in a compact unit, which is built directly into a panel board. The faceplates of pneumatic and electrical control devices are very similar to each other. The transition to process control systems with newer display technologies is resulting in the disappearance of single-loop controllers. 12.4.6

Digital Control Systems

Since the 1970s the hardware used in the implementation of process controls has evolved from pneumatic analog technology to electronic analog technology to microprocessor-based controls. Although pneumatic and electronic analog process

12.4 Regulatory Control Engineering

controls may still be in service in older facilities, almost all of the newer plants now utilize microprocessor technology in implementing process control. Because of rapid advances in computer technology and the increasing computation power available on a single microprocessor, the hardware is becoming more powerful in terms of available functionality. Excellent descriptions of control system hardware is available in Refs. 4, 6, and 7 as well as in the technical literature provided by the major vendors such as Honeywell, ABB–Bailey, Fisher Controls, and Siemens and also in the specialized trade shows, publications, and books sponsored by the Instrument Society of America. From the perspective of polymer reaction engineering, there are four types of digital control hardware, which are listed in order of increasing functionality, complexity, and cost: 1. Single-loop controllers are a stand-alone microprocessor-based version of the older analog single-loop controller (see Section 12.4.5). They can operate independently or be part of a distributed control system and can execute the standard PID algorithm but also increasingly cascade control and gain scheduling. 2. Programmable logic controllers (PLCs) were the ﬁrst digital technology to successfully compete with conventional technology in industrial control applications. They were originally developed to replace hard-wired relay logic. PLCs are extremely important in batch control applications, because they are ideally suited to program large and complex batch sequencing cycles using ladder diagrams and a higher-level programming language. Because of their logic handling capability PLCs can also be used to implement process interlocks, which are designed to prevent process conditions that would unduly stress equipment (perhaps resulting in minor damage) or lead to oﬀ-speciﬁcation product. 3. Personal computer (PC) controllers are appropriate for the control of a laboratory or small- to medium-size pilot plant installations. The typical PC is augmented by the appropriate input/output (I/O) interface software. Software tools are available for the implementation of the previously discussed PID controllers using the PC. Advanced control algorithms can also be executed at the PC level. The PC also provides graphic capability for the monitoring of the performance of the control loop and can also provide data historian capability, which can be very important for polymer process development. 4. The distributed control system, the process control architecture referred to as a DCS, was introduced in the mid-1970s and has since become the standard for large installations. The main concept behind the DCS is that microprocessorbased nodes are interconnected by a digital communications network often referred to as the ‘‘data highway’’. A typical DCS installation handles seamlessly process I/O from the ﬁeld, including interfaces with process analyzers (that is, gas chromatographs), which are important in advanced control applications. The DCS can be interfaced with PLCs, with operator stations for process monitoring and control including alarm management, with engineering workstations for the system conﬁguration activities, and also with a set of host computers which can provide product quality information, data compression, reporting, and archival capabilities, as well as advanced control calculations.

651

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12 Measurement and Control of Polymerization Reactors

12.4.7

Actuators

Actuators govern the ﬂow of energy and mass streams in the process [25]. Through the actuators the controller output results in changes to the controlled variable (temperature, pressure, ﬂow, level, and so on) and thus the control loop is closed. Actuators consist of the control positioner (for example, a valve positioner) and the actuator element. The control positioner transfers the signal output of the controller in a controlled movement to bring the actuator element into a new position. Examples of actuators are rheostats to set and control electrical current, valves, ﬂaps, and metering pumps for liquids. The actuator, which is most frequently encountered in chemical process engineering, is the diaphragm valve (Figure 12.29). The input actuating pressure lies on one side of a diaphragm and exercises a force on it; the spring works against this force. The actuating pressure shifts the position of the diaphragm so as to create equilibrium of forces with the spring. The diaphragm is connected with the valve stem; the stem (and the plug) therefore make a stroke proportional to the actuating pressure. Between the plug and the valve seat a variable annular gap opens or closes, thus resulting in variable ﬂow. The seat and the plug are in a housing, which is solidly anchored and also holds the stem. For high-temperature service the housing is cooled with ﬁns in order to protect the bushing packing. With dangerous material one uses valves with sealed bellows and therefore no bushing is needed. Additional equipment, for example position controllers or limit switches, can be attached to the valve as necessary. In the pressure-free state the valve stem goes naturally to a terminal position. In Figure 12.30(a), as the pneumatic controller output signal increases, increased pressure on the diaphragm compresses the spring, thus pushing the stem in and closing the valve further. Such a valve is termed ‘‘air-to-close’’ (A–C). By reversing either the plug/seat or the spring/air inlet orientation, the valve becomes ‘‘air-toopen’’ (A–O) as in Figure 12.30(b). Normally the choice of A–C or A–O valve is based on safety considerations. One chooses the way the valve should operate (full ﬂow or no ﬂow) safely in case of a transmitter failure. Standard diaphragm valves have a diaphragm area from 200 to 1000 cm 2 and a stroke of 15–30 mm [1]. Very large valves are equipped with diaphragms of up to 3000 cm 2 and have a 100 mm stroke. These large valves have large power requirements and operate with 3 bar pressure. In addition a valve positioner is needed. To achieve higher strokes, specialized cylinders are used. The valve stem, plug, and seat (called the ‘‘trim’’) have diﬀerent forms: ball, plug, disk, or gate valves [7, 25]. Plugs are normally fabricated as rotationally symmetric parabolic cones. There are numerous other plug shapes, which can be selected for each individual case. The ﬂow rate as a function of stem position, or lift, l, is called the valve characteristic, f ðlÞ. A valve characteristic may be linear, quick opening, or equal percentage. The plug shape, seat, and cage primarily determine the valve characteristic, the sensitivity to contamination, and ﬂutter avoidance. To size a valve for liquids we may return to a modiﬁed form [Eq. (98)] of the equation given for restrictors [7, 12, 25].

12.4 Regulatory Control Engineering

Diaphragm with disk

Actuator springs

Control air Diaphragm casing

Yoke

Valve stem Stuffing box Plug

Seat

Direction of flow Valve body Fig. 12.29.

Diaphragm valve [1].

qv ¼ CV f ðlÞ

rﬃﬃﬃﬃﬃﬃ Dp d

ð98Þ

Here qv is the volumetric ﬂow rate, CV is the valve coeﬃcient, Dp ¼ p1 p2 is the pressure drop across the valve, and d is the relative liquid density (speciﬁc gravity) [Eq. (99), where r is the ﬂuid density at operating temperature and rw0 is the water density at a standard temperature and pressure (usually 4 C and 1 bar)]. d¼

r rw0

ð99Þ

The valve characteristic 0 a f ðlÞ a 1 is given by Eq. (100), where 0 a l a 1 is the lift and 20 a R a 50 is a valve design parameter [7].

653

654

12 Measurement and Control of Polymerization Reactors Control air

Diaphragm chamber

Spring Diaphragm chamber

Control air

Valve stem

Plug

(b) Air-to-open valve Seat Diaphragm valves with diﬀerent rest states: (a) air-to-close; (b) air-to-open [1].

(a) Air-to-close valve Fig. 12.30.

8 l; >
: l1 R ;

for a linear valve for a quick opening valve

ð100Þ

for an equal percentage valve

In USCS units the valve coeﬃcient, CV , is the full open ﬂow rate in gal min1 of water at 60 F (15.6 C) that will ﬂow through the valve with a 1 psi (6.89 kPa) pressure drop. In SI units the valve coeﬃcient (called KV ) is the full open ﬂow rate in m 3 h1 of water at 20 C that will ﬂow through the valve with a 1 bar pressure drop. International control valve standards are listed in Ref. 25. For gases the pressure ratio p2 = p1 at the valve must not be less than about 0.53; otherwise sonic velocity, c, is achieved in the restriction [10]. At the sonic velocity the throughput cannot be increased as the valve cross-section continues to open [10]. The pressure inside the valve must be lower than the liquid vapor pressure; otherwise ﬂashing or cavitation occurs. Besides diaphragm valves, one frequently uses butterﬂy valves (particularly for gases in lines with large diameter), which are constructed with a stop (Figure 12.31) or without (Figure 12.32). Butterﬂy valves without a stop (so-called regulating ﬂaps; see Figure 12.32) do not seal completely; their leakage ﬂow is within 5– 10% of the expected value when the valve is fully open.

12.4 Regulatory Control Engineering

Fig. 12.31.

Butterﬂy valve with a stop [1].

Fig. 12.32.

Butterﬂy valve without a stop [1].

655

656

12 Measurement and Control of Polymerization Reactors

With large nominal sizes and higher pressure, gate valves are used. It is advantageous that they have only a low-pressure drop and are insensitive against contamination. Gate valves have one large stroke; one cannot equip them with a diaphragm control drive, but the stem moves with gear racks, piston drive units, or electric motors.

12.5

Advanced Control Engineering

The polymerization reactor is usually at the heart of the manufacturing process, impacting both downstream processing and ﬁnal customer-related polymer properties. The following factors have contributed to the industrial signiﬁcance of polymer reactor control:

the need to improve ﬁxed-asset productivity by optimizing reactor yield and uptime; the trend toward shorter manufacturing campaigns for the diﬀerent polymer grades manufactured in the same reactor, or toward more frequent on-line product transitions to reduce product inventories and hence working capital; global competition, which imposes tough requirements for polymer grade uniformity; safety and environmental considerations regarding the stable operation of a potentially thermally unstable reaction. The measurement and control techniques discussed in the previous sections, many of which are generic in nature, are not always adequate to meet the following practical considerations speciﬁc to a polymerization reactor, which pose additional challenges to the successful application of a control strategy:

Polymerization reactors are well known both theoretically and experimentally to exhibit multiple steady states [26, 27] and in some cases they may also exhibit oscillations in terms of monomer conversion and polymer particle diameter [28]. In other cases it may be necessary to choose an open-loop unstable state as the reactor operating point. Furthermore, polymerization reactors can be highly exothermic and result in reactor thermal runaway. On-line measurements of polymer architecture such as composition, molecular weight, and degree of branching which were discussed in Sections 12.2.8 and 12.2.10, are not always available and for some polymer systems may be simply unavailable. In many cases the control engineer may have to rely on polymer properties inferred from infrequent laboratory analysis of reactor samples or from laboratory analysis of the ﬁnal polymer after it has experienced signiﬁcant post-reactor processing, thus introducing large dead times in the control loop. The relationship between reactor operating conditions, such as monomer conver-

12.5 Advanced Control Engineering

sion, temperature, residence time, polymer composition, and viscosity, and the customer-related ﬁnal properties, such as tensile strength, elongation at break, and processability in an injection molding machine, may not always be well deﬁned. Variability in the polymer isolation process and long-term polymer structure changes such as ‘‘aging’’ may result in the fact that, although in some cases the reactor may be operating ‘‘on-aim’’ within well-deﬁned manufacturing speciﬁcations, the ﬁnal polymer may not process satisfactorily when delivered to the customer. The control system for a polymerization reactor must be suﬃciently robust to handle unmeasured disturbances, which impact polymer reactor operation. These disturbances typically result either from trace amounts of polymerization inhibitors left over after monomer puriﬁcation before the polymerization reaction or from trace amounts of other compounds which may be present in a typical polymerization recipe and which may be aﬀecting the reaction. A polymerization reactor often produces several grades (in terms of composition and viscosity) of the same polymer and therefore the control strategy must be easily adapted to a multi-product plant and in some cases to on-line grade transitions. In the case of a multi-product plant it may be necessary to operate the reactor in terms of rather short campaigns in order to minimize the ﬁnishedproduct inventory and thus the working capital. In these cases the reactor control system must be designed in such a way as to achieve fast startups while minimizing oﬀ-speciﬁcation polymer formation. The purpose of this section is to highlight several control engineering techniques, some of which are known in the literature as ‘‘advanced control’’, which can be deployed to meet some of the challenging aspects of polymer reactor control. These techniques have been extensively discussed in literature reviews such as Refs. 29–32 and in books [18, 19]. The discussion will be facilitated by focusing on the polymerization process shown in Figure 12.33 [33]. Although the process is generic, the ﬂowsheet captures many of the elements of actual free-radical polymerization reactor installations. Monomers A and B are continuously added with initiator, solvent, and chaintransfer agent. In addition, an inhibitor may enter with the fresh feeds as an impurity. These feed streams are combined (stream 1) with the recycle stream (stream 2) and ﬂow to the reactor (stream 3), which is assumed to be a jacketed well-mixed tank. A coolant ﬂows through the jacket to remove the heat of polymerization. Polymer, solvent, unreacted monomers, initiator, and chain-transfer agent ﬂow out of the reactor to the separator (stream 4), where polymer, residual initiator, and chain-transfer agent are removed. Unreacted monomers and solvent (stream 7) then continue on to a purge point (stream 8), which represents venting and other losses and is required to prevent accumulation of inerts in the system. After the purge, the monomers and solvent (stream 9) are stored in the recycle hold tank, which acts as a surge capacity to smooth out variations in the recycle ﬂow and composition. The eﬄuent (stream 2) recycle is then added to the fresh feeds.

657

658

12 Measurement and Control of Polymerization Reactors

Monomer (A) Purge Valve 9

8

Monomer (B) Hold Tank Initiator (I) Reactor

2 Solvent (S) Transfer Agent (T)

1

7

3 Coolant

Coolant Separator 4

Inhibitor (Z) 6 Polymer, Initiator, Transfer Agent Fig. 12.33.

Copolymerization with recycle loop [33].

It has been found that the hierarchical approach summarized in Figure 12.34 is very useful in the successful application of process control in a complex industrial environment. Process knowledge, which is usually captured in an experimentally validated mathematical model, is the cornerstone of a successful control strategy. This is particularly true for polymerization reactors, where the in-depth knowledge of process operation in terms of the eﬀect of operating variables on polymer properties can be used to great advantage in the design of the control system and can result in a much more straightforward (and therefore easy to maintain) strategy than would have been possible otherwise (see, for example, the control strategy discussed in Ref. 30). Process knowledge together with the appropriate sensors, transmitters, and analyzers are the prerequisites for the design of the basic control system to regulate pressure, temperature, level, and ﬂow (PTLF). Only when the elements of the regulatory control system are in place and are properly designed and maintained can the control engineer attempt, in increasing order of complexity, the implementation of more advanced regulatory control strategies, multivariable model-based control algorithms, and on-line scheduling and optimization strategies to compute set points for the regulatory controls. In many instances advanced control applications have failed in an industrial environment not because

12.5 Advanced Control Engineering

SCHEDULING AND OPTIMIZATION MODEL BASED CONTROL ADVANCED REGULATORY CONTROL REGULATORY CONTROL (P,T,L,F) SENSORS, TRANSMITTERS, ANALYZERS PROCESS Fig. 12.34.

The process control hierarchy.

the algorithms were necessarily faulty but because the basic regulatory control system performed poorly, either because of inadequate design or because one of the critical measurements (that is, a process analyzer) was poorly maintained. 12.5.1

Feedforward Control

The traditional PID feedback controller, which was discussed previously (see Section 12.4.3.4), is very widely used, because it requires minimal process knowledge for its design. In particular, a mathematical model of the process is not required although it can be quite useful for appropriate tuning. Furthermore, if process conditions change, the PID controller can be retuned to maintain satisfactory performance. A properly tuned PID controller can be quite robust in maintaining good steady-state operation in the face of unmeasured disturbances. However, since control action can only occur if a deviation occurs between the set point and the measured variable, perfect control is not possible. Therefore feedback control fails to provide predictive control action to compensate for the eﬀects of known disturbances. A more serious limitation, which is particularly important for polymer reactor control, is that the controlled variable cannot always be measured on-line. Feedforward control was developed to counter some of these limitations. Its basic premise is to measure the important disturbance variables and then take corrective compensatory action based on a process model. The quality of control is directly related to the ﬁdelity and accuracy of the process model. Two implementations of feedforward control which are widely used in polymer reactor control will be discussed, namely feedforward control design based on steady-state models, and

659

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12 Measurement and Control of Polymerization Reactors

ratio control. Furthermore, the powerful combination of feedforward and feedback control will be considered, because it utilizes the best of both approaches: feedforward control works by reducing the eﬀects of measured disturbances, and feedback control provides the necessary ‘‘trim’’ to compensate for the eﬀect of model and measurement inaccuracies as well as measurement error. Steady-state Model Feedforward Control To illustrate this approach the polymerization process previously described in Figure 12.33 is considered. The presence of the recycle stream introduces disturbances in the reactor feed which perturb the polymer properties. The objective of the feedforward control is to compensate for these disturbances by manipulating the fresh feeds in order to maintain constant feed composition and ﬂow to the reactor. Feedforward control of the recycle allows the designer to separate the control of the reactor from the rest of the process. As shown in Ref. 33, the feedforward control equations were obtained by writing component material balances around the recycle addition point. For example, for monomer A this balance is given by Eq. (101). 12.5.1.1

qn3 ¼ qna1 þ ya2 qn2

ð101Þ

This equation is then solved in Eq. (102) for the fresh feed of monomer A since it is desired to keep the goal ﬂow of monomer A to the reactor (qna3 ) constant. qna1 ¼ qna3 ya2 qn2

ð102Þ

The recycle composition ( ya2 ) is typically measured by online gas chromatographs, which may have signiﬁcant time delays. If a faster response time of the analyzer is required, an infrared or Raman spectroscopy probe may be used. As discussed in Section 12.2.8 ﬂow qn2 is typically measured and controlled by manipulating the recycle valve to maintain the desired inventory in the feed tank. Any disturbances in the recycle composition or ﬂow will cause variations in the fresh feed in order to keep the reactor feed constant. Similar feedforward controllers are implemented for monomer B and solvent, which are also present in the recycle stream. As shown in Refs. 22 and 33, the performance of the feedforward control allows perfect compensation of disturbances that can arise, for example, from a stepchange in the purge ratio, so that the reactor polymer characteristics (composition, molecular weight) are unaﬀected. Without the presence of the feedforward controllers the reactor dynamics and hence its control can be aﬀected directly by the presence of three lags in series (reactor, separator, hold tank) and thus become unnecessarily more complex. Ratio Control Ratio control is a form of feedforward control which is widely used in the chemical industry and has proven very useful in polymerization reactor control. As is evident from its name, its purpose is to keep the ratio of two process variables at a 12.5.1.2

12.5 Advanced Control Engineering

Monomer B/A Ratio Setpoint

RC

FC

FT Monomer (B)

FT Monomer (A) Fig. 12.35.

Ratio control.

given value; hence it can be deployed when the objective is to maintain a certain proportion (or stoichiometry) of reactants in the reactor. The implementation of ratio control can be described by referring back to the polymerization process in Figure 12.33 and to the discussion in Ref. 7. Typically, ﬂow controllers are designed for each of the reactor feed streams (for example, monomer, initiator, chain transfer) and each one of these controllers has a set point, which is dependent on the particular polymer being made. However, when ratio control is implemented as shown in Figure 12.35, one of the reactor feed streams (monomer A in this case) is chosen as the reference stream. The measured ﬂow rate of monomer A is then transmitted to the ratio station RC, which multiplies the signal by the desired ratio KR (typically determined by the polymer chemist) to calculate the set point ysp for the ﬂow controller of monomer B. Dynamic simulations for the polymerization reactor shown in Figure 12.33 and reported in Ref. 33 showed that the selection of ﬂow ratios as manipulated variables reduced the interactions among the output variables such as composition and molecular weight. It is important that the ﬂow controller for monomer B be tightly tuned with short settling times so that when there are disturbances in the ﬂow of monomer A, any transient mismatches between monomer A and B ﬂows are minimized. 12.5.2

Cascade Control

Cascade control is also widely used in the chemical process industries and especially in cases where there may be nonlinear behavior in the dynamics of the con-

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12 Measurement and Control of Polymerization Reactors

TC

Polymerizer Coolant TT Polymerizer Feed

Feed Heat Exchanger

Fig. 12.36.

Conventional temperature control of an adiabatic polymerizer.

trol loop. It also addresses the main drawback of conventional feedback control, namely the fact that control action only occurs where the controlled variable deviates from the set point. Unlike feedforward control, which requires that disturbances be explicitly measured and a model be available to calculate controller output, cascade control introduces an additional measurement and an additional feedback controller. The secondary measurement is typically located so that it recognizes the upset conditions sooner than the controlled variable. The concept of cascade control has been traditionally very much used for eﬀective reactor temperature control. An example of the implementation of cascade control is shown in Figures 12.36 and 12.37. In many instances polymerization reactors are operated adiabatically. In the case shown in Figure 12.36, in which only traditional feedback control is used, measurement of the reactor temperature is used to manipulate the heat exchanger ﬂow to cool the reactor feed so that the reactor adiabatic temperature rise is adequate to remove the heat of polymerization. This conventional scheme may regulate reactor temperature satisfactorily but disturbances that occur in the feed line may result in a rather sluggish response of the temperature controller. Polymer properties are very sensitive to temperature excursions and in many cases this sluggish response of the temperature control loop may not be acceptable. Cascade control as shown in Figure 12.37 resolves the problem by introducing an additional measurement, namely the temperature of the reactor feed, and an additional controller. The cascade control structure has the following characteristics:

The output signal of the primary (frequently referred to as the ‘‘master’’) reactor temperature control loop serves as the set point of the secondary (frequently referred to as the ‘‘slave’’) reactor feed temperature control loop.

12.5 Advanced Control Engineering

Master TC Slave

TC Polymerizer

Coolant TT TT Polymerizer Feed

Feed Heat Exchanger

Fig. 12.37.

Cascade temperature control of an adiabatic polymerizer.

The two feedback controllers are nested with the secondary control loop located inside the primary control loop. There is one manipulated variable but two sensors and two controlled variables. The secondary controller responds rapidly to any temperature disturbances in the reactor feed line and provides improved reactor temperature control. As discussed in Ref. 7, a similar cascade control scheme can be implemented in the case where the reactor is jacketed and the reactor temperature is controlled by manipulating the cooling medium (typically water) inlet stream. In this case the additional measurement is the temperature of the jacket, which is compared with a set point provided by the master reactor temperature controller. The resulting error signal is the input to the controller for the cooling water makeup. It is obvious from the preceding discussion that for a cascade control system to function eﬀectively, the secondary control loop must be selected and tuned so as to have a faster response than the primary controller. According to Ref. 7, the secondary controller is normally a P or PI controller with derivative action hardly used. The primary controller is typically a PI or PID controller. 12.5.3

Feedforward–Feedback Control

The combination of feedforward and feedback control provides a very powerful practical strategy for the control of polymer properties such as composition and molecular weight. Typically it is still fairly diﬃcult to have on-line direct measurements of polymer composition, so the control design has to incorporate the avail-

663

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12 Measurement and Control of Polymerization Reactors

able oﬀ-line reactor sample composition measurements obtained at the laboratory, typically using IR or NMR techniques. Similarly, despite advances in size exclusion chromatography/gel permeation chromatography technology (SEC/GPC), on-line SEC/GPC is not routinely available for most industrial polymer reactor control applications. In many cases the control engineer has therefore to rely on oﬀ-line measurements of molecular weight. Typically measures of molecular weight used in control applications are the inherent viscosity and/or the melt index. The latter is very common in polyoleﬁn production and recent advances in capillary rheometry, as reported in the vendor literature, provide the capability of continuous measurement of melt index and viscosity during polymer production using on-line or at-line instrumentation. It is also very important to establish appropriate set points and speciﬁcations for the inherent viscosity and melt index by relating them to the underlying molecular weight distribution, as shown for example in Ref. 34. Returning to the polymerization process shown in Figure 12.33 and following the discussion in Ref. 30 (which dealt with an emulsion polymerization reactor) as well as the previously discussed concept of the process control hierarchy, it is very important to use process understanding (typically captured in dynamic simulations using experimentally validated models) in the design of the control system for polymer composition and molecular weight. For example, it was shown for a speciﬁc case, which is however typical of a wide class of continuous polymer reactors in industrial practice, that the reactor temperature control is crucial because inherent viscosity is extremely sensitive to temperature variations. As discussed above (see Section 12.5.2), cascade control can be eﬀectively deployed to control reactor temperature within very tight speciﬁcation limits. Achieving good reactor temperature control can become particularly challenging in multi-product semibatch polymerization reactors, because physical properties of the reactor contents vary from run to run and within a run and the standard PID controllers used in a cascade design may not be able to perform satisfactorily over the entire range of operation required. In these cases more advanced temperature control strategies based on adaptive control [7] or model-based control can be eﬀectively used, as shown, for example, in Ref. 35 for an industrial reactor and in Ref. 36 for a laboratory-scale one. Moreover, adaptive cascade control strategies can provide better performance without the need for retuning than a traditional PI cascade control system in the case of jacketed stirred tank reactors in which multiple products are produced and the overall heat-transfer coeﬃcient is unknown and can vary signiﬁcantly as a result of fouling, as shown, for example, in Ref. 37. An additional diﬃculty in the control of polymer properties is that in some cases the control problem is multivariable, in the sense that there are interactions between the molecular weight and composition loops and therefore when a manipulated variable is chosen to control molecular weight it may also aﬀect composition. It is important to use process knowledge to validate the selection of manipulated variables. For example, for the polymerization reactor shown in Figure 12.33, process simulations showed that one way to decouple polymer quality control is to take advantage of the fact that polymer composition is naturally very sensitive to changes in reactor feed composition but inherent viscosity is relatively insensitive

12.5 Advanced Control Engineering

to reactor feed composition changes. As discussed in Ref. 33, there exists a much more formal way for feedback controller design. It consists in creating an approximate linear multivariable model from the nonlinear polymer reactor model using step test data and then using the techniques of relative gain array (RGA) and singular value decomposition analysis (SVD) (as described, for example, in Ref. 7) to determine the best pairings of controlled and manipulated variables for robust multi-loop control. A typical control strategy for polymer composition, which can be implemented for the reactor shown in Figure 12.33, is illustrated in Figure 12.38. The measurements used are the ﬂows of the fresh monomer feeds, the recycle ﬂow, the monomer composition of the recycle feed and the total monomer reactor feed provided by two on-line gas chromatographs, and the polymer composition provided by laboratory analysis of reactor samples. The supervisory control consists of three levels implemented in a cascade fashion:

Gas Chromatograph Feedback

Feed Forward Monomer FC (A)

Polymer Composition Feedback

FT Purge Valve 8

9 Monomer FC (B)

FT Hold Tank

Initiator (I)

Solvent (S)

FC

FT 2

FC

FT

XT

FT

XT

1

3

7

Reactor

Lab

Transfer Agent FC (T)

XT

FT 4

Inhibitor (Z)

Separator

6 Polymer Fig. 12.38.

Copolymer composition control strategy.

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12 Measurement and Control of Polymerization Reactors

1. Feedforward controller, previously discussed, which maintains the total monomer feed rate and monomer A feed composition at the appropriate set points for the speciﬁc polymer grade being produced. The feedforward controller can be easily extended to n monomers by specifying the total monomer reactor feed rate and the feed composition for n 1 monomers. 2. Gas chromatograph feedback controller, which uses the velocity algorithm for digital PID control, discussed in Section 12.4.3.4 to calculate ﬂow correction factors for the monomers from the gas chromatograph measurement of the actual monomer feed composition. This controller provides the necessary integral action so that the oﬀset between the actual reactor monomer feed composition and its set point, which may be caused by ﬂowmeter inaccuracies or other unmeasured disturbances, is minimized. 3. Polymer composition feedback controller, which updates the set points for the reactor monomer feed composition based on the laboratory analysis of a reactor sample. This controller thus provides the necessary integral action so that the oﬀset between the measured composition of the reactor sample and the polymer grade composition is minimized. It is important to note that these feedforward and feedback controllers have been designed hierarchically, in the sense that each level in the structure will not activate unless the levels below it are functioning properly. Furthermore, in practice extensive data validation checks must be incorporated so that robust performance can be assured even when the gas chromatograph or laboratory analysis measurements may be unavailable or faulty. The control of inherent viscosity shown in Figure 12.39 uses the same approach as the composition control. Depending on the polymer grade being manufactured, the initiator or the chain-transfer agent may be used to control emulsion viscosity. An inherent viscosity feedback controller adjusts automatically the set point of the monomer to transfer agent ratio controller based on the measured viscosity value, (Section 12.2.7) and provides the necessary integral action so that the diﬀerence between the reactor inherent viscosity and the polymer grade inherent viscosity speciﬁcation is minimized. As in the case of composition control, extensive data and controller output checks must be incorporated in any practical implementation to provide robust performance. 12.5.4

State Estimation Techniques

In the design of the composition and viscosity feedback controllers it is very important to establish whether the polymer reactor dynamics need to be taken explicitly into account. The choice of sampling frequency balances the requirements for good quality control versus the need to minimize analytical costs. Usually, when the reactor residence time is much shorter than the sampling frequency, integral control is appropriate, because the time between measurements is usually suﬃcient for the eﬀect of an adjustment to a process variable set point to be com-

12.5 Advanced Control Engineering

Monomer to Transfer Agent Ratio Controller Monomer FC (A)

Polymer Inherent Viscosity Feedback Controller

FT Purge Valve

Monomer (B)

FC

9

FT

8

Hold Tank Initiator FC (I)

Solvent FC (S)

FT 2 FT

1

7

Reactor 3 Lab

Transfer Agent FC (T)

XT

FT 4

Inhibitor (Z)

Separator 6 Polymer

Fig. 12.39.

Inherent viscosity control strategy.

plete within this interval. In other cases the sampling dead time introduced by the periodic analysis of polymer concentration, polymer composition, and molecular weight may necessitate the incorporation of on-line state estimators of polymer properties. Reaction calorimetry aims to measure heat released from a polymerization in order to infer monomer conversion and polymerization rate (as reviewed, for example, in Refs. 8, 38, and 39). Careful measurement and balancing of mass and energy ﬂows are necessary for success of this technique. For example, the commercial Mettler–Toledo RC1 jacketed reactor acts as a calorimeter supplying mass balance, polymerization heat generation, and transport data. On-line estimation may also be accomplished using ﬁrst-principles polymerization kinetic models implemented on-line in the form of an extended Kalman ﬁlter (EKF) (as illustrated for example in Refs. 8 and 40–42). It should be pointed out that the choice of techniques for on-line estimation of polymer properties is still an active area of research and is very much dependent on the speciﬁcs of the polymer chemistry and available on-line instrumentation.

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12 Measurement and Control of Polymerization Reactors

12.5.5

Model Predictive Control

The previously discussed single-loop and appropriately chosen multi-loop feedforward and PID feedback control strategies may not be adequate for the eﬀective control of polymer properties, particularly in the case when the polymerization reactor exhibits strongly nonlinear dynamic behavior, or when there are strong interactions between the controlled variables, or when there are constraints on the manipulated variables. From the advanced process control techniques such as internal model control (IMC), inferential control, and adaptive control that have been developed by the academic process control community for these tough multivariable control problems, model predictive control (MPC) has reached the stage where it is having a signiﬁcant impact on industrial practice. MPC algorithms are rapidly becoming imbedded in the distributed control system (DCS) software libraries, which facilitates their use. As reported in a 2003 survey [43], by the end of 1999 there were at least 4500 industrial MPC applications worldwide, mainly in oil reﬁneries and petrochemical plants. The structure of MPC is shown in the block diagram of Figure 12.40 [7]. A mathematical model of the process is used to predict the current values of the output (controlled) variables. The model is usually implemented in the form of a multivariable linear or nonlinear diﬀerence equation. It is typically developed from data collected during special plant tests consisting of changing an input variable or a disturbance variable from one value to another using a series of step-changes with diﬀerent durations, or more advanced protocols such as the pseudo random-binary sequence described in Ref. 7. The residuals (that is, the diﬀerence between the pre-

Set-point calculations Set points (targets)

Prediction

Predicted Outputs

Control calculations

Inputs

Process outputs Process

Inputs Model

Residuals Block diagram for model predictive control [7]. D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, Copyright 8 2003 John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.

Fig. 12.40.

Model outputs

-+

12.5 Advanced Control Engineering

dicted and actual output variables) serve as a feedback signal to the prediction block and are used in two types of control calculations that are performed at each sampling instant, namely set point calculations and control calculations. A unique feature of MPC is that inequality constraints can be incorporated in both the set point and control calculations. In practice inequality constraints arise as a result of physical limitations on plant equipment such as pumps, control valves, and heat exchangers. The set points for the control calculations are typically calculated from an economic optimization of the process based on a steady-state model. Typical optimization objectives can include maximizing a process function, minimizing a cost function, or maximizing a production rate. The objective of the control calculations in the control block is to determine a sequence of control moves (changes in the manipulated variables) so that the predicted response moves to the set point in an optimal manner, for example by following a reference trajectory. The calculated control actions are implemented as set points to regulatory control loops. A detailed explanation of the diﬀerent design choices that are necessary for the eﬀective implementation of design of the MPC controller is available in Refs. 7 and 22. It is also essential to point out that the quality of MPC is very strongly dependent on the availability of a reasonably accurate process model that can capture the interactions between input, output, and disturbance variables. Although MPC control has become an established technology for tough control problems in petrochemical plants, its application in polymer reactor control is currently transitioning from purely academic studies using simulated examples to applications in industrial reactors. Referring to the polymerization process previously described in Figure 12.33 [33], several academic researchers (for example, in Refs. 44–46) have designed and implemented MPC controllers. These controllers were used successfully for plant startup, minimization of oﬀ-grade product during grade transitions, and regulation around a set point. Recently reported industrial applications of MPC in polymer reactors, for example in Refs. 47 and 48, have focused on the transition control requirements, which typically consist in completing the transition in either a minimum time period or with minimum amount of oﬀspeciﬁcation product. MPC is well suited to the grade transition problem [49], which often translates into aggressive manipulated variable moves, often against manipulated variable limits or plant constraints. Depending on the speciﬁc process reported, beneﬁts have also been achieved in product consistency or increased production rate. 12.5.6

Batch and Semi-batch Control Operation and Variability Batch reactors are the most common reactor used in polymerization engineering. They may vary in size from a ﬁve-gallon pilot unit to a 30 000-gallon (or greater) production size [19]. Removal of the heat of polymerization is accomplished by circulating coolant through a jacket or by reﬂuxing monomer and solvent. The main advantage of batch reactors is the ﬂexibility to accommodate multiple products. 12.5.6.1

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12 Measurement and Control of Polymerization Reactors

They are well suited for low-volume products and for products for which there are numerous grades (as in specialty polymers), because each batch can be made according to its own recipe and operating conditions without the costs incurred when a continuous reactor is shut down and restarted. Process control of batch reactors must address the main disadvantage of batch reactors versus continuous ones, namely variability within a batch and/or variability from batch to batch. This variability is particularly important in free-radical batch polymerization, where the time of formation of a single chain is only a very small fraction of the batch time and therefore inhomogeneity results from the fact that polymer chains are formed under very diﬀerent conditions during the course of the batch. This is especially signiﬁcant for composition control in a free-radical batch copolymerization reactor where, unless special control strategies are deployed, polymer chains formed early in the reaction may contain a higher fraction of the more reactive monomer than the chains formed later in the reaction (compositional drift). On the contrary, in step growth polymerization (polyamides, polyesters), where the growth time of an individual chain is approximately the batch time, the eﬀects of the changing reaction environment, and hence within-batch inhomogeneities, are much less of an issue, since all chains will see the same changing environment [19]. Operation in semi-batch reactor mode is very common in polymer reaction engineering practice. Typically, one way to address the issue of compositional drift in free-radical batch copolymerization is to operate the reactor under the so-called ‘‘starved feed’’ policy. In this case the monomer feed rate is automatically adjusted to maintain a constant rate of reaction, as inferred for example by reactor pressure (Section 12.2.2). In this starved feed operation the reaction environment is maintained constant during the batch and therefore the monomer composition in the reactor feed is equal to the desired polymer composition. It is also possible to implement more sophisticated control strategies during the batch by establishing operating trajectories for initiator addition, monomer addition, and/or reactor temperature to achieve desired polymer properties in minimum time, to maximize productivity, or to tailor the polymer molecular weight distribution. This is typically accomplished by solving oﬀ-line an optimization problem using a kinetic model of the process as shown, for example, in Refs. 50–54. These essentially open-loop trajectories constitute a form of feedforward control and are then implemented as part of the batch sequential logic and recipe management system using ladder logic and binary logic diagrams as shown in Ref. 7. If monomer conversion and molecular weight information is available during the batch (for example, through on-line densitometry, energy balance estimation, or on-line gel permeation chromatography) it can be incorporated as part of a feedback adaptive predictive control strategy, as for example in Ref. 55, to maintain the molecular weight at a desired value, while bringing the reaction to a speciﬁed monomer conversion in minimum time by manipulating initiator feed rate and coolant jacket temperature. In addition, if a fundamental model of the process is available including reaction kinetics and an energy balance, control moves during the batch can be calculated by solving a nonlinear dynamic optimization problem within the context of the previously discussed model predictive control to account

12.5 Advanced Control Engineering

for the wide variety of constraints typically encountered in batch systems. This approach has been described [56] and applied recently [57] to a set of commercial polymerization reactors exhibiting challenging dynamic behavior that prevented conventional control from delivering optimum manufacturing performance. Statistical Process Control In many cases of batch and semi-batch polymerization control there are no on-line measurements of polymer quality (for example, polymer composition, molecular weight) during the batch and these measures of end-use properties are only available at the end of the batch. In this case recipe modiﬁcations from one run to the next are common. The minimal information needed to carry out this type of runto-run control is a static model relating the manipulated variable to the quality variables at the end of the batch. As pointed out in Ref. 7, this model can be as simple as a steady-state (constant) gain relationship or a nonlinear model that includes the eﬀects of diﬀerent initial conditions and the batch time. The philosophy of statistical process control can be very useful in this case, since the polymer quality variable (for example the Mooney viscosity in elastomer manufacture) can be plotted for each successive batch on a Shewhart (x-bar) chart with the upper and lower control limits placed at three standard deviations above and below the target. A point outside the control limits means that the batch is out of control and the batch recipe and possibly the sequence logic must be adjusted for the next batch. If the quality variable for the batch is within the control limits, no control action is taken to prevent manipulations of the batch process based on stochastic variations within it. Very often in DCS-operated batch polymer reactors the primary process variables such as pressure, temperature, level, and ﬂow (Section 12.2.1–12.2.4) are recorded during the batch as well as the quality variables at the end of the batch. However, it may be very diﬃcult to obtain a kinetic model of the polymerization process due to the complexity of the reaction mechanism, which is frequently encountered in the batch manufacture of specialty polymers. In this case it is possible to use advanced statistical techniques such as multi-way principal component analysis (PCA) and multi-way partial least squares (PLS), along with an historical database of past successful batches to construct an empirical model of the batch [8, 58, 59]. This empirical model is used to monitor the evolution of future batch runs. Subsequent unusual events in the future can be detected during the course of the batch by referencing the measured process behavior against this ‘‘in-control’’ model and its statistical properties. It may therefore be possible to detect a potentially bad batch before the run is over and to take corrective action during the batch in order to bring it on aim. 12.5.6.2

12.5.7

Future Trends

Trends in industry, which reinforce the importance of measurement of polymer reactor process control, include the increasing emphasis by customers on receiving

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12 Measurement and Control of Polymerization Reactors

a more uniform product with desired property speciﬁcations from their polymer suppliers; the increasing computing power provided by DCS manufacturers; and the availability inside the DCS of advanced control modules such as the MPC so that the implementation of nonlinear multivariable control is easier and faster. As new polymer reactors are designed (for example, to manufacture polymers with tailor-made properties using novel metallocene catalysts or living radical polymerization technology), it would be beneﬁcial to incorporate process control considerations in the process design phase, so that the controllability of the plant can be established before it is actually constructed. Furthermore, it is expected that the on-line control not only of average polymer properties but also of polymer distributions such as the particle size distribution, as shown for example in Ref. 60, and the branching distribution will become important, together with recipe management, production scheduling, and production optimization. The instrumentation and control methodologies that will be needed to be deployed to meet these needs are a challenging and vibrant area of investigation for academic researchers and industrial practitioners alike.

Notation [61, 62]

Symbols Symbol

Units

A a a b c c CD Cp Cv Cd CV d d d d di E e eF f F

m2 m s2 1 C 2 C 3 C m s1 – J mol1 K1 J mol1 K1 – gal min1 psi1=2 m various – – various Pa (¼ N m2 ) various various – N (¼ kg m s2 )

area acceleration resistance coeﬃcient based on calibration resistance coeﬃcient based on calibration resistance coeﬃcient based on calibration speed of sound drag coeﬃcient heat capacity at constant pressure heat capacity at constant volume discharge coeﬃcient valve coeﬃcient in USCS units diameter disturbance variable relative liquid density (speciﬁc gravity) diﬀerential operator discrete deviation or error modulus of elasticity controller error, ysp y ﬁltered error valve characteristic force

Notation

g gc gc h HðtÞ I Kc K cu Kp KV l l m M Mn Mw Mz n p Pu PB PD qm qn qv r R R R s t T t u v v V x xðtÞ xi xn x y yðtÞ y

gravitational acceleration gravitational conversion factor in SI units gravitational conversion factor in USCS units height Heaviside step function radiant intensity controller gain ultimate controller gain process gain valve coeﬃcient in SI units length valve lift mass magnitude of step-change in input variable number average molecular weight weight average molecular weight z-average molecular weight number of measurements pressure ultimate period proportional band polydispersity index mass ﬂow rate molar ﬂow rate volume ﬂow rate radius resistance valve resistance coeﬃcient equal percentage valve parameter sample standard deviation Celsius temperature thermodynamic temperature time manipulated deviation variable velocity velocity average volume input deviation variable continuous input variable discrete measured value discrete input variable sample mean controlled deviation variable continuous output variable mole fraction

m s2 1 (¼ kg m s2 N1 ) lb m ft s2 lb f m – W m2 various various various m 3 h1 bar1=2 m – kg various kg mol1 kg mol1 kg mol1 – Pa (¼ N m2 ) s % – kg s1 mol s1 m 3 s1 m W s m2 – various C K s various m s1 m s1 m3 various various various various various various various –

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12 Measurement and Control of Polymerization Reactors

Y yn ysp z

expansion factor discrete output variable set point space coordinate

– various various m

Greek Symbol a a b g_ Dp Dt Du Dy e y k m m r rw0 s s X tD tF tI t t

SI unit

ﬁlter factor kinetic energy correction factor diameter ratio, d2 =d1 shear rate or rate of deformation pressure drop time interval change in input deviation variable change in output deviation variable absorption coeﬃcient dead time ratio of speciﬁc heats, Cp =Cv viscosity population mean mass density mass density of water at reference temperature population standard deviation surface tension summation operator controller derivative time ﬁlter time constant controller reset time shear stress time constant

– – – s1 Pa (¼ N m2 ) s various various m 2 kg1 s – Pa s various kg m3 kg m3 various N m1 – s s s Pa (¼ N m2 ) s

Indices 0 1 2 B c D F f I i

refers refers refers refers refers refers refers refers refers refers

to inlet, reference, standard, or in front of to datum, upstream, or outlet of tank 1 to surface, downstream, in throat, or outlet of tank 2 to buoyancy to conversion factor or controller to drag or derivative control to ﬁlter to ﬂoat to integral control to ith sample

References

m max min n p s sp t u v w W x g_

refers to mass refers to maximum refers to minimum refers to nth sample, tank number, or moles refers to pressure or process refers to steady state value refers to set point refers to temperature or tube refers to ultimate refers to volume refers to water refers to weight bar above symbol refers to average dot above symbol refers to time rate of change

References 1 A. Echte, Chapter 4: ‘‘Meß-, Steuer-

2

3

4

5

6

7

8

und Regelungstechnik’’, Handbuch der Technischen Polymerchemie, WileyVCH, Weinheim, 1993. R. P. Benedict, Fundamentals of Temperature, Pressure, and Flow Measurements, 3rd ed., WileyInterscience, New York, 1984. D. M. Considine, Process/Industrial Instruments and Controls Handbook, 5th ed., McGraw-Hill, New York, 1999. B. G. Lipta´k, Instrument Engineers’ Handbook, 4th ed., Process Measurement and Analysis, Volume I, CRC Press, New York, 2003. D. M. Levine, P. P. Ramsey, R. K. Smidt, Applied Statistics For Engineers and Scientists Using Microsoft Excel and MINITAB, Prentice Hall, 2001. R. H. Perry (Editor), D. W. Green (Editor), J. O. Maloney (Editor), Chemical Engineers’ Handbook, 7th ed., McGraw-Hill, New York, 1997. D. E. Seborg, T. F. Edgar, D. A. Mellichamp, Process Dynamics and Control, John Wiley & Sons, New York, 2003. O. Kammona, E. G. Chatzi, C. Kiparissides, ‘‘Recent developments in hardware sensors for the on-line monitoring of polymerization reactions’’, J. Macromol. Sci., Rev.

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13

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15

Macromol. Chem. Phys., 1999, C39, 57. E. A. Avallone (Editor), T. Baumeister (Editor), Marks’ Standard Handbook for Mechanical Engineers, 10th ed., McGraw-Hill, New York, 1996. W. L. McCabe, J. C. Smith, P. Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, New York, 1993. R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley & Sons, New York, 2002. Flow of Fluids through Valves, Fittings, and Pipe, Technical Paper 410, Crane Co., 300 Park Ave., New York, NY, 1988. F. Rodriguez, C. Cohen, C. Ober, L. A. Archer, Principles of Polymer Systems, 5th ed., Taylor and Francis, New York, 2003. M. P. Vega, E. L. Lima, J. C. Pinto, ‘‘In-line monitoring of weight average molecular weight in solution polymerization using intrinsic viscosity measurements’’, Polymer, 2001, 42, 3909. O. Elizalde, J. R. Leiza, J. M. Asua, ‘‘On-line monitoring of all acrylic emulsion polymerization reactors by Raman spectroscopy’’, Macromolecular Symposia, 2004, 206, 135.

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12 Measurement and Control of Polymerization Reactors ´ jo, C. 16 M. M. Reis, P. H. H. Arau

17

18

19

20

21

22

23

24

25 26

27

Sayer, R. Giudici, ‘‘Comparing near infrared and Raman spectroscopy for on-line monitoring of emulsion copolymerization reactions’’, Macromolecular Symposia, 2004, 206, 135. F. J. Wyzgoski, P. L. Rinaldi, E. F. McCord, M. A. Stewart, D. R. Marshall, ‘‘Poly(n-butyl acrylate-cocarbon monoxide-co-ethylene) characterization by high-temperature two-dimensional NMR at 750 MHz’’, Macromolecules, 2004, 37, 846. F. J. Schork, P. B. Deshpande, K. W. Leffew, Control of Polymerization Reactors, Marcel Dekker, New York, 1993. F. J. Schork, ‘‘Reactor operation and control’’, in Polymer Reaction Engineering, C. McGreavy (Editor), Kluwer Academic Publishers, Dordrecht, 1993. A. K. Hipp, G. Storti, M. Morbidelli, ‘‘Acoustic characterizations of concentrated suspensions and emulsions. 1. Model Analysis’’, Langmuir, 2002, 18, 391. A. K. Hipp, G. Storti, M. Morbidelli, ‘‘Acoustic characterizations of concentrated suspensions and emulsions. 2. Experimental validation’’, Langmuir, 2002, 18, 405. B. A. Ogunnaike, W. H. Ray, Process Dynamics, Modeling, and Control, Oxford University Press, New York, 1994. C. L. Smith, Digital Computer Process Control, International Textbook Company, Scranton, PA, 1972. J. G. Ziegler, N. B. Nichols, ‘‘Optimum settings for automatic controllers’’, Trans. ASME, 1942, 64, 759. Control Valve Handbook, 3rd ed., Fisher Controls International, 2001. A. K. Adebekun, K. M. Kwalik, F. J. Schork, ‘‘Steady-state multiplicity during solution polymerization of methyl methacrylate in a CSTR’’, Chem. Eng. Sci., 1989, 44, 2269. W. H. Ray, C. M. Villa, ‘‘Nonlinear dynamics found in polymerization processes – a review’’, Chem. Eng. Sci., 2000, 55, 275.

28 G. R. Meira, ‘‘Forced oscillations in

29

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continuous polymerization reactors and molecular weight distribution control. A survey’’, J. Macromol. Sci – Rev. Macromol. Chem., 1981, C20, 207. M. Embirucu, E. L. Lima, J. C. Pinto, ‘‘A survey of advanced control of polymerization reactors’’, Polymer Eng. Sci., 1996, 36, 433. J. P. Congalidis, J. R. Richards, ‘‘Process control of polymerization reactors: an industrial perspective’’, Polym. Reaction Eng., 1998, 6, 71. M. A. Dube´, J. B. P. Soares, A. Penlidis, A. E. Hamielec, ‘‘Mathematical modeling of multicomponent chain growth polymerizations in batch, semi-batch, and continuous reactors. A review’’, Ind. Eng. Chem. Res., 1997, 36, 966. W. H. Ray, ‘‘Process modeling for polymerization process operation and control’’, presented at the Tutorial on Polymerization Reactor Control, AIChE Annual Meeting, November, 2003. J. P. Congalidis, J. R. Richards, W. H. Ray, ‘‘Feedforward and feedback control of a solution copolymerization reactor’’, AIChE J., 1989, 35, 907. K. C. Seavey, Y. A. Liu, N. P. Khare, T. Bremner, C.-C. Chen, ‘‘Quantifying relationships among the molecular weight distribution, non Newtonian shear viscosity, and melt index for linear polymers’’, Ind. End. Chem. Res., 2003, 42, 5354. R. W. Chylla, D. R. Haase, ‘‘Temperature control of semi-batch polymerization reactors’’, Computers Chem. Eng., 1993, 17, 257. G. Defaye, N. Regnier, J. Chabanon, L. Caralp, C. Vidal, ‘‘Adaptivepredictive temperature control of semi-batch reactors’’, Chemical Engineering Science, 1993, 48, 3373. D. Tyner, M. Soroush, M. C. Grady, ‘‘Adaptive temperature control of multi product jacketed reactors’’, Ind. End. Chem. Res., 1999, 38, 4337. H. U. Moritz, ‘‘Polymerization calorimetry: a powerful tool for reactor control’’, in Polymer Reaction Engineering, Vol. 248, VCH, New York, 1989.

References 39 T. F. McKenna, S. Othman, G.

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Fevotte, A. M. Santos, H. Hammouri, ‘‘An integrated approach to polymer reaction engineering: a review of calorimetry and state estimation’’, Polym. Reaction Eng., 2000, 8, 1. K. B. McAuley, J. F. MacGregor, ‘‘On-line inference of polymer properties in an industrial polyethylene reactor’’, AIChEJ., 1991, 37, 825. M. F. Ellis, T. W. Taylor, K. F. Jensen, ‘‘On-line molecular weight distribution estimation and control in batch polymerization’’, AIChEJ., 1994, 40, 445. M.-J. Park, S.-M. Hur, H.-K. Rhee, ‘‘Online estimation and control of polymer quality in a copolymerization reactor’’, AIChEJ., 2002, 48, 1013. S. J. Qin, T. A. Badgwell, ‘‘A survey of industrial model predictive control technology’’, Control Eng. Practice, 2003, 11, 733. B. R. Maner, F. J. Doyle III, ‘‘Polymerization reactor control using autoregressive-plus Voltera-based MPC’’, AIChEJ., 1997, 43, 1763. ¨ zkan, M. V. Kothare, C. L. O Georgakis, ‘‘Control of a solution copolymerization reactor using multimodel predictive control’’, Chem. Eng. Sci., 2003, 58, 1207. R. Bindlish, J. B. Rawlings, ‘‘Target linearization and model predictive control of polymerization processes’’, AIChE J., 2003, 49, 2885. H. Seki, M. Ogawa, S. Ooyama, K. Ahamatsu, M. Ohshima, W. Yang, ‘‘Industrial application of nonlinear Model Predictive Control to polymerization reactors’’, Control Eng. Prac., 2001, 9, 819. D. Bartusiak, R. W. Fontaine, C. O. Schwanke, A. R. Gomatam, ‘‘Nonlinear Model Predictive Control of a polymerization reactor’’, presented at the Tutorial on Polymerization Reactor Control, AIChE Annual Meeting, November, 2003. M. Takeda, W. H. Ray, ‘‘Optimalgrade transition strategies for multistage polyoleﬁn reactors’’, AIChE Journal, 1999, 45, 1776. I. Saenz de Buruaga, P. D.

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Armitage, J. R. Leiza, J. M. Asua, ‘‘Nonlinear control for maximum production rate of latexes of welldeﬁned polymer composition’’, Ind. End. Chem. Res., 1997, 36, 4243. G. Maschio, T. Bello, C. Scali, ‘‘Optimization of batch polymerization reactors: Modelling and experimental results for suspension polymerization of methyl methacrylate’’, Chem. Eng. Sci., 1992, 47, 2609. G. Maschio, T. Bello, C. Scali, ‘‘Optimal operation strategies to control the molecular weight distribution of polymer products’’, Chem. Eng. Sci., 1994, 49, 5071. I. M. Thomas, C. Kiparissides, ‘‘Computation of the near optimal temperature and initiator policies for a batch polymerization reactor’’, Can. J. Chem. Eng., 1984, 62, 284. D. Tyner, M. Soroush, M. C. Grady, J. R. Richards, J. P. Congalidis, ‘‘Mathematical modeling and optimization of a semi-batch polymerization reactor’’, Proceedings IFAC Symposium on Advanced Control of Chemical Processes, Pisa. Italy, 2001, 3, 983. W. E. Houston, F. J. Schork, ‘‘Adaptive predictive control of a semi batch polymerization reactor’’, Polym. Process Eng. 1987, 5, 119. T. Peterson, E. Hernandez, Y. Arkun, F. J. Schork, ‘‘A nonlinear DMC algorithm and its application to a semi-batch polymerization reactor’’, Chem. Eng. Sci., 1992, 42, 737. J. W. Wassick, D. Coffey, B. Callihan, ‘‘Nonlinear Model Predictive Control of a commercial polymerization semi-batch reactor’’, presented at the Tutorial on Polymerization Reactor Control, AIChE Annual Meeting, November, 2003. P. Nomikos, J. F. MacGregor, ‘‘Multi-way partial least squares in monitoring batch processes’’, Chemometrics and Intelligent Laboratory Systems, 1995, 30, 97. T. Kourti, P. Nomikos, J. F. MacGregor, ‘‘Analysis, monitoring, and fault diagnosis of batch processes using multiblock and multiway PCA’’, J. Proc. Cont., 1995, 5, 277.

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Crowley, ‘‘Hybrid model based approach to batch-to-batch control of particle size distribution in emulsion polymerization’’, Computers Chem. Eng., 2003, 27, 1153. 61 I. Mills, T. Cvitas, K. Homann, N.

Kallay, K. Kuchitsu, Quantities, Units and Symbols in Physical Chemistry (The Green Book), Blackwell Science, 1993. 62 J. Inczedy, T. Lengyel, A. M. Ure, Compendium of Analytical Nomenclature (deﬁnitive rules 1997), 3rd ed., Blackwell Science, 1998.

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13

Polymer Properties through Structure1 Uday Shankar Agarwal

Since polymers display an incredible range of properties, their potential for applications is ever increasing. The link between polymer structure and the resulting properties has long been realized, and emerged as the concept of tailor making of polymers. In earlier times, developments were largely targeted on achieving the desired properties through new monomers, copolymer and blend compositions, physical and chemical additives, and control of polymer molecular weights, as well as through structural changes derived from processing. In recent years, the increased expertise in anionic polymerization and the emergence of controlled radical polymerization techniques have led to relatively easy access to several precisely controlled topologies such as those of telechelic, block, graft, star-shaped, and several other branched homoploymers and copolymers with controlled molecular weights. For example, phenomenal growth has been recorded in the area of synthesis of block copolymers, as well as in examination of their self-assembling characteristics leading to unique control of nanoscale morphology. This is providing tools for development of properties suitable for applications extending from those requiring mechanical toughness to electronic and biological uses. Another rapidly developing area is nanocomposites, where the attainable range of properties is further expanded by incorporating inorganic and organic nanoparticles in polymeric matrices. In this chapter we aim to demonstrate the relationship of the structure of polymers with their thermal, solution, and rheological behavior. Besides providing a general review of such behavior, we will emphasize some of the recent developments in these areas.

13.1

Thermal Properties of Polymers

The thermal behavior of polymers is diﬀerent from that of simple compounds in that, on heating, the transition of polymers from solids to liquids occurs not at 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

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13 Polymer Properties through Structure

a sharp temperature, but over a range of temperatures. Such behavior is at least partly attributable to the mixture of chain lengths of molecules that make up a polymeric material. In addition to a crystalline solid phase that gains liquid-like mobility at the corresponding crystalline melting point, polymeric materials display an amorphous phase that undergoes a gradual transformation to a liquid-like state over a temperature range called the glass transition temperature. 13.1.1

Crystalline and Amorphous Polymers

Crystallization of polymers is important because it imparts several highly useful properties such as strength, toughness, stiﬀness, and solvent and chemical resistance. A polymer can crystallize when it is cooled from a melt, or concentrated from a diluted solution. This is because the approaching chains may fold and pack into a regular, long-range, three-dimensional, positional and orientational order characterized by a certain unit cell of that polymer. Figure 13.1 shows the

Fig. 13.1. Arrangement of polyethylene molecules in the orthorhombic unit cell of dimensions a ¼ 7:41 A˚, b ¼ 4:94 A˚, c ¼ 2:55 A˚ (from C. W. Bunn, in Fibers from Synthetic Polymers, ed. R. Hill, Elsevier, Amsterdam, 1953).

13.1 Thermal Properties of Polymers

Fig. 13.2. Helical conformation of an isotactic vinyl polymer crystalline chain (from F. Rodriguez, Principles of Polymer Systems, 2nd Ed., McGraw Hill Intl., Auckland, 1982).

unit cell in a crystal of a commercially important crystalline polymer: polyethylene. Here, the chains are extended in planar zigzag (trans) conformation, which corresponds to low energy. However, steric hindrances can lead to other chain conformations in crystals. For example, in the presence of bulky substituents, the vinyl polymer chains rotate to take up a rotated helical conformation (Figure 13.2). If a given polymer lacks a tendency to get ordered, the chains stay randomly coiled and entangled during solidiﬁcation (Figure 13.3). Such polymers (for example, poly(methyl methacrylate)) are called amorphous. Crystallinity is best detected by wide-angle X-ray diﬀraction (WAXD, Figure 13.4), where the peak positions provide information about the repeat distances in the crystal lattice. Unlike small molecules that display sharp peaks during WAXD, even crystallizable polymers display broad WAXD peaks (Figure 13.4). This is because polymers are never

~10 nm

Fig. 13.3.

Lack of long-range order in an amorphous polymer.

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13 Polymer Properties through Structure

fully crystalline material

amorphous material diffraction intensity

semicrystalline polymer 10

diffraction angle (deg)

32

Fig. 13.4. WAXD patterns of fully crystalline solids, amorphous solid/melts, and semicrystalline polymers.

100% crystalline due to entanglements between chains, and are therefore called semicrystalline. The extent of crystallization (xc ) can vary from nearly 100% for polytetraﬂuoroethylene or linear polyethylene to nearly zero for noncrystallizable polymers such poly(methyl methacrylate). The extent of crystallization is determined by considering the variation of properties (r, such as density, speciﬁc heat, and electrical resistivity) between amorphous phase polymer and crystalline phase polymer to be linearly additive [Eq. (1)]. r ¼ ð1 xc Þra þ xc rc

ð1Þ

13.1.2

Inﬂuence of Polymer Structure on Crystallizability of Polymers

Since crystallization requires that the participating chains to come close together, the spatial regularity and the packing possibilities of the polymers are the most important factors in determining their crystallizability. For example, isotactic and syndiotactic polypropylene are stereoregular, and show a higher tendency to crystallize than atactic polypropylene (Figure 13.5). Randomly arranged or bulky pendent groups reduce this tendency because they hinder the close alignment of the polymer molecules that is necessary for intermolecular interactions. For example,

13.1 Thermal Properties of Polymers

isotactic

syndiotactic

atactic The regularity of a polypropylene chain structure decides its crystallizability. The isotactic and syndiotactic polypropylene crystallize, but atactic polypropylene does not. Fig. 13.5.

poly(vinyl acetate) with large pendent groups (aOaCOaCH3 ) is amorphous, but crystallizes easily when hydrolyzed to poly(vinyl alcohol), which has small pendent groups (aOH). Crystallizability is also induced by existence of functional groups capable of intermolecular hydrogen bonding (such as amides, urethanes, urea, and so on) or functional groups capable of polar interactions (such as aCl and aCN). Block copolymers have large segments of regular structure capable of participating in crystallization, and hence show a higher tendency to crystallize than random copolymers. 13.1.3

The Glass Transition Temperature

An amorphous polymer can exist in the glassy (hard, brittle, and stiﬀ ) or the rubbery (soft) state, depending on the temperature being lower or higher than the socalled glass transition temperature (Tg ). The glass-to-rubber transition at Tg is associated with an increase in free volume resulting in an increase in cooperative rotational and translational motion of the chain segments (20–50 atoms), as the thermal energy overcomes the intermolecular restraints imposed by secondary bonding forces between polymer chains. This results in marked changes, such as an increase in the number of possible chain conformations and an increase in the free volume. This is manifested as jumps in the thermal expansion coeﬃcient (Figure 13.6) and speciﬁc heat (Figure 13.7), and a drop of a few orders of magnitude in the mechanical modulus (Figure 13.8). Even as the chain segments gain mobility above Tg , the chain entanglements persist, imparting rubberlike elasticity (Figure 13.8). With further increases in temperature and the associated energy and motion of the molecules, translation/slip of the entire molecules becomes possible. A consequence of Eq. (1) is that the changes at Tg are less drastic in semicrystalline polymers as the crystallites impart rigidity, even at temperatures higher than Tg .

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13 Polymer Properties through Structure

melt Specific volume

amorphous solid crystalline solid Tg Tm Temperature

Fig. 13.6. Volume expansion (dilatometry) in amorphous and crystalline polymers with increasing temperature.

13.1.4

Inﬂuence of Polymer Structure on Tg of Polymers

The Tg of a polymer is important as it deﬁnes the maximum temperature at which an amorphous solid polymer still displays structural rigidity. For a given polymer structure, the Tg increases with an increase in molecular weight (M) [Eq. (2), where Tg; y is the Tg of a polymer with inﬁnite molecular weight, and K is a constant]. Tg is aﬀected by structural features of the polymer chains, such as chain stiﬀness and intermolecular forces.

Heat flow (W/g)

50

Tg

Tm

Tc

100

150

200

250

300

Temperature (°C) Heat ﬂux during heating of poly(ethylene terephthalate) (PET) at a constant rate in a diﬀerential scanning calorimeter (DSC), showing changes in speciﬁc heat at Tg , crystallization exotherm at Tc , and melting endotherm at Tm . Fig. 13.7.

13.1 Thermal Properties of Polymers

glass + crystals rubber + crystals

Log G (MPa)

Tg

rubber (entanglements) Tm

liquid

Temperature (°C) Fig. 13.8. Inﬂuence of temperature on mechanical modulus of a semicrystalline polymer. The broken line indicates the corresponding curve at higher crystallinity.

Tg ¼ Tg; y

K M

ð2Þ

For example, while low Tg values are exhibited by ﬂexible chains such as polyethylene with easy rotation about the backbone, a higher Tg results from chain stiﬀness such as that resulting from cyclic structures in the backbone (as in BPA polycarbonate). This concept is used in enhancing the Tg of PET by partial replacement of the aliphatic glycol in it with an alicyclic glycol, thereby making the polymer more suitable for high-temperature applications [1]. While bulky side groups (for example, in poly(a-methylstyrene)) enhance the Tg because of the resulting steric hindrance, long and ﬂexible side groups decrease the Tg as they enhance the separation between polymer backbones and introduce additional chain-ends. Secondary attractive forces (for example, polarity) enhance the Tg . The extent of syndiotacticity in polymers such as PMMA can be inﬂuenced by the method of polymerization, and in-turn inﬂuence the Tg [2, 3]. When a random or alternating copolymer is made up of non-interacting monomers, its Tg can be calculated from the weight fractions (wi ) and glass transition temperatures (Tgi ) of its components, by means of Eq. (3). 1 w1 w2 ¼ þ Tg Tg1 Tg2

ð3Þ

When a polymer is desired to be ﬂexible at the temperature of use, this eﬀect (Eq. 3) can be used to plasticize a polymer internally (that is, reduce its Tg ) by incorporating comonomers such as vinyl acetate and vinyl chloride. In the case of block or graft copolymers, microphase separation of the blocks often leads to separate glass transitions of the corresponding blocks [4, 5]. The case is similar for immiscible blends. Plasticization can be achieved externally by addition of low molecular weight, high-boiling organic compounds (plasticizers) that can enhance

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13 Polymer Properties through Structure

inter-backbone distance and drastically reduce Tg . For example, dioctyl phthalate is often added to turn the otherwise rigid plastic poly(vinyl chloride) into a ﬂexible material [6]. 13.1.5

The Crystallization Temperature and the Melting Point

When heated beyond its Tg , a crystallizable polymer undergoes crystallization that is detected calorimetrically as an exotherm (Figure 13.7). The rate of crystallization ﬁrst increases with temperature, reaching a maximum at the so-called crystallization temperature, Tc . With a further increase in temperature, the solid undergoes a ﬁrst-order transition into a melt at Tm (Figures 13.7, 13.8), corresponding to zero free energy change on melting [Eq. (4)]. Tm ¼

DHm DSm

ð4Þ

The Tm is high for polymers with a lower DSm; this is due, for example, to chain stiﬀness in PEEK (poly(ether ether ketone), Tm ¼ 395 C) or to speciﬁc interactions that persist into melts (nylon-6, Tm ¼ 228 C). Tm can also be high for polymers exhibiting high DHm values, for example through regular and frequent polar interactions. Random addition of a comonomer leads to reductions in both the Tm and the extent of crystallization, while long homopolymer segments in block and graft copolymers or in immiscible blends may crystallize into their respective crystalline forms and display two melting points. Heating a polymer above Tm to a melt allows its fabrication into the desired shape in processing operations such as injection molding, extrusion, ﬁber spinning and blow-molding. During cooling from melt, a semicrystalline polymer undergoes crystallization at temperatures below Tm but above Tg . Since crystallization involves translation from a highly disordered melt to a highly ordered state under viscous conditions, the extent, size, and perfection of crystallization depends on crystallization conditions such as the rate of cooling and the temperature of crystallization. For example, rapid cooling often results in a large number of crystals, as well as reductions in the extent and perfection of crystallization. On the other hand, crystallization at temperatures just below Tm for long periods results in larger and more perfect crystals. 13.1.6

Tuning Polymer Crystallization for Properties

Crystallization plays a major role in determining the thermal and mechanical behavior of polymers as it permits polymers to retain rigidity at temperatures exceeding the Tg . Crystallization also enhances the barrier properties and solvent resistance, as small molecules cannot diﬀuse through crystalline domains. Crystallization of polymers can be tuned, and thus can provide very useful control over

13.1 Thermal Properties of Polymers

the subsequent processing and ﬁnal properties. For example, crystallization from dilute solutions can lead to disentanglement and high drawability in polyethylene, producing ﬁbers with a very high modulus and strength [7]. Crystallization and morphology development in polyethylene during its formation on heterogeneous catalysts leads to a ‘‘nascent’’ state with special characteristics such as a high Tm compared to molded polyethylene. One hypothesis attributes this to the inﬂuence of strain resulting from the temperature gradient at the polymerization site, in a manner somewhat similar to strain-induced extended chain crystallization from stirred solutions [8, 9]. Polymer molecules being formed at the heterogeneous catalyst fragment surface experience strong elongational ﬂow, which could inﬂuence the nascent crystallization process [10, 11] Innovations in polyethylene reactor technology are directed toward controlling the morphology and properties of the polymerization product [12]. One expectation is that polyethylene as a nascent reactor product can be directly drawn into high-modulus, high-strength ﬁbers similar to those produced via the solution route [13]. Crystallization in step-growth polymers such as polyesters and nylons is known to assist their subsequent solid-state polymerization because exclusion of reactive end-groups from crystalline domains enhances their eﬀective concentration in the amorphous domains [14, 15]. However, the condensation reaction between the last fraction of end-groups may be hindered by crystallization [16, 17]. The possibility and rate of crystallization can also be enhanced by processes that enhance orientation, such as shearing and ﬁber drawing [18]. For example, partial replacement of terephthalic units with isophthalic units in PET reduces crystallinity, so that no crystallization in seen in 70:30 random poly(ethylene terephthalate-co-ethylene isophthalate) under quiescent conditions. However, heating its amorphous ﬁber above its Tg under a moderate tensile force results in rapid stress-induced crystallization [19]. The reduction in crystallization by copolymerization has been employed to enhance drawability of melt-spun polyester and polyamide ﬁbers [20]. On the other hand, a high rate of crystallization during cooling from melts is desired, as it determines the cycle time and thus the suitability of a polymer for very important shape-forming processes such as injection molding. The crystallization rate is easily enhanced by reducing the molecular weight of polymers, but that is also detrimental to the mechanical properties. Therefore, nucleating agents are often used to enhance the crystallization rate. The activity of several heterogeneous nucleating agents is related to oriented deposition of polymer chains on the surfaces of inorganic particles [21]. Nucleating agents such as talc, by virtue of their highly oriented crystalline surfaces, induce epitaxial crystallization of polymers such as PE and PET [22]. The activity of organic compounds (salts) such as sodium benzoate for PET crystallization is related to formation of ionic chain-ends that aggregate in polymer melts to form nucleating clusters [23]. The crystallization enhancement that occurs on replacing a small fraction (< 1%) of the ester segments in PET with amide segments was attributed to self-assembly of the amide segments [24]. These homogenous nucleators may oﬀer an advantage in impact properties. Their incorporation in PET can be carried out by simple solid-state modiﬁcation reactions, thus avoiding the degradation reactions accompanying melt-mixing

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13 Polymer Properties through Structure

Fig. 13.9. DSC cooling scans of (i) PET homopolymer (½h ¼ 0:62 dL g1 ); (ii) PET chemically modiﬁed in the solid-state with ethylene diamine (EDA) vapor to replace 2.2 mol% of ester functionality with amide functionality (nitrogen content ¼ 0.31%, PET–EDA-1: ½h ¼ 0:5 dL g1 , PET–EDA-3:

½h ¼ 0:61 dL g1 ); and (iii) PET melt mixed with 0.3% talc. The modiﬁed samples undergo crystallization at a higher temperature (that is, sooner during the cooling process). (Reprinted from Polymer, Volume 43, page 5709, Copyright 2002, U. S. Agarwal, G. de Wit, P. J. Lemstra, with permission from Elsevier).

methods [25]. Figure 13.9 shows the enhanced tendency to crystallize of the thusmodiﬁed PET during cooling from melt. The optical clarity of polymers generally decreases with the appearance of crystallinity. Semicrystalline polymers are generally opaque because of the scattering due to the diﬀerence in the refractive indices of the crystalline and the amorphous domains. However, enhancement of the nucleation density with suitable nucleating agents can lead to reduction of spherulite sizes (below the wavelength of light) and hence to transparent semicrystalline polymers. For example, the recently developed derivatives based on d-sorbitol dissolve in a polypropylene melt and are thus easily dispersed. On cooling, these additive molecules aggregate, and provide nucleation sites for ﬁne PP spherulites [26]. More recently, nanoparticles (such as clay) and carbon nanotubes are being explored as potential nucleating agents [27]. 13.1.7

Morphology of Crystalline Polymers

Early X-ray diﬀraction work in the 1920s indicated the polymers to be semicrystalline, with crystallites dispersed in an amorphous matrix. The longest dimension of the crystallites in polymers is 5–50 nm, which is a fraction of the length of the extended polymers (up to 5 mm). Based on this, the crystalline morphology was conceived to be of the ‘‘fringed micelle’’ type, where the long polymer chains pass successively through the lengths of several crystalline and amorphous domains (Figure 13.10). Later work showed that crystalline domains are most often made up of ﬂat rib-

13.1 Thermal Properties of Polymers

Fig. 13.10. Fringed micelle model of semicrystallinity in polymers. (Reprinted from Paul J. Flory, Principles of Polymer Chemistry. Copyright 1953, Cornell University and Copyright 1981 Paul J. Flory. Used by permission from the publisher, Cornell University Press).

bons like ‘‘lamellae’’, within which the chains are largely folded (insets a and b of Figure 13.11) and oriented along the smaller-thickness direction (@10 nm). The lamellae grow to dimensions of approximately 1 mm by addition of segments to a ‘‘fast-growing’’ direction (inset a of Figure 13.11). The growth in the thickness direction is by a not-so-regular stacking of multiple lamellae, with some interlamellar links (inset b of Figure 13.11). The crystal thickness increases with the temperature of crystallization, or with subsequent annealing at a higher temperature. During cooling from melt or concentration from solution, polymers form spherulites which are spherical aggregates of lamellae (Figure 13.11). These spherulites

Fig. 13.11. Growth of a spherulite with lamellar growth. (a) Lamellae grow as chain segments (parallel to lamellar thickness direction) are added successively in the ‘‘fast growth’’ direction. (b) Lamellae are stacked

somewhat irregularly, with some interlamellar links (adapted from G. Strobl, The Physics of Polymers, Springer, Berlin, 1996 and J. M. Schultz, Makromol Chem. Makromol Symp. 1988, 15, 339).

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13 Polymer Properties through Structure

L

L1 L 11

Undeformed crystals

(a) Phase changes, twinning

(b) Tilt, slip & twist

(c) Cracks formed, some chains pulled out of crystals, more tilt, slip, twist

(d) Fibrils formed

Increasing deformation of single crystals Fig. 13.12. Mechanical deformation (drawing) of spherulitic structures breaks the constituent lamellae into blocks and tie molecules. The blocks tilt, with the chains aligning along the load axis, and make up highly oriented micro-

ﬁbrils. (From A. Peterlin, in Macromolecular Reviews, Volume 1 (ISBN 0470-68245-0). Copyright 1967, John Wiley. Used by permission from the publisher, John Wiley & Sons, Inc.).

grow radially from a central nucleus, to dimensions ranging from about 0.1 mm to a few millimeters, until they meet the neighboring spherulites. Polymer chains are oriented tangentially around each nucleus. On deformation of such material under stress (for example, during ﬁber drawing), the spherulitic structure is destroyed and crystalline rearrangements lead to polymer chain orientation coinciding with the stress direction (Figure 13.12). This imparts a high modulus and strength to the ﬁbers in the axial direction. 13.1.8

Tailoring Polymer Properties through Modiﬁcation, Additives, and Reinforcement

The properties of polymers are determined by the nature and composition of the structural units, as well as the molecular architecture. Polymers are most often made from the corresponding monomers. In comparison with developing new monomers and polymerization methods, a less expensive route to tailor-making of polymers is through blending, copolymerization, architecture control during polymerization, post-polymerization chemical modiﬁcation, and additives. Detailed discussions on such possibilities are available in several reviews [37–42]. Here we will brieﬂy highlight two recent developments in the areas of block copolymers and polymeric nanocomposites.

13.1 Thermal Properties of Polymers

FCC

PCC

micelle

HEX

cylindrical micelle

F surface ( ) gyroid ( )

resicle

P surface ( ) LAM NLAM

PLAM

Fig. 13.13. Self-organization of block copolymers into various possible morphologies (from S. Forster, T. Plantenberg, Angew. Chem. Int. Ed. 2002, 41, 688).

New Morphologies through Block Copolymers Traditionally, well controlled block copolymers were prepared by living anionic polymerization. However, the advent of controlled radical polymerization techniques [28, 29] has led to more facile synthesis of block copolymers in large quantities. The sizes of the blocks are controlled by the ratio of monomer to initiator, which thus also provides control over their relative sizes. Since the blocks are covalently bonded, their phase separation on the macroscopic scale is prevented. However, a repulsive interaction between the constituent blocks can lead to chain segregation, and thus self-organization into a variety of periodicities in the range of 10–20 nm [4, 5] (Figure 13.13). Self-assembling behavior of polymers leads to interesting characteristics that can be gainfully employed in improvement of mechanical properties [30]. Shell long ago commercialized PI–PS–PI triblock copolymers in which the glassy PS domains provided physical crosslinks, and thus elastomeric behavior, even in the absence of chemical crosslinks [31]. Other morphologies may ﬁnd applications in catalysis, membranes, electro-optics, production of nanoparticles, absorption/release of drugs, and so on. For example, selective removal of one block from a bicontinuous morphology (gyroid, Figure 13.13) leads to a nanoporous 13.1.8.1

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13 Polymer Properties through Structure

structure with a network of interconnected nanochannels (diameter 20–30 nm). Metal plating of these channels can provide a very high internal surface area for catalytic activity [32]. Homopolymers and block copolymers have also been used to design novel materials through their capacity to organize organic and inorganic nanomaterials [33, 36]. Polymeric Nanocomposites Conventionally, high-modulus, high-strength ﬁbers are used as reinforcing materials in polymer composites, and ﬁllers such as CaCO3 , talc, silica, glass and carbon black are used to reduce resin cost or modify electrical conductivity, dimensional stability, abrasion resistance and other properties. In recent years, enhancement of polymer properties by composite formation with nanoparticles has developed as an area of intensive research. Nanostructured materials such as nanoclay, graphite nanosheets, carbon nanotubes, and polyhedral silsesquioxane (POSS) are being explored as potential nanoscale reinforcing agents in polymeric matrices [43, 44]. In particular, single-wall carbon nanotubes (SWNTs) are unique among nanoparticles due to their large aspect ratio and their mechanical as well as electronic and conductive properties. For example, the modulus and strength of nanocomposites with 1 wt.% SWNT compare with those of conventional ﬁber composites with 10 wt.% carbon ﬁbers [45]. Incorporation of 1 wt.% SWNT in PMMA is shown to result in a seven-fold increase in the strain to fracture [46]. In addition, several electrical, thermal, barrier, and crystallization properties are enhanced. For example, threshold concentrations for electrical conductivity of SWNTs in polymers have been reported to be as low as 0.1% [47]. Crystallization in PP, nylon-66, and PPT, and thermal stability in PET, PPT, PBT, and nylon-66 have been shown to be enhanced by incorporation of nanoparticles. One critical issue is how to achieve a good dispersion of the nanoparticles in the polymeric matrix. This is often diﬃcult by melt-mixing in the high-viscosity polymers. Better dispersion of nanoparticles often requires either solvent processing, or mixing with monomers followed by in-situ polymerization [48–50]. 13.1.8.2

13.2

Polymer Conformation and Related Properties

A large variety of polymer properties are attributable directly not to their chemical nature, but to their macromolecular constitution – that is, the long chains. In particular, several properties of polymers are related to, and can thus be estimated from, the conformational characteristics of the long chains. 13.2.1

The Chain Conformation

Polymer molecules mostly exist as random coils in solutions and melts. Their largest dimension is much smaller than the fully extended chains, but several times

13.2 Polymer Conformation and Related Properties

3

l

2 1

n

R Fig. 13.14.

End-to-end distance of a polymer coil made up of n segments, each of length l.

larger than the dimensions based on polymer density. The size of a polymer molecule is often described by its characteristic dimension: the root mean square endto-end distance, ðr 2 Þ 1=2 . Representing a polymer molecule as a chain made up of n bondsof lengthl (Figure 13.14), it can be shown that Eq. (5) applies, where 1 cos y 1 þ cos j and gðjÞ ¼ are 1 for a hypothetical freely jointed f ðyÞ ¼ 1 þ cos y 1 cos j chain, and are greater than 1 for real chains where the bond angle is ﬁxed at y and rotation about the bonds is restricted unequally at some value of j between 0 and 2p (Figure 13.15). r 2 ¼ hR Ri ¼ f ðyÞgðjÞa 0 nl 2

ð5Þ

The factor a 0 is 1 for the unperturbed coil deﬁned in this way, and is larger (or smaller) when a polymer molecule is expanded (or compressed) due to polymer– solvent interactions. Thus, for an idealized freely jointed chain in theta solvent, we have r 2 ¼ nl 2 , and the corresponding mean square radius of gyration S 2 ¼ nl 2 =6. When one accounts for the steric eﬀects that prevent distant chain segments from overlapping (excluded volume eﬀect), the dependence of ðr 2 Þ 1=2 is predicted to be n 3=5 (closer to experimental observation), rather than n 1=2 [Eq. (5)]. This polymer coil size is much larger than that based on polymer density, and hence markedly inﬂuences the viscosity behavior of polymers.

θ

θ

φ

Possible restriction on bond angles (y) and on rotation about the bonds at some values of f between 0 and 2p.

Fig. 13.15.

693

694

13 Polymer Properties through Structure

Fig. 13.16. Segments of a polymer chain in a liquid lattice. (Reprinted from Paul J. Flory, Principles of Polymer Chemistry. Copyright 1953 Cornell University and Copyright 1981 Paul J. Flory. Used by permission from the publisher, Cornell University Press).

13.2.2

Solubility of Polymers

Flory [51] and Huggins [52] were the ﬁrst to examine the case of overlapping chains in solutions. They represented the chains of n segments by placing n beads on the interconnected cells of a lattice (Figure 13.16), and the other cells were occupied by solvent molecules. Statistical analysis of the possible conformations allows calculation of the entropy, while the enthalpic interactions between the polymer segments and solvent molecules are represented by the so-called w parameter, which is temperature-dependent: w < 0 represents good solvents, and w ¼ 12 represents the so-called y-condition where Gaussian statistics prevails. The corresponding free energy change for mixing per site is given by Eq. (6), where f is the fraction of polymers made up of n segments each. DGm f ¼ ln f þ ð1 fÞ lnð1 fÞ þ wfð1 fÞ n kB T

ð6Þ

This allows the calculation of osmotic pressure by Eq. (7), where c is the mass concentration of the polymer of molecular weight M. cR g T 1 2 p¼ lnð1 fÞ þ 1 f þ wf M n

ð7Þ

Equating the derivative of the mixing free energy [Eq. (6)] with respect to f allows

13.2 Polymer Conformation and Related Properties

χ

n1 < n 2

χc

n1

C n2

φc T 1/2

θ-line

φ

Fig. 13.17. Phase separation in polymer–solvent systems predicted by Flory–Huggins theory. The system exhibits UCST behavior: that is, phase separation at temperatures below a critical point C. Coexistence curves for two diﬀerent molecular weights are shown.

the calculation of two f values (for w values larger than a critical value wc ) corresponding to minimum energy, and thus the coexistence curve (Figure 13.17) that describes separation of polymer solutions into two phases at low temperatures (w ¼ A þ ðB=TÞ > wc ) [53]. The molecular weight dependence of wc ð¼ 0:5 þ n1=2 Þ yields a downward shift of the curve in Figure 13.17, indicating an increasing tendency toward phase separation with increasing molecular weight of the polymers. Extension of the calculations to the case of polymer–polymer blends [51, 52] suggests incompatibility at w values greater than wc ¼ 2=n for two polymers with an equal number of segments. This strong tendency for phase separation of polymers when no speciﬁc polymer–polymer enthalpic interactions (for example, favorable hydrogen bonding) are present is attributed to the only small increase in entropy on mixing of high molecular weight polymers. A detailed discussion on blending of polymers and the resulting morphologies and properties is beyond the scope of this review; the interested reader is referred to more specialized texts [54–56]. 13.2.3

Dilute Solution Zero-shear Viscosity

Rheology is the science of deformation and ﬂow. Viscosity (h) is a measure of the stress (t) required to make a ﬂuid deform (ﬂow) at a desired strain rate [Eq. (8), where g_ ¼ dvx =dy; Figure 13.18]. tyx ¼ hg_ yx

ð8Þ

695

696

13 Polymer Properties through Structure

V vx

y

Fig. 13.18. Simple shear ﬂow between two parallel plates separated by a distance y, and moving with relative velocity V ¼ yg_yx .

A unique characteristic of polymers is that even a small concentration (c) of a high molecular weight polymer can signiﬁcantly enhance the viscosity of a solution (h) as compared to the viscosity of the solvent (hs ). This is because the expanded polymer coils in slowly deforming dilute solutions behave as rigid spheres with a radius of the order of ðr 2 Þ 1=2 , resulting in a large polymer volume fraction f [Eq. (9)] with the corresponding solution viscosity (Einstein, for f f 1) given by Eq. (10). 4 f ¼ pðr 2 Þ 3=2 ðcNA =MÞ 3 h ¼ hs ð1 þ 2:5fÞ

ð9Þ ð10Þ

A polymer’s capacity to enhance solution viscosity is described in terms of its intrinsic viscosity ([h]): ½h ¼ lim c!0

h hs chs

ð11Þ

Combining Eqs. (9)–(11) with Eq. 2.1, and realizing that n @ M, one ﬁnds that [h] has an M 1=2 dependence on polymer molecular weight. In reality, the factor a 0 [Eq. (5)] depends to a small extent on M, resulting in the Mark–Houwink relation [Eq. (12)]. ½h ¼ KM a

ð12Þ

The constants K and a are obtained by ﬁtting experimental data for a given polymer–solvent system. The intrinsic viscosity [h] is conveniently measured by observing the time for ﬂow of a certain volume of solution (and solvent) through a capillary, and thus provides an easy indication of the molecular weight from the tabulated Mark–Houwink constants [57]. Polymers as Dumbbells More details of the rheological behavior of polymer solutions can be evaluated by representing a polymer molecule as a dumbbell: that is, two beads connected by spring (connecting vector Q , Figure 13.19). Their contribution to the ﬂuid 13.2.3.1

13.2 Polymer Conformation and Related Properties

v2

Q

F(c)

v1

Fig. 13.19. Microscopic deﬁnition of polymer contribution to stress on a plane, as the stress exerted by the dumbbell bead above the plane on the bead below the plane.

stress [Eq. (10)] on any plane comes from the tensile or compressive spring force (F ðcÞ ¼ HQ, H being the spring constant) transmitted through the springs [58] [Eq. (13), where n is the number concentration of the springs]. t hs g_ ¼ ðnkB Td þ nhQF ðcÞ iÞ

ð13Þ

For example, a linearly elastic dumbbell model representation (Figure 13.19) of polymer molecules predicts Eq. (14) to apply, where l ¼ x=4H is a time constant and x is the bead friction factor. h ¼ hs þ nkB Tl

ð14Þ

When it is desired to represent a real polymer as a dumbbell, l can be estimated from the intrinsic viscosity measurement by means of Eq. (15). l¼

½hhs M NA kB T

ð15Þ

Polymers as Chains of Beads and Springs A more realistic representation of polymer chains is the Rouse model [59], which considers a polymer molecule to be a linear chain of N free-draining beads interconnected by springs (each of time constant lH ¼ x=4H), and predicts Eq. (16) to apply. 13.2.3.2

½h ¼

NA kB TlH N 2 1 3 hs M

ð16Þ

697

698

13 Polymer Properties through Structure

This implies that [h] has a linear dependence on molecular weight. This is not consistent with the experimental observations [Eq. (12)]. When hydrodynamic interaction and excluded volume eﬀects between chain segments are considered, the corresponding Zimm model [60] correctly predicts the @M 1=2 dependence [Eq. (17)]. ½h@

NA 3=2 3 N l M

ð17Þ

13.2.4

Viscosity of Concentrated Solutions and Melts

The models discussed in Section 13.2.3 for dilute polymer solutions all predict [h] to be independent of concentration; that is, they predict linear dependence of viscosity on concentration. When the concentration of polymer in solution is higher than the overlap concentration (c @ 1=½h @ M 1=2 ), the random coils get entangled with each other (f b 1). This is responsible for a faster than linear increase in viscosity with concentration (Figure 13.20), which often follows the scaling relation of Eq. (18) [61]. That is, if h=hs is plotted against c=c or ½hc, then the curves of various molecular weights can be superimposed. h c ¼ F ¼ Gð½hcÞ hs c

ð18Þ

While accounting for the concentration eﬀects by analyzing a single chain placed in an ‘‘eﬀective medium’’ made up of the other chains, the dependence expressed

12 10 8 viscosity 6 (cP) 4

dilute

C* semidilute

2 0

0

20

40 60 80 100 120 140 concentration (mg/mL)

Fig. 13.20. Concentration dependence of viscosity of synapse (molecular weight 18.55 kDa) solution is linear at c < c , c @ 50 mg mL1 . Redrawn from Malvern application note no. MRK511-01 (http://www.particular.ie/ m%20wt%20protein%20nano%20zs.pdf ).

13.2 Polymer Conformation and Related Properties

in Eq. (19) was obtained [62], indicating Rouse-like [Eq. (16)] linear dependence on N at intermediate concentrations (Figure 13.20). c 2l 6 N h @ hs 1 þ 24

ð19Þ

The transition from the Zimm regime [Eq. (17), ½h @ N 1=2 ] at dilute concentrations to Rouse-like behavior [Eq. (16), ½h @ N] at the intermediate concentrations is related to the hydrodynamic screening at high concentrations: the hydrodynamic interactions between segments become negligible because the local velocity around the beads falls quickly. At even higher molecular weights and concentrations, the viscosity in concentrated solutions and melts goes from a linear to a 3.4 power dependence on M [Eq. (20); Figure 13.21]. h @ c 4--5 M 3:4

ð20Þ

This behavior is described by conﬁnement of a polymer molecule to its own tube whose walls are deﬁned by the network of surrounding entangled chains (Figure 13.22) [63, 65]. Hence, chain motion (including stretching and relaxation) is permitted only curvilinearly to the tube (reptation), while motion of the chain perpendicular to the tube is hindered. During ﬂow, the relaxation of chain stretching along the tube axis is instantaneous, while a nonrandom orientation of tube segments contributes to the stress. 13.2.5

Nonlinear Polymers

Modern synthesis methods have allowed preparation of polymers with a great variety of branching architectures, such as random, comb, uniform and nonuniform star, brush, and dendritic [66, 67]. It is understandable that the conformations and hence several properties of these structures can vary remarkably among linear polymers of similar compositions and molecular weights. The compression of the branched molecule compared to linear molecules of same molecular weight can be quantiﬁed in terms of the factors deﬁned in Eqs. (21) and (22), where the subscripts b and l represent the branched and the linear polymers. g ¼ ðS 2 Þb =ðS 2 Þl

ð21Þ

g 0 ¼ ½hb =½hl

ð22Þ

The factors g and g 0 can often be related, depending on the branching architecture. For example, for star polymers with p equal arms [68, 69], the relationship is given by Eq. (23).

699

700

13 Polymer Properties through Structure

Fig. 13.21. Variation of polymer melt viscosity with molecular weight. (from C. W. Macosko, Rheology: Principles, Measurements and Applications, Copyright 1994 VCH. Used by permission from the publisher, John Wiley & Sons, Inc.).

g¼

3 2 ¼ ½0:2624 þ ð1 0:2624Þðg 0 Þ 1:088 ðg 0 Þ 0:6087 p p2

ð23Þ

Such characteristics of several other branched architectures have been examined [70]. The molecular compression behavior by branching allows high loading of the polymeric additives without the excessive viscosity enhancement associated with polymers. As we shall discuss in Section 13.3.6, the rheological eﬀects of

13.2 Polymer Conformation and Related Properties

Fig. 13.22. Conﬁnement of a polymer chain’s motion in its own tube deﬁned by a ﬁxed network of surrounding chains. Back and forth motion in the tube leads to tube renewal and

thus to the loss of the original tube orientation and stress (adapted from M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, Clarendon, Oxford, 1986).

branching are more substantial and are put to use in strongly ﬂowing polymer melt systems. 13.2.6

Rigid Rod-like Polymers

Some polypeptides and para-linked synthetic aromatic polymers take up elongated forms, forming rodlike structures (rather than ﬂexible coils). Their solution viscosity displays a strong dependence on the polymer molecular weight according to Eq. (24), where L and R are the length (@M) and the radius, and rb is the bulk density of the rodlike molecules [64]. ½h ¼

2ðL=RÞ 2 45rb ðlnðL=2RÞ 0:8Þ

ð24Þ

At higher concentrations (number concentration > 1=L 3 ), the motion of rodlike polymers is highly hindered in the directions perpendicular to the rod axis, and the viscosity increases spectacularly with mass concentration (r) and molecular weight [Eq. (25)] [64]. h z r 3 M 6 =lnðNÞ

ð25Þ

701

702

13 Polymer Properties through Structure

Due to this eﬀect, the increase in molecular weight during their step-growth polymerization leads to a reduction in the reaction rate of such polymers [71], and provides the possibility of controlling the kinetics and molecular weight distribution by orienting ﬂow [72]. Spontaneous liquid-crystalline ordering of these polymers at still higher concentrations allows their processing into very strong ﬁbers [73].

13.3

Polymer Rheology

In Section 13.2 we discussed the equilibrium conformations of polymers and their inﬂuence on the polymeric behavior. We shall now discuss the rheological properties of polymers associated with their very special characteristic that their conformation changes under imposed ﬂows. 13.3.1

The Viscous Response: Shear Thinning

One important characteristic of polymers is their shear thinning behavior: the viscosity decreases with increasing rate of deformation (Figure 13.23). This behavior at g_ @ l1 is attributed to the partial disentanglement of the chains as they get extended in the ﬂow direction. Since the relaxation time increases with molecular weight [for example, l @ M 1:5 at low concentrations; Eqs. (12), (15)] and with polymer concentration, shear thinning begins at lower shear rates for polymers with

Fig. 13.23. Shear thinning behavior of an ABS polymer melt at three diﬀerent temperatures. Broken lines are power law ﬁts and solid lines represent the Cross model. (from C. W.

Macosko, Rheology: Principles, Measurements and Applications, Copyright 1994 VCH. Used by permission from the publisher, John Wiley & Sons, Inc.).

13.3 Polymer Rheology

higher molecular weights and at higher concentrations. The shear thinning behavior is responsible for the processibility of many polymer melts. For example, high strain rates of extrusion are employed lest they should undergo stress and thermal degradation, and the circumferential shear (due to screw rotation) during extrusion reduces the viscosity suﬃciently to enable axial ﬂow (due to the pressure developed) [74]. Empirically, the shear thinning behavior is most conveniently represented by the power law relationship expressed by Eq. (26) [75] (Figure 13.23) or, more appropriately by a four parameter Cross model [Eq. (27), where ho and hy correspond to the low (zero) and the high shear plateaus, and l and n ð< 1Þ correspond to the location of transition and the slope on a log–log plot (Figure 13.23)] [76]. h ¼ Kðg_ Þ n1

ð26Þ

h hy ¼ ½1 þ ðlg_Þ 2 ðn1Þ=2 ho hy

ð27Þ

It is interesting to examine the ability of the simple molecular models (Section 13.2.2) to represent the shear thinning phenomena. The linearly elastic dumbbell and the Rouse models represent chain extension (unlimited at increasing g_), yet these predict the viscosity to be independent of shear rate. The ﬁnitely extensible nonlinear elastic (FENE) dumbbell model considers the connecting spring to stiﬀen inﬁnitely as the extension approaches the maximum possible value Q o . This permits prediction of the shear thinning behavior starting at (lg_ @ 1) [58]. Similarly, when one accounts for hydrodynamic interaction, the linearly elastic dumbbell model also predicts shear thinning according to Eq. (28). h hs ¼

nkB Tlð1 xÞ 1 þ 2xð1 x=2Þðlg_ Þ 2

ð28Þ

13.3.2

Normal Stresses during Shear Flow

In contrast to simple ﬂuids under shear that generate stresses parallel to the direction of shear, polymeric ﬂuids also generate normal stresses perpendicular to the ﬂow direction [Eqs. (29), (30), where N1 and N2 are called the ﬁrst and the second normal stress diﬀerence, respectively, and are dependent on the shear rate]. N1 is positive, corresponding to a tendency of the polymeric ﬂuid to push apart the two plates between which the ﬂuid is sheared (Figure 13.24). Such a tendency of polymeric ﬂuids is manifested in interesting phenomena such as die-swell and rod-climbing (Figure 13.25). txx tyy ¼ N1

ð29Þ

txx tzz ¼ N2

ð30Þ

703

704

13 Polymer Properties through Structure

manometers at various radial positions stationary cone fluid rotating flat plate

Fig. 13.24. When a polymeric ﬂuid is sheared between a cone and a plate, stresses are generated that are also normal to the ﬂow direction, as is evident from the monomeric ﬂuid levels.

Fig. 13.25. Normal stresses in shearing polymeric ﬂuids are responsible for their swelling on exit from a die.

13.3 Polymer Rheology

v

Fig. 13.26. Uniaxial extensional ﬂow generated between a ﬁxed plate and a plate moving at desired velocity (v), while the entire geometry is held in a constant-temperature bath. Measurement of the sample dimension and the required tension force allows calculation of the elongational viscosity.

13.3.3

Extensional Thickening

A ﬂow that converges or diverges is called extensional or compressional (Figure 13.26). For example, stretching of a ﬁlament creates extensional ﬂow with the corresponding velocity gradient (g_xx ¼ dvx =dx) resulting in tensile stress [Eq. (31), where he is called the extensional viscosity]. txx tyy ¼ he g_xx

ð31Þ

With an increase in ﬂow time or strain at low g_xx, he increases to its asymptotic ‘‘steady-state’’ value given by Eq. (32). he ¼ 3h

ð32Þ

However, many polymers show strain hardening, that is, an increase in he beyond this asymptotic value at high strain rates (Figure 13.27). Such behavior of high molecular weight polymers is responsible for stabilizing stretching processes involved in the production of ﬁlms and ﬁbers.

705

706

13 Polymer Properties through Structure

Fig. 13.27. Uniextensional extensional viscosity (hþ u , open symbols) and shear viscosity (hþ , closed or half-ﬁlled symbols) of low-density polyethylene (LDPE) as functions of ﬂow time after inception of the steady

extensional rate (_e) or shear rate (g_), respectively. (from C. W. Macosko, Rheology: Principles, Measurements and Applications, Copyright 1994 VCH. Used by permission from the publisher, John Wiley & Sons, Inc.).

13.3.4

The Elastic Response Ideal Elastic Response Polymeric materials often tend to behave as springs (that is, they have a tendency to retract on stretching), thus displaying some degree of elasticity. An ideal elastic material responds instantaneously to application or removal of stress, with the strain (g) being proportional to the stress (Hookean), independently of the strain rate [Eq. (33)]. Here, the constant G is the modulus of the elastic material. 13.3.4.1

t ¼ Gg

ð33Þ

Rubberlike Elasticity Elastic polymers contain a crosslinked network to restrain the gross mobility of their chains. When the chain segments between crosslinks are deformed aﬃnely in a macroscopic sample due to application of an external force, the resulting decrease in chain entropy brings in a springlike retractive force (with spring constant 3kB T=r 2 ) [77]. Generalization to a three-dimensional deformation to account for cross-sectional area changes corresponding to an axial strain of ða 1Þ results in 13.3.4.2

13.3 Polymer Rheology

Fig. 13.28. Stress–extension curve for a sample of natural rubber compared with the ideal rubber prediction [Eq. (35)] (Reprinted from Paul J. Flory, Principles of Polymer

Chemistry. Copyright 1953 Cornell University and Copyright 1981 Paul J. Flory. Used by permission from the publisher, Cornell University Press).

the retractive force per unit deformed cross-sectional area according to Eq. (34), where G ¼ Ne kB T, and Ne is the number of crosslinked chains per unit volume. 1 txx ¼ G a 2 a

ð34Þ

The a 2 dependence indicates that the chains become progressively harder to stretch. As seen in Figure 13.28, this simple model based on (inﬁnitely extensible) Gaussian chains fails to explain the extreme stiﬀening behavior at very high strains near the maximum possible chain extension. 13.3.5

The Viscoelastic Response

Elastic behavior in polymers is rubberlike and thus not Hookean, and viscous behavior is shear thinning and thus not Newtonian. Further, real polymers display characteristics that are a combination of such elastic and viscous responses. This is best illustrated by the Maxwell model [75, 78], which considers the viscoelastic

707

708

13 Polymer Properties through Structure

G

τ, γ1

η

τ, γ2

τ, γ Fig. 13.29. The Maxwell model for a viscoelastic material considers an elastic spring and a viscous dashpot connected in series.

behavior to be a series combination of the Hookean elasticity as represented by a linear spring and the Newtonian viscous behavior as represented by a dashpot (Figure 13.29). The stress (t) being the same for the spring and the dashpot, and the total strain (g) being the sum of the strains of the spring and the dashpot, one obtains Eq. (35) or, substituting l ¼ h=G, Eq. (36). dg dg1 dg2 1 dt t ¼ þ þ ¼ dt G dt h dt dt tþl

dt dg ¼h dt dt

ð35Þ ð36Þ

In Figure 13.30, we compare the creep and stress relaxation responses of such a material with the responses of an ideal elastic solid and a viscous Newtonian ﬂuid. It is easily realized from the creep behavior that a small value of h and large value of G (and thus a small value of l) correspond to the viscoelastic response approximating to the viscous response. Further, the creep response of the viscoelastic material at short times (for example, instantaneously) closely approximates to the elastic behavior, because viscous ﬂow occurs only over time. At long times, the accumulated viscous response represents the overall deformation. Similarly, the stress relaxation behavior of the viscoelastic material suggests that l is a measure of time taken for stress to relax to 1/e of the original stress, and hence this is called the relaxation time of the material. The eﬀect of material characteristics l and process time (t) are thus easily put together in the dimensionless parameter called the Deborah number De, as in Eq. (37).

13.3 Polymer Rheology

1 γ

τo / η

τ /τo 1/e

τo /G 0

time (a)

0

λ

time

(b)

(a) Creep behavior (strain response to an applied step-stress to ) and (b) stress relaxation behavior (stress response to step-strain go ). The behavior of an elastic solid Fig. 13.30.

De ¼ l=t

(broken lines with long dashes), a viscous liquid (broken line with short dashes, and dotted line), and a Maxwell viscoelastic material (solid line) are compared.

ð37Þ

De > 1 and De < 1 correspond to predominantly elastic and viscous responses respectively. De determines the level of shear stress that can relax during the melt processing, and hence the residual chain orientation, morphology, and properties in the solid-state article fabricated by melt processing. The Maxwell model [Eq. (36)] can also be expressed in the integral form of Eq. (38) [75]. That is, the stress can be looked upon as the sum of every incremental strain g_ ðt 0 ÞDt 0 multiplied by the corresponding exponentially decreasing modulus 0 Geðtt Þ=l . t¼

ðt

0

Geðtt Þ=l g_ ðt 0 Þ dt 0

ð38Þ

y

Linear Viscoelasticity in Dynamic Oscillatory Flow When a sinusoidally varying strain (Figure 13.31) of frequency o and a small amplitude gmax is imposed on a ﬂuid according to Eqs. (39) and (40), the corresponding stress also oscillates [Eq. (41)], albeit with a phase delay (d, Figure 13.31) after the imposed strain rate that can be decomposed into two strain-dependent terms [Eq. (42)], of which the ﬁrst is the elastic contribution in phase with the strain [Eq. (39)], and the second is the viscous contribution in phase with the strain rate [Eq. (40)]. 13.3.5.1

gyx ¼ gmax sinðotÞ

ð39Þ

g_yx ¼ g_ max cosðotÞ

ð40Þ

tyx ¼ tmax cosðot dÞ 0

ð41Þ 00

tyx ¼ G gmax sinðotÞ þ G gmax cosðotÞ

ð42Þ

709

710

13 Polymer Properties through Structure

τ m ax

δ γ

max

τ, γ 0

2

4

6

ωt

8

10

Fig. 13.31. Imposed sinusoidal strain (solid line), and the corresponding sinusoidal stress (broken line) response of a viscoelastic material.

G 0 ¼ tmax cos d=gmax and G 00 ¼ tmax sin d=gmax are called the storage modulus and the loss modulus, corresponding to the elastic and viscous contributions, respectively. For the Maxwell model [Eq. (36)], their dependence on frequency is ðolÞ 2 ol 00 . With an increase in frequency, given by G 0 ¼ G 2 and G ¼ G 1 þ ðolÞ 2 1 þ ðolÞ a polymer behaves increasingly like a solid (storage/elastic), G 0 ! G: that is, the storage modulus equates to the elastic modulus. This single-mode Maxwell model correctly represents the observed G 0 @ o 2 and G 00 @ o dependence in the terminal zone (long time, small o), and a crossover at o ¼ 1=l (Figure 13.32). However, real polymers display not a single time constant and a single modulus, but a spectrum

G', G''

crossover ω defines (1/ λ)

log (ω) Frequency response of a viscoelastic material. Storage modulus (G 0 , solid line) and loss modulus (G 00 , broken line) are shown. Fig. 13.32.

13.3 Polymer Rheology

5

10

Gi (Pa) 3

10

branched

linear

1

10

-3

10

1

-1

10

λi (S)

Fig. 13.33. Relaxation spectra for linear and branched polyethylene (PE). The spectrum of the branched PE includes a relaxation time that is 10 times longer than the longest relaxation time of linear PE. This is consistent with the

10

3

10

long relaxation times predicted for branched PE. (redrawn from S. H. Wasserman in Metallocene Catalyzed Polymers, Eds. G. M. Benedikt, B. L. Goodall, BF Goodrich, Brecksville, 1998).

of relaxation times (li ) and modulus parameters (Gi ) corresponding to diﬀerent modes of relaxation. The Maxwell model can then be generalized by considering the stress to be a sum (superposition) of the contributions of each mode of relaxation, as represented by Eq. (43), where Gi and li can be found by carrying out a multiparameter ﬁt of the stress response to step-strain or dynamic strain. t¼

ðt y

X 0 ðGi eðtt Þ=li Þg_ ðt 0 Þ dt 0

ð43Þ

i

These values of Gi and li constitute the relaxation spectrum of a polymer, and are useful in correlating the viscoelastic behavior of polymers with their molecular characteristics, such as long-chain branching in polyethylene (Figure 13.33) [79]. 13.3.6

Inﬂuence of Polymer Branching Architecture in Bulk Polymers

Polymer chemistry has played a very important role in providing polymeric material with pre-designed molecular architectures. In particular, the branching architecture has been seen to greatly inﬂuence the rheological behavior of polymer melts as the reptation of the arms is severely suppressed. Relaxation of the arms can take place only by arm withdrawal, which is exponentially slowed with increasing arm length [63]. It is only after suﬃcient movement of the arms that the motion of branching points (otherwise hindered by the network around them) can

711

712

13 Polymer Properties through Structure

slowly take place (Figure 13.34) [80]. Some spectacular eﬀects of these entanglements in long-chain branched industrial polyethylene (LDPE) are the following: very long relaxation times (Figure 13.33), greater extensional thickening (Figure 13.27), and greater shear thinning (Figure 13.35) [81, 82]. The extensional thickening is critical for imparting the characteristics necessary for ﬁlm blowing, blowmolding and foaming applications of polyoleﬁns, and the increased shear thinning allows easier extrusion at high shear rates. Eﬀorts have been made to enhance processibility of polyesters by introducing branching, and lower melt and solution viscosities are obtained. However, the solid-state properties were found to be poor [83]. Hyperbranched polymers are compact, highly branched, three-dimensional macromolecules with a high density of end-groups [84]. Their compact structure results in inherently low viscosity, as evidenced by the value of the Mark–Houwink constant a being less than 0.5. Hyperbranched aliphatic polyesters have been applied as toughners for epoxy thermosets without a signiﬁcant increase in viscosity [85]. Hyperbranched polymers can be used as rheology control agents during the processing of polymers, for example to obtain a dramatic decrease in viscosity [86], as lubricants [87], or for sharkskin elimination [88]. Dendrimers are a relatively new class of highly and regularly branched polymers with well deﬁned numbers of branching layers and a low molecular weight between branches. Dendritically branched polymers may have somewhat higher molecular weights, yet lower intrinsic viscosity, than corresponding regular starshaped polymers, due to their dense internal structure [89]. They can display high elasticity [90], and this makes them potentially interesting rheology modiﬁers. The large number of peripheral terminal groups contribute to their solubilities, and oﬀer potential for loading high functionalities when utilized as additives, for example for solubility enhancement, ﬁber dyeability, crystallization suppression, and suchlike [91, 92]. Dense grafting of side chains onto linear backbones, and homopolymerization of macromonomers, are both used to synthesize macromolecular brushes. Steric repulsion of the side chains forces the main chain into an extended wormlike conformation, resulting in liquid-crystalline phases, and lower dynamic shear moduli than linear ﬂexible coils in concentrated solutions [93, 94]. Densely grafted polymeric brushes on sliding surfaces have been found to reduce friction, and therefore have potential for providing biolubrication for artiﬁcial implants [95]. 13.3.7

Polymers as Rheology Modiﬁers

Due to their large coil size and hence their great inﬂuence on the shear and elongational behavior of solvents, polymers are used as rheology-modifying additives in applications ranging from fuels, lubricants, coatings, and sprays to enhanced oil recovery and turbulent drag reduction [96]. For example, macromolecular coils of poly(alkyl methacrylate) and ethylene–propylene copolymers that expand with temperature are used to oﬀset the solvent viscosity decrease upon heating (Figure

13.3 Polymer Rheology

Fig. 13.34. Relaxation in a branched H-polymer melt involves retraction of arms by ﬂuctuations, followed by crossbar reptation as branch points move along the dilated tube (Reprinted with permission from T. C. B. McLeish et al., Macromolecules, Volume 32, Page 6734. Copyright 1999).

complex viscosity (Pa-s)

frequency (rad/s) Fig. 13.35. Complex viscosity of linear (broken curve) and long-chain branched (solid curve) polymers. Long-chain branching results in stronger shear thinning properties.

713

714

13 Polymer Properties through Structure

Fig. 13.36. Temperature eﬀect on intrinsic viscosity of ethylene–propylene copolymers in toluene. The calculated hydrodynamic volume of EP-5 (Mw ¼ 322 000, ethylene 80 mol%) increases from 5:6 1018 to 43:7 1018

cm 3 /molecule as the temperature increases from 10 C to 50 C. (Reprinted with permission from A. Sen, I. D. Rubin, Macromolecules, Volume 23, Page 2519. Copyright 1990, American Chemical Society).

13.36), resulting in ﬂatter viscosity–temperature curves [97]. This kind of application in automotive lubricating oils is termed viscosity index improvement [96]. Extensional thickening by polymers is responsible for the use of polymeric additives as anti-misting agents to resist ﬂuid breakup into minute droplets [98, 99]. Another application of polymers is as drag-reducing agents where a few parts per million of polymeric additives can locally enhance the elongational viscosity in order to dampen turbulence, thereby enhancing ﬂuid ﬂow by as much as 100%. At the high deformation rates during such applications, polymers can undergo chain scission and thus lose their eﬃciency in rheology modiﬁcation [100]. Enhancement of shear stability against such chain scission by way of control of chain architecture is of much interest [101]. Once again, developments in living polymerization techniques in recent decades have provided access to new polymeric topologies. Some of these have now been optimized suﬃciently to yield polymers of high enough molecular weight to be of interest for rheology modiﬁcation [102]. 13.3.8

Rheological Control with Block Copolymers

Nanophase separation of block copolymer melts introduces interesting rheological behavior. For example, the transient elongational viscosity behavior of a triblock copolymer of styrene and oleﬁns was found to be strongly dependent on the initial orientation of the cylindrical domains [103]. While conventional processing of polymers involves melting, block copolymers such as PS–b–PnBA display melt-like behavior simply on application of pressure at room temperature; this is due to pressure-induced miscibility of the blocks (baroplastics [104]). Precipitation and gelation of long-chain paraﬃns in waxy crude oils hinders their ﬂow and hence their recovery and processing. Flow characteristics can be im-

13.4 Summary

proved with addition of a diblock copolymer with a crystalline polyethylene block and an amorphous poly(ethylene–propylene) block which self-assembles in the oil. The former block provides nucleation sites for crystallization of the paraﬃns, and the latter provides steric stabilization for the wax crystals [105]. 13.3.9

Polymer-like Structures through Noncovalent Associations

As compared with the all-covalently linked monomers forming polymeric structures, weak noncovalent associating interactions between macromolecules/ oligomers also provide a possibility of forming chainlike structures. Examples of such interactions are van der Waals and Coulomb interactions, hydrophobic interactions, hydrogen bonds, and ionic bonds. Use of reversible associations like these have been successful in tailoring rheological behavior. For example, polymers with ionically associating groups have been described as useful drag-reducing agents, as their shear stability is enhanced due to sacriﬁcial breaking of the reversible associations at high shear, and reassociation in appropriate low-shear conditions [106, 107]. Hydrophobically associating polymers (HEURs) are used for controlling the rheological behavior of industrial coatings, since they provide suﬃcient shear viscosity through network formation [108]. As compared to solutions of high molecular weight polymers that show divergent trends with solvent evaporation, linear polymers and networks can be built by strongly associating 2u-ureido-4-pyrimidone end-groups [109]. High sensitivity of the associating behavior to temperature provides opportunities for on-line tuning of rheological behavior. Introduction of ionic groups or proteins into polymers (forming ionomers) leads to physical associations at the temperature of use [110, 111]. For example, Surlyn (DuPont) is a copolymer of ethylene and methacrylic acid that shows enhanced zero-shear viscosity and elastomeric green strength. Viscoelastic characteristics are also enhanced due to loss of associations at the appropriate dissociation temperature. Ion pair associations are exploited to obtain miscibility in otherwise immiscible polymers [112]. Molecules that are capable of spatial complementarity and weak reversible noncovalent interactions with each other can self-assemble from a less ordered state (such as a solution) to an ordered structure (such as a solid crystal) without human intervention [113]. In this manner, precise tuning of oligopeptide associating structures provides an completely new route to a hierarchy of structures such as tapes, ribbons, ﬁbrils, ﬁbers, and ﬁber networks, which can be useful as materials [114– 116].

13.4

Summary

In this chapter, we have discussed the interrelationships of the molecular structures of polymers with their thermal, solution, and rheological behavior. The ther-

715

716

13 Polymer Properties through Structure

mal response is largely determined by interactions at short distances, such as symmetry and steric eﬀects. On the other hand, the solution and rheological behavior is dominated by topological considerations such as the long-chain structure and branching. Thus, a large variety of polymer properties can be tuned by characteristics such as (co)polymer composition, average molecular weight, molecular weight distribution, branching architectures, reversible associations, and (nano)additives. In the context of the present handbook, it seems appropriate to emphasize once again the role of the emerging polymerization techniques in achieving these characteristics in polymers.

Notation

a c c De f ðyÞ F ðcÞ g; g 0 gðjÞ G G 0 ; G 00 DGm H DHm kB K l L M n n N NA Ne N1 ; N2 p Q R Rg S2 DSm t

Mark–Houwink constant mass concentration of the polymer [kg m3 ] critical overlap concentration of the polymer [kg m3 ] Deborah number factor for coiled chain dimension corresponding to bond angle spring force in the connector of a bead–spring model [N] factors describing compression of branched polymer versus linear polymer factor for coiled chain dimension corresponding to rotation about bonds elastic modulus [Pa] the storage modulus and the loss modulus [Pa] free energy change on mixing [J] spring constant in a bead–spring model heat of melting [J] Boltzmann’s constant [J K1 ] Mark–Houwink constant length of bonds (segments) in polymer chain [m] length of the rodlike molecules [m] molecular weight [kg kmol1 ] number of bonds (segments) in the polymer chain number concentration of the springs number of beads in the bead–spring model Avogadro’s number number of crosslinked chains per unit volume [m3 ] ﬁrst and second normal stress diﬀerence [Pa] number of arms in a star polymer connector (spring) vector in a bead–spring model polymer chain’s end-to-end distance vector [m] universal gas constant [J K1 kmol1 ] mean square radius of gyration [m 2 ] entropy of melting [J K1 ] time [s]

Notation

T Tg Tg; y Tm vx wi xc

temperature [K] glass transition temperature [K] glass transition of a polymer with inﬁnite molecular weight [K] melting temperature [K] x component of velocity [m s1 ] weight fraction of species i extent of crystallization

Greek a a0 g_ g d h hs he [h] y l lH x p r rb tij f j w wc o

extension ratio of an elastic material factor for polymer coil expansion due to polymer–solvent interactions shear rate [s1 ] strain phase delay in dynamic analysis viscosity of a polymer solution [Pa s] shear viscosity of the solvent [Pa s] extensional viscosity [Pa s] intrinsic viscosity of a polymer in solution [m 3 kg1 ] angle between consecutive bonds in a polymer chain [rad] time constant of a polymer [s] time constant of a polymer segment [s] bead friction factor osmotic pressure [Pa] polymer property such as mass concentration, density, speciﬁc heat bulk density of a polymer stress tensor [Pa] polymer volume fraction in solution angle describing rotation of bonds at ﬁxed y [rad] Flory’s interaction parameter for mixing critical value of w for phase separation angular frequency [s1 ]

Acronyms BPA EDA HEUR LDPE PBT PE PEEK PET PI

bisphenol-A ethylene diamine hydrophobically associating polymer (hydrophobically modiﬁed, ethoxylated urethane resin) low-density polyethylene poly(butene terephthalate) polyethylene poly(ether ether ketone) poly(ethylene terephthalate) polyimide

717

718

13 Polymer Properties through Structure

PMMA PnBA POSS PP PPT PS SWNT WAXD

poly(methyl methacrylate) poly(n-butyl acrylate) polyhedral silsesquioxane polypropylene poly(propylene terephthalate) polystyrene single-wall carbon nanotube wide-angle X-ray diﬀraction

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Polymer Mechanical Properties1 Christopher J. G. Plummer 14.1

Introduction 14.1.1

Long-chain Molecules

A polymer is made up of macromolecules, that is, long chains of covalently linked chemical units, called monomer units. In many synthetic macromolecules, these units are identical, and the term ‘‘polymer’’ implies their number per chain exceeds about 10 2 , although it may reach over 10 5 in many instances. Branched and highly crosslinked architectures are also important in practice, but it is above all the presence of long linear monomer sequences that leads to the unique properties of polymers. For conciseness, therefore, the emphasis in the present overview of mechanical properties will be on isotropic, chemically homogeneous polymers composed of ﬂexible linear or lightly crosslinked macromolecules. Long chains of atoms or molecules linked by single covalent bonds are usually ‘‘ﬂexible’’ in that rotation about the bonds is relatively unhindered, so that they have many internal degrees of freedom. The forces that oppose any attempt to constrain the macromolecule, such as maintaining its ends at ﬁxed positions, are therefore essentially entropic in origin. This entropic contribution to their mechanical properties is one of the key features of polymers; it gives rise to the phenomenon of rubber elasticity, for example, as will be discussed in Section 14.2. Another characteristic of polymers is the important role of their glass transition temperature, Tg . Polymers with regular conﬁgurations, such as polyethylene (PE), may crystallize, but generally remain partly amorphous, even at temperatures, T, well below their melting point, Tm . Moreover, polymers with random conﬁgurations, such as atactic polystyrene (aPS), often show little crystallinity at any T and are consequently referred to as ‘‘amorphous’’. In either case, a signiﬁcant proportion of a polymer becomes glassy at T < Tg . The glass transition therefore has a pro1) 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

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14 Polymer Mechanical Properties

Fig. 14.1.

Sketch of the low strain mechanical response of various types of polymer.

found eﬀect on polymer properties, whereas in many low molar mass materials it only has observable physical consequences if solidiﬁcation is associated with extremely rapid cooling rates. Figure 14.1 shows schematically the eﬀect of T on the low strain mechanical behavior of various types of polymer, along with the principal regimes of behavior. Below Tg , unoriented polymers are rigid solids with a Young’s modulus, E, of between about 1 and 4 GPa, whose low strain response is conditioned by the cohesive forces between nonbonded atoms. In organic polymers, these are mainly due to weak Van der Waals interactions so that, even below Tg , the moduli are much lower than in ionic and metallic crystals as well as in three-dimensional covalently bonded structures such as diamond. As T is increased above Tg , E drops by a factor of more than 10 3 in amorphous polymers such as aPS, although the decrease is less marked in semicrystalline polymers, whose Tm is typically very much greater than Tg , and the reinforcing eﬀect of the relatively rigid crystalline regions persists. Remarkably, even in uncrosslinked amorphous polymers, the modulus remains nearly constant as T is increased further. This gives rise to a ‘‘rubbery plateau’’, whose extent depends strongly on the molar mass, M, as indicated by Figure 14.1. Thus, the plateau becomes eﬀectively inﬁnite for an inﬁnite, crosslinked network of chains, but disappears altogether when M is less than some critical molar mass, Mc . Beyond the plateau there is a transition to viscous liquid behavior, a transition that is deﬁned by Tm in semicrystalline polymers. 14.1.2

Simple Statistical Descriptions of Long-chain Molecules

A real macromolecule contains bonds with ﬁxed valence angles, as shown in Figure 14.2(a) for PE. However, as one moves along an instantaneous snapshot of a

14.1 Introduction

Fig. 14.2.

Schematic representations of an isolated polyethylene chain at two length scales.

typical macromolecular chain, taking any bond as a starting point, the correlation between the direction of the skeletal bonds and that of the starting bond decreases rapidly. Thus, taken as a whole, a linear molecule resembles a long, ﬂexible piece of string, as shown in Figure 14.2(b). It may therefore be modeled as a sequence of N freely jointed bonds of length b, in which there is no correlation between the vector corresponding to the nth bond, rn , and rnþ1 . The end-to-end vector R that links one extremity of the chain to the other is then given by Eq. (1).

R¼

N X

ð1Þ

rn

n¼1

The magnitude of R in an unconstrained chain is uncorrelated with its direction, so that its value averaged over a large number of arbitrarily chosen conformations must be zero. Because thermal motion causes a real chain to change its local conformation every few picoseconds, for more familiar time scales it would therefore be perceived as a ‘‘fuzzy ball’’ with hRi ¼ 0 and a time-averaged mean square endto-end distance given by Eq. (2) (rn and rm are uncorrelated so that hrn rm i must vanish for n 0 m).

hR2 i ¼

N X N X n¼1 m¼1

hrn rm i ¼

N X

hrn2 i ¼ Nb 2

ð2Þ

n¼1

For a real macromolecule, the value of b characteristic of the equivalent freely jointed chain is greater than the mean length of the skeletal bonds, lo , owing not only to ﬁxed valence angles but also to the presence of rigid units and steric hindrance to rotation about single bonds. If Nl is the number of skeletal bonds in the chain, hR2 i ¼ Cy Nl lo2 , where the constant Cy is the Flory ‘‘characteristic ratio’’, also called the ‘‘chain stiﬀness’’ [1, 2]. It is of the order of 10 in ﬂexible polymers (7.3 in PE and 9.6 in aPS, for example [3]), but may be much higher for more rigid architectures such as para-linked benzene rings. Cy provides the link between

723

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14 Polymer Mechanical Properties

molecular architecture and general models for macroscopic physical properties governed by chain statistics, often expressed in terms of b ¼ Cy Nl lo2 /Rmax , where R max is the maximum end-to-end distance of the chain (which depends on both lo and the bond angles). In the limit of large N, the freely jointed chain model implies the probability density PðR; NÞ for a given R to be a Gaussian distribution [Eq. (3)], whence the frequent use of the term ‘‘Gaussian chain’’ to refer to idealized ﬂexible linear macromolecules [4]. PðR; NÞ ¼

3 2pNb 2

3/2

3R2 exp 2Nb 2

ð3Þ

This result is a simpliﬁcation in that it does not take into account long-range interactions and solvent eﬀects, for example. It nevertheless provides a working description of conformations in solid and molten polymers, and hence forms the basis for a wide range of theoretical models in polymer physics and mechanics, some of which will be discussed in what follows.

14.2

Elasticity 14.2.1

Deformation of an Elastic Solid

The stress state associated with an elemental cube may be deﬁned in terms of the six independent quantities in Eq. (4), where the ﬁrst subscript refers to the direction of the normal to the plane on which the stress acts and the second subscript refers to the direction of the stress (at equilibrium the net torque is zero so that sij ¼ sji ). 2

sxx

6 sij ¼ 4 sxy sxz

sxy

sxz

3

syy

syz 7 5

syz

szz

ð4Þ

The deformation may likewise be deﬁned in terms of the components of engineering strain [Eq. (5)]. 2

exx

6 eij ¼ 4 exy exz

exy

exz

3

eyy

eyz 7 5

eyz

ezz

ð5Þ

The generalized form of Hooke’s law, which proposes a linear relationship between stress and strain at vanishingly small strains, is then given by Eq. (6).

14.2 Elasticity

eij ¼ Eijkl skl

ð6Þ

In an isotropic solid, there is no coupling between tensile and shear stresses, and Hooke’s law takes the familiar form of Eq. (7), where n is Poisson’s ratio and G is the shear modulus. exx ¼

1 n sxx ðsyy þ szz Þ; E E

2exy 1 g ¼

1 sxy G

ð7Þ

Moreover, it is easily shown that Eq. (8) holds, so that a complete description of the small strain response requires only two independent parameters. G¼

E 2ð1 þ nÞ

ð8Þ

Similarly, the bulk (compression) modulus, K, is given by Eq. (9). K¼

E 3ð1 2nÞ

ð9Þ

It is implicit in such descriptions of small-strain elasticity that the stress is unaﬀected by the deformation, which justiﬁes the assumption of linearity. For the large deformations associated with rubber elasticity and shear yielding in glassy and semicrystalline polymers, however, this is no longer valid, and it is more usual to express deformations in terms of the deformation ratios, l, such that the point ðx; y; zÞ is transformed to ðl1 x; l 2 y; l3 zÞ. If the coordinate system is deﬁned with respect to the deformed system, it is always possible to choose the axes so that they are not rotated by the deformation. The shear components of the deformation are then equal to zero, which permits considerable simpliﬁcation. For example, the incremental work per unit volume of undeformed material is given by Eq. (10), where the fi are the applied forces along the axes of a unit cube of undeformed material [5]. dw ¼ f1 dl1 þ f2 dl 2 þ f3 dl3

ð10Þ

14.2.2

Thermodynamics of Rubber Elasticity

Perhaps the most striking diﬀerence between rubbers and other materials is their capacity for large reversible or nearly reversible deformations at their service temperature, and it is this aspect of their behavior that will be described here. The commercial exploitation of natural rubber developed rapidly with the discovery that crosslinking greatly improves its mechanical properties, giving the ﬁrst of what is now a broad class of materials often referred to as ‘‘elastomers’’. Elasto-

725

726

14 Polymer Mechanical Properties

mers are characterized by ﬂexible chains, Tg well below room temperature, little or no crystallinity in the undeformed state, and light chemical or physical crosslinking (through the presence of rigid glassy or crystalline microdomains, for example). Although E drops signiﬁcantly as T is raised above Tg , K changes relatively little, so that K g E and, from Eq. (9), n A 0:5. Volume changes may hence be considered negligible compared with other types of deformation. This justiﬁes the use of the Helmholtz free energy in the thermodynamic analysis of rubber elasticity, deﬁned by Eq. (11). A ¼ U TS

ð11Þ

where U is the internal energy and S is the entropy. For a reversible change of state, the ﬁrst law of thermodynamics gives Eq. (12), where dW is the work carried out on the system and dQ is the heat transfer into the system, leading to Eq. (13). dU ¼ dQ þ dW ¼ TdS þ dW

ð12Þ

dA ¼ dU TdS SdT ¼ dW SdT

ð13Þ

The work done when a specimen subject to a force f undergoes an incremental elongation dl is dW ¼ fdl. The corresponding change in free energy is given by Eq. (14), from which Eq. (15) follow. dA ¼ fdl SdT qA qA ; S¼ f ¼ ql T qT l

ð14Þ ð15Þ

From the identity of Eq. (16), one obtains Eq. (17), and hence Eq. (18). q qA q qA ¼ qT ql T ql qT l qf qS ¼ qT l ql T qA qU qS qU qf f ¼ ¼ T ¼ þT ql T ql T ql T ql T qT l

ð16Þ

ð17Þ

ð18Þ

Measurements of f ðTÞ at large ﬁxed deformations, such as shown in Figure 14.3, indicate that f ðTÞ A CT for T > Tg , where C is a constant. The second term in Eq. (18) therefore dominates under these conditions and the response is almost entirely entropic [4, 6]. (The increase in f with decreasing T below Tg is due to thermal contraction.)

14.2 Elasticity

Fig. 14.3. Tensile stress at constant extension (350%) as a function of T in a crosslinked rubber (after Ref. 2).

Another characteristic of elastomers is that their temperature increases during rapid deformation, for which dQ A 0. In this case, one can show that Eq. (19) applies, where Cl is the speciﬁc heat at ﬁxed l. qT T qf ¼ ql S Cl qT l

ð19Þ

Given that ðqf /qTÞl > 0 (Figure 14.3), T must indeed increase with l. It may also be shown that the length of an elastomer subject to a constant force should decrease as T is increased, which is again consistent with observation, albeit somewhat counterintuitive, given that crystalline and glassy solids tend to become less rigid on heating [4]. 14.2.3

Statistical Mechanical Approach to Rubber Elasticity

The entropy changes that give rise to rubber elasticity may be modeled in terms of the chain statistics introduced in Section 14.1.2. For a chain whose end-to-end vector is ﬁxed and equal to R, the number of conformations, WðRÞ, that the chain can adopt is proportional to PðR; NÞ. From Eq. (3), one thus obtains Eq. (20), where k is Boltzmann’s constant and So is a constant.

727

728

14 Polymer Mechanical Properties

S ¼ So þ k ln W ¼ So

3kR2 2Nb 2

ð20Þ

Hence, for constant U, the free energy is obtained from Eq. (21). A ¼ TS ¼ A o Tk ln W ¼ A o þ

3kTR2 2Nb 2

ð21Þ

The magnitude of the restoring force on a chain with end-to-end vector R is therefore given by Eq. (22). f ¼

qA 3kTR ¼ qR Nb 2

ð22Þ

The total free energy change when a large deformation is applied to the chain, so that R ¼ ðx; y; zÞ changes to RO ¼ ðl1 x; l 2 y; l3 zÞ, is given by Eq. (23). DA ¼ TDS ¼

3kTðRO 2 R2 Þ 3kTððl12 1Þx 2 þ ðl 22 1Þy 2 þ ðl32 1Þz 2 Þ ¼ 2Nb 2 2Nb 2

ð23Þ

In an elastomer the forces are transferred to individual chains through the crosslinks, which constitute the nodes of a continuous network. If the chains linking these nodes are assumed to be freely jointed chains composed of N links of length b, Eq. (2) implies the spatial separation of the nodes, R, to be given by Eq. (24) in the undeformed state. hR2 i ¼ hx 2 i þ h y 2 i þ hz 2 i ¼ Nb 2

ð24Þ

Because R has no preferred direction, Eq. (25) holds. hx 2 i ¼ h y 2 i ¼ hz 2 i ¼

Nb 2 : 3

ð25Þ

If there are nx chains per unit volume, then assuming aﬃne deformation of the nodes (so that they move as if ﬁxed to a uniform elastic background), it follows from Eq. (23) that the total change in the free energy per unit volume, a, for a deformation l1 ; l 2 ; l3 is given by Eq. (26). Da ¼

nx kTðl12 þ l 22 þ l32 3Þ 2

ð26Þ

From the incompressibility criterion, l1 l 2 l3 ¼ 1. Hence, for a uniaxial extension along x, that is l1 ¼ l, l 2 ¼ l3 ¼ l1/2 , Eq. (27) applies and, from Eq. (10), the

14.3 Viscoelasticity

force f1 acting on a unit cube of undeformed material (equivalent to the nominal or ‘‘engineering’’ stress) is given by Eq. (28). nx kT 2 2 Da ¼ l þ 3 2 l f1 ¼

qw qDa 1 ¼ ¼ nx kT l 2 ql ql l

ð27Þ

ð28Þ

This remarkable result implies that the deformation behavior depends only on the number of network chains per unit volume and on T. The detailed chemistry of the chain is therefore unimportant, with the proviso that the chains must be much longer than the statistical segment length b. For very small deformations, l ¼ 1 þ exx , and sxx is given by Eq. (29). sxx A f1 A 3nx kTexx

ð29Þ

Given that n A 0:5 and hence that E ¼ 3G [Eq. (9)], Eq. (30) follows. G ¼ nx kT

ð30Þ

There are many shortcomings with this simple treatment, and, as illustrated in Figure 14.4, the behavior of real elastomers deviates markedly from Eq. (28), particularly at large l. A particular diﬃculty with Eq. (3) is that it predicts ﬁnite probabilities for jRj > R max, which is clearly unphysical. To account for the ﬁnite extensibility of the chains, and hence for the observed work hardening, it is necessary to chose more realistic expressions for PðR; NÞ, such as the inverse Langevin function or the associated analytical approximations [4]. Moreover, in the case of relatively stiﬀ chains such as DNA or certain polypeptides, the worm-like chain model of Porod and Krattky may be more appropriate than the Gaussian chain. Even if the chain statistics are correctly described, non-aﬃne deformation of the network junctions, long-range inter- and intra-chain interactions (entanglement), speciﬁc interactions such as hydrogen bonding, strain-induced crystallization (observed in natural rubber, for example), the presence of ﬁllers, and so forth may need to be taken into account in real rubbers and elastomers [7–9].

14.3

Viscoelasticity 14.3.1

Linear Viscoelasticity

The discussion has so far implicitly been limited to behavior in the rubbery plateau regime in Figure 14.1, where the conditions for rubber elasticity in amorphous

729

730

14 Polymer Mechanical Properties

Fig. 14.4. Experimental stress–extension ratio curve for natural rubber along with a best ﬁt of the predicted behavior for network described by the inverse Langevin function, and of Gaussian network [Eq. (28)] with the same network parameters (after Ref. 7).

polymers are met, namely long ﬂexible chains, weak intermolecular forces ðT > Tg Þ and anchoring of the chains to form a network (the nature of this anchoring in uncrosslinked polymers will be discussed in Section 14.3.3). The corresponding constitutive expressions [Eq. (28), for example] predict the response to deformation to depend weakly on T and to be independent of the deformation rate. Although this is reasonably consistent with experiment, the situation changes drastically near Tg, for example, where time, t, becomes a very important factor, and the mechanical behavior is said to be strongly ‘‘viscoelastic’’. In a linear viscoelastic material, the moduli and compliances EðtÞ; GðtÞ; DðtÞ, and JðtÞ (tensile and shear modulus and tensile and shear compliance respectively) are functions of t, although they remain independent of stress or strain. Development of constitutive equations for such materials is generally based on the idea that the eﬀects of small increases in stress or strain are additive, which is known as the Boltzmann superposition principle [4, 10, 11]. A strain eðtÞ may be considered to result from a sum of step strains applied at time u and maintained for a time t u [Eq. (31)]. deðuÞ ¼

deðuÞ du du

ð31Þ

14.3 Viscoelasticity

In uniaxial tension, the Boltzmann superposition principle implies Eq. (32). sðtÞ ¼

ðt

Eðt uÞ

y

de du du

ð32Þ

Similarly, for a given stress history, Eq. (33) results. eðtÞ ¼

ðt

Dðt uÞ

y

ds du du

ð33Þ

Linear viscoelastic materials thus retain a memory of their entire mechanical history although, conveniently for the experimentalist, the memory fades with t u, and for practical purposes the lower limit of integration in Eqs. (32) and (33) is often set to zero. Equations (32) and (33) may be used to calculate the response to an arbitrary deformation using EðtÞ or DðtÞ determined from simple static experiments (creep or relaxation). Another important class of experiments involves periodic tensile or shear strains of the type eðtÞ ¼ eo cos ot ¼ Reðeo e iot Þ. In a linear elastic solid deformed in tension, s oscillates in phase with e with amplitude Eeo . In a linear viscous liquid, s ¼ hde/dt and hence varies as heo o sin ot ¼ heo o cosðot þ p/2Þ. In a linear viscoelastic material, however, intermediate behavior is observed [Eqs. (34), where 0 < d < p/2]. eðtÞ ¼ eo cos ot

ð34Þ

sðtÞ ¼ so cosðot þ dÞ It is hence possible to deﬁne two dynamic moduli, the storage modulus, E 0 (or G 0, in the case of a shear experiment), which gives the stress component in phase with the strain, and the loss modulus, E 00 (or G 00 ), which gives the stress component out of phase with the strain. Thus, in tension, Eqs. (35) is obtained, where E 0 ; E 00 , and tan d are given by Eq. (36). sðtÞ ¼ so cosðot þ dÞ ¼ E 0 eo cos ot þ E 00 eo sin ot E0 ¼

so cos d; eo

E 00 ¼

so sin d; eo

tan d ¼

E 00 E0

ð35Þ ð36Þ

Equation (35) is sometimes more conveniently expressed in complex notation as Eq. (37), where the complex modulus E is deﬁned by Eq. (38). sðtÞ ¼ Reðso e iðotþdÞ Þ ¼ ReðE eo e iot Þ E ¼

so ðcos d þ i sin dÞ eo

ð37Þ ð38Þ

731

732

14 Polymer Mechanical Properties

Fig. 14.5.

The Zener model (or ‘‘standard linear solid’’).

Viscoelastic materials are often modeled using linear combinations of ﬁxed Hookean and Newtonian elements (springs and dashpots). Although such models may not fully account for the complexity of real materials, generalizations of this approach have been extensively developed and methods and approximations for transforming between the diﬀerent viscoelastic functions are widely available in the literature [10, 11]. For the sake of illustration, one simple example will be considered here, namely the Zener element, a possible representation of which is shown in Figure 14.5. At very high frequencies most of the deformation in a Zener element is taken up by spring 1 and the limiting modulus Eo is equal to E1 . At very low frequencies, where the stress in the dashpot becomes negligible, the deformation is taken up by spring 1 and spring 2 in series, and the limiting modulus Ey ¼ E1 E2 /ðE1 þ E2 Þ. However, at intermediate frequencies, the behavior becomes strongly inﬂuenced by h [Eqs. (39), where DE ¼ Eo Ey and the relaxation time is given by Eq. (40)]. E ¼ Eo DE t¼h

DE Eo2

1 iot ; 1 þ o2t2

tan d ¼

DEot Ey þ Eo o 2 t 2

ð39Þ ð40Þ

The behavior predicted by Eqs. (39) for values of E1 and E2 appropriate to the glass transition in an amorphous polymer (compare Figure 14.1) is shown in Figure 14.6. The model accounts for the qualitative features of experimentally observed transitions, namely a step-like drop in modulus as o decreases below t1, and a characteristic peak in tan d. However, more complex models involving many relaxation times, that is, a discrete or continuous relaxation time spectrum, are necessary if more quantitative agreement with experiment is to be obtained (for an example of a discrete relaxation time spectrum derived from a molecular model, see Section 14.3.3) [10, 11].

14.3 Viscoelasticity

Fig. 14.6. Predictions of the Zener model for the dynamic behavior in the vicinity of a transition (E1 ¼ 2 10 9 Pa, E2 ¼ 10 6 Pa, t ¼ 0:05 s).

Similar behavior to that in Figure 14.6 is observed if the viscoelastic functions are measured as a function of T at ﬁxed o, because the viscous response, which tends to dominate the relaxation times, is a strongly decreasing function of increasing T, particularly close to Tg . The eﬀect of increasing T is therefore equivalent to that of decreasing o (or increasing t in a static experiment). Thus, in dynamic torsion temperature sweeps on poly(methyl methacrylate) (PMMA), the glass transition at about 120 C is associated with a clear peak in tan dðTÞ. As shown in Figure 14.7, other transitions may also be observed at T well below Tg in this type of experiment. These ‘‘secondary’’ relaxations are important features of both glassy and semicrystalline polymers because they show that molecular motions are not entirely suppressed at T < Tg . Indeed, such motions play an important role in plasticity (see Section 14.4.1). Conventionally, the ﬁrst relaxation encountered as T decreases is labeled a, and may correspond to either the melting/crystallization transition (for semicrystalline polymers) or the glass transition. The secondary relaxations are designated b; g, and so on, in order of their appearance as T decreases further (see Figure 14.7). They may often be assigned to speciﬁc types of molecular motion, such as side-group motions or cooperative motion of several main-chain units, and often show an Arrhenius T dependence. Activation of side-group motions typically requires more energy than main-chain motions. Thus the mainchain relaxation in bisphenol-A polycarbonate (PC) occurs at about 80 C, which is about 230 K below Tg (150 C), whereas relaxation of the aromatic side groups in aPS is observed at 50 C, which is only 50 K below Tg (100 C) [5].

733

734

14 Polymer Mechanical Properties

Fig. 14.7. Dynamic behavior as a function of T of poly(methyl methacrylate) (PMMA) tested in torsion at 10 Hz.

14.3.2

Time–Temperature Superposition

The idea of time–temperature equivalence introduced in Section 14.3.1 is of considerable practical importance because one would often like to predict the longterm response of materials on the basis of experiments carried out on a laboratory time scale. This is to some extent possible in polymers, for which it has been widely veriﬁed that viscoelastic functions determined at diﬀerent T over a ﬁxed range of o or t, slightly adjusted to take into account the eﬀect of T and density r on the elastic response [through Eq. (30)], superpose if the o or t scale is multiplied by a shift factor, aT ðT; Tr Þ, where Tr is some convenient reference temperature. A typical master curve obtained in this way is shown in Figure 14.8 for stress relaxation data from polyisobutene, taking Tr ¼ 66:5 C. The eﬀect is to expand the t or o scale of the measurement carried out at Tr , revealing the whole of the a transition in this case. Williams, Landel, and Ferry (WLF) observed that if Tr is set to Tg , the variation of log aT with T Tr is similar for a wide variety of polymers [10]. They rationalized this in terms of the molecular response, starting with Doolittle’s equation [Eq. (41)] for the viscosity, where A and B are constants. fv is the fractional free volume, equivalent to the ‘‘unoccupied’’ volume divided by the total volume of the polymer (the ‘‘occupied’’ volume includes that necessary to accommodate thermal vibrations). ln h ¼ ln A þ

B fv

ð41Þ

14.3 Viscoelasticity

Fig. 14.8. Stress relaxation data for polyisobutene at diﬀerent temperatures T, as indicated and superposed onto the data for 66.5 C by shifting along the logðtÞ axis (after Ref. 7).

Equation (41) is based on the idea that the greater fv, the greater the molecular mobility (owing to reduced crowding), and the lower h. For T > Tg , fv is given by Eq. (42), where af is the coeﬃcient of thermal expansion of the fractional free volume and fg is the fractional free volume associated with the glass transition. fv A fg þ af ðT Tg Þ

ð42Þ

If the T dependence of all the relaxation times is assumed to be that of h in Eq. (41), the shift factor for the scaled viscoelastic functions is given by Eq. (43), and hence Eq. (44) follows. aT ¼

hðTÞ hðTg Þ

B ðT Tg Þ g 2:303fg C ðT Tg Þ log aT ¼ 1 g1 fg C2 þ T Tg þ T Tg af

ð43Þ

ð44Þ

735

736

14 Polymer Mechanical Properties

Equation (44) is the well-known WLF equation. ‘‘Universal’’ values of the various g g physical parameters in Eq. (44) lead to C1 ¼ 17:44 and C2 ¼ 51:26 K [10]. These g g are of the same order of magnitude as C1 and C2 obtained empirically (34 and 80 K for PMMA, for example), and indeed, time–temperature superposition has been found to work well for a wide range of single-phase polymers, with the proviso that it begins to break down for the relatively fast vibrational modes characteristic of the glassy state [12]. Moreover, although superposition may work for T g Tg , at temperatures above about Tg þ 50 K the shift factors tend to show an Arrhenius dependence rather than following the WLF equation. 14.3.3

Molecular Models for Polymer Dynamics

The starting point for molecular models for polymer dynamics based on the ideas introduced in Section 14.2.3 is the Rouse model for an isolated chain in a viscous medium, in which the chain is taken to behave as a sequence of m ‘‘beads’’ linked by Gaussian springs [Figure 14.9(a)] [13–16]. The chain interacts with the solvent via the beads, and the solvent is assumed to drain freely as the chain moves. Hence, Eq. (22) leads to Eqs. (45), where N 0 is the number of links between adjacent beads, z is a friction coeﬃcient per bead and ri is the position of the ith bead. z

dri 3kT þ ð2ri rix riþ1 Þ ¼ 0 dt N 0 b 2

ð45Þ

Rouse solved the m 1 simultaneous equations [Eqs. (45)] by transforming them into a set of uncoupled equations for the normal modes of motion of the chain.

Fig. 14.9. (a) The bead–spring model for the dynamics of an isolated chain in solution; (b) sketch of the second to fourth normal modes of motion (the circles represent ﬁxed nodes and the arrows indicate coherent motion of the sub-chains deﬁned by the nodes).

14.3 Viscoelasticity

The moduli of a dilute solution containing n chains per unit volume may then be expressed in terms of a discrete relaxation time spectrum, where each relaxation time corresponds to one of the normal modes of motion [Figure 14.9(b)]. For simple shear, this leads to Eqs. (46) and (47), with tp given by Eq. (48) [13–16].

GðtÞ ¼ nkT

m X

et/tp

ð46Þ

p¼1

G ðoÞ ¼ nkT

m X p¼1

o 2 tp2 1þ

o2t

tp ¼ N 0 b 2 z 24kT sin 2

þ inkT 2

p

pp 2ðm þ 1Þ

m X

otp 1 þ o 2 tp2 p¼1

ð47Þ

1 ;

p ¼ 1; 2 . . . ; m

ð48Þ

Because the length of chain corresponding to each Gaussian spring is the shortest unit that can relax in the model, these expressions are only physically meaningful for m g 1. In this limit, Eq. (48) may be replaced by Eq. (49), where zo ¼ z/N 0 is a ‘‘monomeric friction coeﬃcient’’ and N ¼ mN 0 is the total number of statistical segments per chain. tp A

N 02 m 2 b 2 zo N 2b2z ¼ 2 2 o 2 2 6p p kT 6p p kT

ð49Þ

Hence, the longest ‘‘Rouse’’ relaxation time, tR , is proportional to N 2 . For t > tR , the motion of the chain becomes essentially diﬀusive, and the diﬀusion constant of the center of gravity of the chains is given by the Einstein relation, Eq. (50). DR ¼

kT Nzo

ð50Þ

Rouse-like behavior is not in fact observed in dilute solutions, for which it is necessary to take into account the inﬂuence of the chain on the motion of the solvent, and deviations from Gaussian statistics arising from polymer–solvent interactions [17, 18]. These factors are incorporated in the Zimm model, which predicts the diffusion constant to be proportional to N n , for example, where n depends on the solvent quality, in better agreement with experimental data [4, 14]. Indeed, although it was ﬁrst proposed for isolated chains, the Rouse model turns out to be more appropriate to polymer melts, where ﬂexible linear chain conformations are approximately Gaussian and hydrodynamic interactions are relatively unimportant [4, 14– 16]. The Rouse model is nevertheless inadequate to describe the high-frequency response associated with bond rotations and local cooperative motions, important for the glassy state (see, for example, Ref. 4). Moreover, as the chain length increases (or the concentration increases in a solution of long chains), the fact that

737

738

14 Polymer Mechanical Properties

Fig. 14.10.

(a) Schematic representation of entanglement constraints; (b) the ‘‘tube’’ model.

chains cannot cross one another leads to additional constraints on chain motion, known as entanglement, which remain eﬀective at T > Tg . Figure 14.10(a) shows the origin of these constraints schematically for an arbitrary chain in a polymer melt. The dots correspond to the nearest neighbors of the chain, which are assumed for the sake of illustration to intersect the plane of the drawing. Because the chain cannot cross its neighbors, it cannot move very far in directions perpendicular to its local trajectory. Diﬀusion is therefore limited to snake-like motion of the chain along its own contour, called ‘‘reptation’’ [14, 18]. A convenient way of representing this eﬀect in three dimensions is to assume the chain to be trapped in a tube with a primitive path length L, as shown in Figure 14.10(b), so that only motion within or along the tube is possible. For an undeformed Gaussian chain, the tube is also Gaussian, with an end-to-end vector R. The primitive path may thus be represented by NL statistical segments of length a, such that L ¼ NL a and hR2 i ¼ NL a 2 ¼ La. Because hR2 i must also equal Nb 2 , Eq. (51) follows. L ¼ Nb 2 /a

ð51Þ

The tube model forms the basis for detailed theoretical approaches to the dynamics of an entangled polymer melt pioneered by Doi and Edwards [13, 14]. The present discussion will be restricted to a simple description of some of its basic features. Consider, for example, the relaxation behavior of a chain subjected to a step shear strain at t ¼ 0. As shown in Figure 14.11, the applied strain results in a deformation of the tube, and hence of the chain trapped inside it. The ﬁrst relaxation occurs within the tube at times t < te , where te is the Rouse relaxation time for a chain with Ne ¼ ða/bÞ 2 statistical segments, and hence a mean square end-to-end distance equal to a 2 . At t > te , the constraints due to the tube begin to dominate and the only way the stress can relax further relax is for the chain to escape the deformed tube and re-establish a random (unperturbed) conformation. As sketched in Figure 14.11, it achieves this through Brownian motion backward and

14.3 Viscoelasticity

Fig. 14.11. Relaxation according to the tube model (small strain limit): (a) Rouse relaxation; (b) rubbery plateau; (c) to (d) reptation.

forward along the tube. The time necessary for the chain to diﬀuse a distance L and hence relax fully is tD ¼ L 2 /DR , where DR is given by Eq. (50). Hence, Eq. (52) is obtained from Eq. (51). tD ¼

N 3 b 4 zo 6p 2 N 3 ¼ te a 2 kT Ne3

ð52Þ

Because tD , which is known as the ‘‘reptation time’’, is signiﬁcantly longer than te for large N, in Figure 14.11 there is an intermediate regime of roughly timeindependent behavior, which is identiﬁed with the rubbery plateau. Thus, entanglement may be considered to play an equivalent role in this regime to the chemical crosslinks in an elastomer. The greater N, that is, the higher M, the greater the extent of the rubbery plateau (compare Figure 14.1). By analogy with an elastomer [Eq. (30)], the small strain rubbery plateau modulus may be set equal to Ge ¼ ne kT, where ne is the ‘‘entanglement density’’, assumed to play an equivalent role to nx in an elastomer. Ge may therefore be expressed in terms of an entanglement molar mass, Me , according to Eq. (53), where NA is Avogadro’s number. Ge ¼

rNA kT Me

ð53Þ

In terms of the tube model, Me is interpreted as the mass that corresponds to Ne statistical segments. Values of ne and Me deduced from Ge are found to vary widely

739

740

14 Polymer Mechanical Properties

according to chemical structure; ne A 4 10 25 m3 in aPS, which has relatively bulky chains, but is nearly an order of magnitude higher in PC, which has compact, relatively rigid chains [19]. The network and tube models both imply entanglement to be ineﬀective for M < 2Me , so that Mc , the critical mass introduced in Section 14.1.1, may be taken to equal 2Me . As well as deﬁning the threshold of rubber-like behavior above Tg , Mc marks the transition from a regime in which the melt viscosity h m M (Rouse dynamics), to a regime in which h m M 3:4 , characteristic of an entangled melt. The original theory of Doi and Edwards predicts h m M 3 , but better agreement with experiment has since been obtained by incorporating the eﬀects of primitive path ﬂuctuations and constraint release (if all the chains reptate simultaneously, the constraints in Figure 14.10(a) cannot be considered strictly permanent for times of the order of tD ) [4, 5, 14]. Even so, in the limit of very long monodisperse chains, for which the tube eﬀectively remains ﬁxed, or free chains trapped in a crosslinked network, the M 3 dependence is recovered. A strong dependence of the viscosity on M is a general feature of entangled polymers and is of great practical importance, because the choice of M in applications is often dictated by the need to reach a compromise between fracture resistance, which tends to improve with M (see Section 14.4.4) and melt processing, which is facilitated by low viscosities. 14.3.4

Nonlinear Viscoelasticity

Although it is a powerful means of investigating molecular structure and of basic characterization, and provides a general indication of the inﬂuence of M on ﬂow behavior, the restrictions imposed by linear viscoelasticity make it inapplicable to a wide variety of practical problems. Nonlinearity is often associated with the phenomenon of ‘‘shear thinning’’, that is, a reduction of the viscosity with shear rate in steady ﬂow, characteristic of many polymer melts at intermediate shear rates [15]. This contrasts with the Newtonian behavior implied by the Boltzman superposition principle for steady ﬂow [Eq. (54), in a liquid, GðsÞ must vanish as s ! y].

sðtÞ ¼

ðt

Gðt uÞg_ du ¼ g_

y

ðy

GðsÞ ds ¼ g_ h

ð54Þ

0

Newtonian regimes are nevertheless widely observed in polymer melts in the high and low shear rate limits, where the viscosities are designated by hy and ho respectively. This is reﬂected by the empirical expressions widely used in engineering practice to describe the response to steady shear ﬂow, an example being the Cross model [Eq. (55)], which reduces to the well known ‘‘power law’’ of Eq. (56) when hy g h g ho , with n ¼ 1/ðm þ 1Þ between 10 and 20 for most polymer melts. ho h ¼ K g_ m h hy

ð55Þ

14.4 Yield and Fracture

1/n s g_ ¼ so g_o

ð56Þ

However, these simple empirical expressions are far from universal, and fail to account for eﬀects speciﬁc to nonlinear behavior, such as the appearance of ﬁnite ﬁrst and second normal stress diﬀerences sxx syy ¼ N1 ðg_Þ and syy szz ¼ N2 ðg_Þ in steady shear ﬂow. (For a linear viscoelastic material in shear, sxx ; syy and szz are equal to the applied pressure, usually atmospheric pressure.) These may be linked to the development of molecular anisotropy in polymer melts subject to ﬂow, and are responsible for the Weissenberg eﬀect, which refers to the tendency for a nonlinear viscoelastic ﬂuid to ‘‘climb’’ a rotating rod inserted into it, as well as practically important phenomena such as die swell [20]. There has been considerable eﬀort to develop phenomenological models for nonlinear viscoelasticity that provide a comprehensive description of such phenomena by extending the formalism of linear viscoelasticity to include situations where the viscoelastic functions can no longer be considered independent of stress or strain [21]. These are exempliﬁed by the K-BKZ (Kaye-Bernstein, Kearsley, and Zapas) formalism, which includes a ‘‘damping function’’ in order to account for the acceleration of the relaxation rate at large deformations. The popularity of the K-BLZ approach is partly linked to the fact that the Doi–Edwards tube model, with the so-called ‘‘independent alignment’’ approximation, leads to expressions of the same form when generalized to large deformations, providing a molecular basis for its interpretation (in terms of the tube model, a large deformation will signiﬁcantly increase L, giving rise to a supplementary relaxation process called ‘‘chain retraction’’ on a time scale tR , during which L returns to its equilibrium value [13, 14]). Phenomenological models, as well as expressions derived from the generalized Doi–Edwards approach are able to account for many of the observed features of polymer melts, and, although they are cumbersome, numerical techniques are now powerful enough to permit their incorporation into complex ﬂow simulations of direct relevance to industrial practice.

14.4

Yield and Fracture 14.4.1

Yield in Polymers

The stresses and strains of practical importance for polymers at T < Tg or Tm are often relatively large, and, unlike the small strains referred to in the preliminary discussion of the glassy state in Section 14.1.1, they may result in signiﬁcant plastic deformation, in that the associated changes in specimen morphology are not recovered after stress release over laboratory time scales. A plastically deformed polymer with suﬃciently high M is nevertheless able to return to its initial shape

741

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14 Polymer Mechanical Properties

when heated above Tg or Tm, so that such deformation cannot strictly be considered irreversible. As the extent of plastic deformation increases with the overall deformation, pronounced softening occurs, even when adiabatic heating eﬀects are negligible (as will be assumed here). If fracture does not intervene, the force, f , on a specimen tested at constant speed in tension typically reaches a local maximum at a strain of a few percent, which is taken to correspond to a plastic yield stress, sy (typically between 20 and 100 MPa in unoriented polymers). The subsequent yield drop is at least partly geometric in origin, although many glassy polymers also show an intrinsic yield drop [5]. The true stress on the polymer is given by s ¼ A1 o lf , where A o is the initial cross-sectional area of the specimen, so that Eq. (57) holds. df 1 ds s ¼ Ao 2 dl l dl l

ð57Þ

Given that ds/dl is a decreasing function of l in the initial stages of deformation, as shown schematically in Figure 14.12, A1 o f ¼ s/l > ds/dl beyond a certain value of l, and df /dl becomes negative. The resulting instability leads to localized necking in a tensile test, but if deformation continues beyond the yield point, the neck is stabilized by work hardening (df /dl becomes positive again) and propagates to the rest of the specimen at roughly constant f . Global work hardening then takes place, followed by rupture.

Fig. 14.12. Schematic of (a) the intrinsic stress-deformation curve of a polymer and (b) the force per unit area of an undeformed specimen observed as a function of total deformation in a tensile test.

14.4 Yield and Fracture

In an amorphous glassy polymer, work hardening is considered to correspond to stretching of the entanglement network invoked to account for the rubbery plateau above Tg (see Section 14.3.3). This explains the recoverability of the deformation above Tg (in the absence of an applied force, the network retracts to its equilibrium conformation) and it is borne out by the observed correlation between the value of 1/2 l in the neck and the maximum extensibility of the network, lmax A Ne . In semicrystalline polymers, the evolution of the crystalline texture may also contribute to work hardening, because the resolved shear stress on activated slip systems tends to decrease as deformation proceeds at constant stress, as will be discussed further (see Section 14.4.3). The large strain response in the glassy or semicrystalline state is that of a nonlinear viscoelastic solid. However, both engineering and theoretical approaches to plasticity in polymers have largely developed as an independent discipline, in which sy plays a central role, in spite of its somewhat arbitrary deﬁnition (indeed it is not always possible to associate sy with a maximum in the force–deformation curve [5]). This is because in practice the yield point, rather than the ultimate strength, is usually considered to be the failure criterion for ductile materials. For yield to occur at all, the global stress state must contain a deviatory component, so that pure hydrostatic stress states do not result in plasticity in a uniform specimen. Indeed, in many types of material, including metals, yield is usually well described by the Von Mises criterion, in which it is considered to be independent of the hydrostatic pressure, p ¼ ðsxx þ syy þ szz Þ/3. With an appropriate frame of reference, any multiaxial stress state may be expressed as Eq. (58), where s1 ; s2 , and s3 are the principal stresses (compare Section 14.2.1). 2

s1

0

6 s ¼4 0

s2

0

0

0

3

7 05 s3

ð58Þ

The von Mises criterion is then Eq. (59). ðs1 s2 Þ 2 þ ðs2 s3 Þ 2 þ ðs3 s1 Þ 2 b 2sy2

ð59Þ

It may be veriﬁed that sy is equal to the tensile yield stress by setting any two of the principle stresses to zero. For pﬃﬃﬃ comparison, in pure shear, s1 ¼ s2 and s3 ¼ 0, and the yield stress is ty ¼ sy / 3. Moreover, it is easily seen that Eq. (59) is independent of p. In fact, most polymers do show a pressure-dependent yield stress, an observation that is attributed to their relative compressibility, so that a negative value of p signiﬁcantly reduces the space available for the molecules and hence reduces their mobility. However, this can usually be accounted for by replacing Eq. (59) with Eq. (60), where S and m are materials constants [5]. ðs1 s2 Þ 2 þ ðs2 s3 Þ 2 þ ðs3 s1 Þ 2 ¼ S mp

ð60Þ

743

744

14 Polymer Mechanical Properties

14.4.2

Models for Yield

The yield stress in disordered solids is strongly dependent on T and the deformation rate, and eﬀorts have consequently been made to describe it in terms of a Ree–Eyring activated rate process, in which a rheological element ‘‘jumps’’ from place to place by overcoming local energy barriers [5, 22]. For an activation energy Q, and an attempt frequency n0, the jump rate n ¼ n0 expðQ/kTÞ. In the absence of an applied stress, the jumps are in arbitrary directions, and there is no net deformation. However, an applied stress, s, alters the eﬀective energy barrier height for forward (þ) and backward () jumps with respect to the stress axis, so that Eqs. (61) apply, where x is the displacement associated with each jump and A is the effective area of the rheological element. no Q sAx n sAx ¼ exp exp kT 2 kT 2 n sAx n ¼ exp 2 kT

nþ ¼

ð61Þ

The strain rate in the stress direction is proportional to the net jump rate nþ n , leading to Eq. (62), where V ¼ xA typically corresponds to a few repeat units and is interpreted as an activation volume. v sV Q sV A e_O exp exp e_ m sinh 2 kT kT kT

ð62Þ

s is then taken to equal sy when Eq. (62) is satisﬁed for a given strain rate e_, which implies sy m log e_. This simple approach, which may easily be modiﬁed to include a pressure-dependent contribution to the activation energy, is often remarkably successful in predicting the strain rate dependence of sy , as shown in Figure 14.13(a) for PC. However, as shown in Figure 14.13(b), the agreement is more limited in polymers such as poly(vinyl chloride) (PVC). Because the Ree– Eyring model includes no assumptions regarding the nature of the rheological element, changes in the slope of sy versus log e_, such as those in Figure 14.13(b), have been attributed to secondary relaxations in the corresponding range of T and e_. As pointed out in Section 14.3.1, the main secondary relaxation in PC occurs at 80 C, but in PVC it is at about 50 C. This suggests that a complete description of yield may need to involve multiple activated rate processes [23]. Among the better-known eﬀorts to provide a more microscopic description of yield within the general Ree–Eyring framework is the approach of Argon, based on metallurgical models, in which the energy barrier is associated with the elastic displacements necessary to accommodate the elementary shear processes [24]. Another well known, but somewhat diﬀerent, approach is that of Robertson, who

14.4 Yield and Fracture

Fig. 14.13.

sy /T versu logðe_Þ in (a) polycarbonate; (b) poly(vinyl chloride) (after Ref. 23).

postulated that yield takes place when the average polymer conformation is equivalent to that at the glass transition [25]. He adopted a simpliﬁed model for a polymer chain such that skeletal bonds adopt either a high- or a low-energy state with an energy diﬀerence DE. At equilibrium, the proportion of bonds in the highenergy state is given by the Boltzmann factor, so that for a polymer below its Tg . Eq. (63) holds, assuming the conformations at Tg to be substantially frozen in on vitriﬁcation. w¼

expðDE=kTg Þ 1 þ expðDE/kTg Þ

ð63Þ

745

746

14 Polymer Mechanical Properties

The value of w in the presence of an applied stress, wdef , is estimated by assuming the stress to modify the energy barrier to DE sv cos a, where a is the local orientation of the bond with respect to the stress axis and v is the volume swept out during the transition from the low to high energy state. From the expression for wdef , obtained by averaging the eﬀective energy barrier over all possible orientations, a stress-dependent structure temperature, Y, may be deﬁned through Eq. (64). wdef ¼

expðDE/kYÞ 1 þ expðDE/kYÞ

ð64Þ

The WLF equation [Eq. (44)] may be used to calculate the viscosity as in Eq. (65), where hg is viscosity at Tg . "

g

g

C1 C2 Y g C1 hðy; TÞ ¼ hg exp 2:303 g Y Tg þ C2 T

#! ð65Þ

The strain rate is then given by Eq. (66). e_ ¼

s hðy; TÞ

ð66Þ

Robinson’s model has met with some success in regimes of T close to Tg where the role of intramolecular barriers is relatively important. However, for T well below Tg, intermolecular barriers are expected to dominate, so that phenomenological or semi-phenomenological approaches of the Ree–Eyring type are arguably more appropriate to the observed T dependence, as borne out by Figure 14.13. 14.4.3

Semicrystalline Polymers

In amorphous polymers, plasticity is associated with deformation at T < Tg . In semicrystalline polymers, however, there is a further regime Tg < T < Tm , in which the amorphous regions are in the rubbery state and hence no longer contribute directly to yield. Plasticity is therefore dominated by the intermolecular barriers that oppose chain slip in the crystalline regions. This is of considerable practical importance, given that commodity semicrystalline polymers such as PE and isotactic polypropylene (iPP) are typically employed at T > 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.

14.4 Yield and Fracture

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

14 Polymer Mechanical Properties

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

14.4 Yield and Fracture

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

14 Polymer Mechanical Properties

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

14 Polymer Mechanical Properties

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

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3

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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.

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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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14 Polymer Mechanical Properties

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

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

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

15 Polymer Degradation and Stabilization

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<340 nm

POaOH ! PO þ OH

ð10Þ

Cations of transition metals (Cu, Fe, Mn, Co, Ti) which may exist in the polymer as contaminants or as catalyst residues can induce the decomposition of hydroperoxides and contribute to an increase in the oxidation rate at low temperature (Haber– Weiss reaction, Eqs. (11) and (12)). POaOH þ M nþ ! PO þ OH þ Mðnþ1Þþ POaOH þ M

nþ1

þ

! PO2 þ H þ M

nþ

ð11Þ ð12Þ

At high hydroperoxide concentrations, a bimolecular dissociation mechanism [Eq. (13)] between hydrogen-bonded molecules (for instance, between adjacent peroxide groups formed by an intramolecular abstraction reaction in PP) can also occur. 2POaOH ! PO2 þ PO þ H2 O

ð13Þ

Apart from homolytic cleavage, the hydroperoxides can undergo a number of other reactions such as addition to double bonds that may have been initially present in the polymer or formed in the course of degradation [Eq. (14)] and radical-induced decomposition [Eqs. (15) and (16)].

761

15 Polymer Degradation and Stabilization

ð14Þ

P þ POaOH ! PO þ POH

ð15Þ

HO þ POaOH ! PO2 þ H2 O

ð16Þ

a

POaOH þ aHCbCHa ! PO þ aHCaCHa OH

The decomposition of hydroperoxides creates more radicals than are initially formed by initiation, resulting in an autoacceleration of the degradation process (chain branching). Additionally, initiation by hydroperoxides is autoregenerated in the sense that, overall, the same quantity of peroxides is recovered at the end of the decomposition cycle. Polyalkoxy and polyperoxy radicals formed from hydroperoxide decomposition are unstable intermediates, owing to the presence of an unpaired electron which weakens the dissociation energy for bonds in the b-position by some 100 kJ mol1 . Monomolecular b-scission is the main mechanism for a decrease in MW and accounts for the variety of oxygenated products observed during oxidative degradation (see Section 15.4.4.6). Termination Under anaerobic conditions, the alkyl macroradicals can recombine by combination and dismutation [Eq. (17)]. 15.2.1.4

P þ P ! PaP or

PaH þ PðaHÞ

ð17Þ

In the presence of oxygen, hydroperoxy radicals are the most abundant; this is a result of their relative low reactivity in solid polymers. Recombination of peroxy radicals can proceed through a tetroxide intermediate, giving peroxides, polymer alkoxy radicals, and oxygen [Eqs. (18a) and (18b)], or it can take place through a six-membered transition state which would lead to an alcohol, singlet oxygen, and triplet excited carbonyl, which, upon de-excitation, produces the chemiluminescence observed in many oxidative degradations (compare the Russell mechanism, Section 15.3.6). 2PO2 ! fPaOaOaOaOaPg ! fPO þ O2 þ OPgcage !

762

POaOP þ O2

ð18aÞ

2PO þ O2

ð18bÞ

As with smaller radicals, the reactivity of a macroradical much depends on the electron density of the atom bearing the unpaired electron. If reactivity is low, the chance of a radical being trapped before abstraction can occur increases with decreasing reactivity. For radicals formed by degradation, the lifetimes follow the order P < PO < PO2 . Free isolated PO2 in polypropylene may have a half-lifetime as long as 500 s at ambient temperature. To account for this exceptional stability, and in analogy to the post-gel eﬀect in free-radical polymerization, it may be

15.2 General Features of Polymer Degradation

appropriate to include a monomolecular mechanism for radical recombination, in addition to the classical bimolecular scheme [Eq. (19)] [3]. ðP; PO; PO2 Þ ! trapped radicals

ð19Þ

The various steps described by the general auto-oxidative degradation of Scheme 15.1 will be illustrated by the oxidative degradation of polypropylene (Section 15.4.4). 15.2.2

Some Nonradical Degradation Mechanisms

Although the majority of polymers degrade thermally or photochemically by a freeradical process, there are a few examples where covalent bond scission occurs by electron transfer around a six-membered transition state. Thermal degradation of poly(ethylene terephthalate) [Eq. (20)] and side group cleavage in poly(methyl acrylate) and poly(tert-butyl methacrylate) are some examples where b-scission of the alkyl–oxygen bond takes place from a cyclic transition state without free-radical formation [4].

O O

C H2C

H

O

O O

O H

O O

CH

+

O

ð20Þ

O

O O

CH2 Cyclization of side chains may also occur without bond scission as in polyacrylonitrile (see Section 15.4.6.3). 15.2.3

Physical Factors

Polymer degradation is highly dependent on chemical and physical factors. The chemical factors will be examined in relation to thermal degradation (see Section 15.4.2). This section will be focused exclusively on the inﬂuence of physical factors on polymer degradation. Polymer degradation can be observed in solution, in melt or in the solid state. Degradation in solution is principally used with ‘‘model’’ compounds for kinetics studies, whereas degradation in melt is frequently associated with processing con-

763

764

15 Polymer Degradation and Stabilization Tab. 15.1.

Factors relevant to chemical reactions in solid polymers [5].

Principal factor

Inﬂuences

Mobility of reactants

translational and rotational diﬀusion of reacting groups as function of temperature and macromolecular motions (Tg ; Tb ; Tg ; . . .) size and shape of reacting groups free volume and conformation interactions between reacting groups and matrix cage eﬀect macroheterogeneity (gradients of absorbed light, oxygen concentration, . . .) morphological heterogeneity (crystalline and amorphous domains, phase separation in block copolymers and blends) microheterogeneity (distribution of reactive sites, free volume) selective depletion of reactive sites crosslinking, chain scission, change in absorbance change in crystallinity excited energy transfer and migration electron transfer transfer of active sites due to chain reaction

Heterogeneity of the system

Change in matrix structure during reaction Factors other than mass transfer

ditions. Obviously, solid-state experiments are the most relevant to normal polymer weathering conditions. It is also in this state that degradation kinetics reveals its highest degree of complexity, as a result of interplay between physical, physicochemical, and chemical variables. From the chemical viewpoint, elementary reactions should depend on temperature and not speciﬁcally on the state of aggregation. Chemical reactions in the solid do diﬀer, however, from those in the gas or in the liquid phase by the rate of transport of reacting species, a factor which is further exacerbated by the heterogeneity of the reacting medium. Some prevalent factors which may aﬀect chemical reactions in solid polymers are summarized in Table 15.1. Glass Transition Temperature In contrast to small organic molecules, long-chain polymers are characterized not only by the center of mass diﬀusion, but also by small-scale diﬀusion of a few monomer units. It is generally observed that even monomolecular reactions could happen only if these motions are unfrozen. In the wider sense, the dependence of reaction eﬃciency on polymer morphological structures can be described in terms of the free volume concept, and of diﬀusion constants [6]. These molecular characteristics are themselves dependent on the thermal transitions in polymers, the most important of which being the glass transition temperature. The glass transition corresponds to the onset of large-scale motions of long seg15.2.3.1

15.2 General Features of Polymer Degradation

ments of the macromolecules (10–20 monomer units). These liquidlike motions require more free volume, resulting in a larger volumic thermal expansion coeﬃcient above the glass transition temperature (Tg ). Below Tg , intra- and intermolecular rearrangements are strongly hindered. This restricted mobility also limits the diﬀusion of oxygen, which is a controlling factor in oxidative degradation. Reactions which require a notable change in activation volume should depend much on the Tg . This is the case for the Norrish II process, in which the transition state involves a six-membered ring conformation (Section 15.5.5.2). In the photolysis of vinyl ketone polymers and ethylene–carbon monoxide copolymers, it was observed that the quantum yield in solid ﬁlm above the Tg was identical to that in solution, but decreased with temperature below Tg , to become negligible below the b-transition temperature. Reactions which involve free radicals generally require little change in activation volume and should proceed equally well below or above Tg , as is the case for the photo-Fries rearrangement (Section 15.5.5.3) [7]. Chain mobility has important implications for bimolecular reaction eﬃciency. Bimolecular reactions depend on (i) the frequency of encounter and (ii) the concentrations of the reactive species. Upon bond cleavage, macroradicals are formed in pairs and can initiate degradative reactions only if they are suﬃciently ﬂexible to move apart. If the macroradicals are held rigid, as in thermosets or below the Tg , they may recombine by the ‘‘cage’’ eﬀect without any detectable molecular change. Mass transfer limitations are a function of the diﬀusion coeﬃcient for each species. Two situations can be encountered [6]: At least one reactant is a small molecule (or radical). Diﬀusion of small species, such as oxygen or methyl radicals, in solid glassy polymer can be much more rapid than predicted from the bulk viscosity of the medium. The internal viscosity of solid polymers can be very low and bimolecular reactions in the solid state, even below the Tg , can be almost as eﬃcient as in solution. Both reactants are polymeric species. In this case, the eﬀective encounter rate is highly limited by mass transfer and shows a marked increase above the Tg . Below the Tg , all the large-scale motions are suppressed although motion of short segments or of side groups of the main polymer may persist and have an inﬂuence on the rate of degradation. It is observed, for instance, that recombination of macroradicals in PMMA, PS, and PVC can occur well below the Tg . A valency migration mechanism [Eq. (21)], in which the unpaired electron is transferred between neighboring polymers until another macroradical is encountered, has been proposed [6].

a CHa þ aCH2 a ! aCH2 a þ a CHa

ð21Þ

The rate constant for this radical ‘‘relay transfer’’ reaction has the same activation energy as that of the b-relaxation in the corresponding polymer. This result again suggests that the kinetics of macroradical reactions in solid polymers is sensitive to small-scale molecular dynamics characterized by the secondary (b; g; . . .) transitions.

765

766

15 Polymer Degradation and Stabilization

Polymer Morphology Structural organization of a solid polymer can be characterized by three principal supramolecular parameters: the content of the amorphous phase, the degree of chain orientation in this phase, and the degree of crystallinity in semicrystalline polymers. These parameters depend to a large extent on its processing history, and on the glass transition (Tg ) and melting temperature (Tm ). 15.2.3.2

Crystallinity It is generally considered that only the amorphous regions in semicrystalline polymers can participate in the degradation reactions. The chemical inertness of the crystalline phase originates from the close packing of the polymer chains (minimum free volume), from the rigidity of the crystal lattice, and from the lack of oxygen to initiate oxidative degradation. Only small species such as hydrogen atoms can penetrate the crystal lattice. Any macroradical which may be created is virtually trapped in the crystallites as an inert species. From the eﬀects mentioned, it is obvious that degradation stability parallels the degree of crystallinity. In some infrequent circumstances, it has been observed that increasing the degree of crystallinity in a polymer sample may increase the rate of photodegradation. Closer scrutiny of the results has indicated that multiple light scattering by the crystallites, which increases the eﬀective optical length, and hence the number of absorbed photons for degradation, is the proper explanation for this unusual behavior. In addition to the degree of crystallinity, the size of the crystallites can also inﬂuence the polymer stability through the dual action of chain orientation and presence of tie molecules. A quenched LDPE, for instance, has a higher resistance to photooxidation than an annealed sample of the same degree of crystallinity, as a result of smaller crystallite sizes and a greater number of tie molecules [8]. In iPP, a reverse trend has been observed when the rate of oxygen diﬀusion becomes the control step (Section 15.4.4). It has been suggested that oxygen diﬀusion may depend on the type of amorphous phase, which is distinguished in PE as intralamellar, interlamellar, and interspherulitic (Ref. 6, p. 292). Although it is well established that oxidative degradation of semicrystalline polymers initiates and spreads in the amorphous interphase without aﬀecting the crystalline domains, it does change the density and crystalline content through a process known as chemicrystallization. The best documented account of this phenomenon is found in isotactic polypropylene. At the beginning of degradation, chain shortening from bond scission facilitates molecular rearrangement, leading to an increase in crystallinity. For long degradation times the crystallinity decreases, sometimes with a shift toward lower Tm , even if the degradation temperature is too low to allow for a change in molecular conformation [9]. Chain orientation Drawing orients the polymer chains, changes the sample morphology and can inﬂuence the rate of degradation in two ways:

Orienting chains increases the intermolecular interactions, while at the same time decreasing the free volume and segmental mobility. Most of the orientation experiments have been performed with semicrystalline polymers (iPP or HDPE).

15.3 Degradation Detection Methods

In many respects, the behavior of oriented amorphous phases shows qualitative similarity to that of the crystalline phase. The reduction of molecular mobility results in a decrease in the rate constant for chain propagation reactions. A decrease in free volume results in a reduction in the rate of oxygen diﬀusion. Even at the same concentration of absorbed oxygen, an oriented sample retains its mechanical strength better than an isotropic polymer and can survive a longer period of degradation. Hydrogen bonding Polar polymers such as the polyamides, aliphatic or aromatic, are strongly hydrogen-bonded in the solid state. Polymers devoid of polar groups can nevertheless develop hydrogen bonding during the course of degradation as a result of formation of oxidation products (hydroperoxy, hydroxy, carbonyl, and carboxyl groups) which can then interact through the acidic hydrogen and the basic oxygen groups. The eﬀect is similar to crosslinking, with a reduction in segment mobility and a decrease in gas permeability. In addition, hydrogen bonding changes the activation energy for some chemical reactions, such as hydroperoxide decomposition (Section 15.4.4). Externally applied stress Most polymers in practical use are subjected to some type of deformation process. Depending on the amplitude, the duration, and the geometry of these deformations, the polymer morphology may change in a number of complex (although generally predictable) ways, such as chain orientation, shear banding, and crazing. How these stress-induced morphologies interact with environmental degradation factors is, however, less well documented. In photo-oxidation, it is found that the rate of decomposition increased with applied stress. The main factors which aﬀect the rate of decomposition following application of tensile stress are an increase in oxygen diﬀusion and an acceleration of chain scission. It is believed that the latter eﬀect may result from a decrease in radical recombination rate, following the pulling apart of the formed radicals by the applied stress before they can recombine again. Several investigations in the 1970s tended to favor a molecular interpretation for stress-activated chemical degradation (Section 15.7.1). However, recent investigations of polyoleﬁns tend to indicate that chemical evolution does not depend on mechanical stresses, which modify only the consequences of the chemical evolution and not the kinetics of the chemical reactions [10].

15.3

Degradation Detection Methods

A few eﬀects of outdoor weathering, such as discoloration and embrittlement, are readily detectable by visual or manual inspection. Quantitative investigations nevertheless require modern analytical techniques capable of probing material changes at diﬀerent levels, from determining macroscopic bulk properties to un-

767

15 Polymer Degradation and Stabilization Tab. 15.2.

Hierarchy of degradation investigative methodologies.

Scale

Aspect investigated

Methodology

Macroscopic

tensile test, fracture energy measurements optical and electronic microscopy

Chemical

decrease in mechanical properties with time development of surface microcracks change in morphology change in molecular weight distribution stable degradation products

Reactive intermediates

free-radical formation

Microscopic Supramolecular Macromolecular

DSC, TEM GPC oxygen uptake, FTIR, Raman, NMR, impedance spectroscopy ESR, chemiluminescence

derstanding molecular mechanisms. The hierarchy of methodologies which has been applied to degradation investigation is summarized in Table 15.2. Many analytical techniques used to inspect the cited properties are common to the ﬁeld of polymer characterization: vibrational spectroscopy (FTIR, Raman), magnetic resonance spectroscopy (NMR, ESR) and liquid chromatography (GPC, HPLC). A few methods, such as oxygen consumption and chemiluminescence, are more speciﬁc to oxidative degradation. Mechanical tests are frequently used in combination with other analytical tools to asset the eﬀects of degradation on mechanical properties. 15.3.1

Mechanical Tests

Many plastic materials are used for their load-bearing properties, and mechanical testing occupies a dominant position among the diﬀerent criteria for degradation. Typical molecular consequences of degradation processes are chain scission and crosslinking. These eﬀects have a profound inﬂuence on the mechanical properties of the material. In general, a predominance of crosslinking over chain scission increases the tensile strength while decreasing the elongation at break and the viscoelastic ﬂow of the material. The reverse eﬀects are observed for chain scission. The loss of toughness in semicrystalline polymers arises from oxidatve chain scission in the amorphous region, which involves:

scission of tie molecules recrystallization of lower MW polymer fraction following scission densiﬁcation from increasing polarity.

In thermal aging of polypropylene, for instance, a rapid transition from ductile to brittle behavior was observed before the appearance of carbonyl groups in the FTIR spectra. Recent data nevertheless indicate that some of the delay in aCbO buildup a

768

15.3 Degradation Detection Methods

Fig. 15.1. Change in elongation at break, weight-average MW (Mw ), and carbonyl content as a function of degradation time, in the thermo-oxidative degradation of unstabilized iPP.

may result from the loss, during the initial degradation stage, of volatile ketones and carboxylic acids which hence elude detection by FTIR. Gel permeation chromatography reveals the presence during the induction period of chain scission, most likely b-scission of alkoxy radicals, which accompanies loss in mechanical strength when the weight-average molecular weight Mw drops below some value of the order of 10 5 g mol1 (Figure 15.1) [11]. Degradation, particularly photodegradation, is a surface phenomenon which aﬀects essentially only the outer few hundred microns from the surface. Scanning electron microscopy invariably reveals the appearance of multiple cracks and microcracks at the surface of degraded polymers, as in Figure 15.2. The morphology of these cracks depends on the type of polymer and the degradation conditions. Crack formation normally occurs on unstrained material according to the following sequence: Oxidation of the superﬁcial layer ! change in material density ! volume contraction ! diﬀerential deformation ! crack formation Oxidized polymers show an increase in density owing to the presence of polar functions. Densiﬁcation of the degraded material creates internal stresses between the outer layer and the intact interior. Ultimately, cracks will appear when the mechanical strength of the degraded layer falls below the diﬀerentiial tensile stresses. Water absorption from ambient humidity or from rain, followed by surface evapo-

769

Fig. 15.2. Thermally degraded polypropylene samples. Left: macroscopic aspect, showing the interior of a water pipe degraded in hot water (six months, 90 C). Right: microscopic (SEM) view of an HAS-stabilized sample thermally aged in an oven (54 days at 135 C [11]).

770

15 Polymer Degradation and Stabilization

15.3 Degradation Detection Methods

ration, is another source of cyclic mechanical stresses which can favor crack formation in certain polymers. The spatial irregularities in oxidation are another source of diﬀerential stresses between degraded and undegraded domains. Fracture usually occurs from surface cracks. Starting from the degraded layer, these surface microcracks can easily propagate into the internal intact layer by stress concentration at the crack tip. Some mechanical tests are more sensitive to the outer layers than to the bulk properties, and this peculiarity should not be overlooked when interpreting experimental results. Tensile strength of degraded elastomers, for instance, shows a diﬀerent behavior when compared with tensile elongation, because the former depends on the entire cross-section of the material, whereas ultimate elongation is correlated with surface modulus because cracks initiated at the hardened surface immediately propagate through the sample (Ref. 10, p. 557). 15.3.2

Gel Permeation Chromatography

Polymer degradation involves cleavage of bonds, resulting in a decrease in MW, branching, cyclization, and crosslinking. These changes in the MW and MWD, and their evolution with the extent of degradation, can give valuable insights into the degradation mechanism. Before the advent of GPC, diﬀerent MW averages such as Mn ; Mv or Mw , were used routinely in evaluation of polymer degradation processes. Determination of the number-average MW (Mn ) has the major advantage that it is directly related to the inverse of the number of molecular chains per unit polymer mass [Eq. (22), where n i is the polymer molar fraction of molecular mass Mi ]. Mn ¼

X

n i Mi

ð22Þ

In polymer degradation experiments, it is convenient to deﬁne the scission index s as the number of broken bonds per initial macromolecule. Because each mainchain scission produces an additional molecule, the indice s is directly given by the change in number-average MW regardless of the initial MWD and degradation mechanism [Eq. (23), with n representing the number of polymer chains per gram of polymer, and the superscript ‘‘ ’’ referring to the initial conditions]. s ¼ ½n n /n ¼ ½Mn /Mn 1

ð23Þ

Pure chain scission seldom occurs without other competitive radical reactions, except perhaps during the early stage of oxidative degradation. When chain scission and crosslinking happen simultaneously, the polymer molecular weight may increase or decrease depending on the relative importance of each process. For random scission and crosslinking, Flory and Charlesby have shown that a threedimensional network begins to occur at the ‘‘gel point’’, when the crosslink density

771

772

15 Polymer Degradation and Stabilization

k(i,j) random scission Gaussian scission parabolic scission

j=0 Fig. 15.3.

j=i/2

j=i

Some commonly encountered scission probability distribution functions.

reaches one unit per weight average molecule. The soluble fraction (s) after the gel point can be used to determine the relative yield of chain scission to crosslinking during degradation (see Section 15.6.2). Gel permeation chromatography (GPC), particularly in its multidetection version, is now a mature and well-accepted technique for MW characterization. The capability of GPC to determine changes throughout the MWD, in addition to the diﬀerent MW averages, opens new possibilities for polymer degradation studies. The scission probability as a function of position along the molecular chain, for instance, could be inferred from the MWD of degraded polymer [12]. In a given polymer where most of the covalent bonds have on the average the same bond energy, it is expected that the rate of bond scission may not depend on its position along the chain (random scission). Although this is generally true for the thermal and photodegradation of several polymers, other bond scission statistics can also be observed, as shown in Figure 15.3 [12]. The Gaussian scission probability distribution function, with a preference for mid-chain scission, is frequently encountered in shear-induced mechanochemical degradation. The parabolic distribution, on the contrary, indicates a preference for chain-end degradation. This phenomenon has been reported in the hydrolysis of dextran, as a result of chain branching. A fourth situation, not depicted in Figure 15.3, is encountered in some polymers which have weak links at a chain end or a low ceiling temperature. In the former situation, initiation starts at the chain end bearing the weak bond with volatilization of a monomer out of the reaction medium. In the second case, exempliﬁed by polyacetals, initiation also occurs at a terminal position, but the process contin-

15.3 Degradation Detection Methods

ues and the monomers keep being evolved until complete volatilization of the polymer (unzipping). 15.3.3

Fourier Transform Infrared Spectroscopy

Identiﬁcation of the degradation mechanism, and its evolution with time, are mainly based on analysis of the intermediate and ﬁnal chemical products. FTIR proves to be particularly suited to that purpose. The infrared absorption intensity A s is related to the square of the change in dipole moment (m) during molecular vibration according to Eq. (24), where n is the frequency of the band center, and Q, the normal coordinate of the vibration. A s m n ðqm/qQÞ 2

ð24Þ

The dynamic dipole moments of oxidized compounds are high, owing to the polar nature of the oxidation products, and FTIR provides a sensitive and quantitative technique for the investigation of oxidative degradation. Oxidized polymers show changes in the IR absorption spectrum over the whole range of measuring wavelengths. Most noticeable are the appearance of complex overlapping bands in the carbonyl (1600–1800 cm1 ) and hydroxyl (3200–3600

1

absorbance

0.8

0.6

degraded

0.4

undegraded 0.2

0 500

1000

1500

2000

2500

cm-1 Fig. 15.4. Evolution of the ATR-FTIR spectra of an unstabilized commercial iPP, before (—) and after (----) 3 h of thermal treatment at 120 C.

3000

3500

4000

773

15 Polymer Degradation and Stabilization

cm1 ) regions (Figure 15.4). The content of carbonyl groups is an indirect indicator of the number of scission events. The average ratio of the absorption coeﬃcient of hydroxyl (aOaH) to carbonyl ( aCbO) functions is approximately 3.4:1, and this value can be used to estimate the relative proportion of hydroperoxides to other oxidation products of degradation. IR absorption bands in amorphous solids are rather broad, with extensive overlap. Direct identiﬁcation and quantiﬁcation of the numerous oxidation products, most often of similar chemical structures, in a degraded polymer are diﬃcult. More precise conclusions about the nature of absorbing species can be obtained by selective chemical derivatization. Selective modiﬁcation of functional groups with reactive gases, such as SF4 , NH3 , SO2 , or NO, results in a shift in absorption band positions, which can then be compared with model compounds to allow for a better chemical assignment of the absorbing species (Table 15.3) [13]. Chemical derivatization can also be used for structure elucidation in complex mixtures. For instance, dimethyl sulﬁde has been used to diﬀerentiate peracids from other peroxy compounds. The rate of conversion of dimethyl sulﬁde into dimethyl sulfoxide is almost instantaneous with peracids [Eq. (25)], whereas it is slow with sec- and tert-hydroperoxides [Eq. (26)]. a

R R R R

O–OH C

O

CH–O–OH

Tab. 15.3.

+

+

CH3–S–CH3

CH3–S–CH3

R

→

R

→

R R

OH C

+

O

CH3–S–CH3

ð25Þ

O

CH–OH

+

CH3–S–CH3

ð26Þ

O

Some common derivatization techniques for identiﬁcation by FTIR. Derivatization reaction

IR frequency [cmC1 ]

Carboxylic acid

O RaCaOH þ NH3 ! RaCOO

1564

Carboxylic acid

O O RaCaOH þ SF4 ! RaCaF

1848 1841 (a-branching)

Hydroperoxides

RaOaOH þ NO ! RaONO2 1276 þ 1290 the sharp and intense absorption bands of nitrates and nitrites can be used to diﬀerentiate primary from secondary and tertiary compounds

Alcohols

RaOH þ NO ! RaNO2

Aldehydes

b

b

Functional group

b

774

778

NHþ 4

RaCHbO þ 2Ag(NH3 )2 OH ! RCOO þ 2Ag# þ 3NH3 þ H2 O (Tollens’ reagent does not react with ketones)

1550–1555

15.3 Degradation Detection Methods

The carboxyl group, in the presence of hydroxyls, may be diﬀerentiated by reaction with silver triﬂuoracetate [Eq. (27); Ref. 7, p. 76] or with triﬂuoroanhydride [Eqs. (28) and (29)]. aCOOH þ CF3 COOAg ! aCOOAg þ CF3 COOH

ð27Þ

aOH þ ðCF3 COÞ2 O ! aOaCOaCF3

ð28Þ

aCOOH þ ðCF3 COÞ2 O ! aCOOaCOaCF3

ð29Þ

15.3.4

Magnetic Resonance Spectroscopy Nuclear Magnetic Resonance (NMR) NMR is an absolute and unique method of resolving the microstructure of polymer species. This technique has been instrumental in the elucidation of degradation mechanisms in several polymer systems, including poly(vinyl chloride) (PVC). At around its glass transition temperature (81 C), unstabilized commercial PVC starts to discolor with release of hydrochloric acid, according to the global reaction of Eq. (30). 15.3.4.1

aðH2 CaCHClÞn a ! aðHCbCHÞn a þ nHCl

ð30Þ

Because the thermal stability of PVC is substantially lower than one may expect from its nominal structure, numerous studies have been initiated to identify the defect sites responsible for this anomaly. After considerable controversy concerning the role, nature, and importance of possible irregular structures, the most active sites for initiating PVC degradation were ﬁnally identiﬁed by 1 H- and 13 CNMR as internal allylic (Figure 15.5, I, II) and tertiary chlorine structures (III, IV) [14]. It was believed for a long time that head-to-head radical addition to monomers is a major route for formation of labile structures. Kinetic studies, in association with NMR measurements, reveal that formation of internal allylic and tertiary chlorine structures actually proceeds through an intramolecular or intermolecular chaintransfer reaction to polymer [Eqs. (31), (32); VC ¼ vinyl chloride].

HC=CH CH2Cl

CH2 CHCl CH2 CH 2Cl | CH 2 CC1 CH 2 Fig. 15.5.

(I)

HC=CH CHCl

(III)

CH 2 CHCl CH 2 CHCl | CH 2 CC1 CH 2

Major labile structural defects in emulsion PVC.

(II)

(IV)

775

15 Polymer Degradation and Stabilization

CH2 aCHClaCH2 aCH2 Cl P ! aCH2 a CClaCH2 aCHClaCH2 aCHCl ! aCH2 aCClaCH2 a ð31Þ a

CH2 aCHCla þVC þVC P þ aCH2 aCHClaCH2 a ! aCH2 aCClaCH2 a ! ! aCH2 aCClaCH2 a ð32Þ a

776

Electron Spin Resonance (ESR) Most mechanisms for the degradation of polymers during processing, storage, and long-term use include free radicals. The technique of ESR is therefore a logical choice for the study of polymer degradation. Two kinds of free radicals are of key importance in determining degradation kinetics: (i) the carbon-centered macroalkyl CaC aC; and (ii) the oxygen-centered alkylperoxyl ROO, alkoxyl RO, and acylperoxyl R(CO)OO radicals. Most of these species have unique ESR spectra which allow their unambiguous identiﬁcation under favorable conditions. In addition to structure identiﬁcation, the decay signal of the radicals with time provides information about termination rates and reaction order. At room temperature and in solutions, free radicals are too reactive, and their concentrations too low, to be conveniently detected by ESR spectroscopy. To observe and quantify transient radicals, it is customary to transform the unstable species into stable nitroxide radicals with the spin-trapping technique. Spin-trapping of a macroalkyl radical with pentamethylnitrosobenzene, for instance, can be represented schematically by Eq. (33). The structure of the parent radical could be deduced from the ESR spectrum of the spin adducts, using standard rules of spin coupling and experimental coupling constants. 15.3.4.2

H3C H3C H3C

CH3 N=O CH3

H3C +

.R

H3C H3C

CH3 N– R O. CH3 ð33Þ

15.3.5

Oxygen Uptake

Oxygen uptake provides a simple, yet highly sensitive means to monitor oxidative degradation. A weighed amount of polymer is hermetically enclosed with a ﬁxed amount of oxygen under controlled pressure. Oxygen consumption can be monitored by measuring volumetric change, or the decrease in oxygen content by gas chromatography. The ﬁrst method cannot discriminate oxygen consumption from the release of volatile gases (CO, CO2 , H2 O, and so on) and tends to underestimate the amount of oxygen consumed. The second technique beneﬁts from the high

15.3 Degradation Detection Methods

photooxidation

oxygen uptake

thermal oxidation

oxidation rate

induction period

degradation time

Fig. 15.6. Oxygen uptake curves during photo-oxidation and thermal oxidation of unstabilized polymers.

sensitivity and separation capability of gas chromatography which, with proper care, allows determination of oxygen consumption rates down to 10 13 mol g1 s1 . The gas chromatography approach has been used to determine the instantaneous oxidation rate of elastomers even at room temperature, and reliable lifetime can be predicted under ambient conditions without having recourse to extrapolation from accelerated testing [15]. Measurement of oxygen consumption is the starting point for several analyses of polymer oxidation kinetics. The oxygen uptake curve for most polyoleﬁns shows an induction period with a very slow oxidation rate, before reaching a ‘‘steady state’’ with constant slope. The value of this slope has been often interpreted as the limiting oxidation rate of the polymer. The induction period is generally much shorter in photodegradation than in thermal degradation (Figure 15.6). The main diﬀerence stems from the diﬀerent modes of initiation. The activation energy of primary photochemical processes, 8–40 kJ mol1 , is generally much less than the energy required for thermal activation (120–200 kJ mol1 ). Because photodegradation generally occurs at low temperature, photodecomposition of peroxides is slow and this class of products tends to accumulate rather than being decomposed to initiate chain branching reactions as in thermal degradation. During oxygen uptake, part of the absorbed oxygen is transformed into hydroperoxides, while the remaining fractions are found in hydroxyl- and carbonylcontaining compounds formed by competitive reactions of P and POO radicals. The relative importance of each process depends on the oxygen partial pressure, and various kinetic equations have been derived to relate the rate of oxygen absorption (d[O2 ]/dt) to the formation of oxidation products, under low and high oxygen

777

778

15 Polymer Degradation and Stabilization

pressure conditions. Kinetic information, and rate coeﬃcients for propagation and termination, have frequently been obtained from the ﬁt of the oxygen consumption curve at relatively low conversions. This procedure, however, is complicated by the facts that intermediate oxidation products may be orders of magnitude more susceptible to oxidation than the starting material, and that degradation in solid polymers is fundamentally heterogeneous. Therefore, any obtained value can yield only average information about the degradation process. 15.3.6

Chemiluminescence

As with oxygen micro-uptake, chemiluminescence is an ultrasensitive technique for determining the rate of oxidation at the earliest stage of degradation. Oxidative degradation of polymers is generally accompanied by weak photon emission, with a very low quantum yield (approximately 109 ). The light-emitting reaction which accompanies free-radical oxidation is generally attributed to an exoenergetic termination reaction of peroxy radicals according to the Russell mechanism [Eq. (34)].

R R

O C

O. O.

H

R R

O* C

O O

H

O

O

R

R R R

ð34Þ

C=O* +

1O2

+ ROH . H = -460 kJ mol-1

The Russell mechanism requires one of the terminating radicals to be either primary or secondary so that a six-membered transition state can be formed. Such a mechanism may be prevalent in PE and PA, but not in PP, where the chaincarrying radical is tertiary. For this polymer, more complex alternative routes for chemiluminescence-producing reactions have been proposed [Ref. 10, p. 175].

15.4

Thermal Degradation

Industrial plastics are frequently exposed to elevated temperatures during melt processing and in many engineering applications, some of which will be described in Section 15.4.6. The glass transition temperature (Tg ) and the melting point (Tm ) of semicrystalline polymers are the most important thermal characteristics of polymers. These two parameters alone are generally not suﬃcient to predict the mate-

15.4 Thermal Degradation

rial’s service temperature, if thermal stabilization is not properly controlled. Several polymer systems produce toxic low molecular weight compounds when subjected to heat, which can create hazardous conditions if they come into contact with a human body, either by ingestion or by breathing. Flammability of polymers used in construction is another concern when the polymers are exposed to high temperatures. All these factors combined make thermal stability appraisal a necessity in most application developments. 15.4.1

Thermal Stability

The notion of thermal stability is a vague concept, since it depends primarily on the time scale of observation. A PMMA sample, for instance, can be stable at 300 C for a few seconds, but can withstand only a temperature of 150 C if the heating time spans several hours. To be able to compare thermal stability between polymers of diﬀerent structures, it is necessary to rely on some standardized system, such as the temperature of half-decomposition (T1/2 ). The temperature of half-decomposition is deﬁned as the temperature at which the polymer loses half of its weight when heated in vacuo for 30 min. Experimentally, T1/2 can be conveniently determined by thermal gravimetry (TG). From the TG curves obtained at diﬀerent scan speeds, an Arrhenius plot at constant weight-loss ratio is derived. The pre-exponential factor and activation energy determined are then used to calculate T1/2 . 15.4.2

Polymer Structure and Thermal Stability

Intrinsic chemical factors which inﬂuence heat resistance include primary bond strength, secondary or van der Waals interactions, hydrogen bonding, resonance stabilization, the mechanism of bond cleavage, structure regularity, intrachain rigidity, crosslinking, and branching. Owing to the presence of multiple secondary reactions, the eﬀect of chemical structure on degradation kinetics is not easily rationalized from chemical ﬁrst principles. The actual degradation kinetics is highly variable and depends not only on the polymer structure, but also on the reaction conditions: sample size, internal or external unstable structures, and additives. More speciﬁcally, it is experimentally observed that the degradation temperature and the product distribution can be controlled by changing the heating rate of the polymer, as a result of competitive pathways for degradation. Apart from the primary eﬀects cited, diﬀerences in thermo-oxidative resistance can be discerned between diﬀerent stereoisomers of the same compound (PMMA, PVC, and PP). Notwithstanding this complexity, some semiempirical rules could be identiﬁed by analogy with small organic ‘‘model’’ molecules, and by considering that thermal degradation is a homolytic bond scission process initiated by thermally activated molecular vibrations:

779

780

15 Polymer Degradation and Stabilization

The rate of thermal degradation is related to the number of pendent groups present on the polymer chain. Thus polyisobutylene (PIB) degrades faster than isotactic PP, which itself decomposes faster than HDPE. A rigid polymer backbone has less possibility of rearrangement and fragmentation and can withstand higher thermal energy. The heat resistance increases with the number of covalent bonds per repeat unit: a crosslinked or ladder polymer can be broken only after scission of two or more covalent bonds. The bond of lowest energy is the ﬁrst bond to be cleaved. Conversely, polymers with multiple bonds and aromatic structures are less prone to thermal degradation. Poly(para-phenylene), with its rodlike structure composed of highly delocalized pelectron orbitals, satisﬁes most of the requirements for high thermal stability. Possessing T1/2 > 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

15 Polymer Degradation and Stabilization

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

15 Polymer Degradation and Stabilization

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

15.4 Thermal Degradation

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

15 Polymer Degradation and Stabilization

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

15.4 Thermal Degradation

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

15 Polymer Degradation and Stabilization

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

15.4 Thermal Degradation

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

15 Polymer Degradation and Stabilization

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

15.4 Thermal Degradation

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

15 Polymer Degradation and Stabilization

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 <0.5% of the total radiant energy of @1300 W m2 at noon in the southern regions (Figure 15.10). 15.5.3

Photo-oxidation Proﬁle

It has long been recognized that photochemical damage is more severe at the surface than in the sample interior. The reasons for this inhomogeneity are twofold:

15.5 Photodegradation

0.7 0.6

UV-C

UV-A

ozone absorption spectrum

0.5 W.m-2.nm-1

UV-B

top of the atmosphere

0.4 0.3 bottom of the atmosphere

0.2 0.1 0 250

300

nm

350

400

Fig. 15.10. Spectral irradiance of sunlight at the top (—) and bottom (– –) of the atmosphere, in the summer at latitude 41 N. The diﬀerence results from ozone absorption.

Penetration distance of light For transparent systems, the light intensity IðxÞ decreases with penetration distance x according to the Lambert–Beer law according to Eq. (63), where I0 is the incident light intensity, and a the absorption coeﬃcient, which is proportional to the molar absorption coeﬃcient e.

IðxÞ ¼ I0 expðaxÞ

ð63Þ

Because most radiation is polychromatic, the range of available a is broad. For many materials, the propensity for degradation increases in parallel with the absorption coeﬃcient when the light wavelength decreases. As a consequence, the depth of penetration decreases for the wavelengths which are most eﬀective in promoting photodegradation. As the extent of degradation increases, more UVabsorbing substances are formed, which act as surface screeners and further increase the fall-oﬀ of the depth of the degraded layer. For materials which scatter light, such as semicrystalline polymers, and for polymers which contain charges or pigments, the Lambert–Beer law should be replaced by the Kubelka–Munk equation [Eq. (64), with K ¼ absorption coeﬃcient, S ¼ scattering coeﬃcient and R ¼ reﬂectance of the material]. ðK/SÞ ¼ ð1 RÞ 2 /2R

ð64Þ

797

798

15 Polymer Degradation and Stabilization

Penetration distance of oxygen Oxidative degradation is a reaction between oxygen and the polymer. The solubility and diﬀusion coeﬃcient of oxygen in solid polymers are typically one order and two orders of magnitude less, respectively, than in organic liquids. Because of this slow diﬀusion of oxygen, the oxidative rate decreases rapidly with depth from the surfaces. The eﬀect of oxygen diﬀusion can be modeled by Fick’s second law, modiﬁed by the rate of oxygen consumption (k). It is frequently assumed that the rate of oxygen consumption exactly matches the oxygen supply by diﬀusion, that is, d[O2 ]/dt ¼ 0. In this ‘‘steady-state’’ situation, the oxygen concentration proﬁle does not change with time, resulting in a diﬀerential equation [Eq. (65)].

Dðq 2 ½O2 /qxÞ k½O2 ¼ 0

ð65Þ

The oxygen diﬀusion constant, D, depends on the state of aging process, the reactant concentrations, and the sample morphology (orientation, crystallinity). It follows approximately an Arrhenius-type temperature dependence as in Eq. (66), where D0 is a proportionality constant, and R the molar gas constant. ð66Þ

D ¼ D0 expðE a /RTÞ

In the case of iPP ﬁlm, for instance, the values given by Eqs. (67)–(69) have been determined (Ref. 6, p. 169). Small-spherulite iPP:

D ¼ 26:0 exp½43 900/RTÞ cm 2 s1

ð67Þ

Large-spherulite iPP:

D ¼ 3:5 exp½38 500/RTÞ cm 2 s1

ð68Þ

1

ð69Þ

Oriented iPP:

2

D ¼ 2:0 exp½38 500/RTÞ cm s

Free radicals react with oxygen according to reactions (2) and (3) of the simpliﬁed degradation (Scheme 15.1). A simple approximation for k½O2 is given by Eq. (70), where C1 and C2 are constants [34]. k½O2 ¼ C1 ½O2 /fC2 ½O2 þ 1g

ð70Þ

By combining Eqs. (65) and (70), the oxidation distribution can be calculated by numerical methods and compared with experiments using some depth-proﬁling technique (microFTIR on microtome slices, ATR-FTIR, photoacoustic FTIR). The frequently encountered U-shaped curve indicates a ‘‘diﬀusional’’ regime, in which the rate of chemical reactions is limited by oxygen diﬀusion (Figure 15.11). The temperature coeﬃcient of the oxidation rate, 105–125 kJ mol1 , is much higher than the activation energy for oxygen diﬀusion (@40 kJ mol1 ). Thus, by lowering the temperature, one can reach the ‘‘kinetic oxidation’’ regime, characterized by an oxidation rate independent of sample size.

15.5 Photodegradation

0.6

Absorbance (1718 cm-1)

0.5 0.4 0.3 0.2 0.1 0.0 0

50

100

150

200

distance from exposed surface (µm) Fig. 15.11. Typical photodegradation proﬁle for a weakly UV-absorbing, 200 mm-thick polymer ﬁlm.

15.5.4

Inﬂuence of Wavelength: the Activation and Action Spectrum

The actinic eﬀects of UV and visible radiation on material are wavelengthdependent, owing to the spectral selectivity in the absorption of the incident radiation, and the quantum eﬃciency of degradation. The absorption of a photon by a chromophore promotes an electronic transition between two molecular orbitals of diﬀerent energies. Starting from the ground state, a transition to a diﬀerent electronic state can be eﬀected by changing the energy of the photon. It is therefore logical to expect that the irradiation wavelength may have a profound inﬂuence not only on the rate, but also on the mechanism of photodegradation. The concept of quantum yield, deﬁned as the number of degraded molecules per absorbed photon, is widely applied in the photochemistry of small organic molecules. It is diﬃcult to apply to industrial plastics, which are complex mixtures of species with variable photon absorption and diﬀerent photophysical and photochemical properties. Early investigations relied on the notion of an activation spectrum to characterize the eﬀect of the photon wavelength on the extent of degradation. The activation spectrum of a material is deﬁned by (i) the spectral absorption properties of the material; (ii) the emission properties of the light source; (iii) the quantum eﬃciency of the photoinitiated processes; and (iv) the type of degradation being measured. Although simple to measure, the activation spectrum does not consider the amount of incident radiant energy present at each wavelength, and is hence depen-

799

800

15 Polymer Degradation and Stabilization

0.9

3.5

increase in absorbance (left scale)

0.8 0.7

3.0

sample absorbance (right scale)

0.6

lamp irradiance (W.m-2.nm-1, left scale)

0.4

1.5

0.3

absorbance

2.0

0.5

1.0

0.2

0.5

0.1 0.0

2.5

280

290

300

310

320

330 340 350 wavelength [nm]

Fig. 15.12. Activation spectrum of a 700 mm ﬁlm of unstabilized bisphenol-A polycarbonate, based on the increase in absorbance at 310 nm after accelerated aging with a borosilicate-glass

360

370

380

390

0.0 400

ﬁltered xenon lamp. Superimposed are the spectral irradiance of the irradiation source and the UV-absorption spectrum of the polymer (redrawn from data in Ref. [35]).

dent on the speciﬁc light source involved. The action spectrum is introduced to correct for this weakness by explicitly taking into account the eﬃciency of light-induced damage per incident photon, as a function of irradiation wavelength. Relatively few investigations have been performed on polymer action spectra until recently, as a result of the lack of suitable high-intensity, tunable, monochromatic source for irradiation. The use of sharp-cut ﬁlters in pairs with half-height bandwidths of approximately 20 nm does not usually provide suﬃcient wavelength resolution. The availability of large spectrographs now allows the recording of continuous activation and action spectra across the whole wavelength range of interest [34]. An action spectrum is a graph of the reciprocal of the radiant exposure required to produce a given eﬀect (for instance, yellowing index or increase in absorbance at a standard wavelength) at each wavelength (Figure 15.12). All the data in such curves are normalized to the datum at the most eﬃcient wavelength. This kind of plot gives crucial information on the spectral sensitivity of the material. Its knowledge is particularly important for proper comparison between outdoor weathering and accelerated artiﬁcial aging, for which the light source spectrum may diﬀer signiﬁcantly from solar irradiance. In most instances, the action spectrum decreases empirically with increasing irradiation wavelength according to a logarithmic relationship (Figure 15.13B), as given in Eq. (71), with l ¼ irradiation wavelength, and SðlÞ ¼ relative spectral damage [33].

15.5 Photodegradation

30

(A)

radiant energy [J.cm-2]

25

10°C

50°C

20 15 10 5 0 290

300

310

320

330

340

350

irradiation wavelength [nm]

(B)

4

ln(radiant energy /J.cm-2)

3

10°C

2 1 50°C

0 -1 -2 -3 290

300

310

320

330

340

irradiation wavelength [nm] Fig. 15.13. Action spectrum of a 700 mm ﬁlm of bisphenol-A polycarbonate, based on the radiant energy required for a 10% decrease in transmittance at 360 nm: (A) linear plot; (B) semilogarithmic plot.

350

801

802

15 Polymer Degradation and Stabilization

ln SðlÞ ¼ lnðaÞ bl

ð71Þ

Based on absorption and energetic considerations, photochemistry has long relied on the existence of a critical wavelength below which no reaction should occur (Section 15.5.1). The question then arises of the limit of applicability of the logarithmic relationship when the irradiation wavelength is much greater than the absorption maximum. 15.5.5

Photodegradation Mechanisms

Photodegradation diﬀers from thermal degradation mainly in the mode of initiation. All the other mechanistic steps – bond scission, propagation, chain branching, and termination – are similar to thermal degradation. Indirect diﬀerences may exist, however, between thermal and photodegradation as a result of variations in experimental conditions, such as a higher temperature for the former mode of aging which may result in a lower concentration of dissolved oxygen, a lower rate of peroxide decomposition, or a change in material morphology. Photoinitiation It is practical to distinguish two types of initiation: primary, when the radiation is directly absorbed by the functional group of the polymer; and secondary, when the radiation is absorbed by certain of the oxidation products (peroxides, carbonyls). For saturated polymers, such as polyoleﬁns, secondary initiation is the predominant mode of initiation, particularly in the latter stages of deterioration. Formation of excited states following photon absorption constitutes the ﬁrst step in any photochemical reaction. When light is absorbed by a chromophore, either ‘‘in-chain’’ or ‘‘external’’, the energy is used to promote an electron from the ground state to an excited state. The excited state can relax back to the ground state with release of extra energy in the form of a photon or heat. Photochemistry begins if the energy stored in the excited state is used to drive a chemical reaction. The rate at which this reaction occurs depends on the energy content (the exciting wavelength), the chemical nature of the absorbing chromophore, and its environment. The chemistry of the excited state in polymers has been studied with model compounds. In a real environment, however, the presence of additives, impurities, residual catalysts, and so on can exert a decisive inﬂuence not only on the degradation rates but also on the degradation mechanisms. The reactions are further complicated by the occurrence of additional chromophores which are formed during photodegradation. Following the diversity of impurities which can be encountered, and the complexity of ensuing reactions, there has been much controversy in the literature about the true photoinitiation mechanisms. This debate, however, is essentially academic, since the initiation mechanism does not inﬂuence the global rate of photo-oxidation. Although acting as an external agent, oxygen deserves a special situation, owing to its presence in any practical weathering situation. Oxygen can be easily con15.5.5.1

15.5 Photodegradation

verted to the singlet state ( 1 Dg or 1 Sþ g ) by absorption of a photon, or it can be formed during decomposition of secondary compounds. An important mode of photoinitiation occurs via a charge-transfer complex between oxygen and the polymer (for instance, a polyoleﬁn), with formation of hydroperoxides [Eq. (72)].

R2 R1

C

R2 H

+

R1

O2

R2 CH+ -O2 → R3

H.....O2

R3

R3

R1

C

R2 R1

C. .O 2H R3

ð72Þ

R2 R1

C

OOH

R3

In spite of the diversity in photoinitiation mechanisms, two principal modes of reaction can be identiﬁed in most commercial polymers: the Norrish photoprocesses and the photo-Fries rearrangement. The Norrish Photoprocesses One common chromophore frequently encountered in a number of polymers is the carbonyl ketone which shows a weak p–p transition band centered at 280 nm but extending into the UV-B region. Polymers containing this chromophore undergo two types of photochemical reaction (Scheme 15.2): 15.5.5.2

the Norrish type I reaction, resulting in an a-cleavage with formation of two endpolymeric radicals and carbon monoxide; the Norrish type II reaction, which involves an intramolecular H-abstraction via a cyclic six-membered transition state. The photoscission proceeds by means of a short-lived triplet state and a biradical intermediate. Photo-Fries Rearrangement A second large class of polymers contains phenyl moieties, either in the backbone (PPO, PC, PET, PSU) or as a side group (PS). Another common phenyl group photoreaction, when subjected to UV radiation, is the photo-Fries rearrangement (Scheme 15.3). Upon absorption of a photon, the n- or p-orbital of the chromophore is promoted to a singlet p excited state. Bond scission occurs primarily at aromatic ether CaO bonds, and causes rearrangement or degradation of the polymer backbone. Because the photo-Fries rearrangement proceeds in a ‘‘caged’’ environment, it is independent of free volume and is almost independent of Tg . The photo-Fries mechanism can be promoted with light in the region above 300 nm (Figure 15.14), and accounts for the yellowing of the polymer observed at long wavelengths. Chain scission, on the other hand, is promoted by light at shorter wavelengths. Because photo-Fries products are easily photodegraded, only small 15.5.5.3

803

804

15 Polymer Degradation and Stabilization

Carboxyl in pendent groups sh rri No

–CH2–CH–CH2–CH–CH2 – R

C

O R

C

No rris

O

.

–CH2–CH–CH2–CH–CH2–

I

hI

R I

Carboxyl in backbone

rr No

– CH2– CH2– C– CH2– CH2– CH2–

No rr

O

I ish

ish

C

–CH2–CH2 R

II

C

+

R– C

.

C

Scheme 15.2.

CH3

– CH2– CH2– C– CH3 O

CH2 C

R

C

CH3

– O– C– O–

C

C CH3

CH3

hv

CH

* H O

O

CH3 C

CH3

O

– O. .C – O–

C

CH3

CH3

CH3

– OH

C CH3

CH3

C– O–

C CH3

O CH 3 hv

– OH

C CH 3

C O

Scheme 15.3.

OH

+

C

O

.CH2– CH2– CH2–

O

H O Norrish photoprocesses in carbonyl-containing polymers. R

O

R

CH2=C–CH2– R

O

– CH2– CH2– C

CH 2 CH C

hv

O

O

Six-membered transition state CH

.C

+

CH 3 C CH 3

Photo-Fries rearrangement in bisphenol-A polycarbonate.

+

CH2=CH2–

15.6 Radiolytic Degradation 9.E-4 8.E-4 7.E-4

quantum yield

6.E-4 5.E-4 4.E-4 3.E-4 2.E-4 1.E-4 0.E+0 250

260

270

280

290

300

310

320

irradiation wavelength [nm]

Fig. 15.14. Quantum yield for chain scission (–o–), eﬃciency of photo-Fries rearrangement (–k–), and absorption spectrum (--D--) of bisphenol-A polycarbonate [34].

amounts of them remain in aged material, which shows a predominance of sidechain oxidation products. Once photoradicals are formed, they can either recombine in the cage or diﬀuse to the outside for further reactions. The radicals formed can undergo a variety of abstraction and decomposition reactions, resulting in a multitude of chemical products. The propagation and recombination steps are similar to those encountered in thermal degradation, and will not be discussed further here. In the presence of UV > 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

15 Polymer Degradation and Stabilization

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 <100 eV

primary

short tracks <5000 eV

branch tracks >5000 eV delta rays

Fig. 15.15.

Schematic representation of a fast electron track.

created (d-rays). Owing to the augmentation of linear energy transfer (LET) with decreasing kinetic energy, the fast electrons lost most of their kinetic energy near the end of the track, resulting in local regions of highly reactive species called ‘‘short tracks’’, ‘‘blobs’’, and ‘‘spurs’’ (Figure 15.15). In 60 Co irradiation, the primary interaction is Compton scattering between the high-energy photon and an atomic electron. Only part of the photon energy is transferred to the electron and the resultant photon of lower energy is scattered. The process will continue with creation of more scattered electrons and secondary electrons (Figure 15.16). Except for the primary event, the eﬀects of g-rays and accelerated electron irradiation are similar. From a single incoming g-photon or highenergy electron, a shower of secondary electrons is generated which is responsible for most of the ensuing chemical reactions. 15.6.2

Radiation Chemistry

Radiation chemistry is now a well-established area of science that deals with chemical and physicochemical changes produced by the absorption of high-energy radiation (HER) by matter. The subject covers events that occur from the passage of the ionizing particle to the completion of chemical reactions. To encompass the enor-

hv 2 hv 1 hv 0 e-

ehv 3 e-

Fig. 15.16.

Compton scattering and fate of the secondary electrons.

807

808

15 Polymer Degradation and Stabilization

mous time span for the various processes, it is usual to distinguish between the ‘‘physical’’ stage (1017 –1011 s) when energy from the incident particle is deposited into localized regions of space (‘‘spurr’’), the ‘‘physicochemical’’ stage (1012 – 103 s) during which reactive species (ions, quasi-free electrons, excited molecules, hot radicals) are formed and react within the ‘‘spurs’’, and the ‘‘chemical’’ stage (109 –103 s in solution, several days in solid polymers) where stable species (radicals, trapped electrons, cations) diﬀuse and react outside the clusters. The standard unit of absorbed dose is the Gray (Gy), deﬁned as the energy imparted by the high-energy radiation to a mass of matter equivalent to 1 J kg1 (1 Gray ¼ 100 rad). Molecular changes are characterized by a G factor, in units of mmol J1 , deﬁned as the event yield per 100 eV of absorbed energy. Although much discussion has occurred in the past on the relative importance of radical and ionic reactions, it is now established that the major chemical changes in irradiated polymers are accounted for by free-radical reactions [37]. In the early stage of the reactions, the reactive species are concentrated in ‘‘spurs’’ and particle ‘‘tracks’’ in a manner similar to their parent ionized or excited molecules. The kinetics at this point must take into account the inhomogeneous distribution of the radicals, before they can diﬀuse away. Ionizing radiation is unique in the sense that reactions can be initiated randomly at any temperature. Cryogenic temperatures, at 4 K and below for instance, have been used extensively to prolong the macroradicals’ lifetime for ESR measurements. Apart from the mode of initiation, all other material changes brought about by high-energy radiation are governed by radical reactions, in perfect analogy to those generated by other means, such as photochemical, thermal, or mechanochemical degradation. As a matter of fact, despite the enormous diﬀerence in HER energy (10 6 eV) and molecular binding energy (@5 eV), the chemical eﬀects of HER can best be compared with those of UV light with energy in the 5–20 eV range. The eﬀects of ionizing radiation depend greatly on the structure of the polymer, the temperature, and the nature of environment. A 50% loss in ultimate elongation (a common measure of the eﬀect of irradiation), for instance, can vary from doses as low as 3.5 kGy for PTFE, to more than 4000 kGy for PS, polyimide, or PEEK (Table 15.6). The unusual radiation sensitivity of PTFE is attributed to the unique stability of perﬂuoro macroradicals which favors chain scission over crosslinking. In the presence of air, these ﬂuorine-containing radicals are converted into peroxy radicals which degrade readily into low MW fragments. PTFE can be crosslinked by HER when irradiated in an inert atmosphere above its melting point (330–340 C). Polymers containing phenyl groups owe much of their radiation resistance to excited-state energy transfer to the benzene rings, which act as excitedstate quenchers (see Section 15.6.3). Although ‘‘energy transfer’’ is the widely accepted protection mechanism in the HER degradation of aromatic compounds, it does not explain why styrene does not show the same protective eﬀect as polystyrene. One alternative suggestion is that H atoms resulting from the primary eﬀect of radiation are added to the aromatic rings, and are no longer able to produce secondary macroradicals by abstraction.

15.6 Radiolytic Degradation Tab. 15.6. Relative radiation stability in air of major commercial polymers (based on a 50% decrease in ultimate elongation).

Polymer

Dose [kGy]

Polytetraﬂuoroethylene Polytriﬂuorocholorethylene Poly(methyl methacrylate) Polycaprolactam Isotactic polypropylene High-density polyethylene Poly(vinyl chloride) Poly(ethylene terephthalate) Poly(triethylene glycol dimethacrylate) Low-density polyethylene Polyurethanes Melaminoformaldehyde resin Polycarbonates Polystyrene Epoxy resin ED-10 Epoxy resin ETZ-10 Polyimides

10 30 300 600 1000 1000 1500 2000 2000 3000 3000 4000 5000 5000 15000 30000 100000

The two fundamental processes that result from radiochemical reactions are chain scission and crosslinking, characterized by Gs and G x , respectively. If Gs < 4G x , branched polymers can be formed and may eventually evolve into a three-dimensional network structure. Based on the assumptions that:

the initial polymer MWD follows a random distribution, cross-linking occurs by H-linking, and crosslinking and scission occur with random spatial distribution (without clustering),

Charlesby and Pinner have shown that the sol fraction(s) should follow Eq. (74), where R [kGy] is the absorbed dose [37]. s þ s 0:5 ¼ 0:96 10 5 /ðR Mw G x Þ þ ðGs /2G x Þ

ð74Þ

The Charlesby–Pinner plot of (s þ s 0:5 ) as a function of 1/R gives a straight line, with slope m 1/G x , and an intercept on the ordinate equal to Gs /2G x . Departure of the plot from a straight line may originate from deviation from a random distribution, or from chemical heterogeneities in copolymers. Regardless of the shape of the curve, the incipient dose for gelation is determined, by deﬁnition, from the sol curve at s þ s 0:5 ¼ 2 (Figure 15.17).

809

15 Polymer Degradation and Stabilization 1980 kGy

2

870 kGy

50%

s + √s

810

1

38 kGy

100% SAN

70%

1.0 0.72 Gs/2Gx = 0.36

0

0 ∞

2000

10-3

(kGy/R)

2.10 -3

1000

(R/kGy)

500

Charlesby–Pinner plot of irradiated compatible PMMA–SAN blends. The sol and gel fractions refer solely to the SAN component in the mixture and are corrected Fig. 15.17.

400

for grafting to PMMA. Large curvatures of the plot for blends are interpreted as a result of chemical interactions between the two components (redrawn from Ref. [38]).

15.6.3

Radiolysis Stabilization

It has been known since the early days of radiation chemistry that some simple organic compounds, such as benzene, halogenated hydrocarbons, nitriles, amines, and alcohols, can protect the polymer from the deleterious eﬀects of high-energy radiation. Many of these ‘‘antirad’’ substances interfere at some stage with the radiolytic degradation scheme, as depicted in Table 15.7, to reduce damage to the plastics. The most eﬃcient present-day ‘‘antirad’’ agents are antioxidants which act essentially in the chemical stage by scavenging free radicals in a similar way to that in the other types of degradation. Aromatic compounds are highly eﬃcient at quenching excess energy of excited states formed by geminal recombination. Because most commercial antioxidants have aromatic rings in their structures, they can also act as primary absorbers by diverting the radiation energy into harmless vibrational energy, as in Eqs. (75) and (76). g; e

P ! fPþ þ e gspur ! P ! 2R

P þ Q ! P þ Q

ð75Þ ð76Þ

Aromatic polymers owe much of their radiation resistance to this ‘‘internal protection’’ eﬀect [40]. An eﬃcient method of radiation protection would be to blend the

15.6 Radiolytic Degradation Tab. 15.7.

811

Polymer protection scheme during radiolysis.

Radiolysis stage

Physical or chemical event

Protection means

Protection mechanism

Physical Physicochemical

energy absorption molecular ionization

sheets of lead or concrete positive ion scavengers

geminate recombination

electron scavengers

dissociative states

electronic excited-state quenchers

cleavage of CaH bonds

H atom donors

diﬀusion and abstraction reactions of mobile H atoms reactions of macroradicals

H atom acceptors

decrease in radiation intensity transfer of one electron to polymer cation without subsequent excitation acceptance of ejected electron and lower probability of dissociative recombination divergence of excitation energy into heat or longerwavelength electromagnetic radiation transfer of H atom to macroradicals (reparation mechanism) acceptance of H atoms, preventing formation of more radicals reaction with macroradicals, formation of stable species decrease in oxygen diﬀusion React with molecular oxygen under irradiation Disrupt oxidation chain reaction by converting peroxy radicals into stable products catalytic destruction of peroxy radicals

Chemical

reactions of macroradicals with oxygen benzophenone derivatives

chain reaction of peroxy radicals (radio-oxidation)

radical scavengers surface coatings oxygen absorbers Antioxidants

peroxide decomposers

degradative polymer with a radiation-resistant one. Energy transfer (Fo¨rster type) is eﬃcient only at short range and the eﬀect is most noticeable with compatible blends, such as PMMA and SAN (Figure 15.17). 15.6.4

Applications

High-energy radiation can penetrate deeply into organic materials, and can initiate reactions at low temperature without added chemicals or catalyst. These unique features are exploited in an increasing number of industrial applications, particularly in the biomedical ﬁeld, in which chemical contamination or thermal degradation should be avoided [41]. Radiation sterilization of medical commodities, one of the early achievements of radiation engineering, continues to increase its market share to the detriment of standard methods such as chemical sterilization with di-

812

15 Polymer Degradation and Stabilization

ethyl ether, or heat treatment. In the examples mentioned, the value of the ﬁnished products is generally high and the cost of radiation processing does not enter into consideration. In other applications, where the product is inexpensive radiation processing can still be economically viable if the quantity of radiation energy required is low. Radiation treatment of food and radiation-induced grafting or crosslinking of certain plastics belong to the latter category. Food is commonly irradiated at low doses in the 0.1–10 kGy range, in ordinary boxes or containers. Food preservation by irradiation is gaining acceptance for an ever-increasing number of agrochemical products such as spices, vegetables, and processed meat. This technique is currently viewed as the most eﬀective of the available alternatives (cold storage, heat treatment, fumigants). Radiation Sterilization Disposable medical products are required by legislation to be sterilized prior to use. Among the four basic methods of sterilization – heat, ethylene oxide, gamma irradiation ( 60 Co), and electron beam techniques – the last has steadily increased its market share, largely at the expense of the diethyl ether method as a result of regulatory concerns about the quantity of chemical residuals. The Federal Drug Administration (FDA) in the US allows a device to be labeled as sterile if less than one device out of a batch of one million can show biological contamination. The standard dose to achieve this level is 25 kGy. Medical plastics that need to be sterilized must withstand at least this dose. The data given in Table 15.6 can be used as an initial estimate in material selection. Materials which are likely to degrade, such as PTFE or POM, should be avoided. Polymers containing aromatic groups have much greater radiation resistance than those with an aliphatic structure. Similar improvements can be obtained with crosslinks, and most thermosets and elastomers can withstand at least one radiation sterilization. The use of antioxidant additives can signiﬁcantly oﬀset the eﬀects of radiation, and even a radiation-sensitive polymer such as polypropylene can withstand several radiation sterilization cycles when properly stabilized. 15.6.4.1

Controlled Degradation and Crosslinking The reactions of chain scission, crosslinking, and grafting initiated by ionizing radiation have found many important applications in the plastics and rubber industries. Although high-energy radiation is destructive for most materials, this propensity can be usefully exploited in the controlled degradation of several polymers, such as PTFE, PEO, PP and cellulose. The most well-known example of degradation application is the manufacture of PTFE powders. Undegraded PTFE is too tough and slippery to grind. After submitting the polymer to a high dose range (500–1000 kGy), the degraded PTFE becomes brittle and can be converted to ﬁne particles while conserving its low surface energy. The PTFE powder is used as a solid lubricant, or formulated as grease for high-temperature applications. Radiation crosslinking is another way to tailor properties to speciﬁc applications. High-energy radiation provides a clean and cost-eﬀective method to achieve this means. The technique has been applied in the crosslinking of high-purity medical 15.6.4.2

15.7 Mechanochemical Degradation

products, such as gloves and condoms, orthopedic ultra-high MW PE hip joints, and biocompatible hydrogels. Other important industrial applications of radiation crosslinking include the production of heat-shrinkable products, wires, cables, and tubing.

15.7

Mechanochemical Degradation [42] 15.7.1

Initiation by Mechanical Stresses Eﬀect of Tensile Stress on Chemical Reactivity The input of external energy, of either thermal or electromagnetic origin, is well known to the organic chemist as an eﬃcient way to promote chemical reactions. Mechanical energy, although ubiquitous in everyday life, is rarely considered as a bona-ﬁde source of chemical activation. The main reason probably stems from the ineﬃciency of this mode of excitation for small molecules where most of the applied mechanical energy is readily dissipated into heat and the work of bulk deformation, leaving little available as chemical potential. In polymer systems, on the contrary, mechanochemical reactions are ubiquitous as a result of the unique propensity of macromolecules to store free energy upon deformation and to sustain a high level of stress for a suﬃciently long time for chemical reactions to occur. Mechanochemical degradation is ubiquitous in macromolecular systems and can be encountered in practically any ﬁeld involving high MW polymers. The formation of free radicals by mechanical stress has been detected during polymer processing, analysis, weathering, and gel swelling. Chain scission has also been observed in applications as diverse as drag reduction, rubber mastication, and mechanochemical synthesis. It was known for centuries that the application of stress under certain conditions, such as the combing of wool, the mastication of rubber, or the kneading of dough, can improve the physical properties of a few materials which were later recognized as being polymeric. Following the application of stress, a molecular system will attempt to minimize its total free energy by a series of conﬁgurational rearrangements, such as kink annihilation, chain disentanglement, and orientation. According to molecular mechanics, the excess energy of stress (Es ) in a chemical system comes from several sources, namely nonvalence and coulombic interactions, torsional deformation, and angular and valence-bond strain energies. Among the diﬀerent modes of molecular deformation, bond-angle distortion and valence-bond stretching have the most profound inﬂuence on the electronic density distribution, and hence on the chemical reactivity. In the region of stresses of around 80 GPa, the valence angles for CaC bonds will become deformed from their equilibrium positions whereas bond stretching and rupture occur at some still-higher values in the vicinity of 740 GPa. Although stress-induced chemical reactivity can be explained only by quantum mechanical calculations, as was done for the acceleration eﬀect of water 15.7.1.1

813

814

15 Polymer Degradation and Stabilization

during crack propagation of silica glasses, a qualitative picture can be gained based on mechanical arguments. In carbocyclic compounds, Es originates principally from valence-angle deformation and it is well known that ring strain could aﬀect the rate of some chemical reactions in a drastic way. For example, if the transition state necessitates distortion of the valence-bond angle from y0 to y , the valence bond-angle contribution to the activation energy is given by Eq. (77), where ky is the elastic constant for bond-angle deformation. E ang ¼ 12ky ðy0 y Þ 2

ð77Þ

If, however, the same bond angle is already distorted from y0 to y under mechanical stress, the activation energy will be given by Eq. (78), resulting in a diﬀerence of DE ang [Eq. (79)]. E ang ¼ 12ky ðy y Þ 2 DE ang ¼

1 2ky ½ðy0

ð78Þ 2

2

y Þ ðy y Þ

ð79Þ

In solid polymers, rate enhancement under stress has frequently observed, for example in UV-photo-oxidation of natural silk, polycaprolactam, and poly(ethylene terephthalate). However, quantitative interpretation is diﬃcult in these systems due to interdependence of several stress-dependent factors such as the rate of oxygen diﬀusion or changes in polymer morphology which supersede the elementary molecular steps. Similar experiments in the ﬂuid state showed unequivocally that ﬂow-induced stresses can accelerate several types of reaction, the best studied being the hydrolysis of DNA and polyacrylamide. In these examples, hydrolysis involves breaking of the ester OaPO and the amide NaCO bonds. The tensile stress stretches the chain, and therefore facilitates the formation of a transition state in which the bond length is increased for hydrolysis, thus enhancing the rate constant for the process. Other documented examples of stress-induced chemical reactions are the acylation of cellulose and the addition of 4-hydroxy-2,2,6,6tetramethylpiperidine-1-oxy to rubber during mastication. Breaking Strength of a Covalent Bond Initiation in mechanochemical degradation involves homolytic scission, so one basic question would be what level of stress is necessary to separate two chemical moieties which have been attached by a covalent bond. The earliest account for this problem was reported by de Boer in 1936 [42]. The potential of a bond under equilibrium is normally approximated by the Morse potential, given by Eq. (80) where l is the length of the bond, l 0 is the equilibrium separation distance of the atoms, D is the bond dissociation energy, a ¼ ðkf /2DÞ 0:5 is a parameter which deﬁnes the width of the potential minimum, and kf is the bond force constant in the neighborhood of the equilibrium separation. 15.7.1.2

VðlÞ ¼ Dð1 exp½aðl l 0 Þ 2 Þ

ð80Þ

15.7 Mechanochemical Degradation Morse potential Morse potential for a stressed bond

1.0

mechanical potential

V(l)/D

0.8

0.6

0.4

0.2

D'

0.0

-0.2

-2

-1

0

1

l-l0

2

3

4

5

Fig. 15.18. The Morse potential of a bond under equilibrium (----) and in the presence of an applied force equal to 0.6 fb (----).

If the bond is under tension with a constant force fext pulling on either end, the potential energy V 0 ðlÞ will be decreased by an amount equivalent to the work performed by the mechanical force over the stretching distance from the equilibrium position [Eq. (81); Figure 15.18]. V 0 ðlÞ ¼ VðlÞ ðl l e Þ; where C is a proportional factor

ð81Þ

The potential function V 0 ðlÞ has a minimum at l00 > l 0 , in accord with the intuitive expectation that the bond separation should increase in the presence of a tensile stress. The new activation energy (D 0 ) required to break the stressed bond could be calculated from the principle of virtual work performed on the bond in going from l00 to l b (Figure 15.19). Rate of Stress-activated Chain Scission From a chemical viewpoint, bond scission under stress is a particular case of a unimolecular dissociation reaction whose rate is enhanced by mechanical stress. As such, it could be treated with Eyring’s transition-state theory [Eq. (37)], which permits one to bring the treatment of rate processes within the scope of thermodynamic arguments. By combining de Boer’s thermodynamic formulation and the transition-state theory, Tobolsky and Eyring in 1943 developed the rate theory for thermally activated fracture of polymeric threads. When put into an Arrhenius15.7.1.3

815

15 Polymer Degradation and Stabilization 0.4

0.2

f(l)/Da

816

l0'

0.0 0

lb 1

lf 2

3

4

5

l-l 0 -0.2

-0.4

Fig. 15.19. Derivative of the Morse potential. The hatched area corresponds to the bond energy under stress.

like form, Eq. (82) was obtained, where k c is the rate constant for bond scission, U0 , the thermal energy for bond rupture, and c the molecular stress. k c ¼ A exp½fU0 f ðcÞg/kB T

ð82Þ

Several attempts to relate the rate for bond scission (k c ) to the molecular stress have been reported over the years. The simplest and still the most popular is a linear relationship [Eq. (83)] between the decrease in activation energy and molecular stress [42]. f ðcÞ ¼ bs

ð83Þ

The factor b has the dimension of volume and is identiﬁed as the activation volume for the reaction. 15.7.2

Extrusion Degradation

To achieve useful properties, polymers are generally compounded with a variety of additives, transformed into granulates, before being processed into the ﬁnal shape for commercial exploitation. During each step, the material is exposed to a high shear-rate and temperature. The two major sources of chemical degradation during processing are therefore mechanical stresses, which tend to decrease with increased temperature as with the melt viscosity, and thermal reactions. As a result of excessive pressure within the equipment in the course of processing, oxygen

15.7 Mechanochemical Degradation

from the air generally has no access to the polymer melt. Except during the initial stage, when dissolved air can be used up at high rate, most of the degradation occurs under oxygen-deﬁcient conditions. This feature has been veriﬁed during multiple extrusion of polypropylene [44]. In this experiment, as in other similar experiments, the concentrations of oxidation products detected by FTIR were very low. On the other hand, changes in the MW and MWD were always observed during extrusion. Because the MWD is of the utmost importance for the rheological properties, it is important to know the molecular parameters which may inﬂuence the rates of chain scission and crosslinking. Oxidative b-scission of alkoxy radicals is relatively unimportant in extrusion, as a result of lack of oxygen and low activation energy (59 kJ mol1 ). The majority of chain scissions are caused by the b-cleavage of alkyl radicals, with activation energies of 84–117 kJ mol1 . Cleavage of a secondary alkyl radical or a tertiary radical produces a vinyl (H2 CbCHR) or vinylidene group (H2 CbCRR 0 ), respectively. A kinetic model for PE crosslinking and scission involving alkyl radicals under processing conditions has been developed, based on and activation energy of 146 kJ mol1 for chain scission, and 96 kJ mol1 for crosslinking. As a result of the diﬀerence in activation energies, crosslinking tends to dominate at low temperatures and scission at high temperatures. Crosslinking reactions are attributed to the addition of alkyl radicals to vinyl groups, either initially present during synthesis or formed by thermal scission. 15.7.3

Applications

One of the earliest industrial applications of mechanochemical degradation was in the mastication of rubber, to reduce the MW of the latex to a level which could be processed. Rubbers vulcanized with a conventional high sulfur/accelerator system are known to be more resistant to fatigue than peroxide-cured rubbers and rubbers cured by sulfur-free or low-sulfur systems. This behavior was associated with the presence of polysulﬁdic (aCaSn aCa) crosslinks in the conventional vulcanizate [45]. Because the SaS bonds are more labile than the CaC bonds (see Figure 15.18), it is postulated that they can be easily broken under fatigue, resulting in the release of local stresses during bond rearrangement. Mechanochemical degradation creates free macroradicals in pairs, practically without any side reactions, and most potential applications of this technique are centered around the formation and subsequent reactions of these reactive species. Elongational ﬂow-induced degradation breaks polymer chains exactly at their center [42, 46]. This remarkable propensity is being explored in the author’s laboratory as a simple means of obtaining well-deﬁned block copolymers. Polymerization and

–C–C– 410 kJ·mol-1 Fig. 15.20.

–C–S–C–

–C–S–S–C–

–C–(S)n–C–

350 kJ·mol-1

300 kJ·mol-1

260 kJ·mol-1

Bond dissociation energies in vulcanized rubber.

817

818

15 Polymer Degradation and Stabilization

synthesis of block or graft copolymers by mechanical forces have been the subjects of several reviews and books [47]. Although mechanochemical synthesis of inorganic materials has enjoyed wide industrial application, no similar development has yet been witnessed with plastic materials.

15.8

Control and Prevention of Aging of Plastic Materials

It has been shown in the preceding sections that raw polymers are highly susceptible to degradative oxidation. The success of plastic materials, which ﬁnd applications in practically any aspect of life, relies heavily on the performance of polymer stabilizers, 70% of which are used for polyoleﬁns. According to their principal protection activity, common polymer stabilizers are conventionally classiﬁed as antioxidants, photoantioxidants, photostabilizers, metal deactivators, antiozonants, and heat stabilizers for PVC. 15.8.1

Antioxidants

Antioxidants prolong the useful lifetime of the polymer by trapping free radicals (‘‘primary’’ antioxidants) and by decomposing hydroperoxides (‘‘secondary’’ antioxidants) formed during the course of degradation. Radical Antioxidants Since polymer degradation is initiated by free radicals, the most eﬀective approach in both heat and light stabilization would be to reduce the number of reactive radical species by scavenging. Secondary aromatic amines and substituted phenol derivatives are among the most popular radical-scavenging stabilizers. The stabilization action of these compounds is based on the formation of aminoxy and phenoxy radicals, respectively, which can scavenge propagating radicals formed during degradation. Amine additives are converted into conjugated secondary products which absorb visible light. For this reason, hindered phenols are preferred in applications where discoloration is undesirable. Typical phenolic antioxidants are shielded with bulky alkyl substituents in the 2-, 4- and 6-positions to reduce the reactivity of phenoxy radicals toward hydrogen atom abstraction reactions on polymer and mutual coupling. The principal mode of action of hindered phenols is the transfer of a hydrogen atom to a propagating alkyl (R) or alkylperoxy (ROO) radical [Eq. (84)]. 15.8.1.1

ROO

+ HO

R'

ROOH

+

O

R'

ð84Þ

15.8 Control and Prevention of Aging of Plastic Materials

The phenoxy radicals are relatively long-lived and can undergo self-disproportionation, recombination with alkylperoxy radicals, or isomeric rearrangement followed by recombination. The resulting compounds may have some stabilization activity. Propionate-type hindered phenols constitute a special class in this category: during reaction, the phenol is transformed into phenolic cinnamates, which are known to be eﬃcient chain-breaking antioxidants [Eq. (85)].

2

OH

O

O

+

O

O

CH –CH2–C–OR

CH 2–CH 2–C–OR

O CH2–CH 2–C–OR

OH O CH=CH–C–OR

ð85Þ

Hindered Amine Stabilizers (HAS) With a few exceptions, such as hydroxybenzoate derivatives, most conventional phenolic antioxidants suﬀer from low photostabiliy and could not be used for photostabilization. This problem was solved with the introduction and development of hindered amine light stabilizers (HAS) [49]. Because this class of compounds was originally developed for photostabilization, they are frequently referred as ‘‘hindered amine light stabilizers’’ or HALS. HAS are commercialized under a variety of structures, most of which contain one or several 2,2,6,6-tetramethylpiperidine moieties linked together to form polynuclear species. The structure of a typical commercial HAS, Chimassorb2 944, is shown in Figure 15.21. The protection mechanisms of HAS are based on a complex series of chemical transformations, with many unresolved details. The primary step includes oxidation in situ of secondary or tertiary HAS into nitroxides aNO by the various oxidants present during oxidative degradation [Eq. (86)]. The substituent can be H, CH3 , OR, or COCH3 , and the oxidative species 3 O2 , 1 O2 , O3 , POO, POOH, PC(O)OO, or PC(O)OOH. 15.8.1.2

a

a

a

aNaR þ oxidative species !! aNO þ products

ð86Þ

Stable nitroxide radicals are traditionally considered as the principal species which account for the free-radical scavenging activity of HAS. Nitroxide radicals can trap alkyl, alkylperoxy and acylperoxy radicals. The probability of direct radical scaveng-

819

15 Polymer Degradation and Stabilization

H N

N N

N N CH3

N

CH3

CH2

N Fig. 15.21.

(CH2)6

CH3

n

Structure of a typical commercial hindered amine stabilizer (HAS).

ing, such as in Eq. (87), is nevertheless limited as a result of the higher rate for competitive reactions of R with dissolved oxygen and by self-recombination. a

a

aNO þ R ! aNOR

ð87Þ

As with the formation of peracids, it is suggested that scavenging occurs inhomogeneously and is enhanced in regions of accumulated hydrogen-bonded complexes, owing to the formation in situ of R [Eq. (88)]. a

a

a

820

f aNO HOORg ! f aNO; HO; ROg ! f aNO; R 0 g

ð88Þ

The formation of hydrogen-bonded associates with hydroperoxides or phenolic additives is favored by the resonance-stabilized dipolar structure of nitroxides represented by Eq. (89).

N

O

+

N

–

ð89Þ

O

The transformed hydroxylamines and dialkylhydroxylamine ethers are assumed to play a key role in the mechanism of stabilization of HAS, through a regenerative inhibition cycle known as the Denisov cycle [Eqs. (90a)–(90c)] [50].

NO–CH–CH 2– + ROO

NO + –CH=CH– + ROOH

ð90aÞ

NO +

C=O + ROH

ð90bÞ

NO + RO–O–C–CH2–

ð90cÞ

Tertiary hydroxylamine ethers, the main constituent in degraded polypropylene, lead preferentially to the formation of alkyl peroxides [Eq. (90c)].

15.8 Control and Prevention of Aging of Plastic Materials

There is increasing evidence that other regeneration mechanisms are involved, such as scavenging of alkylperoxy or peracyl radicals by hydroxylamines and alkylhydroxylamines according to Eq. (91). O.

NO–CHR1R2 + ROO. (RCOO.) →

N+

O →

OCHR1R2 .... RO.

NO. + O=CR1R2 + ROH (RCOH) O

ð91Þ Another proposed mechanism of action involves long-range energy transfer from polymer–oxygen exciplexes to HAS or HAS–oxygen complexes [49]. This process is particularly eﬃcient in inhibiting photoinitiation which takes place essentially at the crystalline/amorphous interface, where the HAS is concentrated. Basicity of HAS Secondary and tertiary HAS are relatively basic and may catalyze hydrolysis of some polycondensation polymers, such as polycarbonate. HAS lose their activity if they come into contact with strong acids, and hence cannot be used to stabilize PVC or in formulations which contain halogenated ﬂame retardants. HAS with an acylated amino group ( aNCOCH3 ) has a lower pK a and can better resist an acidic environment.

a

Thermal stabilization with HAS As a result of its protection mechanism, HAS are particularly eﬀective in photodegradation. Recently, it has been assessed that HAS can also be used for thermal degradation. In this case, however, HAS should be used in combination with hindered phenols because many degradation products of HAS, which are eﬃcient antioxidants, can be formed only with light. Peroxide Decomposers Peroxides are responsible for the autoacceleration stage during oxidative degradation of polymers. Hydroperoxide decomposers, which cause the reduction of hydroperoxides to alcohols, comprise two main chemical classes: organic sulfur and trivalent phosphorus compounds, generally used in combination with phenolic antioxidants. The decomposition mechanism is complex and involves radical and nonradical processes. During the course of decomposition, a mixture of oxidation products, generally endowed with additional peroxidolytic properties, are formed. For instance, the decomposition of a thioether in the presence of alkyl hydroperoxides can be represented schematically by a series of reactions [Eq. (92)] which involve thiyl RS, sulﬁnyl RS(O), sulfonyl RS(O)2 , and perthiyl RSS radicals. 15.8.1.3

821

15 Polymer Degradation and Stabilization

822

½RaOaCaðCH2 Þ2 a2 S ! ½RaOaCaðCH2 Þ2 a2 SO ! RaOaCaðCH2 Þ2 aSOOH þ H2 CbCaCaOaR ! ! SO2 ; H2 SO4 ; . . .

ð92Þ

Sulfur dioxide and sulfuric acid, both of which are also active hydroperoxide decomposers, are formed at the ultimate stage of transformation. Organic phosphites are very eﬀective hydroperoxide decomposers. Their peroxidolytic activity results from a sacriﬁcial transformation of phosphite (trivalent) to phosphate (pentavalent), according to the general reaction scheme in Eq. (93). RaOOH þ PðOR 0 Þ3 ! ROH þ ObPðOR 0 Þ3

ð93Þ

15.8.2

Photostabilizers

Outdoor weathering is the most common source of polymer degradation. The most direct and eﬃcient way to avoid photodegradation would be to prevent the photons from reaching the polymer either by applying a paint or coating to the surface, or by blending the polymer with strong light-absorbing particles such as carbon black. This procedure, however, is inapplicable for situations where aesthetic appearance or transparency of the material is essential. UV absorbers The use of organic compounds with a high absorption coeﬃcient in the UV but transparent to the visible, known as UV absorbers, should act in a similar fashion to paints, by protecting the polymer while conserving the optical clarity of the material. Paints and coatings themselves are not stable to solar radiation and are often compounded with UV absorbers to prolong their protection eﬃciency. To be useful, UV absorbers must strongly absorb at wavelengths harmful to the polymer, must harmlessly dissipate the energy that they absorb, and must persist in the matrix for the expected lifetime of the article. Most commercial UV absorbers are derivatives of benzophenone, benzotriazole, triazine, oxanilide, or cyanoacrylate with aromatic substituents. All have a highly delocalized p-electron structure with a high extinction coeﬃcient (e ¼ 10 000–40 000 L mol1 cm1 ) in the near-UV region (290–350 nm). The photostability of benzophenone-based UV absorbers is conferred by intramolecular proton transfer in the excited state, and most likely by a charge-separation mechanism in the case of a cyanoacrylate structure.

O

H

O

hv O

R

heat

O

H

O

Φ O

R

Φ

CN COOR

Energy dissipation mechanism for a protic (left) and a cyanoacrylate (right) UV absorber.

Scheme 15.5.

hv

Φ

heat

Φ

+

CN COOR

-

15.8 Control and Prevention of Aging of Plastic Materials

Quenchers Quenchers are used to capture the excess energy of excited chromophores [Eqs. (94) and (95)], which is then dissipated as vibrational energy, before they can initiate harmful reactions.

P ! P ! radicals

ð94Þ

P þ Q ! P þ Q

ð95Þ

Eﬀective quenchers are based on nickel complexes (phenolate, dithiocarbamate, dithiophosphate) and are used almost exclusively in polyoleﬁns at a concentration of 0.2–0.5%. Nickel complexes impart a green coloration to plastic articles and their importance was signiﬁcantly reduced with the introduction of hindered amine light stabilizers (HALS). 15.8.3

PVC Heat Stabilizers

In spite of its inherent thermal instability (see Section 15.3.4), PVC possesses several attractive properties, such as economy of production and processing, and the ease of variation of its properties, with appropriate blending with other polymers or additives, from hard and tough materials to elastomeric ones. Currently, PVC ranks second only to polyoleﬁns in terms of worldwide production among industrial polymers. This technical success is the result of considerable research in the ﬁeld of degradation and stabilization of this polymer since its introduction as a commodity plastic in the middle of the 20th century. The low thermal stability of PVC originates from the presence of labile structures and of the autocatalytic deleterious eﬀect of the hydrochloric acid evolved. Thermal stabilizers for PVC consist principally of metal carboxylates and organotin compounds (primary stabilizers), used in combination for preventive and curative functions. As with degradation, uncertainties continue to exist in the exact stabilizing mechanisms of these additives. There is evidence that organotin derivatives stabilize PVC by substituting the labile allylic chorine with a more thermally stable thioether group [Eq. (96)]. ðC4 H9 Þ2 SnðSCH2 COOaC8 H17 Þ2 þ aCH2 aCHbCHaCHClaCH2 a ! aCH2 aCHbCHaCHðSCH2 COOaC8 H17 ÞaCH2 a þ ClaSnðC4 H9 Þ2 ðSCH2 COOaC8 H17 Þ

ð96Þ

Hydrogen chloride is bound with metallic (Zn, Ca, Ba) organic acid salts with formation of the metal chloride and the corresponding free fatty acids. The formation of polyene sequences can be prevented by combination reactions with thiol or maleate derivatives. Although eﬃcient in blocking the degradation, most of the inor-

823

824

15 Polymer Degradation and Stabilization

ganic stabilizers leave toxic residues and current research is focused on the development of new, less polluting, organic stabilizers. 15.8.4

Other Classes of Stabilizers Metal Deactivators Very often, thermoplastics contain minute amounts of metallic compounds which originate from polymerization catalysts, contaminated ﬁllers, polymerization or processing equipment, or metal contact (wire and cable insulators) during the use of the polymer. The interactions between the polymer and metallic substances are complex, but generally result in the accelerated aging of the material. Most metal deactivators are bifunctional stabilizers with phenolic and nitrogen, or phenolic sulﬁde and phosphite, moieties in their structure, and act by a chelating action which reduces the harmful eﬀects of the metal ions. 15.8.4.1

Antiozonants Diene-type rubbers are particularly sensitive to ozone attack and should be protected by antiozonants. Common antiozonants are aromatic diamines capable of direct ozone scavenging by adduct formation. 15.8.4.2

15.9

Lifetime Prediction

Most of the additives used to protect the polymer from degradation are eventually consumed during the stabilization processes. In addition to chemical loss, the stabilizers can also be depleted by precipitation, migration to the surface, and extraction. For all these reasons, the concentration of additives diminishes with time up to a point where no protection eﬀect could be observed. In general, the stabilization lifetime increases with the amount of added stabilizer, up to an optimum determined by the ﬁnite solubility of the additives in bulk polymers, or by secondary reactions which may have adverse eﬀects on the stabilization mechanisms (Figure 15.22). Excluding disposable devices, most plastic materials are expected to perform reliability for many decades. In general, outdoor weathering of stabilized samples is too slow to be useful for quality control or for formulation development. As a remedy for this ﬂaw, a number of accelerated weathering tests have been devised. Depending on the type of degradation, accelerated tests can be obtained by increasing the temperature (thermal degradation), or increasing the light intensity (photodegradation) and/or the partial oxygen pressure (oxidative degradation). Owing to its economic relevance, photoaccelerated aging is enjoying the most dramatic development, with numerous commercial systems based on a UV source or mirrors designed to mimic the action of the Sun, with a several-fold increase in irradiation intensity. Most of the time, results of accelerated aging show poor correlation not

15.9 Lifetime Prediction

0%

0.01%

0.1%

0.2

carbonyl index

0.15 0.5%

0.1

0.05

0 0

100

200

300

400

500

600

irradiation time [h] Photo-oxidation of commercial polypropylene, in the absence and in the presence of the indicated concentrations of hindered amine stabilizer. Fig. 15.22.

only with outdoor weathering, but also between diﬀerent devices, or for identical devices between diﬀerent laboratories. This lack of correlation is not entirely unexpected, in view of the wide variability inherent in weathering conditions: biological disturbances, pollution, climate (temperature, humidity, wet periods); and changes in the spectrum and intensity of the source with aging, and in the test duration. In addition to the external factors, more fundamental reasons for deviations could be traced back to the bimolecular reactions between intermediate species, whose concentrations in real conditions are too low to interact, or to diﬀusional processes of the reactants (oxygen, water, stabilizers) that become limiting factors in accelerated conditions. By using conditions which mimic exact weathering conditions in terms of spectral distribution, temperature, and oxygen uptake, but with a diﬀerent photon intensity, it has been assessed that accelerated aging in polyoleﬁns is not mechanistically diﬀerent from natural weathering. Therefore, an extrapolation procedure between these two modes of degradation is possible, providing all the dynamics (diﬀusion) factors could be kept under control. Identifying and controlling all the dynamics factors in a real system is nevertheless a challenging task. To account for the poor reproducibility in accelerated aging, statistical approaches have been proposed, such as the nonparametric statistics which relies on performance comparison with a control material. The control material should be as similar as possible to the tested polymer in terms of structures, size, and shape, and its weathering durability should be well established in the speciﬁed exposure test. One useful approach to reduce error analysis was to compare a test property, not as a function of time or absorbed radiation, but relative to another

825

826

15 Polymer Degradation and Stabilization

process going on in the aging material. The objective sought is to have the rates of the two processes occur in the same relationship relative to one another under a wide range of experimental conditions, from natural to accelerated test conditions. The process of interest should be easy to determine and sensitive to low levels of degradation, to minimize the extrapolation range. In thermal aging of nitrile rubber, for instance, it has been found that the ultimate tensile elongation shows good correlation with the rate of oxygen consumption over the experimentally accessible temperature range for mechanical measurements, which was 96–71.5 C. At lower temperatures, the time to failure was inconveniently long for measurement. Oxygen consumption, on the other hand, could be determined by gas chromatography all the way down to room temperature. Based on indirect evidence that the chemistry underlying oxygen consumption is responsible for changes in modulus and elongation values, a lifetime of 100 years was predicted for the elastomer in ambient conditions. It should be noted that an Arrhenius plot extrapolation, based solely on the mechanical properties at elevated temperatures, predicted a lifetime of 50 000 years, which is some 500 times longer than the correctly estimated value [15].

15.10

Conclusions

This chapter has given a general description of some recent advances in the ﬁeld of polymer degradation and stabilization. Despite a considerable volume of work and equally impressive progress in the comprehension of the molecular mechanisms which underlie polymer degradation, many challenging problems are still open to further investigation. The ever-increasing complexity of commercial plastics, which includes the use of copolymers, polymer blends, and diﬀerent combinations of additives in their formulations, adds another degree of diﬃculty to the already complex problem of polymer degradation. Each type of interaction between the diﬀerent components in the formulation, both polymeric and nonpolymeric, may inﬂuence diﬀerently the degradation process. In particular, the antagonistic and synergistic eﬀects between stabilizers in additive blends need better mechanistic comprehension. Two main areas where major advances can be expected in the future include integral characterization of the degraded polymer, and realistic modeling of degradation kinetics. Most degradation and stabilization mechanisms reported in the literature are inferred from the observation of a few end products (generally carbonyl and hydroxyl compounds, which can be easily monitored by FTIR). A comprehensive characterization of the starting material, and of the degradation products at diﬀerent stages of aging, is still lacking in most investigations. Such a complete chemical identiﬁcation, which appears to be feasible with the modern analytical techniques available, is a prerequisite to establish a correlation between eﬀects of speciﬁc structures on the behavior of signiﬁcant degradation intermediates.

Notation

The inhomogeneity of reaction sites, inherent in solid-state reactions, may lead to distributions of reactivity and to time-dependent rate constants. These eﬀects should be recognized and taken into account when performing kinetics analysis. This is consequential to the problem of stabilization, since additives need to diﬀuse toward degradation centers for the stabilization process to be operative. Some heterogeneous reaction models aimed toward a better description of radical chain processes in solid polymers have appeared in the literature. One of these models, the infectious spreading reaction, has been examined in Section 15.4.5. Among other advances in solid polymer degradation, one can cite:

the concept of ‘‘polychronal kinetics’’ which was introduced to explain the stepwise conversion of reactive species under isothermal conditions [6]; and the ‘‘spongy micelle’’ microreactor model in which the amorphous phase in a semicrystalline polymer is depicted as formed by ‘‘granule-core’’ zones, surrounded by ‘‘oriented domains’’ close to the crystallites [51]. By applying diﬀerent chemical kinetics scheme to each zone, with allowance for ‘‘interzone’’ radical migration, the autooxidation curve of polyoleﬁns can be faithfully reproduced. It is found that the induction period for oxygen consumption corresponds to the homogeneous oxidation stage, whereas the (exponential) autoaccelerated stage may be interpreted as a result of heterogeneous oxidation spreading of low MW peroxides. This model refutes the interpretation of chain branching as a source of oxidation autoacceleration. All these theories need to be conﬁrmed by experimental veriﬁcation and tested for their general applicability to diﬀerent polymer systems.

Notation

As a D De E Ea Es E ang DE 0 DH h I0 IðxÞ K k

infrared absorption intensity [cmmmol1 ] constant of the Morse potential [m1 ] oxygen diﬀusion constant [cm2 s1 ] bond dissociation energy [kJmol1 ] tensile modulus [GNm2 ] energy of activation [kJmol1 ] excess energy of stress [kJmol1 ] valence-bond angle contribution to the activation energy [kJmol1 ] molecular energy diﬀerence at absolute zero between the activated complex and the reactant [J] enthalpy of reaction [kJmol1 ] Planck’s constant 6.62561034 [Js] incident light intensity [Wm2 ] light intensity [Wm2 ] Kubelk–Munk absorption coeﬃcient [cm1 ] rate of reaction [s1 ] (ﬁrst-order), [Lmol1 s1 ] (second order)

827

828

15 Polymer Degradation and Stabilization

kf kB kc ky l l0 Mi Mn Mv Mw N n n ni pd pi pr Q qz qA R R R r S SðlÞ s T T1/2 Tg Tm ti t1/2 U0 VðlÞ x

bond force constant in the neighborhood of the equilibrium separation of two chemical moieties [Nm1 ] Boltzmann’s constant 1.3801023 [JK1 ] rate constant for bond scission [s1 ] elastic constant for bond-angle deformation [Nrad1 ] length of a covalent bond [nm] equilibrium separation distance of the atoms in a bond [nm] molecular weight of ith polymer [gmol1 ] number-average molecular weight [gmol1 ] viscosity-average molecular weight [gmol1 ] weight-average molecular weight [gmol1 ] total number of amorphous domains number of polymer chains per gram of polymer [g1 ] number of oxidized domains polymer molar fraction ‘‘dead’’ or oxidized fraction of polymer ‘‘infectious’’ oxidizing fraction of polymer remaining unoxidized fraction of polymer normal coordinate of vibration [nm] molecular partition function of the transition state molecular partition function of the reactant reﬂectance molar gas constant 8.314 [Jmol1 K1 ] absorbed dose of radiation [Gy or Jkg1 ] rate coeﬃcient for spreading [s1 ] Kubelka–Munk scattering coeﬃcient [m1 ] relative spectral damage scission indice temperature [ C] temperature of half-decomposition [ C] glass transition temperature [ C] melting point [ C] induction period [ C] half-life [s] thermal energy for bond rupture [kJmol1 ] Morse potential [J] penetration distance [m]

Greek a b l l max e

absorption coeﬃcient [cm1 ] activation volume for a reaction [m3 mol1 ] irradiation wavelength [nm] wavelength of light absorption peak [nm] extinction coeﬃcient [Lmol1 cm1 ]

Notation

m n y c

dipole moment [Cm] frequency of band center [cm1 ] angle [rad] molecular stress [GNm2 ]

Acronyms ATR-FTIR ESR FDA FTIR iPP GPC HAS HDPE HER LDPE LET HAS HOMO LUMO MIM MW MWD NMR PA PAN PC PEEK PET, PETP PIB PMMA PMMA–SAN POM PP PPO PPP PS PSU PTFE PVC SEM TG VC

attenuated total reﬂection FTIR electron spin resonance Federal Drug Administration Fourier Transform infrared spectroscopy isotactic polypropylene gel permeation chromatogaphy hindered amine stabilizer high-density polyethylene high-energy radiation low-density polyethylene linear energy transfer hindered amine light stabilizer highest occupied molecular orbital lowest unoccupied molecular orbital metal injection molding molecular weight molecular weight distribution nuclear magnetic resonance spectroscopy polyacrylamide polyacrylonitrile polycarbonate poly(ether ether ketone) poly(ethylene terephthalate) polyisobutylene poly(methyl methacrylate) PMMA–poly(styrene-co-acrylonitrile) polyoxymethylene polypropylene poly(phenylene oxide) poly(para-phenylene) polystyrene poly(ethersulfone) polytetraﬂuoroethylene poly(vinyl chloride) scanning electron microscopy thermal gravimetry vinyl chloride

829

830

15 Polymer Degradation and Stabilization

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3 4

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7

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33

34

Waroquier, E. Schacht, J. Am. Chem. Soc., 2001, 123, 10650. M. R. Nyden, J. Gilman, Comp. Theor. Polymer. Sci., 1997, 7, 191. J. J. P. Stewart, J. Comp. Chem., 1989, 10, 221. D. A. Gallagher, Scientiﬁc Computing & Automation, June 1996. T. A. Bogaevskaya, T. V. Monakhova, Yu. A. Shlyapnikov, Vysokomol. Soyed., 1972, 14A, 1552. P. Gijsman, J. Hennekens. Polym. Degr. Stab., 1993, 42, 95. S. Commereuc, D. Vaillant, J. L. Philippart, L. Lacoste, J. Lemaire, D. J. Carlsson, Polym. Degr. Stab., 1997, 57, 175. S. Verdu, J. Verdu, Macromolecules, 1997, 30, 2262. P. Eriksson, T. Reitberger, B. Stenberg, Polym. Degr. Stab., 2002, 78, 183. M. Celina, G. A. George, N. C. Billingham, in Handbook of Polymer Degradation, S. Halim Hamid, ed., Marcel Dekker, New York, 2000, p. 159. F. Gugumus, Polym. Degr. Stab., 1996, 52, 159; Polym. Degr. Stab., 1996, 53, 161. T. P. Wampler, J. Anal. Appl. Pyrol., 2004, 71, 1. M. Saule, S. Navarre, O. Babot, W. Maslow, L. Vertommen, B. Maillard, Macromolecules, 2003, 36, 7469. J. P. Riggs, in High Performance Polymers and Composites, J. A. Kroschwitz, ed., John Wiley, New York, 1991, pp. 20–65. B. A. Mathew, R. Mastromatteo, Metal Powder Report, 2002, 57, 20. W. C. McCaffrey, D. G. Cooper, M. R. Kamal, Polym. Degr. Stab., 1998, 62, 513. R. L. Feller, Accelerated Aging: Photochemical and Thermal Aspects, Research in Conservation Series, The Getty Conservation Institute, 1994. H. W. de Bruijn, in Handbook of Polymer Degradation, S. Halim

References

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36 37

38 39 40 41

42 43

Hamid, ed., Marcel Dekker, New York, 2000, p. 611. A. L. Andrady, N. D. Searle, L. F. E. Crewdson, Polym. Degr. Stab., 1992, 35, 235. H. Zweifel, Chimia, 1993, 47, 390. T. Q. Nguyen, Ph.D. thesis, No. 243, EPFL, Lausanne, 1976; T. Q. Nguyen, T. Ga¨umann, Radiat. Phys. Chem., 1977, 10, 263. A. Charlesby, Atomic Radiation and Polymers, Pergamon, Oxford, 1960. T. Q. Nguyen, H. H. Kausch, J. Appl. Polym. Sci., 1984, 29, 455. V. S. Ivanov, Radiation Chemistry of Polymers, VSP, Utrecht, 1992, p. 202. I. Kaetsu, in Radiation Processing of Polymers, A. Singh, J. Silverman, eds., Hanser, Munich, 1992, pp. 149– 186. T. Q. Nguyen, H. H. Kausch, Adv. Polym. Sci., 1992, 100, 73–182. S. N. Zhurkov, V. E. Korsukov, J. Polym. Sci.: Polym. Phys., 1974, 22, 881.

44 V. A. Gonza´lez, G. Neira-Vela´zquez,

45 46

47

48 49 50 51

J. L. Angulo-Sa´nchez, Polym. Degrad. Stab., 1988, 60, 33; H. Zweifel, Polymer Durability, ACS Adv. Chem. Ser., 1996, 249, 375. M. S. S. Coker, G. Scott, H. A. A. Sweis, Polym. Degr. Stab., 1982, 4, 333. T. Q. Nguyen, R. Porouchani, H. H. Kausch, in Flexible Chain Dynamics in Elongational Flow, T. Q. Nguyen, H. H. Kausch, eds., Springer Verlag, Berlin, 1999, pp. 185–258. A. Casale, R. S. Porter, Polymer Stress Reactions, Academic Press, New York, 1978. J. Pospisil, S. Nespurek, Polym. Degr. Stab., 1995, 49, 99. F. Gugumus, Polym. Degr. Stab., 1993, 40, 167. E. T. Denisov, Dev. Polym. Stab., 1980, 3, 1. G. E. Zaikov, Handbook of Polymer Degradation, S. Halim Hamid, ed., Marcel Dekker, New York, 2000, p. 437.

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Thermosets1 Rolf A. T. M. van Benthem, Lars J. Evers, Jo Mattheij, Ad Hoﬂand, Leendert J. Molhoek, Ad J. de Koning, Johan F. G. A. Jansen, and Martin van Duin 16.1

Introduction 16.1.1

Thermoset Materials

The world of thermoset materials is characterized by a high degree of both diversity and complexity, often diﬃcult for non-experts to oversee [1]. Let us ﬁrst ask ourselves what thermoset materials have in common. Polymeric materials are divided into two classes, thermoplastic and thermosetting materials. The term ‘‘thermoplastic’’ refers to the ability of these polymers to become plastic, that is, to melt and ﬂow upon heating. In terms of viscoelasticity, a thermoplastic material can be hard and elastic at room temperature, but it behaves like a liquid when heated to a certain higher temperature. The expression ‘‘thermoset’’ or ‘‘thermosetting’’ is sometimes erroneously understood to be exclusively a material which ‘‘sets’’ (becomes hard and elastic) upon heating, but there are many examples of thermoset materials which have ‘‘set’’ without a temperature trigger, for example through irradiation (see Section 16.8.5). For the discussions in this chapter, the term ‘‘thermoset’’ will merely refer to the inability of this class of polymeric materials to behave like a liquid when heated. A thermosetting material can be hard and elastic at room temperature and be heated above its glass transition temperature to become ‘‘softer’’, and deformable. The deformation, however, will be transient rather than permanent: the softened material is still fully elastic and behaves like a rubbery material.

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

834

16 Thermosets

16.1.2

Networks

The molecular origin of thermoset behavior lies in the formation of a polymeric network. Unlike thermoplastic materials, which consist of individual macromolecules that can move relative to each other over inﬁnite distances in the melt, the polymeric chains in a thermosetting material are attached to each other and can therefore not move any more relative to each other over greater distances. They are bound together through so-called crosslinks and form an inﬁnite, threedimensional structure. A consequence of the presence of junctions between the polymeric chains, other than the inability to behave like a liquid and ﬂow, is the inability of this material to dissolve in another medium such as a solvent. Solvent or plasticizer molecules can migrate into the network structure, to swell it, but this is limited to a maximum determined by the crosslink density. The ideal network structure can be envisaged as a three-dimensional array of crosslink points, each crosslink point being connected to at least three other crosslink points via linear polymer segments, which are called elastically active network chains. In practice non-ideal network elements are also present, such as loops or dangling ends (Figure 16.1). Network density, or crosslink density, is expressed as the concentration of either the crosslink joints or the elastically active network chains (those chains that are part of the inﬁnite structure and attached to crosslink junctions at both ends) per unity of volume of the unswollen material.

EAN Cross-link

Dangling end

Fig. 16.1. Network nomenclature: crosslinks, elastically active network chains (EAN), dangling ends, and loops.

Loop

16.1 Introduction

16.1.3

Advantages

There are two obvious advantages in the material property of a thermoset compared to a thermoplastic material: the network structure will ensure a higher resistance toward unwanted penetration of chemicals (solvents, dyes, water, acids, bases, and electrolytes) in use, and secondly, a thermoset has a high-temperature stability as it retains its shape and (some of ) the properties at elevated temperatures. These are not the main advantages of a thermoset material, however. The wide processing window of thermosets has given them the present position in the performance materials market. A thermoset can be processed easily (shaped, cast, molded, poured) as a liquid; the liquid is transformed into the ﬁnal thermoset material after the processing, through a series of chemical reactions. The ﬁnal properties of the material are thus determined by the network formation reactions performed in situ. As a consequence, the chosen molecular weight of the starting material can be arbitrarily low. The viscosity, in relation to the molecular weight of the (short) polymeric chains, is decreased strongly as well. In contrast, a thermoplastic material derives its properties from the high molecular weight of its polymer chains which allows the chains to entangle with each other. This high molecular weight in most cases results in rather high melt viscosities for these polymers. A high processing viscosity is a disadvantage with respect to both processing speed (ﬁlling the mold, ﬂowing out) and wetting phenomena: dispersion of pigments and ﬁllers and even ﬂow-out over the walls of the mold or the substrate. 16.1.4

Curing Resins

In thermoset nomenclature the low molecular weight polymers (oligomers) are referred to as resins. Apart from their relatively low molecular weight, typically between a few hundred and a few thousand grams per mole, resins are characterized by the presence of reactive groups in the chains, ideally at least two per molecule. These reactive groups form the chemical handles to connect the polymer chains together through covalent crosslink bonds, via a chemical reaction. The process of crosslinking is mostly referred to as ‘‘cure’’ or ‘‘hardening’’. When the reaction proceeds, the gel point is deﬁned as the moment in time when the ﬁrst molecule of inﬁnite dimensions is formed in the polymer mixture. The viscosity of the mixture then approaches inﬁnity and in this state the material is referred to as gel. The curing reaction can still proceed beyond this point to increase the crosslink density. 16.1.5

Functionality

The number of reactive groups present in one molecule is referred to as functionality. This term can reﬂect both on a monomer and on a polymer. For example, ethylene glycol is a two-functional monomer for a polyester (see Section 16.6),

835

836

16 Thermosets

and a linear epoxy resin with two glycidyl groups on each end is a two-functional polymer (Section 16.4). An unsaturated polyester (Section 16.7) or EPDM rubber (Section 16.9) can have much higher functionalities, that is, unsaturations along the polymer chain. Functionality is an important concept to understand the role of a polymer chain in the ﬁnal network. For example, a two-functional polymer of a certain molecular weight can only be built in as an elastically active network chain of that same length (at maximum, provided that it reacts at both groups). A three-functional polymer chain of similar molecular weight, containing a branch in the chain, can give a higher crosslink density because it has an internal network junction as well. The junctions in the network can originate from a branching point already present in the polymer, or from chemical reaction between polymer strands. 16.1.6

Formulation

The mixture of the diﬀerent constituents of a formulation for a thermosetting material is always rather complex and referred to as a ‘‘system’’. Systems can be classiﬁed as either 1K (one component) or 2K (two components). In the ﬁrst case, all the (co)reactive ingredients of the formulation are already mixed, but the network formation is only started when it is triggered chemically or physically, for example by a raise in temperature, by electromagnetic radiation, or by exposure to air (oxygen) or moisture. In the latter case the two components will immediately start their chemical reaction once they are mixed with each other, with or without an additional inﬂuence from outside (such as a rise in temperature to speed up the crosslinking reaction). Independently of the system’s reactivity (1K/2K), the combination of reactive groups required to form the network can range from very simple to highly complex. Reactions can be between co-reactive groups of diﬀerent chains of one resin type, or between the reactive groups on one resin type with corresponding reactive groups of another compound, that compound either being a chemical (monomer) or another resin (polymer) that is added to the ﬁrst resin. In some resins only one kind of reactive group A is present which is reactive with itself (A þ A ! AA); in most resins, however, A is not reactive with itself and requires the presence of an additional compound (B) of complementary reactivity to give a chemical bonding reaction (A þ B ! AB). Dual-cure reactions also exist (A þ B þ C þ D ! AB þ CD) between mixtures of resins of diﬀerent reactivities. In the next sections the most important resin classes will be presented. Systems can be formulated from just one one of them but also from a wide, almost inﬁnite, variety of combinations of these resins. The systems are made more complex with the addition of additives to control the chemical reaction, either to enhance it (catalysts, accelerators, initiators) or to slow it down or inhibit it (stabilizers, inhibitors). Besides additives that control the network formation, other additives can also be used to inﬂuence the ﬁnal properties: ﬁllers, pigments, plasticizers). Formulation of thermosetting systems is a technol-

16.1 Introduction

ogy of its own and requires a deep insight into both the complex chemical reactions and the ﬁnal network structure in order to steer toward the desired material properties. 16.1.7

Production

It is highly important to note, however, that the production of the resins themselves is in most cases far from as straightforward as would be expected for a simple low molecular weight polymer. Unlike the (linear) thermoplastic polymers, resins are designed to react to form networks; they contain several reactive groups and often are branched polymers. Either by uncontrolled reaction or uncontrolled branching, polymer molecules with a much higher molecular weight than intended, or even polymer networks, could already be formed during the production stage (‘‘gelation’’). This should be avoided at all times, as the production facility could be plugged and highly damaged in such a case. In each of the following section on the resin types a short description is given of the (mostly batchwise) production technology associated with them, and the measures for controlling unwanted, premature, reactions. 16.1.8

General Areas of Application

The areas of application of thermosetting materials are in coatings, rubbers, inks, adhesives, laminates, press moldings, composite (glass-ﬁlled) moldings, mineral wool, and electronic materials. A general overview of the main application areas of the resins discussed in the following sections is given in Table 16.1 [1]. They are found in a wide variety of markets (household, automotive, building and construction, electronics, paper and press) and have a collected industrial volume of several tens of millions of tons.

Tab. 16.1.

General overview of thermoset applications.

Resin

Coating

Phenolic Amino Epoxy Alkyd Saturated polyester Unsaturated polyester Acrylate Rubber

þ þ þ þþ þþ þ þþ

Structural parts

þ þ

Wood and paper composites

Adhesives

Inks

Elastic rubbers

þþ þþ

þ

þ

þ

þ þ

þþ

þ þ

þ

þ þþ

837

838

16 Thermosets

16.2

Phenolic Resins 16.2.1

Introduction

Historically, phenol–formaldehyde resins (phenolic resins, phenoplasts) are the oldest synthetic thermoset materials: as early as in 1910 the ﬁrst ones (‘‘bakelite’’ to their inventor Baekeland) were introduced. Who does not remember the black telephones and light switches that were popular in the 1930s and 1940s? Phenolic resins are based on hydroxy-aromatic compounds (phenol and phenol derivatives) and aldehydes [2]. Furfural is occasionally used as aldehyde; but by far the most widely used aldehyde is formaldehyde. Formaldehyde can be administered as an aqueous solution (formalin), polymeric solid (paraformaldehyde) or solid adduct with ammonia (hexamethylenetetramine, HMTA). The resins are formed by a step-growth mechanism in aqueous solution. Thermoset materials can be made from phenolic resins themselves, requiring an acid or base catalyst or additional formaldehyde source, or in combination with various other resins. An exhaustive list is not given here, but examples of combinations with all the resin types in the next sections are known (especially amino resins, Section 16.3; epoxy resins, Section 16.4; alkyd resins, Section 16.5; and saturated polyester resins, Section 16.6).

16.2.2

Chemistry

The reactive group involved on the hydroxy-aromatic compound is not the hydroxy group itself but the carbon atoms on the ortho (2,6) and para (4) positions relative to the hydroxy group (attached to carbon atom 1). Unsubstituted phenol can therefore react with formaldehyde from three diﬀerent positions independently, so should be regarded as a three-functional monomer. If the phenol derivatives have substituents on the ortho and/or para positions, their functionality is decreased accordingly. For example (see Scheme 16.1), cresol (methylphenol) is two-functional when the methyl group is in the 2 or 4 position relative to OH (o-cresol and pcresol, respectively) but three-functional when in the 3 position (m-cresol). Further, commonly used two-functional substituted phenol derivatives are p-tert-butylphenol and p-tert-nonylphenol. The aliphatic groups on these monomers decrease the solubility of the resins in water but increase the solubility (compatibility) of these resins with more hydrophobic solvents (hydrocarbons) and resins. The aldehyde group of formaldehyde can react twice with an aromatic carbon atom of a phenolic monomer and should therefore be regarded as a two-functional monomer. After the ﬁrst reaction, the carbonyl group has been changed into a hydroxy group. After the second reaction, the hydroxy group is either etheriﬁed or substituted for a carbon–carbon bond. The reaction pathways are displayed

16.2 Phenolic Resins

OH

OH

OH

OH

839

OH

2

6

HO

4

Phenol

m-Cresol

o-Cresol

p-Cresol

Resorcinol

OH

OH Nonylphenol p-tert-Butylphenol

Scheme 16.1.

Some phenolic monomers.

in Scheme 16.2 Starting from three-functional phenol and formaldehyde, a branched structure is obtained in which the phenol moieties are linked together by formaldehyde-derived bridges. These bridges can comprise either one methylene group (‘‘methylene bridge’’) or two methylene groups connected by an ether oxygen atom (‘‘ether bridge’’). One formaldehyde molecule is needed to form a methylene bridge; two are needed to form an ether bridge. The relative amounts

OH

OH

OH

OH

HO

OH OH

OH

OH

HO

OH

HO

ether bridges

methylene bridges OH

OH

OH

OH

OH O

Scheme 16.2.

OH

OH

OH

Resol resin

Phenolic resin formation.

OH O

Novolac resin

840

16 Thermosets

of these bridges are determined during the resin synthesis by the reaction parameters. Two reaction parameters are of speciﬁc importance: pH and molar ratio. When phenol and formaldehyde are dissolved together in water, a pH of about 3–4 is reached (phenol is slightly acidic). After prolonged heating, hardly any reaction is noticeable. The reaction requires a catalyst, either an acid or a base, to reach a sufﬁcient speed. Depending on the ratio between phenolic monomers and aldehyde, the phenolic resins are classiﬁed as either resol or novolac types. Resols In a resol synthesis reaction, formaldehyde is used in excess over the phenolic monomer. In the case of phenol (P), typical P/F (formaldehyde) ratios are between 1:1 and 1:3. Therefore the ready resin will still contain reactive groups derived from single-sided reacted formaldehyde: hydromethylene groups (‘‘methylol groups’’). Care must be taken not to let the reaction run too far, because the mixture can form a gel. Acid catalysis, for example, is therefore not applicable, because the reaction will be out of control (proceed spontaneously toward the gel point). With weak-base catalysis, however, it is possible to reach a more or less stable situation (at fairly low temperature, below 60 C) in which most formaldehyde molecules have reacted with phenols to form methylols, but hardly to form bridges. Catalysts often used are alkali and alkaline earth oxides and hydroxides (such as potassium hydroxide, calcium oxide). ortho-Methylols are slightly preferred over paramethylols. The higher the pH, the higher the para content and the more ether bridges are transformed into methylene bridges. If a higher molecular weight is desired, the reaction can be allowed to proceed further in a slow and controlled manner at 60 C, until the right viscosity is reached. A higher molecular weight can be obtained more easily without risk of gelation by using two-functional phenol monomers, partly (together with phenol) or exclusively. Resols can be cured to thermoset materials themselves by adding an acidic catalyst and/or increasing the temperature. 16.2.2.1

Novolacs Novolac resins are obtained by reacting formaldehyde with an excess of the phenolic monomer (P/F > 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 de´cor 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

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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

Properties and Applications

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 de´cor 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

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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

<10 (liquid) 65–75 95–105 125–135 145–155

[a] Molar ratio of building blocks in product. Applicable to Taﬀy process for the 1001 type. [b] Molar ratio of monomers in Advancement process.

O

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16 Thermosets

OH

HNu

O

R

Nu

R

O O

R

O

OH

O

R

O

O OH

Scheme 16.12. Reactions of glycidyl groups with compounds containing active hydrogen groups, such as carboxylic acids.

the more hydroxyl groups are present in relation to the glycidyl groups. Epoxy resins can be cured themselves by addition of a (Lewis) acidic or basic catalyst. Tertiary amines and imidazolines are especially eﬀective catalysts. They serve as a nucleophilic ring-opening initiator for a cascade of epoxy–epoxy reactions, resulting in ether linkages between the polymer chains. The glycidyl groups can also undergo ring-opening reactions with a wide variety of active hydrogen-containing compounds, in close analogy to the synthesis reaction where the aromatic hydroxyl groups acted as active hydrogen donors (see Scheme 16.12). Depending on the reactivity of the active hydrogen donor, a catalyst may be needed to eﬀect a suﬃciently fast cure reaction.

O O

H R1

N

H R2

R1

OH

N

O

H

Primary amine

R2

Secondary amine OH

R2 O

O O

R2

OH R1

N

O

R2

Tertiary amine Scheme 16.13.

Reaction of amine groups with glycidyl groups, formation of tertiary amines.

Amines generally do not need a catalyst, as the reaction with epoxy resins occurs spontaneously at room temperature. Therefore, amine curing agents can be used

16.4 Epoxy Resins

for epoxy resins only in two-component (2K) formulations. Low molecular weight polyamines such as diethylenetriamine and triethylenetetramine are frequently used as such. Because of their toxicity, they are also modiﬁed with amide groups to lower their volatility. To their advantage, the functionality of these amines is quite high. Not only do they contain several nitrogen groups in a small molecule, but each NH counts as an active hydrogen donor in the reaction with epoxy groups. A primary amine group, NH2 , is two-functional: after one reaction with a glycidyl group a secondary amine group, NH, remains which can undergo an additional reaction with a glycidyl group, as shown in Scheme 16.13. Phenolic resins can react in two ways with epoxy resins. Using the right catalyst, the aromatic hydrogen groups can react with glycidyl groups, in analogy to the synthesis of higher epoxy resin oligomers. This applies to both resol and novolac type resins (see Section 16.2.2). Methylol groups, as present in resol resins, can also react with the aliphatic hydroxyl groups of bisphenol-A based epoxy resins, with formation of ether linkages. Carboxylic acids and anhydrides also can be used for curing epoxy resins. Carboxylic anhydrides can react spontaneously at room temperature with the aliphatic hydroxyl groups of bisphenol-A based epoxy resins, with formation of carboxylic acid groups. The carboxylic acid groups can then react further with the glycidyl groups (see Scheme 16.12), at higher temperatures and using the right catalyst such as a quaternary ammonium or phosphonium halide. Polyester resins (see Section 16.6) can be used as a source of carboxylic acid functional polymers to cure the epoxy resins. 16.4.3

Production

Three variations on the synthesis of epoxy resins from bisphenol-A and ECH exist to date, diﬀering in the ratio between these monomers and the viscosity of the product resins, as determined by the removal of the sodium chloride formed. Standard Liquid The diglycidyl ether of bisphenol-A (DGEBPA) is made with a large excess of ECH (6 equiv.) to minimize formation of higher oligomers. Bisphenol-A and ECH are charged together into the reactor. Powdered base (NaOH) is added slowly to this mixture at 60 C to make the reaction proceed, while salt (NaCl) is formed as a byproduct. After the exothermic reaction is ﬁnished the temperature is raised to 100 C while water (also a by-product of the reaction) is distilled oﬀ under reduced pressure, and then the temperature is raised further to 120 C to drive the reaction to completion. The salt formed and residual base are removed from the product through washing with water. Finally all remaining volatile components, including the excess ECH, are removed under vacuum (30 mbar). This technical grade (average n ¼ 0:16) contains approximately 90% DGEBPA (n ¼ 0), 9% dimer (n ¼ 1), and 1% trimer (n ¼ 2). 16.4.3.1

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854

Taﬀy Process With a much lower excess of ECH, and in the presence of a catalyst, it is possible to make higher oligomers directly in a procedure similar to that for the standard liquid, called the Taﬀy process. Bisphenol-A and a stoichiometric amount of NaOH (10% aqueous solution) are charged into the reactor, and ECH is added with vigorous stirring at 45 C. The reaction mixture is heated to 95 C and is kept at this temperature for another 1–2 h. The product, a viscous liquid at this temperature, is washed with water to remove the salt and catalyst, dried, let down over a cooling belt and ﬂaked. Because the viscosity of the product is a limiting factor in the water washing step, this process is applicable to lower oligomers only, maximum n ¼ 4, for example the 1001 type. 16.4.3.2

Advancement Process When even higher oligomers are desired, it is important to remove the salts from the reaction before the ﬁnal molecular weight is obtained. This is achieved in the Advancement process (see Scheme 16.14). In this process the reactor is charged with standard liquid resin, free of salts. Fresh bisphenol-A, in a speciﬁc molar ratio to the DGEBPA for the desired Mn , and an appropriate catalyst are added. The polymerization is eﬀected exclusively by consecutive reactions of the hydroxyl groups of the bisphenol-A with the glycidyl groups of the DGEBPA. It is carried out at higher temperatures than the Taﬀy process, well above the glass transition temperatures of the targeted polymer; see Table 16.2. After the desired viscosity has been reached, the product is let down over a cooling belt and ﬂaked to glassy pellets. Unlike the Taﬀy process, in which oligomers with every value of n are formed (n ¼ 0; 1; 2; 3; 4; 5 . . .), the Advancement process yields polymers with predominantly (> 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

Properties and Applications

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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16 Thermosets

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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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

861

862

16 Thermosets

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 <2 mm) and surfactants must be used. Alternatively to the use of additional surfactants, incorporation of surface-active groups, such as polyethylene glycol derivatives, in the alkyd resin is also possible; it is sometimes even more desirable not to have free surfactant molecules in the resin from a paint quality point of view. The production of an alkyd emulsion with built-in surfactants starts with a conventional alkyd polyesteriﬁcation process, in which acid- or hydroxy-functional surfactants are added along with the other monomers. Care must be taken to keep the viscosity as low as possible to facilitate the next step. After the polycondensation reaction is ﬁnished, the resin is cooled to <100 C and water is slowly added to the hot resin to form a w/o emulsion. With increasing water content, an inversion to an o/w emulsion will take place. The inversion is accompanied by a large increase in viscosity. After the inversion more water can be added to reach the right viscosity at room temperature. 16.5.5.1

16.6

Saturated Polyester Resins 16.6.1

Introduction

Thermosetting polyester resins are almost as old as alkyd resins. The ﬁrst attempts to make polyesters derive from the second half of the 19th century. In the beginning of the 20th century more and better-deﬁned polyesters were made by Vorlander and later by Smith. Most of the polyester resins were based on glycerin and aliphatic diacids as fumaric and adipic acids. From these times the ﬁrst application of polyester resins in paints and coatings was developed. Besides the modiﬁcation with fatty acids (alkyd resins), which are cured by air, thermosetting resins were developed which are cured by crosslinkers or other resin types. The choice of monomers, more speciﬁcally the ratio of polyacids to polyalcohols, determines the type of end groups, for a polyester these being either acid aCOOH

16.6 Saturated Polyester Resins

or hydroxy aOH. These functional end groups can react with epoxy resins (see Section 16.4), isocyanates, phenolic resins (Section 16.2), and melamine resins (Section 16.3). 16.6.2

Chemistry

Polyester resins consist of polyfunctional acids and polyalcohols [6]. These compounds esterify at higher temperatures with evolution of water (Scheme 16.19). The chemical equilibrium for this polycondensation reaction is unfavorable. It is important that water is removed from the reaction mixture continuously, shifting the equilibrium to the right (see Chapter 4). The temperature for this reaction lies normally between 200 and 260 C and the choice is dependent on the stability of the raw materials used. This direct esteriﬁcation reaction is far the most used chemistry for saturated polyester resins. Polycondensation via transesteriﬁcation, as in Scheme 16.20 and primarily used in the synthesis of thermoplastic polyesters, is only occasionally used for resins. By varying the type and the molecular ratio of the reactants, the molecular weight and structure of the polymer, and the type and number of functional end groups, can be varied. Excess of acid functional raw materials gives an acid functional polymer. Excess of polyfunctional alcohols gives a polymer with hydroxyl end groups. If branching is desired, polyfunctional raw materials are used. Mostly three- or four-functional polyhydroxyl compounds are used. In practice a large variety of raw materials are available for the formulator

O

O HO

R1

HO

OH

(n+1)

n O

O HO

R1

OH

R2

O

Scheme 16.19.

O

R2

R1

n General esteriﬁcation scheme for OH-polyester resins.

O

O HO

R1

CH3O

OH

R2

(n+1)

R1

OCH3

n O

O HO

OH + 2n H2O

O

R2

O

R1

OH

+ 2n CH3OH

n Scheme 16.20.

General transesteriﬁcation scheme for OH-polyester resins.

863

864

HO

16 Thermosets

HO

OH

HO

OH

OH HO

1,3-propanediol

2-methyl-1,3-propanediol

neopentylglycol

OH OH

trimethylolpropane OH

HO

OH

HO

HO

OH

1,2-propanediol

buteneglycol

OH

ethyleneglycol HO

OH OH

OH

HO

HO

OH

O

pentaerythritol O

1,4-cyclohexanedimethylol Scheme 16.21.

2,2-dimethyl-1,3-propanediol hydroxypivalic ester

Some polyol monomers used in polyester resins.

O

O CO2H

O

O

CO2H

O

O CO2H CO2H

CO2H terephthalic acid

isophthalic acid

phthalic anhydride

CO2H

HO2C

O O

CO2H adipic acid

HO2C

trimellitic anhydride

O

CO2H

CO2H

hexahydrophthalic anhydrid

cyclohexane-1,4-dicarboxylic acid sebacic acid Some polyacid monomers used in polyester resins.

Scheme 16.22.

to design the polymer molecule. Schemes 16.21 and 16.22 give an overview of the most frequently used polyol and polyacid monomers, respectively. In principle, the molecular weight distribution can be calculated, but for the more complicated formulations computer aid is needed. The choice of diols depends on which properties the resin would get. Rigid and

16.6 Saturated Polyester Resins

cyclic diols will increase the Tg , melting point, and hardness of the polyester resin. Long and ﬂexible diols will decrease the Tg and melting point, and improve the ﬂexibility. Diols which are substituted on the b-position toward the hydroxyl groups, such as neopentyl glycol, are normally more resistant against hydrolysis and photooxidation. Besides a wide variety of diacids, several cyclic anhydrides are available as well. Anhydrides react faster than the corresponding acids and require less water to be distilled oﬀ. Aliphatic diacids will lower the Tg and melting point, but give more ﬂexibility. Aromatic diacids give a higher Tg and hardness to the coating ﬁlm and are more resistant toward hydrolysis. Aliphatic diacids impart less UV absorbance to the resin and therefore enhance the UV resistance of coatings made from these resins. From aliphatic diacids, crystalline or semicrystalline polyester resins can also be made. 16.6.3

Production

Most saturated polyester resins are produced in the melt phase. At temperatures of about 200–260 C and in the presence of a catalyst the diacids and the diols are esteriﬁed by removal of water. The high temperature during the synthesis is normally suﬃcient to remove the water eﬃciently from the reactor by distillation, under reduced or normal pressure. To remove the last percentages of water and to complete the reaction, a stronger vacuum is applied at the end of the reaction. In cases where the resins are to be supplied in a solvent anyway, azeotropic distillation can be used to remove the water more eﬃciently and at lower temperatures. Figure 16.3 describes the reactor design for polyester resin production, which can also be applied to alkyd resin production (see Section 16.5). The product resin can be either cooled and discharged via a cooling belt in case a solid delivery form is required, or thinned with solvents. Monitoring the Reaction The conversion and progress of the esteriﬁcation reaction are controlled by analyzing samples that are taken from the mixture. For the application of thermosetting resins the amount of end groups is very important and has to be measured during production. The concentration of end groups (aCOOH and aOH) is measured by titration; several titration methods have been developed. The concentration is expressed as the ‘‘acid value’’ or ‘‘hydroxyl value’’ in mg KOH per gram resin (by back-titration in the case of aOH). The removal of water by distillation is not 100% eﬃcient and some diols, diacids, small oligomers, and side products of the reaction at high temperature can be removed from the reactor with the water. In order to reach the right speciﬁcations, small amounts of either diols or diacids can be added (‘‘corrections’’), as even in the early stages of the reaction a misbalance of the OH/COOH groups can be detected. Another important speciﬁcation is the viscosity of the melt or of the solution, as an indication for the molecular weight. Both parameters are plotted for a typical polyester resin in Figure 16.4. 16.6.3.1

865

866

16 Thermosets

Fig. 16.4.

Evolution of acid number and viscosity with time during polyester resin synthesis.

16.6.4

Properties and Applications

The ratio of ‘‘hard’’ to ‘‘soft’’ monomers determines the Tg of the polymer. The reactivity during the curing reaction and network formation is determined by the overall amount of end groups and the overall functionality. The main application of these types of polyester resins is in thermosetting coatings. Thermosetting paints are formulated by mixing polyester resins (in solid form, melt, or solution) with a crosslinker, pigments, and additives. These paints will be cured to a coating ﬁlm, generally with stoving. Surface coatings based on saturated polyester resins distinguish themselves from other resins by good adhesion on metals, good ﬂexibility (ability to be deformed together with the substrate after painting), and hardness. Their resistance to hydrolysis is rather poor; their outdoor durability is modest in comparison to acrylic binders but superior to epoxy resins. Saturated polyester resins can be applied in all kinds of paint types. The major application ﬁelds today include coatings for beverage and food cans, and coated steel coils. Saturated polyester resins further play a dominant role in the binders for thermosetting powder coatings, described in the next paragraph. 16.6.5

Powder Coatings

Powder coatings are, as the name suggests, paints in solid, powdery form [7]. All the ingredients are solid materials, which are mixed in a hot-melt compounding

16.6 Saturated Polyester Resins

Fig. 16.5.

Powder coating production.

process (extrusion). Figure 16.5 depicts the powder coating preparation process. Powder paints consist of a binder (polymer, crosslinker and curing catalyst), pigments, and additives. All these solid ingredients are mixed thoroughly in a premixer and fed into an extruder. The mixture is molten and homogenized in a short period of time, typically in half a minute, to prevent premature crosslinking reactions. After the extrudate is cooled down, it is crushed and milled to the required particle size, which is dependent of the application but is normally in the range of about 50–100 mm. Application Powder paints are commonly applied by electrostatic spraying on metal substrates. The powder particles are charged inside the spray gun (Corona or Tribo) and adhere to the earthed metal object. Powder coating is ideal for coating complicated three-dimensional objects because the powder particles can easily reach recessed surfaces or even the backside of surfaces, following the electrical ﬁeld lines from the gun to the substrate. Powder paint which does not reach the object or disassociates from the surface (‘‘overspray’’) can easily be gathered, recycled, and re-used without loss of quality (Figure 16.6). The substrate, covered with a layer of powder paint, is subsequently heated in a hot-air oven. The powder particles melt, coalesce, and ﬂow out into a ﬁlm. In some special cases the powder paints can be applied by dipping. The object is preheated and dipped into a ﬂuid bed of powder paint particles. The powder particles adhere to the preheated surface of the object and melt into a ﬁlm. Further 16.6.5.1

867

868

16 Thermosets

Fig. 16.6.

Powder coating spraying and recycling system.

heating causes the crosslinking reactions in the ﬁlm to give the required thermoset properties. The processes of melting, ﬂowing, and curing take place in the same cycle and this makes it diﬃcult to obtain a completely ﬂat ﬁlm before the cure prohibits further ﬂow. Powder coatings are therefore characterized by a certain visual ‘‘orange peel’’ eﬀect. Crosslinking Although many examples exist of thermosetting powder coatings (using mainly polyamide, polyethylene, and polypropylene), most powder paints are thermosetting systems, which are cured at much higher temperatures (typically 180–200 C) than the extrusion temperature (90–110 C). Because of these high curing temperatures, powder coatings ﬁnd their application mainly on metal surfaces. The ﬁrst thermosetting powder coating systems were based on bisphenol-A epoxy resins, which were cured by either amide/amine or carboxylic anhydride crosslinkers (for the chemistry, see Section 16.4.2). The epoxy-based powder coatings show excellent application properties, such as ﬂow, mechanical strength, and chemical resistance. However, they have limited exterior durability and they show a severe tendency toward yellowing. These disadvantages have greatly stimulated the development of powder coatings based on saturated polyesters with carboxylic end groups. When cured with bisphenol-A resins, these polyester resins result in coatings with much improved exterior durability with less yellowing. When cured with triglycidyl isocyanurate (TGIC) excellent outdoor durability is obtained and polyester resin coatings are used, for example, in the building industry (fac¸ades). The mutagenic properties of TGIC have stimulated the search for safer crosslinkers, such as b-hydroxyalkylamides, which are now taking over the market. Hydroxyl-functional polyester resins are also used; these polyesters are cured with aliphatic (for outdoor use) or aromatic (indoor use) polyisocyanates. 16.6.5.2

16.7 Unsaturated Polyester Resins and Composites

Advantages The advantages of powder coating compared to the more classic type of coatings in solution are clear: they are 100% solvent-free and thick layers up to 500 mm can be applied in one step. Moreover, they are very cost-eﬃcient owing to a very high application eﬃciency, close to 100%, as over-sprayed powder can be easily recycled. Last but not least, their mechanical properties are usually equal or superior to those of their solvent-borne counterparts. Powder coating can also be favorably combined with UV curing (see Section 16.8.5.2). In that case, the powder coating can also be applied to heat-sensitive substrates such as wood, paper, and plastics because the powder can melt and ﬂow from about 100 C. In addition, the melt/ﬂow-out phase is easily separated (hot air or infrared radiation) from the cure phase (irradiation with UV light) resulting in a good ﬂow. Binders for UV-curable powder coatings are either based on acrylated saturated polyester resins (see Section 16.8.3.2) or on unsaturated polyesters (Section 16.7) in combination with vinyl ether crosslinkers (Secheme 16.34). 16.6.5.3

16.7

Unsaturated Polyester Resins and Composites 16.7.1

Introduction

Unsaturated polyester resins (UP resins) cover a broad range of products serving a wide variety of market segments. They can be considered the most versatile of the thermoset resins. By varying the resin composition, one can produce an amazing range of properties in polyesters. Additionally, polyesters are capable of being processed by a number of diﬀerent methods, varying from simple hand-lay-up contact molding to highly automated compression molding. Glass ﬁber reinforced UP resins are extensively used in building and construction; and in the transportation, electrical, and electronic industries; and for sanitary and domestic applications. The biggest use of these resins is in the construction industry, followed by transportation. The worldwide consumption of UP resins in 2002 was approximately 2.5 million metric tons. 16.7.2

Chemistry

The unsaturated polyester is a macromolecule with an Mn value between roughly 1000 and 3000 g mol1 , obtained from a polycondensation reaction (that is, a polymerization with the formation and elimination of water) between unsaturated- and saturated dicarboxylic acids (or their corresponding anhydrides) and diols [8]. This reaction and the most commonly used raw materials are represented schematically in Figure 16.7. Mono- or trifunctional acids or alcohols or other monomers may be incorporated optionally into the polyester backbone to tailor properties.

869

870

16 Thermosets Unsaturated dicarboxylic acids

Saturated dicarboxylic acids

Maleic anhydride Fumaric acid

Phthalic anhydride Isophthalic acid Terephthalic acid Adipic acid

Glycols

Other

Propylene glycol Dipropylene glycol Ethylene glycol Diethylene glycol Neopentyl glycol Propoxylated bisphenol A

Benzoic acid Ethylhexyl alcohol Trimethylol propane Dicyclopentadiene

POLYCONDENSATION OH + HOOC

Unsaturated Polyester Macromolecule

+ water

Reactive Monomers

Finishing ingredients

Styrene α-Methyl styrene Methyl methacrylate Vinyl toluene Diallylphthalate Triallylcyanurate

Inhibitors

hydroquinone / benzoquinone tert. butyl catechol Stabilizer copper (I) naphthenate Accelerator cobalt- octoate or naphthenate Thixotropic agent fumed silica UV absorber benzophenone derivatives LSE agent paraffin or wax

UP resin Glass fiber reinforcement Chopped rovings Chopped strand mat Woven rovings

Radical initiator Methyl ethyl ketone peroxide Benzoyl peroxide Cumene hydroperoxide tert Butyl perbenzoate

Fillers Calcium carbonate Aluminum trihydrate Pigments / colorants Titanium dioxide Zinc sulfide

RADICAL POLYMERIZATION

Cured UP-based composite Fig. 16.7.

Unsaturated polyester resins: from raw materials to cured composites.

16.7 Unsaturated Polyester Resins and Composites

The preferred unsaturation in the polyester backbone is the fumaric double bond, which results from using either fumaric acid or maleic anhydride as the unsaturated dicarboxylic acid. Under the reaction conditions normally applied to synthesize the polyester, most of the maleic (cis) double bonds isomerize to fumaric (trans) double bonds. The unsaturated polyester macromolecule itself is either a very viscous liquid or a solid. Usually this polyester is dissolved in a reactive monomer containing a double bond capable of copolymerizing through a radical polymerization with the (fumaric) unsaturations in the polyester backbone. The workhorse of the reactive solvents is styrene. The low-viscosity polyester solution (UP resin) thus obtained is formulated with a number of ingredients that provide critical properties, such as storage stability, reactivity, thixotropy, UV resistance, and color, before being sold by the polyester producer. Crosslinking The processor converts the liquid polyester by means of a radical polymerization, in the presence of glass ﬁber reinforcement and/or ﬁllers, into a three-dimensional composite network as displayed in Figure 16.7. This so-called crosslinking or curing reaction takes place between the fumaric unsaturations in the polyester backbone and the unsaturation in the styrene monomer. The reaction is typically initiated by free radicals generated by the decomposition of peroxides. Depending upon the type of peroxide (and optionally the accelerator used) the crosslinking reaction may take place at temperatures as low as 10 C or as high as 160 C. The rate at which these radicals are generated is one of the factors determining how fast a resin gels and cures. The temperature development of a mass of resin against time from the moment of (accelerator and) peroxide addition is an important characteristic of a polyester resin. This is illustrated for a typical curing process at ambient temperature in Figure 16.8. The portion AB is the induction period during which free radicals produced by decomposition of the peroxide are being absorbed by the inhibitor present in the resin. This is the so-called pot life of the resin, during which processing of the part can take place. This time may vary from a few minutes up to 20 h. At point B the inhibitor concentration has become low enough for an appreciable proportion of the free radicals produced to initiate resin polymerization. Due to heat of polymerization, the temperature rises. The exothermic peak D does not necessarily correspond to completion of the curing process. As the resin solidiﬁes during the latter part of the curing process, the rate of reaction is greatly reduced (diﬀusion-controlled). A post-cure at elevated temperatures may be needed to reach full cure. 16.7.2.1

Styrene Emission Actually more than 50% of UP resin consumption still involves open-mold techniques, hand lay-up and spray-up being the most important. During processing and curing, styrene emission is considerable. Styrene (boiling point 145 C) is classiﬁed as a dangerous substance according to EC Directive 67/548/EEC and is labeled with the following R (Risk) and S (Safety) phrases: R10 (ﬂammable), R20 (harmful by inhalation), R36/38 (irritating to eyes, skin, and respiratory tract), S2 (keep out 16.7.2.2

871

872

16 Thermosets Peak temperature

D

Temp (°C) 150

C

35 25

A

Peak time

B 50

Fig. 16.8.

80

Time (min)

A typical curing proﬁle of unsaturated polyester resins at ambient temperature.

of reach of children), S23 (do not breathe gas/fumes/vapor/spray). In 1992 the European Center for Ecotoxicology and Toxicity of Chemicals concluded in a study, Styrene Toxicology, that the carcinogenic potential of styrene is so low that occupational or environmental exposure to styrene is unlikely to present any genotoxic or carcinogenic hazard in humans. Styrene has, however, a proven neurotoxic eﬀect. Occupational exposure to high styrene concentrations may lead to symptoms like drowsiness, headache, and tiredness. As a consequence the threshold limit value (TLV) has been lowered signiﬁcantly since the 1980s in virtually all countries. The UP industry reacted by developing so-called LSE (low styrene emission)- and LSC (low styrene content) resins. An LSE resin contains paraﬃn or wax in such a concentration that upon only a little styrene evaporation it forms a thin barrier ﬁlm on top of the laminate, curtailing styrene emission. However, one aspect that warrants attention is the interlaminar adhesion of parts built up from several laminate layers. Note that these LSE resins are only active in the static state when the surface of the laminate is at rest. During processing, when the surface is regularly disturbed, the eﬀect of the ﬁlm-forming additive is far less. LSC resins have a styrene concentration between 25–35% versus the 40–50% of classical UP resins. Because of the lower styrene content, the evaporation during processing and cure is considerably less. As the processing technique is determining the viscosity range of the resin that can be used, this property cannot be changed. So, keeping the same resin viscosity at lower styrene content implies that the viscosity of the unsaturated polyester must be reduced. This can be achieved by reducing the molecular weight of the unsaturated polyester but, unfortunately, this leads to a strong deterioration of mechanical and chemical proper-

16.7 Unsaturated Polyester Resins and Composites

O

O H2 O O 80-100°C O

O OH

O

OH

OH

O

100-150°C

O Glycols, (un)saturated dicarboxylic acids

Unsaturated Polyester with DCPD end groups Polycondensation Scheme 16.23.

Synthesis of DCPD end-capped unsaturated polyester resins.

ties. Another way is to replace the polar end groups of the polyester by apolar end groups, which show good solubility in styrene. Dicyclopentadiene (DCPD) endcapped polyesters (see Scheme 16.23) are therefore the resins of choice to reach the required properties. Of course, addition of paraﬃn to these resins will further reduce the styrene emission. The most obvious way to reduce styrene emission is to use another monomer. However, there appears to be no real alternative that fulﬁlls all the technical and economical requirements as well as being less toxic, and consequently styrene is at most partially replaced in some cases. Other elements which alone, or in combination, will contribute to less styrene emission/exposure in the workplace are:

use of specially designed spray guns (for example, ﬂow coaters), which reduce the surface area of the resin that is sprayed, and consequently reduce emission; a change from open mold techniques to closed or semi-closed mold techniques; in particular, the vacuum-assisted resin infusion technique, whereby a ﬂexible ﬁlm is used as a counter-mold, is gaining popularity; proper and adequate ventilation; use of robots. Vinyl Ester Resins A special class of unsaturated polyesters, developed for composite applications in corrosive environments, is the so-called vinyl esters and vinyl ester urethane resins. The best-known vinyl ester is made by reacting the diglycidyl ether of bisphenolA (DGEBA; see Section 16.4.2) with a monocarboxylic unsaturated acid such as methacrylic or acrylic acid; see Scheme 16.24. This addition reaction occurs at a temperature of 120–140 C. The vinyl ester urethanes, on the other hand, are synthesized from a low molecular weight (unsaturated) polyester diol, mostly based on bisphenol-A, a diisocyanate, and a hydroxyalkyl (meth)acrylate (see also Schemes 16.30 and 16.31, Section 16.8.3.3). As for standard UP resins, the most commonly used reactive solvent is styrene. Both types of products have only a few ester linkages (which are the points of greatest vulnerability to chemical attack) in the mole16.7.2.3

873

874

16 Thermosets

O

O O OH

O

O

O

O

O Scheme 16.24.

HO

OH

OH O

O

O O

Synthesis of vinyl esters from methacrylic acid and DGEBPA.

cule and hence show improved chemical resistance, whilst the network built up from the terminal (meth)acrylate double bonds results in a tougher, more resilient resin structure. In Table 16.4, the UP, vinyl ester, and vinyl ester urethane resins are classiﬁed in order of chemical resistance. 16.7.3

Production

Unsaturated polyesters are produced batchwise in agitated stainless steel reactors of up to 50-ton capacities (see Figure 16.3). The reactors are equipped with heating jackets and internal heating coils to bring the raw materials to the required reaction temperature (190–230 C). The production starts by charging the raw materials into the reactor by the use of pumps and meters for liquids such as most of the glycols, molten maleic anhydride, and molten phthalic anhydride. Solid raw materials, such as isophthalic acid, are added manually by emptying big bags or automatically by pneumatic transport from silos. All the reactants may be charged in one go except for a few dicarboxylic acids which are somewhat more diﬃcult to esterify. In the latter case a two-stage method is applied in which the ﬁrst step is the reaction of such a dicarboxylic acid (for example, isophthalic or terephthalic acid) with a large excess of glycol until a clear solution with a low acid number is obtained, followed by addition and reaction of the other raw materials. When the reactor is brought up to its reaction temperature the contents are periodically sampled to determine the acid number (a measure of the acid groups that have not yet reacted) and viscosity; this may be done either oﬀ-line or in-line. The synthesis is continued until the speciﬁed acid number and viscosity values are achieved. The distillation column performs the separation of water from unreacted (low-boiling) glycols. During synthesis, inert gas (usually nitrogen) ﬂows continuously through the reactor to prevent gelation due to oxidation and to aid removal of water. An alternative method used to remove the water formed from the reaction mixture is to apply vacuum.

16.7 Unsaturated Polyester Resins and Composites

Reaction times vary from 8 to 30 h, depending on the raw materials used, the degree of condensation or molecular weight desired, and the reaction temperature employed. After cooling, the unsaturated polyester is discharged from the reactor into the so-called thinning tank, which has a preweighed styrene charge in it. The resin is then ﬁnished with additives, tested for quality control, and pumped to storage. 16.7.4

Reinforcement

The majority of the UP resins are used in glass ﬁber reinforced (GRP) applications. Because E-glass oﬀers suﬃcient strength at low cost it accounts for more than 90% of all glass ﬁber reinforcements. It also provides good electrical resistance (hence its name). Sizing is applied to the ﬁlaments immediately after their formation to ease processing, to protect the ﬁbers from breakage during processing, and to create better bonding between the ﬁber and the resin matrix. Although it comprises only 0.25–5% of the total ﬁber weight, sizing is a dynamic force behind the performance of any ﬁberglass product. Sizing chemistry distinguishes each manufacturer’s product and determines the grade of ﬁberglass for diﬀerent processing techniques. Sizing can improve the wettability of the ﬁber during resin impregnation, thereby reducing part-manufacturing time. Coupling chemistry to enhance bonding between the glass ﬁber and the matrix resin can also improve the overall strength of the ﬁnal composite. Glass ﬁlaments are supplied in bundles called strands, rovings, or yarns. A strand is a collection of more than one continuous glass ﬁlament. A roving generally refers to a bundle of untwisted strands wound in parallel to form a cylindrical, ﬂat-ended package, similar to thread on a spool. Single-end roving contains only one continuous strand of multiple glass ﬁlaments. Multiple-end roving contains numerous wound strands. Yarns are collections of ﬁlaments or strands that are twisted together. Rovings, the most common form of glass ﬁber, can be chopped or woven to create secondary ﬁber forms used in composite manufacturing. Mats are non-woven fabrics that provide equal strength in all directions. Chopped mats contain randomly distributed ﬁbers cut to lengths typically ranging from 4 to 6 cm and held together with a chemical binder. These mats provide lowcost polymer reinforcement primarily in hand lay-up, continuous laminating, and some closed-mold applications. Continuous-strand mat is formed by swirling ﬁber onto a moving belt ﬁnished with a chemical binder that holds the ﬁber in place. Continuous-strand mat is primarily used in compression molding, resin transfer molding, and pultrusion applications. Woven rovings are made by weaving the untwisted rovings into cloth. Bidirectional woven fabrics provide good strength in the 0 /90 directions and allow fast composite fabrication. To prevent laminates failing by interlaminar shear between layers of woven rovings, these are generally interspersed with layers of choppedstrand mat.

875

Resin type

UP

UP

UP

UP

UP

UP

UP

UP

VE

1A

1B

2A

2B

3

4

5

6

7A

neopentyl and halogenated neopentyl glycol (min 80 mol%)[c] and a diol with at least one secondary OH group (max 20 mol%)[c] hydrogenated bisphenol A[d] and cyclohexane dimethanol dipropoxy-bisphenol A and/or dipropoxy halogenated bisphenolA epoxidized bisphenol-A and/or epoxidized halogenated bisphenolA

standard glycols[a]

standard glycols[a]

standard glycols[a,b]

methacrylic aid and/or acrylic acid

orthophthalic acid, ethylenedicarboxylic acids ethylenedicarboxylic acids

orthophthalic acid, ethylenedicarboxylic acids orthophthalic acid, ethylenedicarboxylic acids isophthalic acid, HET acid, ethylenedicarboxylic acids isophthalic acid, HET acid, ethylenedicarboxylic acids terephthalic acid, ethylenedicarboxylic acids isophthalic acid, orthophthalic acid, ethylenedicarboxylic acids

standard glycols[a,b] standard glycols[a,b]

Types of acids

Types of glycols

55

55

45

55

50

50

50

45

45

Styrene content [wt.%] max.

110

130

120

120

140

120

85

120

85

˚

Tg [ C] min.

90

110

90

90

110

90

60

90

60

˚

HDT [ C] min.

75

60

50

65

75

50

60

50

60

Tensile strength [MPa] min.

4.0

2.0

1.5

3.0

3.0

1.5

2.0

1.5

2.0

Elongation at break (tension) [ %] min.

Classiﬁcation of UP, vinyl ester, and vinyl ester urethane resins according to their chemical resistance (European standard EN 1321).

Resin group

Tab. 16.4.

130

110

100

110

120

75

90

75

90

Flexural strength [MPa] min.

876

16 Thermosets

VE

8

dialkoxy-bisphenol-A and/or dialkoxy halogenated bisphenol-A (min. 90%), alkoxy (meth)acrylate epoxidized novolac methacrylic aid and/or acrylic acid

ethylenedicarboxylic acids

[a] Standard glycols: ethylene, 1,2-propene, diethylene, dipropene glycols; neopentyl glycol, 1,3-butanediol, 1,4-butanediol and corresponding halogenated glycols. [b] May also contain cyclic unsaturated hydrocarbons. [c] Related to the sum of the diol components. [d] This diol deviates from the one mentioned in EN 13121; the diol mentioned in this table is more commonly used. Note: The chemical resistance of the resins increases on going from group 1 ! group 8.

VEU

7B

50

50

150

120

120

105

75

75

2.5

3.5

130

130

16.7 Unsaturated Polyester Resins and Composites 877

878

16 Thermosets Tab. 16.5.

A comparison of reinforcing ﬁbers.

Reinforcing ﬁber

Tensile strength [MPa]

Tensile modulus [GPa]

Speciﬁc gravity [g cmC3 ]

Properties

Aramide

2800–3000

85–130

1.44

Boron Carbon

3500 2500–3200

400 210–700

2.55 1.75–1.96

Glass: E-glass

2400

72

2.54

S-glass

4500

85

2.49

Polyester

1050

10

1.38

Polyamide

1000

5

1.16

Ultra high molecular weight polyethylene

3400

110

0.975

low density, good speciﬁc properties, moderate cost high modulus, high cost high modulus, electrical conductivity, high cost good strength and processability, low cost high strength and good processability good impact resistance and chemical properties good impact resistance, alkali resistant low density, good impact

Surfacing mat, or veil, is used in conjunction with reinforcing mats and fabrics to provide good surface ﬁnish. It is eﬀective in blocking out the ﬁber pattern of the underlying mat or fabric. Surfacing mats are also used as the inside layers of corrosion resistant composites, providing a smooth, resin-rich surface. Fibers can also be produced from carbon, boron, and aramid materials. Generally these ﬁbers exhibit higher tensile strength and stiﬀness than do their glass counterparts, as can be seen in Table 16.5. However, these specialty ﬁbers are more expensive. Therefore they are typically reserved for applications demanding exceptional ﬁber properties, for which the customer is willing to pay a premium. 16.7.5

Fillers

Fillers not only reduce the costs of composites, but also frequently impart performance improvements which might not otherwise be achieved by the reinforcement and resin ingredients alone. Fillers can improve ﬁre and smoke resistance by reducing organic content in composite laminates. Other important properties that can be improved through the proper use of ﬁllers include water resistance, weathering, surface smoothness, stiﬀness, dimensional stability, and temperature resistance. Calcium carbonate is the most widely used inorganic ﬁller. It is available at low cost in a variety of particle sizes and surface treatments. Aluminum trihydrate is used as a ﬁller when improved ﬁre/smoke performance is required. When exposed to high temperatures this ﬁller gives oﬀ water in a

16.7 Unsaturated Polyester Resins and Composites

highly endothermic reaction, thereby reducing the ﬂame spread and development of smoke. 16.7.6

Processing

Fabricators of unsaturated polyester resins use either open-mold or closed-mold processes. The main open-mold processes are hand lay-up, spray-up, continuous lamination, ﬁlament winding, and centrifugal casting. The most used closed-mold techniques are hot-press molding and resin-transfer molding. Pultrusion and coldpress molding may be considered as semi-open or semi-closed processes. An overview of the processing techniques for UP resins, their possibilities and limitations, is given in Table 16.6. Irrespective of the process used to produce glass-reinforced composites there are three basic principles in selecting materials for a particular end use [9]:

mechanical strength; chemical, electrical, and thermal properties; cost/performance.

Mechanical strength The type of ﬁber, the amount in the composite, and the way in which the individual strands are positioned determine the direction and level of strength that is achieved. There are three generic types of ﬁber orientation:

unidirectional oriented ﬁbers, providing the greatest strength in the direction of the ﬁbers; bidirectional oriented ﬁbers, with some ﬁbers positioned at an angle to the rest, as with woven fabric: this provides diﬀerent strength levels in each direction of ﬁber orientation; multidirectional or randomly oriented ﬁbers: this arrangement provides essentially equal strength in all directions of the ﬁnished part. Chemical, electrical, and thermal properties The chemical building blocks of the polyester, the molar ratio of maleic anhydride to saturated dicarboxylic acid, and the choice and content of the monomer dictate to a large extent the mechanical properties (rigid or ﬂexible), the glass transition temperature (Tg ) or HDT (heat deﬂection temperature), and the thermal and chemical resistance of the cured products. Additives are available for use with many resin systems to provide properties such as color, high surface ﬁnish, UV resistance, and low shrinkage. Cost/performance If a large number of parts are to be made from one set of molds, the lowest unit production cost is likely to be obtained by using compression or injection molding. In these processes, presses, tooling, and automated handling equipment can be applied to speed production cycles and reduce direct labor

879

metal rollers or sheet

single GRP, wood, etc

steel or around plastic liner

ambient to 100 high

60–110 low

55–80 2–25

25–35 2–25, usually 2–10 ambient to 40 high

25–35 2–5

Filament winding

Glass [wt%] Laminate thickness [mm] Cure temp. [ C] Styrene emission Type of mold

Continuous lamination

mat, chopped rovings, rovings mat, woven rovings

Hand lay-up and spray up

steel

hardened steel die

ambient to 100–150 100 high low

50–75 2–25

ambient to 60 low

40–65 2–50

mat, woven rovings, cont. ﬁbers, preform

Resin infusion (vacuum)

double GRP bottom, or metal GRP or light metal; top, foil

low

25–50

25–50 1–10

mat, woven rovings

rovings, mat mat, chopped rovings, woven roving 15–40 2–25

Cold-press molding

Centrifugal Pultrusion casting

Comparison of common composite process considerations.

Type of rechopped inforcement rovings, mat, woven rovings

Tab. 16.6.

ambient to 60 low

mat, woven rovings, cont. ﬁbers, preform 25–50 2–6

matched metal

low

100–170

25–70 1–10

chopped rovings, mat

Hot-press Resin molding transfer (BMC/SMC) molding ( p I 1 bar)

double bottom, GRP or stiﬀ light GRP, metal top, ﬂex. GRP

ambient to 60 low

mat, woven rovings, cont. ﬁbers, preform 25–55 2–6

Light resin transfer molding ( p I 1 bar)

880

16 Thermosets

Typical products

high operatordependent – one smooth surface boats, garden ponds, containers, shelters

low

rooﬁng lights, corrugated sheets

up to 12 m min1 low good

>500 km y1 1–2000

1–1000

moderate

tanks, pipes, tubes

pipes and tubes

moderate low good: inside good: two smooth smooth surfaces

moderate

>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.

16.7 Unsaturated Polyester Resins and Composites

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-

885

886

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

16.7 Unsaturated Polyester Resins and Composites

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

888

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.

889

890

16 Thermosets

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 Acrylate Resins and UV Curing

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

891

16 Thermosets

892

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.

16.8 Acrylate Resins and UV Curing

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

894

16 Thermosets

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

16.8 Acrylate Resins and UV Curing

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

16.8 Acrylate Resins and UV Curing

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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16 Thermosets

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

16.8 Acrylate Resins and UV Curing

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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16 Thermosets

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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16 Thermosets

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
DT, diaryldisulfide

+

S thiophene

Scheme 16.37. Products and generally accepted mechanism for accelerated sulfur vulcanization of EPDM (X ¼ accelerator residue; sulfur substitution at C3(exo) and C9 of ENB is just given as an example).

of conjugated dienes/trienes, do not occur for ENB–EPDM, because of the stability of the tris-substituted unsaturation in ENB and its isolation from other ENB units. Peroxide Curing The mechanism of peroxide crosslinking is shown in Scheme 16.38. Upon heating, the peroxide decomposes into the primary alkoxy radicals, which may further react to secondary radicals. These radicals abstract hydrogen atoms from the EPM chain. In the case of EPM, crosslinks are formed by combination of two macroradicals, whereas in the case of EPDM crosslinks are formed via combination, but also via addition of the macro-radical to the pendent unsaturation of a second EPDM chain. The formation of CaC bonds explains the higher thermal stability of peroxide-cured EPDM in comparison with sulfur-vulcanized EPDM with its labile SaS crosslinks. The reactivity for peroxide cure increases in the series ENB @ DCPD < VNB, because of the decreased steric hindrance at the unsaturation. The eﬃciency of peroxide curing is enhanced by the addition of co-agents, that is, chemicals with two or more unsaturated bonds, which are actually built 16.9.3.2

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16 Thermosets

1/2 RO-OR peroxide decomposition

DT RO

ROH

H-abstraction EPDM

EPDM macro-radical (EPDM ) + EPDM

+ EPDM combination

addition

cross-link EPDM

H-transfer

EPDM

cross-link Products and mechanism for peroxide cure of EP(D)M (DCPD is just used as an example; the formation of tertiary free radicals is chosen arbitrarily). Scheme 16.38.

into the elastic network, whereas the peroxide ‘‘only’’ initiates the crosslinking reaction. Note that in contrast to sulfur vulcanization, the EPDM unsaturation is consumed during peroxide cure. Processing EPDM is ﬁrst mixed with ﬁllers and oil on two-roll mills or batch kneaders, and next the crosslinking chemicals are added at moderate temperatures. Then the EPDM compound may be shaped and crosslinked in a hot (> 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

Properties and Applications

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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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

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

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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

17.2 Fiber Terminology

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

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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.

17.2 Fiber Terminology

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

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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

17.2 Fiber Terminology

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 <10 0 ca. 30 40–60 40–50 40–50 40–50 30–40 ca. 100

[a] T

g ¼ glass transition temperature, Tm ¼ melting temperature, Td ¼ decomposition temperature.

cess is relatively simple and cheap. If a polymer is not melt-spinnable, a solvent must be found which can be used in a solution-spinning process. If the solvent can be easily evaporated, dry spinning can be applied. In many cases it will be necessary to spin into a bath containing a nonsolvent, which results in coagulation of the polymer solution. This is called wet spinning. In a single case – viscose rayon – a chemical reaction taking place in the spinning bath is a further complication. The preference for melt spinning is very strong. Attempts have therefore been made to make solution-spun polymers melt-spinnable. For example, small amounts of solvent/plasticizer can make acrylics melt-processable. Cellulose triacetate can be melt-spun if the residence time is held very short. These attempts have not led to commercial processes, however. If a choice can be made between dry and wet spinning, the trend is to favor wet spinning. This may seem surprising because wet spinning is technically complicated. Organic solvents used in dry spinning are often highly ﬂammable, however, which considerably increases the processing costs. New, advanced ﬁber types are often spun from solution. Aramid, for example, has to be spun from solution because the polymer does not melt, but the developed process strongly deviates from standard wet spinning. The new process is called air-gap spinning, or dry-jet wet spinning. In the case of polyethylene the choice was made to spin ultra-high molecular weight polymer from a dilute solution. This sounds economically unattractive, but the resulting ﬁber is much stronger than a melt-spun product. A new name, gel spinning, was introduced for this process. A survey of the most important ﬁber polymers and the spinning processes applied is given in Table 17.5.

921

922

17 Fibers Tab. 17.5.

The most important polymer ﬁber processes.

Polymer

Spinning process

Polyethylene Polypropylene Acrylic copolymers

gel spinning (dilute solutions of UHMW polymer) melt spinning (or ﬁbrillation of melt-extruded ﬁlm) dry spinning (e.g., DMF solvent) wet spinning (e.g., NaSCN/water solvent) wet spinning or dry spinning (solvent: water) wet spinning rayon: polymer is derivatized during dissolution in NaOH lyocell: direct dissolution in NMMO dry spinning acetate in acetone/water triacetate in DCM/methanol melt spinning melt spinning melt spinning air-gap spinning (solvent H2 SO4 )

Poly(vinyl alcohol) Cellulose

Cellulose acetate

Polyamide 6 Polyamide 66 Poly(ethylene terephthalate) Poly( p-phenylene terephthalamide)

17.3.3

Spinnability

What makes a polymer melt- or solution-spinnable? A theoretical prediction is hardly possible [6]. Laboratory tests are in fact small spinning trials. This is the way most ﬁber producers will test newly developed polymers, or modiﬁcations and new grades of known polymers: on a small spinning machine, which may be a ‘‘one-hole’’ laboratory machine. The reason is that in the initial stage of development often only small amounts of material are available. The ﬁrst aspect of the test is whether the extrusion through the spinning holes proceeds smoothly, without excessive die swell, and without extrudate distortion. Modest problems may be solved by increasing the temperature in melt spinning or reducing the polymer concentration in solution spinning. In more severe cases one must probably conclude that the molecular weight is too high or the molecular weight distribution too broad. Elastic behavior is more problematic in spinning than in most other polymer shaping processes. The second aspect of spinnability is whether the ﬁlaments will break under their own weight or if the spin-line stress is increased by imposing higher spinning speeds. Filament breaks during spinning indicate too low a melt strength. This problem can only be solved by increasing the molecular weight of the polymer and/or the concentration of the spinning solution. Obviously, the use of a high molecular weight polymer grade will be preferred. It gives increased melt strength, and will usually give better yarn properties. Problems with die swell and melt fracture can make a compromise necessary, however. For wet spinning there is an additional factor inﬂuencing spinnability. The thread formation process is complicated: it involves diﬀusion of solvents out of

17.4 Melt Spinning

the ﬁlament and nonsolvent from the spinning bath into the ﬁlament, followed by gelation and/or coagulation. Unstable coagulation can make the spinning ﬁlaments weak. A polymer solution may therefore appear unspinnable until the proper composition of the spinning bath has been discovered. With tongue in cheek, one could say that melt spinning is for engineers while wet spinning is a technique for artists.

17.4

Melt Spinning

The most important melt-spun polymers are polyester (PET), polyamide (PA6 and PA66) and polypropylene (PP). A scheme for a melt-spinning machine is given in Figure 17.6. We will follow the process scheme, from extruder to winder. 17.4.1

Extrusion

Some large-scale processes (for polyester and polyamide 66) use ‘‘direct spinning’’, which means that polymerization and spinning are integrated. In most cases poly-

Fig. 17.6. Scheme for a melt spinning machine (reproduced from Ref. 5): a, extruder; b, central ﬁlter; c, manifold (polymer lines); d, spin-box with pumps and spinnerets; e, cooling; h, interﬂoor tube. In section I

as-spun yarns are wound separately (i). Section II is a typical arrangement for staple ﬁber production: f, deﬂection of bundles; g, combination of bundles, which are laid down in containers.

923

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17 Fibers

mer chips are extruded, however. For melt spinning single-screw extruders are used, no diﬀerent from those applied in other ﬁelds of polymer melt processing. Screw designs may vary considerably, and certainly depend on polymer type and grade, but there are no typical melt-spinning screws. Twin-screw extruders are necessary for blending, compounding, or reactive extrusion, but these operations are not usually applied in melt spinning. Moreover, twin-screw extruders may not easily build up the high pressure necessary to pump the melt through a long distribution system (see Section 17.4.2). And, of course, twin-screw extruders are more expensive than single-screw extruders. Complete de-aeration of the melt is an absolute requirement. Remnant undissolved gas will give small bubbles in the extruded ﬁlaments and cause breaks in the spin-line. Vacuum in the hopper would solve this problem but is technically complicated. Spinning under nitrogen is more common. A proper screw design, especially of the feed zone, and temperature control should ensure suﬃcient deaeration. The size of the extruders for commercial production varies between 90 and 150 mm screw diameter, with outputs of about 100–600 kg h1 . This mass ﬂow should feed many spinning positions. For textile yarns with a low titer, the output per spinneret can be as low as 1 kg h1 and consequently an extruder then should feed at least about 100 spinning positions. For industrial yarns the output per bundle can easily go up to 50 kg h1 , and one would see about ten spinning positions per extruder. 17.4.2

Polymer Lines and Spin-box

Each spinning position should receive polymer with the same time and temperature history because yarn processing and properties depend on molecular weight in particular. Residence times of a few minutes between head extruder and spinning plate are common, and some degradation will always occur. Therefore, polymer line systems (manifolds) have simple splits (into two or three) and the same length between the head of the extruder and all positions. An example is given in Figure 17.7. It is obvious now that the examples in Section 17.4.1 were inaccurate. The textile machine would have 96 positions (3 2 5 ) rather than ‘‘about 100’’, and the industrial yarn machine eight positions (2 3 ). Many producers insert static mixers in their polymer lines, for example before

Fig. 17.7.

A polymer line system for eight spinning positions.

17.4 Melt Spinning

each split in the system. The disadvantages are an additional pressure drop and fouling of the elements. There are special ‘‘redistribution’’ elements however, guiding the stagnant wall layer to the center of the line again, with only a modest pressure drop. Fouling has already been mentioned: it is a very critical issue for spinning machines. Small particles of degraded polymer (gels) will cause ﬁlament breaks. Dead pockets must be prevented and bends in polymer lines are therefore as smooth as possible. A careful cleaning operation must be performed before a spinning machine is started. Not surprisingly, this is often carried out with high molecular weight polyoleﬁns, which are highly viscous, very elastic, not spinnable, but eﬀective as cleaning compounds. In some occasions (PA66) the distribution system can even be completely dismantled in order to be cleaned by burning in an oven at 500 C or higher. Polymer lines can be heated electrically, but temperature control is more accurate when a thermal vapor ﬂuid is used. The standard ﬂuid is a mixture of diphenyl and diphenyl oxide, with a boiling point around 260 C. Double-walled lines are therefore almost standard. The spin-box is a vapor-heated enclosure containing the spinning pumps and the spinning assemblies. Each spinning position has its own metering pump. The pump and spinning assembly are installed close to each other. The spinning assembly is inserted in a hole in the spin-box. Both top loading and bottom loading are possible. 17.4.3

Spinning Pumps

Each spinneret of a spinning machine should be fed with a constant volume ﬂow of polymer melt because otherwise the titer of the resulting yarn bundle would vary. This is done with positive displacement pumps: that is, gear pumps with low tolerances. Their output should be independent of the pressure diﬀerence over the pump. A melt-spinning pump must ensure a constant output up to 400 or 500 bar. Most pumps have one driven gearwheel and one co-rotating wheel, but two, three or more co-rotating wheels are also possible. Having a combination of two (or more) spinning pumps on one driving axis is also a fairly common practice. All spinning pumps on a machine run at the same speed, and thus the linear density of all yarn bundles is the same. The capacity of spinning pumps is indicated by their volume per revolution, usually in cm 3 ; values of 0.5–50 cm 3 are common. They are operated at 10–50 rpm. For example, for a low yarn titer the required volume ﬂow could be 50 cm 3 min1 , which would be delivered by a 2 cm 3 spinning pump running at 25 rpm; or a heavy industrial yarn titer could require a 20 cm 3 pump at 30–40 rpm. Note that volume ﬂows are adjusted while mass ﬂows are prescribed. A recalculation is easy when the density of the melt is known, but in practice a mass ﬂow check is always made.

925

926

17 Fibers

Melt-spinning pumps can be constructed from normal steel. Tolerances are not too critical when melt viscosities are suﬃciently high. Pumps are cleaned in hot ovens at b500 C. 17.4.4

Spinning Assembly

The spinning assembly can be round or rectangular. It contains a ﬁlter package on top of a supporting/distribution plate. The spinning plate is the bottom of the spinning assembly. Filtration For production of ﬁbers, usually having diameters of 10–30 mm, very ﬁne ﬁltration is necessary. The rule of thumb is that particles of more than one-third of the eventual ﬁber diameter will cause interruptions of the spinning or drawing process and should therefore be removed. The particles that must be removed can be solid (catalyst remnants, dust) or gel-like (not fully melted or crosslinked polymer). Solid particles can more easily be removed than gel particles. Pigments or delustrants may be added to the spinning melt. Note that these must almost be submicronic and should easily pass any ﬁlter. Filtration can be applied in all stages of the process: in a central ﬁlter in the polymer melt line, or at the end of the line, in the spinning assembly. Central ﬁlters are common in large-scale melt-spinning processes, for example, staple ﬁber production. Two parallel ﬁlters may be used to enable ﬁlter cleaning without interruption of the process. Rotating ﬁlters with continuous cleaning are also used. Spinning assemblies always contain ﬁlters. In many cases it is the only place where ﬁltration takes place, but there may also be a combination of preﬁltration in a central ﬁlter and a last, and ﬁnest, ﬁltration just above the spinning plate. Filter packages are usually a stack of screens, going from coarse to ﬁne in the streamline direction, on top of a support plate with holes for further distribution of the ﬂow. Screens are plain-weave, or sometimes twill-weave, steel wire fabrics. Their ﬁneness is indicated in mesh (threads per inch), openings per cm 2 , or pore size in microns. For example, 325 mesh ¼ 16 800 opening s cm2 A 40 mm. 500 mesh (25 mm) ﬁltration is common in melt spinning. For even ﬁner ﬁltration, steel nonwoven materials are used, with a ﬁlter ﬁneness going down to 5 mm. An alternative ﬁltration method is a ‘‘sand’’ ﬁlter 1–5 cm thick, on top of supporting screens. Rather than sand, a sharp-edged stainless steel powder would nowadays be used. The alleged advantage of a sand ﬁlter is the breakdown of gel particles, which would pass through a relatively thin screen package. 17.4.4.1

Spinning Plate The bottom of the assembly is the spinning plate: 5–30 mm thick. The number of holes varies enormously. A hosiery yarn may have one or three ﬁlaments, a carpet yarn 60 ﬁlaments, an industrial yarn 350 ﬁlaments. For low ﬁlament numbers per 17.4.4.2

17.4 Melt Spinning

yarn bundle, two or more bundles can be spun from one plate. The number of holes can go up to 1000–10 000 for staple ﬁber production. The size of the spinning plates is chosen accordingly, and the plates can range, for example, from round 60 mm textile plates to rectangular plates 1 m wide for staple ﬁber production. The spinning holes are relatively large: 200–500 mm round is common, <200 mm may be used for textile microtiter yarns, and >500 mm for some industrial yarns. The holes are large in comparison with the eventual diameter of the ﬁlaments. This implies that a considerable draft (see Section 17.4.13.3) is applied in the spin-line, reducing the ﬁlament diameter without however inducing much orientation. It is impossible to manufacture 200–500 mm holes in a melt-spinning plate, which must be 5–30 mm thick to withstand the high spinning pressure. Moreover, such a long narrow hole would give too high a pressure drop. In practice there is always a ‘‘backhole’’ (pre-channel, counterbore), with a diameter of 2–3 mm. The actual spinning capillary is usually short, with an L=D ratio of only 1–2. If possible, spinning holes are made by drilling or punching, because these are cheap operations. Very small and non-round holes, however, must be made by spark erosion, making use of an electrode in the form of the hole. The simplest shape of a hole backhole þ spinning capillary (see Figure 17.8a) – is rarely applied. At the least there is some smoothening of the entrance and especially of the transition region to avoid turbulence ﬂow (Figure 17.8b). Holes may have a tapered section or even a trumpet form (Figure 17.8c). The backhole is always round, but the spinning capillary may be proﬁled. One can imagine how complicated it then is to make a smooth transition zone. The holes are positioned 5–10 mm from each other. This spacing is necessary for the air-cooling process. Cold air must penetrate easily into the bundle, and some mobility of the ﬁlaments should be allowed without ﬁlaments immediately sticking as a result.

a, backhole and spinning hole Fig. 17.8.

Examples of spinning holes.

b, as a, but smoothened

c, trumpet form, tapering

927

928

17 Fibers

Fig. 17.9. Principle of a crossﬂow quenching system (reproduced from Ref. 5): a, spin-box; b, spinneret; c, ﬁlament bundle; d, ﬁlter; e, air supply.

17.4.5

Quenching

The standard quenching or cooling system is crossﬂow, in which cold air (at 15– 70 C) ﬂows from one side, perpendicular to the yarn bundle (see Figure 17.9). The disadvantage of this system is that ﬁlaments close to the blow-box cool faster than those in the front of the yarn bundle (see Figure 17.10a). This can be minimized by adapting the drilling pattern and limiting the number of rows of holes in the cross-direction (Figure 17.10b). An alternative is radial cooling (Figure 17.10c), usually outside to inside, with the disadvantage, however, that a column of hot air will be enclosed in the bundle, again resulting in diﬀerences in cooling across the bundle. Enforced cooling only takes place in the ﬁrst 0.5–1.5 m, whereas the total height of the spin-line is 1.5–6.0 m. Cooling air speeds are in the order of 0.1–1 m s1 . A standard spinning machine with a cooling height of 6 m requires at least a four-story building, with chips handling on the fourth ﬂoor and extruders, spinboxes, and cooling on the third ﬂoor (the spinning ﬂoor). On the second ﬂoor one would only see ‘‘interﬂoor tubes’’, or ‘‘chimneys’’, through which the bundles travel downward and cool further. The ﬁrst ﬂoor is the winding ﬂoor. At high spinning speeds the yarn bundle attracts its own cooling air, by selfsuction. With only a few measures to avoid turbulence and ensure symmetric stability, this can be suﬃcient for textile yarns. For industrial yarns the initial cooling

17.4 Melt Spinning

a, crossflow circular drilling pattern Fig. 17.10.

b, crossflow optimal cooling of all filaments

c, radial cooling

Yarn cooling.

is always enforced, but the remaining necessary cooling air is attracted ‘‘automatically’’ by the bundle. For highly viscous polyester and polyamide 66 yarns there may be no cooling in the ﬁrst 0.1–0.5 m. On the contrary, an electrically heated ring or box, which keeps the ﬁlaments at the spinning temperature, would be positioned around the bundle. The technique is called retarded cooling, and the device is a ‘‘quench collar’’. In this way the spinning plate is kept at an even temperature and the orientation level is reduced as well as the variation in orientation between ﬁlaments. This results in applicability of higher draw ratios, and in higher tenacities of the yarns. The cooling speed depends on the mass ﬂow per hole and the temperature difference between the ﬁlament and the air. Of course, large bundles spun at high speeds require much cooling air; this is a simple energy balance calculation. The limiting step in the cooling process is the formation of a sublayer around the ﬁlaments. Removing or refreshing this sublayer is the key for a fast and even cooling process. The ﬁlaments in a bundle should not touch each other before they are solid, below the glass transition temperature or below the crystallization temperature. Polypropylene and polyamide 66 always crystallize in the spin-line, polyamide 6 and polyester crystallize in the spin-line only at high speeds. 17.4.6

Finish

A yarn without spin ﬁnish cannot be processed; yarn–metal friction is too high, and static charge would build up. A ﬁnish is therefore applied before the yarn ﬁrst touches guides, rolls, and so on. Finishes can have sophisticated compositions, but two components are always present: lubricants and antistatics. In water-based ﬁnishes emulsiﬁers are a third essential component. Finishes are speciﬁc for certain

929

930

17 Fibers

yarn types: a polyamide ﬁnish cannot be used on polyester or polypropylene; a ﬁnish for polyester textile cannot be used on a polyester tire yarn. Finishes must be adapted to the polarity of the ﬁber polymer. They must spread over the surface within a very short time. Another criterion is that the ﬁnish should not interfere with later applications. For example, components which are necessary in a textile ﬁnish could ruin the rubber adhesion of a tire cord. Knowledge of ﬁnishes used to be proprietary, with most ﬁber producers using their own ﬁnishes and keeping the compositions secret. Commercial producers have developed good general-purpose ﬁnishes, however, which have found wide application. Only a few larger ﬁber producers still develop their own ﬁnishes, especially for critical processes. One feature has remained: still, ﬁber producers do not reveal which ﬁnish is used or how it is applied. Spin ﬁnishes often used to be solutions in white spirit, but for environmental reasons this is no longer the case. Emulsions in water, or ﬁnishes without a diluent – so-called neat oils, with components of a suﬃciently low viscosity – are used instead. Application of ﬁnish requires special applicators. Fairly old-fashioned is a ﬁnish roll, a porous ceramic roll which constantly takes up ﬁnish from a bath, which is then transferred to the yarn that just touches the surface of the roll (see Figure 17.11a). More modern applicators accurately supply the required amount of ﬁnish via a small gear pump, to ‘‘stift’’ or ‘‘block’’ applicators, through a hole or slit. The yarn is spread as a ribbon over the applicator and takes up the ﬁnish (Figure 17.11b,c). At higher yarn speeds the even distribution of ﬁnish becomes increasingly diﬃcult. An applicator with multiple slits or a ﬁnish wheel (a roll with small holes under the yarn path) may then be applied. Finishes of a diﬀerent nature, so-called after-oils, may be applied at the end of the spinning and drawing process to improve the yarn for a speciﬁc application, for example to produce a wrinkle-free textile fabric, to make carpet yarns antistatic

a, finish roll Fig. 17.11.

b, stift applicator

Application of ﬁnish.

c, block applicator

17.4 Melt Spinning

and/or non-soiling, or to enhance PVC or rubber adhesion for industrial yarns. There is a distinct overlap with other ﬁnish-like aftertreatments such as sizes for weaving and dips for tire cords. 17.4.7

Spinning Speed

Falling under its own weight a spinning bundle would reach a speed of 300– 400 m min1 . In practice, spinning speeds of at least 500 m min1 are required to obtain a suﬃciently high tension in the spin-line. Much higher speeds can be enforced, as long as the ﬁlaments do not break under the constantly increasing tension. Speeds of 4000–8000 m min1 have become common practice, especially for polyester. This is called high-speed spinning (HSS), in contrast to LSS (< 1000 m min1 ). The air friction can only build up modest forces in the ﬁlaments. Even at high speeds the molecular orientation level remains limited, and further drawing is always possible, or even necessary. For further information on HSS see Ref. 7 and Section 17.4.14. 17.4.8

Winding

An old-fashioned winder, operating at 500 m min1 , was still a simple piece of equipment. Modern high-speed winders, operating at 6000 m min1 or more, are ‘‘high-tech’’ devices, the area of specialists. In order to make packages with a good cylindrical shape, the traverse system of the yarn must be very rapid and smooth, especially at the reversal points. A second problem is how to handle the very great weights of the packages, for example, four packages of 30 kg each, wound on one spindle, at >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

946

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-

17.5 Solution Spinning

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

948

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

17.5 Solution Spinning

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

952

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

953

954

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

955

956

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

957

958

17 Fibers

Fig. 17.22.

Lyotropic behavior of PPTA in sulfuric acid.

Fig. 17.23.

Orientation of aramid in spinning hole and air gap.

17.7 High-modulus, High-strength Fibers

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 <1 dtex. The aramid process as described is a wet spinning process, but a fairly strange one. Extrusion, polymer lines, and ﬁltration are like those in melt spinning. The heart of the spinning process, spinneret–air gap–spinning bath or ‘‘coagulator’’, is very speciﬁc for aramid, but would also be found on machines for other lyotropic systems. The section after the coagulation again deviates from a normal wet spinning process. Since the ﬁlaments are not deformable, high tensions can be applied but drawing is not necessary. The construction material for PPTA in the ‘‘melt-spinning section’’ can be a normal steel because 100% sulfuric acid is not very corrosive. In places with high shear (extruder screw, spinning pumps), nitrated steel, stainless steel, or even ceramic inserts will be used, however. As soon as the acid becomes diluted, corrosion becomes very severe. Spinning plates are therefore made of gold/platinum or with capillary inserts of this material. The construction materials for the spinning bath and washing sections are stainless steel, PVC, PP, or glass. Technora There is one aramid product with a deviating composition. Teijin’s Technora (3) is produced from terephthaloyl dichloride (TPC) and a 50:50 mixture of two aromatic diamines: PPD (as in PPTA) and 3,4 0 -diaminodiphenyl ether. This ‘‘kinked’’ monomer has the advantage that the polymer remains dissolved in the polymerization solvent (NMP/CaCl2 ). The solution can be spun directly, but is not liquid-crystalline! After washing and drying the yarn is drawn at high temperature (> 500 C) to a high ratio (> 10). In fact Technora is an example of superdrawabil-

959

960

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 High-modulus, High-strength Fibers

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

961

962

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

17.7 High-modulus, High-strength Fibers

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,

17.7 High-modulus, High-strength Fibers

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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Removal of Monomers and VOCs from Polymers1 Marı´a 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

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

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

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18 Removal of Monomers and VOCs from Polymers

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

18.2 Polymer Melts and Solutions

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

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18 Removal of Monomers and VOCs from Polymers

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.

18.2 Polymer Melts and Solutions

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

18 Removal of Monomers and VOCs from Polymers

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

18 Removal of Monomers and VOCs from Polymers

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

18 Removal of Monomers and VOCs from Polymers

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.

18.4 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

18 Removal of Monomers and VOCs from Polymers

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

18.4 Waterborne Dispersions

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

18 Removal of Monomers and VOCs from Polymers

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

18 Removal of Monomers and VOCs from Polymers

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 Sa´nchez, 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

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

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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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19 Nano- and Microstructuring of Polymers

(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

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19 Nano- and Microstructuring of Polymers

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

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19 Nano- and Microstructuring of Polymers

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).

19.3 Single-exposure Photoembossing

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

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19 Nano- and Microstructuring of Polymers

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

19.3 Single-exposure Photoembossing

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

19 Nano- and Microstructuring of Polymers

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

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19 Nano- and Microstructuring of Polymers

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).

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19 Nano- and Microstructuring of Polymers

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

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19 Nano- and Microstructuring of Polymers

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 Sa´nchez, 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 Sa´nchez, 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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1027. Y. Xia, J. A. Rogers, K. E. Paul, G. M. Whitesides, Chem. Rev., 1999, 99, 1823. P. Ehbets, H. P. Herzig, P. Nussbaum, P. Blattner, R. Da¨ndliker, Appl. Opt., 1995, 34, 2540. D. C. Flanders, Appl. Phys. Lett., 1983, 42, 492. X. T. Li, A. Natansohn, P. Rochon, Appl. Phys. Lett., 1999, 74, 3791. Y. B. Boiko, V. S. Solovjev, S. Calixto, D. J. Lougnot, Appl. Opt., 1994, 33, 787. S. T. Han, Y. L. Tsao, R. M. Walser, M. F. Becker, Appl. Opt., 1992, 31, 2343. D. Schertler, N. George, Appl. Opt., 1999, 38, 291. H. J. Levinson, Principles of Lithography, SPIE, Washington, 2001. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberﬁeld, Nature, 2000, 404, 53. A. K. Bates, M. Rothschild, T. M. Bloomstein, T. H. Fedynyshyn, R. R. Kunz, V. Liberman, M. Switkes, IBM J. Res. Dev., 2001, 45, 605.

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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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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

20.2 Process Analysis

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

20.2 Process Analysis

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 Chemical Analysis for Polymer Engineers

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

20 Chemical Analysis for Polymer Engineers

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.

20.3 Polymer Analysis

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

20 Chemical Analysis for Polymer Engineers

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)

20.3 Polymer Analysis

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

20 Chemical Analysis for Polymer Engineers

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

20 Chemical Analysis for Polymer Engineers

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

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20 Chemical Analysis for Polymer Engineers

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

20.3 Polymer Analysis

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

20 Chemical Analysis for Polymer Engineers

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).

20.3 Polymer Analysis

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

1037

1038

20 Chemical Analysis for Polymer Engineers

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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20 Chemical Analysis for Polymer Engineers

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-

20.3 Polymer Analysis

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

1041

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20 Chemical Analysis for Polymer Engineers

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

20 Chemical Analysis for Polymer Engineers

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

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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.

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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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15

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19

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21

22

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.

27

28

29

30

31

32

33

34

35

36

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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21

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.

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

SR post-processing

1048

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

21.2 Polymer Processes in Supercritical Carbon Dioxide

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

21 Recent Developments in Polymer Processes

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

21.2 Polymer Processes in Supercritical Carbon Dioxide

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

21 Recent Developments in Polymer Processes

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

21.2 Polymer Processes in Supercritical Carbon Dioxide

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

21.2 Polymer Processes in Supercritical Carbon Dioxide

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

21 Recent Developments in Polymer Processes

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-

21.2 Polymer Processes in Supercritical Carbon Dioxide

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.

1060

21 Recent Developments in Polymer Processes

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

21.2 Polymer Processes in Supercritical Carbon Dioxide

1061

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

21 Recent Developments in Polymer Processes

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

21 Recent Developments in Polymer Processes

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].

21.3 Ultrasound-induced Radical Polymerization

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

21 Recent Developments in Polymer Processes

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.

21.3 Ultrasound-induced Radical Polymerization

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

21 Recent Developments in Polymer Processes

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)

21.3 Ultrasound-induced Radical Polymerization

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

21 Recent Developments in Polymer Processes

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

21.3 Ultrasound-induced Radical Polymerization

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

21 Recent Developments in Polymer Processes

[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].

21.3 Ultrasound-induced Radical Polymerization

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-

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21 Recent Developments in Polymer Processes

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

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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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F. de Jong, Recl. Trav. Chim. Pays-Bas, 1994, 113, 533. M. Ibisi, B. Brown, J. Acoust. Soc. Am., 1967, 41, 568. J.-L. Luche, Synthetic Organic Chemistry, Plenum, New York, 1998. G. J. Price, P. J. West, P. F. Smith, Ultrason. Sonochem., 1994, 1, S51. M. W. A. Kuijpers, M. F. Kemmere, J. T. F. Keurentjes, Annual AIChE Meeting, 2001, paper 341B. G. J. Price, P. F. Smith, Polymer, 1993, 34, 4111. P. Kruus, J. A. G. Lawrie, M. L. O’Neill, Ultrasonics, 1988, 26, 352. G. J. Price, Chemistry & Industry, 1993, 75. N. Erolan, F. Arisan, H. C ¸ atalgilGiz, Angew. Makromol. Chem., 1995, 226, 53. P. Kruus, Ultrasonics, 1987, 25, 20. D. Feldman, Polymer News, 1995, 20, 138. P. G. Jessop, W. Leitner, Chemical Synthesis using Supercritical Fluids, Wiley-VCH, Weinheim, 1999. M. W. A. Kuijpers, Physics and Chemistry of Ultrasound Polymer Reaction Engineering, PhD thesis, Technische Universiteit Eindhoven, Eindhoven, 2004. M. F. Kemmere, M. W. A. Kuijpers, L. J. M. Jacobs, J. T. F. Keurentjes, Macromol. Symp., 2004, 206, 321. H. C. Chou, J. O. Stoffer, J. Appl. Polym. Sci., 1999, 72, 797. S. K. Ooi, S. Biggs, Ultrason. Sonochem., 2000, 7, 125. G. Cooper, F. Grieser, S. Biggs, J. Colloid Interface Sci., 1996, 184, 52. C. Zhang, Q. Wang, H. Xia, G. Qiu, Eur. Polym. J., 2002, 38, 1769. H. C. Chou, J. O. Stoffer, J. Appl. Polym. Sci., 1999, 72, 827. S. Biggs, F. Grieser, Macromolecules, 1995, 28, 4877. M. Bradley, M. Ashokkumar, F. Grieser, J. Am. Chem. Soc., 2003, 125, 525. H. C. Chou, W. Y. Lin, J. O. Stoffer, Polym. Mater. Sci. Eng, 1995, 72, 363. H. C. Chou, Y. B. Liu, O. C. Sitton,

161

162

163

164

165 166

167

168 169 170 171

172

173

174

175 176 177

178

J. O. Stoffer, Polym. Mater. Sci. Eng., 1993, 69, 376. Y. Liao, Q. Wang, H. Xia, X. Xu, S. M. Baxter, R. V. Slone, S. Wu, G. Swift, D. G. Westmoreland, Ultrason. Sonochem., 2002, 9, 151. Q. Wang, H. Xia, Y. Liao, X. Xu, S. M. Baxter, R. V. Slone, S. Wu, G. Swift, D. G. Westmoreland, Polym. Int., 2001, 50, 1252. A. V. Machado, J. A. Covas, M. van Duin, J. Appl. Polym. Sci., 2001, 81, 58. P. A. R. Glyn, B. M. E. van der Hoff, P. M. Reilly, J. Macromol. Sci.Chem. A6, 1972, 1653. B. M. E. van der Hoff, C. E. Gall, J. Macromol. Sci. A, 1977, 11, 1739. G. Madras, S. Kumar, S. Chattopadhyay, Polym. Degr. Stab., 2000, 59, 73. M. W. A. Kuijpers, P. D. Iedema, M. F. Kemmere, J. T. F. Keurentjes, Polymer, 2004, 45, 6461. A. M. Basedow, K. H. Ebert, Adv. Polym. Sci., 1977, 22, 83. J. A. Odell, A. Keller, J. Polym. Sci. B, 1986, 24, 1889. G. J. Price, P. J. West, Polymer, 1996, 37, 3975. I. W. Hamley, Block copolymers, in Encyclopedia of Polymer Science and Technology, John Wiley, New York, 2002. T. J. Mason, Sonochemistry, The Uses of Ultrasound in Chemistry, CRC Press, Cleveland, 1990. H. Fujiwara, J. Tanaka, A. Horiuchi, Polymer Bulletin, 1996, 36, 723. G. Schmidt-Naake, M. Drache, M. Weber, Macromol. Chem. Phys., 2002, 15, 203. H. Fujiwara, Polymer Bulletin, 2001, 47, 247. S. Fortini, F. Lavanchy, Th. Meyer, Macromol. Symp., 2004, 206, 79. F. Lavanchy, S. Fortini, Th. Meyer, Organic Process Research Development, 2004, 8(3), 504. P. Nising, Th. Meyer, Ind. Eng. Chem. Res., 2004, 43, 7220.

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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

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

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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