Handbook of Multiphase Polymer Systems

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Author: Abderrahim Boudenne | Laurent Ibos | Yves Candau | Sabu Thomas

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Handbook of Multiphase Polymer Systems Volume 1

Editors

ABDERRAHIM BOUDENNE, LAURENT IBOS, YVES CANDAU Université Paris-Est, Centre d’Etude et de Recherche en Thermique, Environnement et Systèmes, Créteil, France

AND SABU THOMAS Centre for Nanoscience and Nanotechnology, Mahatma Gandhi University, Kottayam, Kerala, India

A John Wiley & Sons, Ltd., Publication

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This edition first published 2011 C 2011 John Wiley and Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom. Library of Congress Cataloging-in-Publication Data Handbook of multiphase polymer systems / editors, Abderrahim Boudenne ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-0-470-71420-1 (cloth) – ISBN 978-1-119-97203-7 (ePDF) – ISBN 978-1-119-97202-0 (oBook) 1. Polymeric composites. I. Boudenne, Abderrahim. TA418.9.C6H3426 2012 547 .7–dc22 2011011524 A catalogue record for this book is available from the British Library. Print ISBN: 9780470714201 ePDF ISBN: 9781119972037 oBook ISBN: 9781119972020 ePub ISBN: 9780470714201 Mobi ISBN: 9781119972891 Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

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List of Contributors

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Foreword

xxiii VOLUME 1

1 Physical, Thermophysical and Interfacial Properties of Multiphase Polymer Systems: State of the Art, New Challenges and Opportunities Sabu Thomas, Abderrahim Boudenne, Laurent Ibos and Yves Candau 1.1 Introduction 1.2 Multiphase Polymer Systems 1.2.1 Polymer Blends 1.2.2 Polymer Composites 1.2.3 Polymer Nanocomposites 1.2.4 Polymer Gels 1.2.5 Interpenetrating Polymer Network System (IPNs) 1.3 A Short Survey of the Literature and Applications 1.4 Book Content 1.4.1 Modeling and Computer Simulation of Multiphase Composites: From Nanoscale to Macroscale Properties 1.4.2 Morphological Investigation Techniques 1.4.3 Macroscopic Physical Characterization 1.4.4 Life Cycling 1.5 Future Outlook, New Challenges and Opportunities References 2 Macro, Micro and Nano Mechanics of Multiphase Polymer Systems Alireza S. Sarvestani and Esmaiel Jabbari 2.1 Introduction 2.2 Unentangled Systems 2.2.1 Microscopic Structure 2.2.2 Macroscopic Properties 2.2.3 Results 2.3 Entangled Systems 2.3.1 Microscopic Structure 2.3.2 Macroscopic Properties 2.3.3 Results 2.4 Conclusion Acknowledgements References

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3 Theory and Simulation of Multiphase Polymer Systems Friederike Schmid 3.1 Introduction 3.2 Basic Concepts of Polymer Theory 3.2.1 Fundamental Properties of Polymer Molecules 3.2.2 Coarse-Graining, Part I 3.2.3 Ideal Chains 3.2.4 Interacting Chains 3.2.5 Chain Dynamics 3.3 Theory of Multiphase Polymer Mixtures 3.3.1 Flory-Huggins Theory 3.3.2 Self-consistent Field Theory 3.3.3 Analytical Theories 3.3.4 An Application: Interfaces in Binary Blends 3.4 Simulations of Multiphase Polymer Systems 3.4.1 Coarse-Graining, Part II 3.4.2 Overview of Structural Models 3.4.3 Overview of Dynamical Models 3.4.4 Applications 3.5 Future Challenges Acknowledgements References 4 Interfaces in Multiphase Polymer Systems György J. Marosi 4.1 Introduction 4.2 Basic Considerations 4.3 Characteristics of Interfacial Layers 4.3.1 Role of Thermodynamic Factors 4.3.2 Role of Kinetic Factors 4.3.3 Relationship Between Interfacial Structure and Mechanical Response 4.4 Interface Modifications: Types and Aims 4.4.1 Interlayers of Controlled Morphology 4.4.2 Interlayers of Modified Segmental Mobility 4.4.3 Interlayers for Improving the Compatibility of the Phases 4.5 Interlayers of Modified Reactivity 4.6 Responsive Interphases 4.6.1 Non-reversibly Adaptive Interphases 4.6.2 Smart Reversibly Adaptive Interphases 4.7 Methods of Interface Analysis 4.8 Conclusions References 5 Manufacturing of Multiphase Polymeric Systems Soney C. George and Sabu Thomas 5.1 Introduction

31 31 32 32 33 34 36 38 39 39 43 49 55 56 56 58 62 65 70 70 71 81 81 82 83 85 87 88 89 90 92 93 100 101 101 103 105 111 112 123 123

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5.2

5.3

5.4

5.5

5.6

5.7

Manufacturing Techniques of Polymer Blends 5.2.1 Solution Blending 5.2.2 Latex Blending 5.2.3 Freeze-drying 5.2.4 Mechanical Blending 5.2.5 Mechano-chemical Blending 5.2.6 Manufacturing of Polymer Blends Using Supercritical Fluids Manufacturing Techniques of Polymer Composites 5.3.1 Hand Layup Process 5.3.2 Spray Layup Process 5.3.3 Vacuum Bag Molding 5.3.4 Resin Transfer Molding 5.3.5 Pultrusion 5.3.6 Filament Winding Process 5.3.7 Reaction Injection Molding 5.3.8 Rotational Molding Manufacturing Techniques of Nanocomposites 5.4.1 Solution Intercalation 5.4.2 In Situ Intercalative Polymerization Method 5.4.3 Melt Intercalation or Melt Blending Method Manufacturing Techniques of Polymer Gels 5.5.1 Microgels 5.5.2 Aerogels 5.5.3 Xerogels 5.5.4 Nanostructured Gels 5.5.5 Topological Networks 5.5.6 Hydrogels Manufacturing Techniques of Interpenetrating Polymer Networks (IPNs) 5.6.1 Full IPNs 5.6.2 Sequential IPNs 5.6.3 Simultaneous Interpenetrating Networks (SINs) 5.6.4 Latex IPNs 5.6.5 Thermoplastic IPNs 5.6.6 Semi-IPNs 5.6.7 Pseudo-IPNs Conclusion and Future Outlook References

6 Macro, Micro and Nanostructured Morphologies of Multiphase Polymer Systems Han-Xiong Huang 6.1 Introduction 6.1.1 Polymer Blends 6.1.2 Polymer and Its Blend Nanocomposites 6.2 Morphology Development Mechanisms of Multiphase Polymer Systems During Processing 6.2.1 Initial Morphology Development in Polymer Blending 6.2.2 Deformation and Breakup of Droplet

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123 124 125 126 126 132 132 133 134 134 135 136 138 139 140 140 141 141 142 143 143 143 145 145 146 146 148 149 150 151 152 153 154 154 156 156 157 161 161 161 162 164 164 168

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6.2.3 6.2.4

6.3

6.4

Coalescence of Droplet Intercalated, Exfoliated, and Dispersed Mechanism of Organoclay During Melt Mixing Material-Relevant Factors Affecting the Morphology 6.3.1 Viscosity of Components 6.3.2 Elasticity of Components 6.3.3 Interfacial Tension 6.3.4 Compatibilization 6.3.5 Composition 6.3.6 Nanoparticles Processing-Relevant Factors Affecting the Morphology 6.4.1 Flow Field Types 6.4.2 Chaotic Mixing 6.4.3 Mixing Sequence 6.4.4 Processing Parameters Nomenclature Acknowledgements References

7 Mechanical and Viscoelastic Characterization of Multiphase Polymer Systems Poornima Vijayan P., Siby Varghese and Sabu Thomas 7.1 Introduction 7.2 Polymer Blends 7.2.1 Ultimate Mechanical Properties and Modeling 7.2.2 Dynamic Mechanical Properties 7.2.3 Impact Properties 7.2.4 Nanostructured Polymer Blends 7.3 Interpenetrating Polymer Networks (IPNs) 7.3.1 Modeling of Mechanical Properties of IPNs 7.4 Polymer Gels 7.5 Polymer Composites 7.5.1 Mechanical Properties of Polymer Macrocomposites 7.5.2 Mechanical Properties of Polymer Microcomposites 7.5.3 Mechanical Properties of Polymer Nanocomposites 7.5.4 Mechanical Modeling of Polymer Nanocomposites 7.6 Conclusion, Future Trends and Challenges References 8 Rheology and Viscoelasticity of Multiphase Polymer Systems: Blends and Block Copolymers Jean-Charles Majesté and Antonio Santamar´ıa 8.1 Introduction 8.2 Morphology of Polymer Blends 8.2.1 Morphology Characterization 8.2.2 Effect of Rheological Parameters on Morphology

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8.3

8.4

8.5 8.6

8.7

8.8

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Microrheology of Droplet Deformation 8.3.1 Breakup 8.3.2 Coalescence Rheology of Polymer Blends 8.4.1 Specificity of Blend Rheology 8.4.2 Blending Laws and Viscoelasticity Models 8.4.3 Low Frequency Viscoelastic Behavior of Polymer Blends Microphase Separated Block Copolymers 8.5.1 Ordered State and Morphologies in Block Copolymers: The Case of SEBS Triblock Dynamic Viscoelastic Results of SEBS Copolymers 8.6.1 Low and Intermediate Frequency Viscoelastic Behavior 8.6.2 Thermorheological Complexity 8.6.3 Specific Mechanical Relaxation at Low Frequencies Flow-induced Morphological Changes 8.7.1 Order–order Transition and Flow Alignment in Block Copolymers 8.7.2 Flow Alignment in a SEBS Copolymer Capillary Extrusion Rheometry Results of Block Copolymers 8.8.1 General Results of Styrenic Block Copolymers 8.8.2 Viscosity and Flow Instabilities in SEBS Copolymers Summary References

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318 318 321 322 322 326 335 339 339 340 340 341 343 345 345 345 347 347 350 354 354

9 Thermal Analysis of Multiphase Polymer Systems György J. Marosi, Alfréd Menyhárd, Géza Regdon Jr. and József Varga 9.1 Introduction 9.2 Thermo-optical Microscopy 9.3 Differential Scanning Calorimetry 9.4 Temperature Modulated Differential Scanning Calorimetry 9.5 Micro- and Nanothermal Analysis 9.6 Thermal Gravimetric Analysis and Evolved Gas Analysis 9.7 Conclusions References

359 360 365 373 376 378 380 381

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Thermophysical Properties of Multiphase Polymer Systems Abderrahim Boudenne, Laurent Ibos and Yves Candau 10.1 Introduction 10.2 Thermophysical Properties: Short Definitions 10.3 Measurement Techniques 10.3.1 Methods for the Measurement of One Property 10.3.2 Methods for the Simultaneous Measurement of Several Parameters 10.4 Thermophysical Properties of Polymers and Composite Systems 10.4.1 Neat Polymers (Unfilled Systems) 10.4.2 Thermophysical Behavior of Composites 10.5 Summary References

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Electrically Conductive Polymeric Composites and Nanocomposites ˇ Igor Krupa, Jan Prokeˇs, Ivo Kˇrivka and Zdeno Spitalsk´ y 11.1 Introduction 11.2 Theory 11.2.1 Percolation Models 11.3 Electrically Conductive Fillers 11.3.1 Carbon Black 11.3.2 Metallic Fillers 11.3.3 Metallized Fillers 11.3.4 Graphite 11.3.5 Carbon Nanotubes (CNT) 11.3.6 Conducting Polymers 11.3.7 Fillers Coated by Conducting Polymers 11.4 Effect of Processing Conditions on the Electrical Behavior of Composites 11.4.1 Blending of Polymeric Composites 11.4.2 Effects of the Secondary Processing Steps on Conductivity 11.4.3 Effect of Polymer Characteristics on the Electrical Conductivity of Composites 11.4.4 Crystallinity Effect 11.4.5 Effect of Polymer–Filler Interaction 11.4.6 Multiphase Morphology of Polymers and Its Influence on the Conductivity of Composites: Multipercolation Effect 11.5 Applications 11.5.1 EMI 11.5.2 ESD 11.5.3 Electrically Conductive Adhesives 11.5.4 Conductive Rubbers 11.5.5 Semi-Conductive Cable Compounds 11.5.6 Fuel Cells 11.6 Resistance Measurements 11.6.1 Two-Probes Method 11.6.2 Four-Probes Method 11.6.3 Van der Pauw Method 11.6.4 Spreading Resistance of the Contacts 11.6.5 Contact Resistance References

425 425 426 427 432 432 435 438 438 443 449 451 452 452 453 454 454 455 456 457 457 457 458 458 458 458 458 459 460 465 468 470 472

VOLUME 2 12

Dielectric Spectroscopy and Thermally Stimulated Depolarization Current Analysis of Multiphase Polymer Systems Polycarpos Pissis, Apostolos Kyritsis and Daniel Fragiadakis 12.1 Introduction 12.2 Dielectric Techniques 12.2.1 Introduction 12.2.2 Broadband Dielectric Spectroscopy (DS) 12.2.3 Thermally Stimulated Depolarization Current (TSDC) Techniques

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12.3

12.4

12.5

12.6

13

Copolymers and Interpenetrating Polymer Networks Based on Poly(alkyl acrylate)s and Poly(alkyl methacrylate)s (Mixing and Phase Separation) 12.3.1 Introduction 12.3.2 Poly(butyl acrylate)-Poly(butyl methacrylate) Sequential Interpenetrating Polymer Networks 12.3.3 Poly(butyl acrylate)-Poly(methyl methacrylate) Interpenetrating Polymer Networks and Copolymers 12.3.4 Poly(ethyl methacrylate)-Poly(hydroxyethyl acrylate) Copolymers 12.3.5 Concluding Remarks Rubber/Silica Nanocomposites (Interfacial Phenomena) 12.4.1 Introduction 12.4.2 TSDC Studies 12.4.3 Broadband DS Studies 12.4.4 Concluding Remarks Polymer Nanocomposites with Conductive Carbon Inclusions (Percolation Phenomena) 12.5.1 Introduction 12.5.2 Analysis of DS Data in Terms of the Dielectric Function 12.5.3 Analysis of DS Data in Terms of ac Conductivity 12.5.4 Concluding Remarks Conclusion Acknowledgements References

Solid-State NMR Spectroscopy of Multiphase Polymer Systems Antonio Mart´ınez-Richa and Regan L. Silvestri 13.1 Introduction to NMR 13.2 Phases in Polymers: Polymer Conformation 13.3 High Resolution 13 C NMR Spectroscopy of Solid Polymers 13.3.1 Chemical Shift 13.3.2 Polyolefines 13.3.3 Polyesters 13.3.4 Carbohydrates 13.3.5 Conducting Polymers 13.3.6 Polymer Blends 13.3.7 Interactions Between Polymers and Low-molecular Weight Compounds 13.3.8 Miscellaneous Polymers 13.4 Additional Nuclei 13.5 NMR Relaxation 13.5.1 NMR Relaxation in the Study of Polymer Blends 13.5.2 NMR Relaxation in the Study of Copolymers 13.5.3 NMR Relaxation in the Study of Polymer Composites 13.5.4 NMR Relaxation in the Study of Polymers for Drug Delivery 13.6 Spin Diffusion 13.7 Concluding Remarks References

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486 486 488 493 496 498 499 499 500 503 506 507 507 507 510 512 512 513 513 519 520 522 525 525 526 526 530 530 531 535 536 536 538 539 542 543 543 544 546 546

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ESR Spectroscopy of Multiphase Polymer Systems Srećko Valić, Mladen Andreis and Damir Klepac 14.1 Introduction 14.2 Theoretical Background 14.3 Copolymers 14.4 Grafted Polymers 14.5 Blends 14.6 Crosslinked Polymers 14.7 Semi-Interpenetrating Networks (SIPNs) 14.8 Composites 14.9 Nanocomposites 14.10 Other Polymer Multiphase Systems 14.11 Conclusion References

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XPS Studies of Multiphase Polymer Systems Mohamed M. Chehimi, Fatma Djouani and Karim Benzarti 15.1 Introduction 15.2 Basic Principles of X-ray Photoelectron Spectroscopy 15.2.1 Photoionization 15.2.2 Surface Specificity of XPS 15.2.3 Spectral Examination and Analysis 15.2.4 Quantification 15.2.5 Determination of Overlayer Thickness 15.2.6 Instrumentation 15.3 Applications of XPS to Polymeric Materials 15.3.1 Polymer Grafts 15.3.2 Colloidal Particles 15.3.3 Epoxy Adhesives 15.3.4 Conductive Polymers 15.3.5 Polymer Blends 15.3.6 Composites 15.3.7 Interpenetrating Polymer Networks 15.3.8 Random and Block Copolymers 15.4 Conclusion Glossary References Light Scattering Studies of Multiphase Polymer Systems Yajiang Huang, Xia Liao, Qi Yang and Guangxian Li 16.1 Introduction 16.2 Light Scattering Technique 16.2.1 Scattering from Multiphase Polymer Systems 16.2.2 Experiment 16.2.3 Intensity Calibration

551 551 555 560 569 569 571 571 573 575 576 578 579 585 585 586 586 587 587 592 592 597 599 600 605 609 613 619 624 627 628 629 630 631 639 639 640 640 642 645

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16.3

16.4 16.5

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Phase Behavior of Multiphase Polymer Systems Studied by SALS 16.3.1 Thermodynamics 16.3.2 Phase Separation Dynamics 16.3.3 Reaction-induced Phase Separation 16.3.4 Phase Behavior of Polymer Blends Under Shear Flow 16.3.5 Multi-scale Approaches in Studying the Phase Behavior of Polymer Blends On-line Morphological Characterization of Polymer Blends Light Scattering Characterization of Other Multiphase Polymer Systems 16.5.1 Gelation 16.5.2 Crystallization References

X-ray Scattering Studies on Multiphasic Polymer Systems Z. Z. Denchev and J. C Viana 17.1 Introduction 17.2 Theoretical Background 17.2.1 Microfibrillar Reinforced Composites (MCF): Definition and Preparation 17.2.2 Clay-containing Polymer Nanocomposites 17.2.3 The use of WAXS and SAXS in Characterization of Polymers 17.3 Studies on Multiphase Polymer Systems 17.3.1 Polyamide 6/montmorillonite Nanocomposites 17.3.2 Microfibrillar Composites (MFC) 17.3.3 Immiscible Polymer Blends 17.3.4 Non-conventional Molding of PP Nanocomposites 17.3.5 Stretching of Nanoclay PET Nanocomposite 17.4 Concluding Remarks Acknowledgements References Characterization of Multiphase Polymer Systems by Neutron Scattering Max Wolff 18.1 Introduction 18.2 Method of Neutron Scattering 18.2.1 Scattering Experiment 18.2.2 Born Approximation 18.2.3 Elastic and Quasielastic Scattering 18.2.4 Scattering at Small Momentum Transfer 18.3 Experimental Techniques 18.3.1 Production and Detection of Neutrons 18.3.2 Instrumentation 18.3.3 Grazing Incidence Small Angle Scattering 18.3.4 Comparison of SANS and GISANS for Crystalline Systems 18.4 Recent Experimental Results 18.4.1 Polymer Dynamics 18.4.2 Contrast Variation 18.4.3 Effect of Shear 18.4.4 Near Surface Crystallization of Micelles

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646 646 647 649 651 655 656 658 658 660 663 669 669 671 671 672 673 679 679 681 693 695 697 699 700 700 705 705 705 706 708 710 713 716 716 719 724 726 729 729 730 730 733

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Conclusion Acknowledgements References

Gas Diffusion in Multiphase Polymer Systems Eliane Espuche 19.1 Introduction 19.2 Gas Transport Mechanisms in Dense Polymer Films: Definition of the Transport Parameters 19.3 Multiphase Polymer Systems for Improved Barrier Properties 19.3.1 Introduction 19.3.2 Dispersion of Impermeable Spheres Within a Polymer Matrix 19.3.3 Influence of the Shape of the Dispersed Impermeable Phase: Interest of Oriented Polymer Blends and of the Nanocomposite Approach 19.3.4 Nanocomposites Based on Lamellar Nanofillers 19.3.5 Multilayers 19.3.6 Active Films 19.3.7 Comparison of the Different Ways Used to Improve Barrier Properties 19.4 Multiphase Polymer-based Systems for Improved Selectivity 19.4.1 Introduction 19.4.2 Organic–inorganic Materials for Gas Separation Membranes 19.5 Conclusion References Nondestructive Testing of Composite Materials Zhongyi Zhang and Mel Richardson 20.1 Introduction 20.2 Failure Mechanisms in Polymer Composites 20.2.1 Matrix Deformation 20.2.2 Fiber–matrix Debonding 20.2.3 Matrix Cracking 20.2.4 Delamination 20.2.5 Fiber Breakage 20.2.6 Combination of Different Failure Modes 20.3 Visual Inspection 20.4 Acoustic Emission 20.5 Ultrasonic Scanning 20.6 Radiography 20.7 Thermography 20.8 Laser Interferometry 20.9 Electronic Shearography 20.10 Optical Deformation and Strain Measurement System 20.11 Summary References

743 743 743 749 749 749 752 752 752 755 757 763 764 766 767 767 767 769 770 777 777 779 779 779 779 780 780 780 781 782 783 785 786 788 790 791 795 795

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Ageing and Degradation of Multiphase Polymer Systems Xavier Colin, Gilbert Teyssedre and Magali Fois 21.1 Introduction 21.1.1 Issues Associated With Material Ageing 21.1.2 Classification of Ageing Types 21.2 Physical Ageing 21.2.1 Ageing Induced by Structural Reorganization 21.2.2 Ageing Induced by Solvent Absorption 21.2.3 Ageing Induced by Additive Migration 21.3 Chemical Ageing 21.3.1 General Aspects 21.3.2 Mechanistic Schemes 21.4 Impact of Multiphase Structure on Ageing Processes 21.4.1 Structural Reorganization 21.4.2 Diffusion Controlled Processes 21.5 Practical Impact of Physical Ageing on Use Properties 21.5.1 Water-Induced Mechanical Damages in Composites 21.5.2 Ageing of Electrical Insulations 21.6 Concluding Remarks References Fire Retardancy of Multiphase Polymer Systems Michel Ferriol, Fouad Laoutid and José-Marie Lopez Cuesta 22.1 Introduction 22.2 Combustion and Flame Retardancy of Polymers 22.2.1 Combustion of Polymers 22.2.2 Flame Retardancy 22.3 Laboratory Fire Testing 22.3.1 Limiting Oxygen Index (LOI) 22.3.2 Epiradiator or ‘Drop Test’ 22.3.3 UL 94 22.3.4 Cone Calorimeter 22.4 Flame Retardant Additives 22.4.1 Hydrated Fillers 22.4.2 Halogenated Flame Retardants 22.4.3 Phosphorus-based Flame Retardants 22.4.4 Nanometric Particles 22.5 Synergistic Effects of Fillers with Flame Retardant Additives 22.5.1 Definition of Synergistic Effects in Flame Retardant Systems 22.5.2 Micronic Fillers and Flame Retardants 22.5.3 Nanometric Fillers and Flame Retardants 22.6 Conclusion Acknowledgements References

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797 797 797 798 799 800 802 803 806 806 816 829 829 831 832 832 833 839 840 843 843 844 844 845 846 846 847 848 848 849 849 850 850 852 854 854 854 856 862 862 863

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Applications of Selected Multiphase Systems Igor Novák, Volkan Cecen and Vladim´ır Pollák 23.1 Introduction 23.2 Construction Applications 23.2.1 Automotive Applications 23.2.2 Marine Applications 23.2.3 Other Applications 23.3 Aeronautics and Spacecraft Applications 23.3.1 Aeronautics Applications 23.3.2 Spacecraft Applications 23.4 Human Medicine Applications 23.4.1 Musculoskeletal and Bone Applications 23.4.2 Dentistry Applications 23.5 Electrical and Electronic Applications 23.6 Conclusion References

865

Waste Management, Recycling and Regeneration of Filled Polymers José-Marie Lopez Cuesta, Didier Perrin, and Rodolphe Sonnier 24.1 Introduction 24.2 Identification and Sorting 24.3 Separation of Components 24.3.1 Mechanical Separation 24.3.2 Dissolution of Resin 24.4 Feedstock Recycling 24.5 Thermal Processes 24.6 Mechanical Recycling of Filled Thermoplastics 24.6.1 Degradation During Reprocessing of Filled Thermoplastics and Influence of Interfacial Agents 24.6.2 Properties of Recycled Filled or Reinforced Thermoplastics 24.6.3 Recycling of Polymer Nanocomposites 24.7 Waste Management of Glass Fiber-reinforced Thermoset Plastics 24.7.1 Waste Management and International Context 24.7.2 Feedstock Recycling by Pyrolysis 24.7.3 Solvolysis or Chemical Recycling 24.7.4 Mechanical Recycling of Glass-reinforced Thermoset Composites 24.8 Conclusion References

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24

25

Nanoparticle Reinforcement of Elastomers and Some Other Types of Polymers James E. Mark 25.1 Introduction 25.2 Fillers in Elastomers 25.2.1 Generation of Approximately Spherical Particles 25.2.2 Glassy Particles Deformable into Ellipsoidal Shapes 25.2.3 Layered Fillers

865 866 866 871 875 882 882 891 895 895 905 907 912 913

921 922 924 924 924 928 929 932 932 935 940 944 944 945 946 947 951 952 959 959 960 960 963 964

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25.3 25.4 25.5 25.6 25.7 25.8 25.9

Index

25.2.4 Magnetic Particles 25.2.5 Polyhedral Oligomeric Silsesquioxanes (POSS) 25.2.6 Nanotubes 25.2.7 Dual Fillers 25.2.8 Porous Fillers 25.2.9 Fillers with Controlled Interfaces 25.2.10 Silicification and Biosilicification 25.2.11 Theory and Simulations on Filler Reinforcement Nanoparticles in Glassy Polymers Nanoparticles in Partially-Crystalline Polymers Nanoparticles in Naturally-Occurring Polymers Nanoparticles in Relatively-Rigid Polymers Nanoparticles in Thermoset Polymers Conclusions Acknowledgements References

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965 965 966 966 967 967 967 968 970 970 971 972 972 973 973 973 981

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Mladen Andreis, Rudjer Boˇsković Institute, Zagreb, Croatia Karim Benzarti, Laboratoire Central des Ponts et Chaussees, Paris, France Abderrahim Boudenne, Université Paris-Est, CERTES EA 3481 – Centre d’Etude et de Recherche en Thermique, Environnement et Systèmes, Créteil, France Yves Candau, Université Paris-Est, CERTES EA 3481 – Centre d’Etude et de Recherche en Thermique, Environnement et Systèmes, Créteil, France Volkan Cecen, Department of Mechanical Engineering, Dokuz Eylul University, Bornova, Izmir, Turkey Mohamed M. Chehim, ITODYS, Univeresity Paris Diderot and CNRS, Paris, France Xavier Colin, PIMM, Arts et Métiers Paris Tech, Paris, France José-Marie Lopez Cuesta, CMGD, Ecole des Mines d’Ales, Ales, France Z.Z. Denchev, Institute for Polymers and Composites, University of Minho, Minho, Portugal Fatma Djouan, ITODYS, University Paris Diderot and CNRS, Paris, France Elian Espuche, Ingénierie des Matériaux Polymères, UMR CNRS 5223, IMP@UCB, Université de Lyon, Université Lyon 1, France Michel Ferriol, LMOPS, UniversitéPaul Verlaine Metz, Sain-Avold, France Magali Fois, Université Paris-Est, CERTES EA 3481 – Centre d’Etude et de Recherche en Thermique, Environnement et Systèmes, Créteil, France Daniel Fragiadakis, Naval Research Laboratory, Washington, DC, USA Soney C. George, Department of Basic Science, Amal Jyothi College of Engineering, Kerala, India Han-Xiong Huang, Laboratory for Micro Molding and Polymer Rheology, South China University of Technology, Guangzhou, China Yajiang Huang, College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Sichuan, China Esmaiel Jabbari, Department of Chemical Engineering, University of South Carolina, Columbia, USA Laurent Ibos, Université Paris-Est, CERTES EA 3481 – Centre d’Etude et de Recherche en Thermique, Environnement et Systèmes, Créteil, France Damir Klepac, School of Medicine, University of Rijeka, Rijeka, Croatia Ivo Kˇrivka, Department of Macromolecular Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic

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Igor Krupa, Polymer Institute, Slovak Academy of Sciences, Dúbravská, Bratislava, Slovakia Apostolos Kyritsis, National Technical University of Athens, Athens, Greece Fouad Laoutid, Materia Nova Asbl, Mons, Belgium Guangxian Li, College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Sichuan, China Xia Liao, College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Sichuan, China Jean-Charles Majesté, Laboratoire de Rhéologie des Matières Plastiques, CNRS, St Etienne, France James E. Mark, Department of Chemistry and the Polymer Research Center, The University of Cincinnati, Cincinnati, Ohio, USA György J. Marosi, Budapest University of Technology and Economics, Budapest, Hungary Antonio Mart´ınez-Richa, Departamento de Quimica, Universidad de Guanajuato, Guanajuato, Mexico Alfréd Menyhard, Jr., Budapest University of Technology and Economics, Budapest, Hungary Igor Novák, Polymer Institute, Slovak Academy of Sciences, Bratislava, Slovakia Didier Perrin, CMGD, Ecole des Mines d’Ales, Ales, France Polycarpos Pissis, National Technical University of Athens, Athens, Greece Vladimir Pollák, Polymer Institute, Slovak Academy of Sciences, Bratislava, Slovakia Jan Prokeˇs, Department of Macromolecular Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic Géza Regdon, Jr., University of Szeged, Szeged, Hungary Mel Richardson, Department of Mechanical and Design Engineering, University of Portsmouth, Portsmouth, UK Antonio Santamar´ıa, Polymer Science and Technology Department, Faculty of Chemistry, University of the Basque Country, San Sebastián, Spain Alireza S. Sarvestani, Department of Mechanical Engineering, University of Maine, Orono, Maine, USA Friederike Schmid, Institute of Physics, Johannes-Gutenberg Universität Mainz, Germany Regan L. Silvestri, Department of Chemistry, Baldwin-Wallace College, Berea, Ohio, USA Rodolphe Sonnier, CMGD, Ecole des Mines d’Ales, Ales, France ˇ Zdeno Spitalsk´ y, Polymer Institute, Slovak Academy of Sciences, Dúbravská, Bratislava, Slovakia Gilbert Teyssedre, Laplace Université Paul Sabatier, Toulouse, France Sabu Thomas, Centre for Nanoscience and Nanotechnology, Mahatma Gandhi University, Kerala, India Srećko Valić, University of Rijeka, Rijeka, Croatia, and Rudjer Boˇsković Institute, Zagreb, Croatia József Varga, Budapest University of Technology and Economics, Budapest, Hungary

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Siby Varghese, Rubber Research Institute of India, Kottayam, Kerala, India J.C. Viana, Institute for Polymers and Composites, University of Minho, Minho, Portugal Poornima Vijayan P, School of Chemical Sciences, Mahatma Gandhi University, Kerala, India Max Wolff, Department of Phyics, Uppsala University, Uppsala, Sweden Qi Yang, College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Sichuan, China Zhongyi Zhang, Advanced Polymer and Composites (APC) Research Group, Department of Mechanical and Design Engineering, University of Portsmouth, Portsmouth, UK

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Foreword

Multiphase polymer systems have been the focus of recent research and have become an important issue from both the industrial and fundamental points of view. The scientific literature devoted to multiphase polymer systems is large and growing as it covers a wide range of materials such as composites, blends, alloys, gels and Interpenetrating Polymer Networks. During the last two decades, major opportunities have appeared due to the possibility of tuning the different relevant length scales with the promise to produce a new generation of materials displaying enhanced physical, mechanical, thermal, electrical, magnetic, and optical properties. In spite of these intensive investigations, there are still many unresolved problems in this field. One of the main issues is the influence of the shape, size and dispersion of the particles in the polymer matrix on the macroscopic behavior of the resulting material. There are many factors which control the dispersion, and one of them is the interaction between the particles and the polymer phase. Describing the interactions between the various components, the physical attributes of polymers and particles, the physical, thermophysical and interfacial properties in a comprehensive universal scheme remains a challenge. This approach requires collecting a large number of experimental data that can be obtained only by using various and complementary experimental techniques. Investigations in this field cover different topics, such as polymer blends and composites and nanocomposites reinforcement, barrier properties, flame resistance, electro-optical properties, etc. Part of these multiphase polymer materials belong to the so-called smart materials which are materials that have one or more properties that can be significantly changed in a controlled fashion by external stimuli. The key to the success of these smart materials hinges on the ability to exploit the potential of nano-structuring in the final product. This book discusses many of the recent advances that have been made in the field of morphological, interfacial, physical, rheological and thermophysical properties of multiphase polymer systems. Its content is original in the sense that it pays particular attention to the different length scales (macro, micro and nano) which are necessary for a full understanding of the structure–property relationships of multiphase polymer systems. It gives a good survey of the manufacturing and processing techniques needed to produce these materials. A complete state-of-the-art is given of all the currently available techniques for the characterization of these multiphase systems over a wide range of time and space scales. Theoretical prediction of the properties of multiphase polymer systems is also very important, not only to analyze and optimize material performance, but also to design new material. This book gives a critical summary of the existing major analytical and numerical approaches dealing with material property modeling. Most of the applications of these smart materials are also reviewed which shows clearly their important impact on a wide range of the new technologies which are currently used in our daily life. Finally, the ageing, degradation and recycling of multiphase polymer systems is not forgotten and some routes are proposed to avoid environmental contamination. The 52 contributors of this book are all leading researchers in their respective fields, and I warmly congratulate the editors Abderrahim Boudenne, Yves Candau, Laurent Ibos and Sabu Thomas for bringing them together to produce this original and important book dealing on multiphase polymer systems.

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Foreword

I am quite convinced that this book will serve as a reference and guide for those who work in this area or wish to learn about these promising new materials.

Dominique Durand Laboratoire de Physicochimie Macromoléculaire, Equipe de Recherche Associée au Centre National de la Recherche Scientifique, Faculté des Sciences, Route de Laval, le Mans, France

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1 Physical, Thermophysical and Interfacial Properties of Multiphase Polymer Systems: State of the Art, New Challenges and Opportunities Sabu Thomas Centre for Nanoscience and Nanotechnology, Mahatma Gandhi University, Kerala, India

Abderrahim Boudenne, Laurent Ibos and Yves Candau Universite Paris-Est, Créteil Centre d’Etude et de Recherche en Thermique, Environnement et Systemes, 61 Av. du Général de Gaulle 94010 Créteil Cedex, France

1.1 Introduction Multicomponent polymer systems find a wide range of applications in each and every phase of our dayto-day life. Continued research has resulted in the development of super performing macro-, micro- and nanostructured polymeric materials. The new emerging fields of micro- and nano-composites have put forward many challenging opportunities for the use of these smart materials. Polymer physicists, chemists, engineers and technologists show great interest in new strategies for developing high-performance multicomponent systems. Recently, polymer nanostructured multiphase systems have gained much interest due to their unique properties. Characterization of the interphase, physical properties and thermophysical properties are crucial for the understanding of the behavior of these smart materials. A comprehensive understanding of these materials is vital for the industrial use of these materials. The main objective of this book is to present a survey of recent advances in the area of multiphase polymer systems covering physical, interfacial and thermophysical properties of these materials. After a short presentation of the different existing multiphase polymer systems, followed by a survey of actual scientific production and of application fields for these materials, we present some of the recent developments in the Handbook of Multiphase Polymer Systems, First Edition. Edited by Abderrahim Boudenne, Laurent Ibos, Yves Candau, and Sabu Thomas. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

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area of multicomponent polymer systems that will be highlighted all through the book. The chapter ends with a summary of unresolved issues, perspectives and new challenges for the future.

1.2

Multiphase Polymer Systems

Multiphase polymer systems are characterized by the simultaneous presence of several phases, the two-phase system being the simplest case. Many of the materials described by the term multiphase are two-phase systems that may show a multitude of finely dispersed phase domains. The term ‘two-component’ is sometimes used to describe flows in which the phases consist of different chemical substances. Multiphase polymer systems in general include polymer blends, composites, nanocomposites, interpenetrating polymer networks (IPNs), and polymer gels. 1.2.1

Polymer Blends

Polymer blends can be considered as a macroscopically homogeneous mixture of two or more polymeric species with synergistic properties. In most cases, blends are homogenous on scales larger than several times the wavelength of visible light. Blends may be either compatible or incompatible. Polymer blends can be broadly divided into three categories: miscible, partially miscible and immiscible blends. A miscible polymer blend is capable of forming a single phase over certain ranges of temperature, pressure, and composition; also it can be thermodynamically stable or metastable, exhibits a single Tg or optical clarity. An immiscible polymer blend means a multiphase system. Although polymer blending is a very attractive way to obtain new materials, most polymers are immiscible and/or incompatible. Reasons for incompatibility are high interfacial tension and poor interfacial adhesion. In general, a miscible blend of two polymers is going to have properties somewhere between those of the two unblended polymers. Whether or not a single phase exists depends on the chemical structure, molar mass distribution and molecular architecture of the components present. The single phase in a mixture may be confirmed by light scattering, X-ray scattering and neutron scattering. Typical dispersed phase morphology of polymer blend is given in Figure 1.1 [1]. 1.2.2

Polymer Composites

Generally a composite is defined as a multi-component material comprising multiple different (nongaseous) phase domains in which at least one type of phase domain is a continuous phase. In polymer composites, at least one component is a polymer. Fillers such as fibers, particulate fillers, wood fibers, glass fibers and minerals are used as reinforcements in polymeric matrices. 1.2.3

Polymer Nanocomposites

A nanocomposite is a composite in which at least one of the phases has at least one dimension of the order of nanometers, or structures having nano-scale repeat distances between the different phases that make up the material. Polymeric nanocomposites prepared from high aspect ratio fillers such as carbon nanotubes, layered graphite nanofillers etc. achieve significant improvements in mechanical and electrical properties at low filler concentrations, compared to conventional composites [2, 3], without a significant increase in density (see Figure 1.2). The polymer nanocomposites could be prepared by solution mixing process, in situ intercalation process, latex compounding techniques, and melt mixing techniques (see Figure 1.3). In the case of layered clay nanocomposites, one can achieve different types of morphologies as shown in Figure 1.4.

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Physical, Thermophysical and Interfacial Properties of Multiphase Polymer Systems Drops

Double emulsion

(toughness, 1 μm surface modification) Fibers

(toughness and stiffness)

Cocontinuous

(strength, thermal expansion)

3

Laminar

(barrier)

Ordered microphases

(high flow, electrical conductivity, toughness, stiffness)

10 nm

Figure 1.1 Dispersed phase morphology of polymer blends. Reprinted from [1]. Copyright (2000) with permission from John Wiley and Sons.

1.2.4

Polymer Gels

Polymer gels consist of a crosslinked polymer network inflated with a solvent such as water. They have the ability to reversibly swell or shrink (up to 1000 times in volume) due to small changes in their environment (pH, temperature, electric field). The swelling behavior of gels is presented in Figure 1.5. 1.2.5

Interpenetrating Polymer Network System (IPNs)

An interpenetrating polymer network (IPN) is a polymer comprising two or more networks which are at least partially interlaced on a polymer scale but not covalently bonded to each other. The network cannot be separated unless chemical bonds are broken. The two or more networks can be envisioned to be entangled in such a way that they are concatenated and cannot be pulled apart, but not bonded to each other by any chemical bond. There are semi-interpenetrating polymer networks and pseudo-interpenetrating polymer

Figure 1.2

Typical morphology of nanocomposites.


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Synthesis Approaches Melt Intercalation: Co-extrusion Functionalization of the NPs*

SWNT Rope

Tailoring the modifier to the polymer promotes favorable interactions SWNT

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O OH

O

n

Organic Modifier

In-situ polymerization with pristine** or functionalized NPs Δ or h ν

Surfactant assisted dispersion of NPs***

Figure 1.3 Synthesis approach to polymer nanocomposites.

networks. A polymer comprising one or more polymer network(s) and one or more linear or branched polymer(s) is characterized by the penetration on a molecular scale of at least one of the networks by at least some of the linear or branched chains. Semi-interpenetrating polymer networks (SIPNs) may be further described by the process by which they are synthesized. These include sequential SIPNs, simultaneous SIPNs, pseudo-interpenetrating networks, etc. A SIPN is prepared by a process in which the second component

d Unmixed

D

Intercalated

Exfoliated

Figure 1.4 Exfoliation and intercalation of clay nanocomposites.

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Physical, Thermophysical and Interfacial Properties of Multiphase Polymer Systems

5

uniformly swollen

non-swollen

partly swollen

Figure 1.5 Behavior of gels: Uniformly swollen, partly swollen and non-swollen gels.

polymer is polymerized or incorporated following the completion of polymerization of the first component polymer, and thus may be referred to as a sequential SIPN. When an SIPN is prepared by a process in which both component polymers are polymerized concurrently, it may be referred to as a simultaneous SIPN. Pseudo-interpenetrating network systems seem to have interpenetrating networks, but actually do not.

1.3 A Short Survey of the Literature and Applications The 20th century saw great progress in the development and use of polymers and polymer composites; today’s broad family of tailor-made materials allows us to realize the latest technological applications; and the future will see polymers used in increasingly innovative ways. Polymers’ and polymer composites’ almost infinite flexibility and affordability mean that only the imagination of designers limits the ways they can be used. Plastics truly deserve the mantle of material of choice for the 21st century. Just as importantly, polymers’ efficiency allows them to make a real contribution to the vital goals of development for several areas in the world. The growth in these polymeric materials continues to outstrip average annual growth in GDP, and this trend looks set to continue [4]. Demand for polymers in all sectors was up in 2000, continuing the growth trend, and with no significant changes in the relative consumption patterns. Unsurprisingly, the Electrical & Electronic sector shows the highest growth, with countless new inventions and applications using polymers and polymer composites materials as an integral material. However, packaging is still the largest user of plastics, representing about 37% of all other sectors in Europe for 2000. The building and construction industry accounted in 2000 about 19% of total consumption as presented in Figure 1.6 and remains one of the main largest users. The applications in the automotive and electrical & electronics sector are also important and increase year after year.

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Handbook of Multiphase Polymer Systems Automotive 7.2%

Other Household/ Domestic 21.3%

Large Industry 5.4% Agriculture 2.6%

Building and Construction 18.9%

Electrical / Electronic 7.3%

Packaging 37.3%

Figure 1.6 Plastics consumption by industry sector Western Europe 2000. R If we look at the numbers of scientific work published (using the data base ISI Web of Knowledge ) by each country in the last decade with outputs ‘Polymers’ with ‘Blends’, or ‘Gels’, or ‘Nanocomposites’ or ‘Polymer Composites’ we can clearly see in Figure 1.7 that the USA is the leader followed by Asian countries. It is also important to notice that in Western Europe the number of scientific works dedicated to polymers and polymer composites is also important and was the third source of publications in the world.

18000 Multiphase Polymer Systems publications

Publications Intensity 1990-2009

16000 14000 12000 10000 8000 6000 4000 2000 0 Spain

Italy

France England India Germany Japan

China

Others

USA

Country Figure 1.7 Publications intensity in main country on polymers, blends, gels, nano- and macrocomposites between 1990 and 2009.

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Physical, Thermophysical and Interfacial Properties of Multiphase Polymer Systems ABCDE-

Polymers Polymer Composites Polymer Nanocomposites Polymer Blends Polymer Gels C

7

6.98% 2.77% E

D

6.13%

A B

72.6%

11.5%

Figure 1.8 Publications area between 2004 and 2009.

However, it remains that during the last five years (2004–2009) over 72% of the scientific works were dedicated to polymers (Figure 1.8). Polymer composites and polymer nanocomposites materials and blends represent respectively 11.5%, 8% and about 6% of publications, the rest relating to gels.

1.4 Book Content This book is intended to deal with most aspects of multiphase polymer systems science. Four transversal main aspects are considered: 1. The modeling of multiphase polymer systems, including theoretical and numerical simulation approaches. 2. The morphological investigation techniques that allow information on the microstructure of these complex systems to be obtained. 3. The physical characterization at macroscopic scale which also brings information on the structure of the material but mainly determines whether the material properties are compatible with its application and use. 4. The life cycling of multiphase polymer systems covering all the steps between the manufacturing process and the recycling. 1.4.1

Modeling and Computer Simulation of Multiphase Composites: From Nanoscale to Macroscale Properties

Predicting the behavior of multiphase polymer composites is an area with vast potential applications, and has attracted a lot of academic work. Chapters 2, 3 and 4 present the state-of-the-art as well as recent applications. The theoretical approach originates with the Flory-Huggins theory and gives rise to modern powerful predictive methods such as the self-consistent field theory (SCF), which has been applied to the calculation of phase diagrams for diblock copolymer blends, or the study of interfacial properties in binary blends [5, 6]. In Chapter 2 the viscoelastic behavior of polymer nanocomposites in the liquid phase is investigated with recent mesoscale models (sticky reptation model) and the results are quantitatively compared to experiment. The simulation approach is more recent but recent advances make it a very promising field. Modeling can be on the microscale (atomistic), the mesoscale (coarse-grain) or macroscale (continuum)

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Recent work focuses on multi-scale modeling in a ‘bottom-up’ approach where properties obtained at a scale are reused on the larger scale, giving these methods real predictive power. Coarse-grain models and field models are reviewed in detail; dynamical as well as static properties can be investigated with these models. These methods have been applied to miscibility of copolymer blends, or the dynamics of demixing in homopolymer blends. The behavior of polymer multiphase composites is largely influenced by interfacial properties. Chapter 4 proposes a review of interphase properties, interface modification techniques and interface analysis techniques. Mastering the effects of interface properties is a challenge both for modeling and for manufacturing developments [7, 8]. Another important factor in understanding and mastering macroproperties of immiscible polymer composites is morphology (Chapter 6). Two important factors are analyzed in detail: evolution during processing, and component individual properties. Morphology development can be influenced by compatibilization techniques which modify interfacial properties.

1.4.2

Morphological Investigation Techniques

Morphological characterization of the multiphase polymer system is extremely important since most of the physical and transport properties (mechanical, electrical, thermal) of polymer systems are determined by the scale of dispersion of the component phases. The length scale (macro, micro and nano) of the dispersion could be monitored by optical light microscopy (phase contrast, polarized), electron microscopy (SEM, TEM), scanning tunneling microscopy, and atomic force microscopy (AFM). Scattering techniques are also very useful for this; they include light scattering, X-ray scattering (SAXS and WAXS) and neutron scattering techniques. TEM, AFM, SAXS and SANS techniques can also be of interest in the investigation of the interphase/interface of multiphase polymer systems. Several researchers have looked carefully into the interphase/interface width of several multiphase systems using these techniques. Spectroscopy is a valuable technique to characterize multiphase polymer systems. These include UV (ultraviolet) spectroscopy, FTIR (Fourier transform infrared spectroscopy), NMR (nuclear magnetic spectroscopy), XPS (X-ray photoelectron spectroscopy), ESR (electron spin resonance spectroscopy). Each technique varies in its sensitivity. For example, NMR spectroscopy and fluorescence spectroscopy can detect heterogeneities in the range of 2–3 nm scale. NMR can generate a lot of information in the nanoscale on the interphase/interface of multiphase polymer systems. XPS can be used for the study of surface chemistry of polymer grafts, colloidal particles, nanocomposites etc, in the nanometer range.

1.4.3

Macroscopic Physical Characterization

Physical properties of polymers and multicomponents systems based on the use of polymers are more strongly dependent on temperature than for other materials such as metals or ceramics. Moreover, the temperature range of use of these systems is thus reduced when compared to other materials. Physical properties of multiphase polymer systems are also closely related to the components’ properties and relative fractions. The structure of the material – for instance the shape, size, dispersion, possible orientation of the dispersed phase – greatly influences macroscopic properties. For instance, crystallization and melting temperatures of nanocomposites are strongly related to the filler content and its dispersion state whereas the thermal expansion coefficient is strongly affected by the alignment of exfoliated platelets as small changes from perfect planar orientation result in significant changes in thermal expansion behavior.

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9

The properties of interest are of two types: bulk properties and phase-transition properties. The bulk properties are mechanical properties, thermophysical properties, electrical properties or permeability. The two main phase-transition properties concern first-order transitions (fusion and crystallization) and the glass transition. All these properties are now accessible thanks to recent progress in experimental techniques that allow measurements in the three physical states over extended ranges of temperature and pressure, including in the vicinity of the critical point (at least in the case of gases and liquids). The possibility of achieving a complete and reliable characterization of multiphase polymer systems is also one reason for the recent and increasing use of these materials. The knowledge of these properties is important for the development of new systems with properties adapted to one particular application, and also for obtaining information on the microstructure of the material. Thus, macroscopic characterization of multiphase polymer systems is important for engineers in industry and for researchers in the field of material science. Thermophysical properties are important in industry for the knowledge of thermal transport properties and thermal stability of materials. They include thermal conductivity and diffusivity, specific heat, melting and crystallization temperature and enthalpies, coefficient of thermal expansion. Thermal analysis techniques allow information to be obtained on the structure of polymers and multi-components systems and on their phase transitions. Optical techniques allow, for instance, the characterization of the organization of crystalline regions in polymers. Differential Scanning Calorimetry (DSC) is a powerful method for the study of phase transition and for the measurement of specific heat capacity; the recent development of Modulated DSC and the enhanced specifications of recent DSC and MDSC devices have also increased the investigation field of this technique. Chapter 9 will extensively present a survey of thermal analysis applied to the characterization of multi-phase polymer systems. Chapter 10 is devoted to the study of thermophysical properties of multiphase polymer systems (characterization methods, thermophysical behavior, and modeling) [9]. Mechanical behavior is of paramount importance in the design of advanced multiphase polymer materials for many applications in different engineering fields such as aerospace and the automotive industry or civil engineering. The stress–strain behavior, tensile strength, yield strength, elongation at break, hardness, impact behaviors (both notched and unnotched), tear properties, abrasion characteristics and flexural properties are very often determined for the comprehensive understanding of the behavior of multiphase polymer systems. Dynamic mechanical properties are also extremely important for the time-dependent dynamic applications of multiphase polymer systems. Materials reduced to nano-scale can suddenly show very different properties compared to what they exhibit in the macro-scale, enabling unique applications The rheological behavior of multiphase polymer systems is of great importance for the understanding of the flow behavior of these materials. Viscosity-shear rate relationships are fundamental basic data to evaluate the processability of these materials. Although multiphase polymer systems are pseudoplastic, they might show complex behavior such as Newtonian character, yield stress, thixotropic and rheopectic characteristics. Additionally, phase separation, gelation and vitrification can be very well monitored using sensitive rotation rheometers. Very often, rubber modified thermoplastics and other polymer/polymer blend systems show structure build-up at low shear which eventually get destroyed at higher shear forces. The exfoliation/intercalation in polymer nanocomposites could be well understood by careful rheological measurements. Chapters 7 and 8 will be devoted respectively to mechanical characterization and rheology of multiphase polymer systems [10, 11]. The mechanical reinforcement of polymers using nanoparticles will be exposed in Chapter 25. Most polymers behave like electrical insulators. A lot of works was devoted in the last 20 years to the development of highly-conducting polymer systems. Different ways were investigated: synthesis of conducting polymers (such as polypyrrole), mixing of common polymers with metal powders, graphite or, more recently, carbone nanotubes to obtain conducting composites. The challenge is to reach electrical conductivity values minimizing the amount of conducting material mixed with the polymer. The use of conducting nanoparticles as filler in a polymeric matrix is the most efficient solution as very small electrical percolation thresholds are observed in these systems. The study of the electrical behavior of insulating

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polymer systems requires the use of characterization techniques specific to dielectric materials, such as broadband dielectric spectroscopy or thermostimulated depolarization currents. Electrical signals obtained by these techniques are the signature of macromolecular chain movements or of interface effects. The large frequency and temperature ranges covered by these techniques allow a lot of information to be obtained on the microstructure of multiphase polymer systems. Chapter 11 will present a synthesis of literature results concerning the development of conductive polymer systems and of the associated characterization techniques and most commonly used models. Dielectric properties will be investigated in Chapter 12 [12]. Diffusion and transport in multiphase polymer systems have attracted a lot of interest recently (Chapter 19). This is due to the fact that multicomponent polymer systems find enormous application in separation technologies, including dialysis, restricted gas transport, pervaporation, gas sorption and vapor sorption. Several factors such as microstructure, dispersion, miscibility, compatibility, interphase adhesion, degree of crosslinking, orientation of the filler particles, phase co-continuity and so on influence the transport process in multiphase polymer systems. Additionally, diffusion and transport could be used as an excellent tool to characterize the morphology and microstructure of multiphase polymer systems. 1.4.4

Life Cycling

Nowadays, the field of applications of multiphase polymer systems is very broad (civil engineering and buildings, electronics, medicine, transports, packaging. . .). Chapter 23 will give a survey of the existing applications of multiphase polymer systems. Moreover, the continuous development of new materials permits new applications, like fire retardancy applications, which are very important in transport and buildings. This particular topic will be investigated in Chapter 22. Nanoreinforcement of plastics leads to the use of small amounts of fillers to obtain a strong increase of mechanical properties (see Chapters 25 and 6) such as the elastic modulus, even if the use of nanoparticules might cause some health problems. Application allows defining shape and size of the manufactured object and components to be used. All these parameters often impose the use of a particular manufacturing technique. These manufacturing techniques, presented in Chapter 5, are numerous. Classical industrial techniques like extrusion and injection are particularly suitable for the processing of multiphase systems based on the use of thermoplastics. The recent development of new techniques like resin transfer molding has brought new applications concerning processing of fiberreinforced thermoset composites, particularly in the aeronautics and space industries. Finally, the choice of the manufacturing process greatly influences macroscopic properties of the final material (interfaces between components, non-isotropic properties). Therefore, a strong link exists between applications and the manufacturing process. A good knowledge of ageing processes is also required to determine the lifetime of a given object. The study of ageing processes is quite complex as it requires the use of characterization techniques at different length scales and the development of realistic procedures of accelerated ageing (see Chapter 21). Non-destructive testing techniques are of great interest as they can provide an on-line control of the manufactured products and also an early detection of defects, delaminations, etc, during the normal use of a material (see Chapter 20). Recycling of multiphase polymers is a major issue in order to limit the effect of industrial production on the environment. Recycling will be considered in Chapter 24. Possibilities of recycling are mainly dependent on the components used and on the manufacturing process. Thus the possibilities of recycling have to be considered from the beginning of the development of a new material [13].

1.5

Future Outlook, New Challenges and Opportunities

In the area of multiphase polymer systems (blends, composites, nanocomposites, IPNs, gels) several questions still need to be answered. Concerning polymer blends, more sophisticated techniques are needed to probe the

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11

very early stages of the phase separation process. Although NMR and fluorescence spectroscopy offer some answer to this, we need to have more fast and easy techniques. Similarly for complex systems, we need to look very carefully at the phase separation mechanisms, spinodal or nucleation and growth. There are many complex polymer systems where both mechanisms operate. In the area of compatibilization of polymer blends by reactive route, we need to go a long way to characterize the chemistry of the interfacial chemical reactions. For many polymer–polymer blends systems compatibilized by the reactive route, the chemistry of several interfacial reactions has to be well understood. In order for this we need to go for selective extraction followed by NMR and FTIR spectroscopies. Another important challenge in the area of polymer blends is the online monitoring or in situ monitoring of morphology development during processing. Of course some success has been made in this direction using model systems which are transparent. However, for real polymer systems, we need to undertake more in-depth studies. In this respect, knowledge of the thermophysical properties of polymers over extended ranges of temperature and pressures and in different gaseous environments is undoubtedly necessary to improve the use and lifetime of end-products made of such polymers.In the area of recycling of polymer blends wastes and polymer products, several problems exist which are very important for dealing with waste disposal problems. For effective recycling of polymer blend wastes, we need to have multifunctional compatibilizers which can convert a mixture of a variety of polymer blend materials into useful value-added materials for a number of different applications. However, these techniques have to be more environment-friendly.In the area of composites, the incorporation of naturally-derived macro-, micro- and nanofillers such as cellulose, chitin and starch has recently captured a lot of interest. Since these fillers are derived from waste biomass, the process is highly green and environment-friendly. For polymer nanocomposites, one of the biggest challenges is to reach an excellent dispersion of nanoparticles in the polymer matrix. Although nanoclay is a very efficient reinforcing filler in many polar polymer matrices including nylon, so far we have not been able to achieve good dispersion of the nanoclay in polyolefins (PP, PE) which are one of the plastics with the higher tonnage. More efficient surfactants have to be designed for the excellent dispersion of clay in nonpolar polymer systems such as PP, PE, etc. The extent of intercalation versus exfoliation could not be quantified in many nanoclay-filled polymer systems. We need to have more sophisticated techniques for the exact quantification of the exfoliation process. More environment-friendly and efficient mixing techniques have to be developed for the mixing of nanofillers with various polymer matrices. Of course, strong progress has been made in the use of supercritical carbon dioxide for the processing of various polymer nanocomposites. The orientation of nanoplatelets, such as clay, carbon nanotubes and graphite in the polymer matrix is a major challenge. Perfect alignment of the nanoplatelets will provide excellent properties, particularly gas barrier properties. We need to develop special extrusion techniques, application of magnetic and electrical fields for the orientation of the nanoplatelets. The in situ monitoring of the flow of the polymer nanocomposites during manufacturing requires in-depth research. We still have to go a long way for the successful use of polymer nanocomposites for many commercial applications although some progress has been made in this direction. We should also examine carefully the toxicity aspects of many nanofillers and nanocomposite materials; once the nanofillers enter into our body, elimination will be very difficult. Therefore one has to take extreme care during the handling of the nanofillers in mixing and processing operations. The legal and ethical issues of nanostructured materials have to be addressed carefully. In the area of IPNs, we have made significant progress for the development of nanostructured IPNs. However, the morphology and phase separation mechanisms of nanostructured IPNs have not been well understood. The design of porous networks based on IPNs has received a lot of attention recently. We believe that such porous IPN materials may find potential applications in separation techniques as chromatography supports, as well as membrane catalysis, and more generally in chemistry in confined medium as nanoreactors. However, such new applications have to be explored in detail. There exists much interest for the development of biopolymer-based IPNs for various applications.

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In the case of polymer gels, the development of smart polymer gels with optimum mechanical properties for drug delivery offers enormous opportunites in the medical field. Spatial inhomogeneities present in ionic and nonionic hydrogels depend systematically on network density and on the degree of swelling. The precise quantitative estimation of these inhomogeneities needs more sophisticated and careful analysis. Mechanical instabilities such as buckling, wrinkling, creasing, and folding are commonplace in both natural and synthetic gels over a wide range of length scales. Several advancements have been made on the spontaneous folding behavior of the highly swellable confined nanoscale (thickness below 100 nm) gel films [14]. In fact, the regular self-folding is originated from periodic instabilities (wrinkles) caused by swelling-initiated stresses under confined conditions. Furthermore, folded gel structures can be organized into a regular serpentine-like manner by imposing various boundary conditions on micro-imprinted surfaces. It is important to add that this demonstration of uniform gel to mechanically mediate morphogenesis has far-reaching implications in the creation of complex, large-area, 3D gel nanostructures

References 1. C. W. Macosko, Morphology development and control in immiscible polymer blends, Macromol. Symp. 149, 171–184, 2000. 2. Y. Konishi and M. Cakmak, Structural heirarchy developed in injection molding of nylon 6/clay/carbon black nanocomposites, Polymer, 46, 4811–4826, 2005. 3. N. Nugay and B. Erman, Property optimization in nitrile rubber composites via hybrid filler systems, J. Appl. Polym. Sci., 79, 366–374, 2001. 4. Association of Plastics Manufacturers in Europe, An analysis of plastics consumption and recovery in Western Europe 2000, 2002/GB/04/02, Published Spring 2002. 5. M. Doxastakis, Y. -L. Chen, O. Guzmán, J. J. de Pablo, Polymer-particle mixtures: depletion and packing effects, J. Chem. Phys., 120, 9335, 2004. 6. M. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett., 72, 2660–2663, 1994. 7. G. E. Fantner, O. Rabinovych, G. Schitter, P. Thurner, J. H. Kindt, M. M. Finch, J. C. Weaver, L. S. Golde, D. E. Morse, A. Lipman E, IW, Rangelow, P. K. Hansma, Hierarchical interconnections in the nano-composite material bone: Fibrillar cross-links resist fracture on several length scales, Compos Sci Technol, 66, 1205, 2006. 8. G. Fantner, E. Oroudjev, G. Schitter, L. S. Golde, P. Thurner, M. M. Finch, P. Turner, T. Gutsmann, D. E Morse, H. Hansma, P. K. Hansma, Sacrificial bonds and hidden length: Unraveling molecular mesostructures in tough materials, Biophys J., 90, 1411–418, 2006. 9. R. Tlili, A. Boudenne, V. Cecen, L. Ibos, I. Krupa, Y. Candau, Thermophysical and electrical properties of nanocomposites based on ethylene-vinylacetate copolymer (EVA) filled with expanded and unexpanded graphite, International Journal of Thermophysics, 31(4–5), 936–948, 2010. 10. L. Elias, F. Fenouillot, J. C. Majeste, G. Martin, G., J. Cassagnau, Migration of nanosilica particles in polymer blends, Polym. Sci. : Part B: Polymer Physics 1976–1983, 46, 2008. 11. J. John , D. Klepac, M. Didovi, C. J. Sandesh, Y. Liu, K. V. S. N. Raju, A. Pius, S. Valic and S. Thomas, Main chain and segmental dynamics of semi interpenetrating polymer networks based on polyisoprene and poly(methyl methacrylate), Polymer, 51, 2390–2402, 2010. 12. M. Micusik, M. Omastova, I. Krupa, J. Prokes, P. Pissis, E. Logakis, C. Pandis, P. Potschke and J. Pionteck, A comparative study on the electrical and mechanical behaviour of multi-walled carbon nanotube composites prepared by diluting a masterbatch with various types of polypropylenes, Journal of Applied Polymer Science, 113(4), 2536–2551, 2009. 13. A. Conroy, S. Halliwell and T. Reynolds, Composite recycling in the construction industry, Composites: Part A, 37, 1216–1222, 2006. 14. S. Singamaneni, M. E. McConney and V. V. Tsukruk, Swelling-induced folding in confined nanoscale responsive polymer gels, ACS Nano 4(4), 2327–2337, 2010.

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2 Macro, Micro and Nano Mechanics of Multiphase Polymer Systems Alireza S. Sarvestani Department of Mechanical Engineering, University of Maine, Orono, Maine, USA

Esmaiel Jabbari Department of Chemical Engineering, University of South Carolina, Columbia, USA

2.1 Introduction Flow behavior of colloidal suspensions in complex fluids has been the subject of intense research [1–10]. The widespread use of suspensions in industry has for a long time provided motivation for these investigations. Early theoretical research was based on hydrodynamic models. For example, the low shear rate viscosity of a suspension with low volume fraction, , of solid spherical particles can be estimated by the often-quoted Einstein equation [11] η = ηm (1 + 2.5)

(2.1)

where ηm represents the linear viscosity of the matrix. Underlying these theoretical models is the assumption of continuum viscoelasticity; i.e. the colloidal particle is saturated by the dispersing medium and is large enough such that the non-hydrodynamic contributions such as Brownian motion, surface forces, and van der Waals interactions between particles are negligible. As a result, models derived from hydrodynamic theories are inherently independent from the size of filler particles and the nature of interfacial bonding. It is therefore not surprising that these equations appear to explain only the results of micron-sized particles [5, 12, 13]. They are indeed entirely inadequate to elucidate the results obtained from the reinforcing fillers of colloidal and sub-colloidal size such as silica and graphitized carbon black in non-Newtonian fluids [12, 14, 15].

Handbook of Multiphase Polymer Systems, First Edition. Edited by Abderrahim Boudenne, Laurent Ibos, Yves Candau, and Sabu Thomas. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

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Reduction of the filler size to the order of nanometers can lead to substantial differences in rheology and dynamics of filled polymer liquids compared to complex fluids, reinforced with micron sized particles [14–19]. In particular, polymer composites reinforced with sub-micron fillers, often referred to as polymer nanocomposites, exhibit a significant enhancement in viscoelastic properties compared to microcomposites, at similar filler volume fractions. In frequency sweep tests, this dramatic increase in viscoelasticity is generally manifested at very dilute concentration of filler particles and by the appearance of a secondary plateau for viscoelastic moduli at low frequency regimes [14–16, 20–22]. It has been shown that the mechanical properties of polymer nanocomposites can be greatly modified by changing surface properties of nanoparticles [18, 19, 23]. The extremely large surface area provided by nanoparticles can intensify the effect of particle–particle and/or polymer–particle interactions compared to other reinforcing mechanisms such as hydrodynamic effect. Therefore, the unique properties of polymer nanocomposites are generally rationalized as arising from strong interparticle affinity [24–27] or the interaction between the particle surface and surrounding matrix [28–30]. In the case of strong filler–filler interactions, it is believed that the material response is influenced by the breakdown and reformation of filler agglomerates during mechanical loading [20, 22, 28, 31, 32]. On the other hand, when polymer–filler interactions dominate, it is shown that the suspension viscoelasticity is controlled by the dynamics of stick-slip motion of the polymeric chains on and close to the filler surface [33–37]. Contrary to extensive experimental studies, theoretical models to quantitatively elucidate the reinforcement mechanism in polymer nanocomposites are scarce. Semi-empirical models are proposed for aggregated dispersions based on the concept of fractals [31, 32]. A few rheological models are also presented for conditions where polymer–particle interaction is the dominant reinforcing mechanism [34, 37–39]. The purpose of this chapter is to review our recent progress in constitutive modeling of macroscopic rheological behavior of multiphase polymer systems with strong interaction between fillers and polymer chains (i.e. no filler agglomeration) [40–42]. We have proposed two classes of models corresponding to two different physical regimes: (i) unentangled regime, where the chain length is short (below the entanglement threshold) and topological interactions between chains are unimportant, and (ii) entangled regime, where the chain length is well above the entanglement threshold and the chain motion is severely restricted by topological constraints imposed by entanglement with other macromolecules. Our goal is to identify the relevant physics which control the small-scale structure and kinetic behavior of the composite and apply this information to predict composite macroscopic properties in a series of generic constitutive models. This problem attracts significant practical interest in the field of nanorheology, tribology, heat transfer, and MEMS, due to the promising properties of polymer nanocomposites.

2.2

Unentangled Systems

When the concentration of polymer chains in a suspending medium is below the overlapping concentration, the polymer molecules do not entangle with each other and chain dynamics is governed entirely by friction with solvent or isotropic monomer–monomer friction. For such dilute polymer solutions, generalized beadspring models for the polymer chains can properly describe the kinetics of polymeric liquids [43]. The springs are representative of the elastic entropic tensile forces, while the beads play the role of centers for application of friction forces. The resultant Maxwell-type constitutive equations can provide quantitative description of the concentration and molar mass dependences of terminal relaxation time, terminal modulus and viscosity [43, 44]. In the presence of adhesive filler particles, the influence of reversible filler–polymer interaction (attachment/detachment kinetics) can be easily captured by assuming the attachment point as a region of enhanced friction [38, 39]. This additional friction coefficient is proportional to the corresponding energy of adsorption.

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Macro, Micro and Nano Mechanics of Multiphase Polymer Systems

15

The structural configuration of adsorbed polymer chains in a good solvent can be described by de Gennes’ theory of reversible adsorption from dilute solutions [45, 46]. It has been shown by many theoretical and experimental studies that the reversible adsorption takes place when the contact energy between the filler surface and each monomer is weak and less than thermal energy, k B T . When the binding energy is somewhat larger than k B T , adsorption becomes irreversible and the chain freezes on the interactive surface [47]. We use the predictions of reversible adsorption theories to study the equilibrium configuration of the polymer layer on the surface of filler particles.

2.2.1

Microscopic Structure

Consider an ensemble of non-entangled flexible polymer molecules and a random distribution of monodisperse non-aggregated rigid spherical particles. The schematic equilibrium configuration of an adsorbed chain on the surface of a particle with radius R f is shown in Figure 2.1. The chain can reversibly adsorb on the colloidal surface and form a polydisperse succession of loops, tails, and sequences of bound monomers (trains). We only consider the case of dilute particle concentration, where no polymer chain may bridge several colloids. Each chain with N monomers (i.e. the lattice sites in Flory-Huggins theory [48]) of size a, occupies a spherical volume with a radius comparable with the Flory radius R F = a N 3/5 in solution. Here, it is assumed that the size of an adsorbed polymer is equal to that of an unperturbed chain in the bulk. This assumption is supported by recent molecular simulation studies for systems with short range (on the scale of one monomer) monomer–surface interactions in the order of k B T [49]. If the interaction range is large and the strength of attraction is relatively high, chains may deform and flatten on the solid surface. This effect is more pronounced for very short chains where the entropic effect is weak [50].

RF

Rf

Figure 2.1 Schematic diagram for equilibrium configuration of an adsorbed polymer chain on a particle surface. The adsorbed chain consists of loops, tails and sequences of bonded monomers.

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The effect of surface curvature on polymer adsorption has been studied by Aubouy and Raphaël [51]. Their scaling approach shows that when R f > R F , the curvature of the particle is not relevant and adsorption is similar to that on a flat surface. The adsorption would be enhanced by the effect of surface curvature only at the limit of R F R f , i.e. when the particles are very small and/or the polymer chains are very long. Although possible, here we do not consider this limiting case, since for solution of non-entangled polymer chains considered in this study, such a small particle size is not practically relevant. In order to describe the structure of the adsorbed and fully equilibrated polymer layer on the filler surface, we use a simplified version of de Gennes scaling theory for reversible adsorption from dilute solutions under good-solvent condition [52]. The configuration of an adsorbed layer is determined by competitive surface attraction and chain entropic interactions. Let us assume that the loops are extended to an average thickness, D, from the filler surface. Since the monomer density is spread over a distance D, we have a f ∼ = D

(2.2)

where f is the fraction of monomers in direct contact with the solid surface. Assuming that the conformational entropy and energetic attraction with the surface are the only factors that determine configuration of the adsorbed layer, the free energy per chain, , can be written as [53] ∼ = kB T

RF D

5/3 − f N E ad

(2.3)

Minimizing the free energy with respect to D yields f ∼ =

E ad kB T

3/2 (2.4)

Equation (2.4) holds true only if the condition of weak monomer–surface coupling is satisfied, i.e. E ad < k B T . Note that even under the limit of weak adsorption energy, the entire chain can be strongly adsorbed due to the many monomer contacts with the surface, i.e. f N E ad k B T . The chain relaxation dynamics is affected by frictional interactions between monomers and particles. The total friction coefficient as a result of the hydrodynamic force acting on the i th monomer is [39] (ξ )i = ξ1 , (ξ )i = ξ 0 ,

i th monomer is adsorbed th

i monomer is not adsorbed

(2.5a) (2.5b)

where ξ1 is the friction coefficient due to monomer–particle interaction and ξ 0 represents the friction coefficient corresponding to self-diffusion of a single monomer and accounts for its friction with solvent molecules and/or other non-adsorbed monomers. One expects that the surface friction for solid–fluid interfaces to exceed the bulk friction factor in many real systems. Here, it is assumed that this condition is satisfied and hence ξ1 > ξ 0 . Since a fraction f of the monomers in an adsorbed chain is in contact with the particle surface, the total friction coefficient affecting the entire chain is given by ξa = N ( f ξ1 + (1 − f )ξ0 )

(2.6)

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17

For weakly attractive surfaces, chains are partially adsorbed to the surface and can exhibit their 3-D Rouse dynamics [54]. Considering that the diffusion coefficient of the adsorbed chain is Da = kξBaT , the corresponding relaxation time is ξa τa ∼ = τ f ( f χ + (1 − f )) = R 2F kB T

(2.7)

ξ where τ f ∼ = R 2F k BfT stands for relaxation time of a free chain and χ = ξ10 . According to the self-similar grid structure theory [45], the adsorbed layer can be modeled as a semi-dilute solution of the polymer with continuously varying local concentration of monomers such that at any distance, r , from the surface, the local blob size is equal to r . Therefore, the equilibrium thickness of the layer is on the order of R F . In addition, the equilibrium number of chains in the layer per unit filler surface area is estimated to be ξ

1 0 ∼ = 2 RF

(2.8)

The polymer-filler junctions are transient and their density fluctuates with thermal agitation or the effect of flow. Since the interaction energy is short range, we can assume that the adsorption–desorption process takes place only between those chains which are located at close vicinity of the filler surface with total surface density equal to 0 . If a shows the surface density of adsorbed chains at any instant, the corresponding number of free chains which are able to participate in the adsorption-desorption process is given by p

f =

1 − a R 2F

(2.9)

The rate of attachment and detachment can be shown by the following kinetic equation: ∂ a 1 = ∂t τads

1 1 − a − a 2 τdes RF

(2.10)

where τads and τdes are the characteristic times of adsorption and desorption of the chains, respectively. The energy expenditure for detachment of an adsorbed polymer molecule is equal to f N E ad . In the presence of an applied deformation rate, the detachment process is favored by the resultant entropic tension exerted by the chains. Considering this effect, the time constants associated with attachment and detachment of the interfacial polymers follow the Arrhenius type relation τdes = A exp τads

f N E ad − δ Fa kB T

(2.11)

where Fa is the chain entropic force, δ is an activation length on the order of the displacement required to detach the bound chain from the particle surface, and A is a constant. Desorption of a bound monomer with weak and short-range interaction with the adsorbing surface can be considered as a local process. This takes place by diffusing a distance on the order of the equilibrium size of the first blob in contact with the wall [55]. According to the self-similar grid structure theory [45], the size of the first blob in contact with the particle surface is on the order of the monomer size. Therefore, δ, the total displacement required to separate the entire chain with f fraction of adsorbed monomers is a ≤ δ ≤ R F .

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Macroscopic Properties

The classical Maxwell model is used to describe viscoelasticity of the composite [44]. At any instant, a representative chain is either adsorbed to the particle surface or it is free. The total volume average stress in the polymer matrix is therefore σ = ψσ a + (1 − ψ)σ f

(2.12)

where σ a and σ f are the stress contribution of the adsorbed and free chains, respectively, and ψ shows the volume fraction of the polymer–particle interfacial zone (with thickness ∼ R F ) in the matrix. Since the particles are homogeneously dispersed, all statistical properties corresponding to any arbitrary representative mesodomain of the composite body are assumed to be statistically homogeneous. Hence, the multipoint statistical moments of any order are shift-invariant functions of spatial variables for any ergodic field and, therefore, the ensemble averaging could be replaced by volume averaging [56]. The contribution of polymer chains to the stress tensor is given by Kramers expression σ a = 3k B T Na σ f = 3k B T N f

Ra Ra R2 F Rf Rf

(2.13a) (2.13b)

R 2F

where Na and N f represent the number density of the adsorbed and free chains, respectively. Ri (i = a, f ) is the chain end-to-end vector and · · · shows the ensemble average. Assuming that the number density and surface density of interfacial chains follow the relation N ∼ = /R F , in steady state situation, we can write Na p = A exp Nf

f N E ad − δ Fa kB T

(2.14)

p

where N f shows the number density of free chains within the interphase zone. In their simplest forms, the constitutive relations for the evolution of rate dependent stresses produced by the chains can be expressed by Maxwell (upper-convected) equations τa σˆ av + σ av − G a I = 0 τ f σˆ vf + σ vf − G f I = 0

(2.15a) (2.15b)

where I is the identity tensor and G i is the stiffness (i = a, f ). Here, σˆ designates the upper-convected − σ · L e f − L Te f .σ , where L e f = h()∇v is the effective derivative of the stress tensor given by σˆ = ∂σ ∂t velocity gradient tensor and v is the velocity field. Here, h() accounts for the hydrodynamic interaction between particles with volume fraction . The contribution of the hydrodynamic effect is controlled by shape and volume fraction of the particles [11]. At low filler concentrations, it is typically represented by h() = 1 + ζ where pre-factor ζ accounts for the geometry of particles.

(2.16)


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2.2.3

19

Results

In this section, the proposed model is used to predict the steady state response of a polymer solution under 2-D shear flow, reinforced with spherical rigid particles (ζ = 2.5). The velocity gradient tensor in this case can be represented by ⎛

0 ∇v = ⎝ 0 0

⎞ 0 0⎠ 0

γ˙ 0 0

(2.17)

where γ˙ is the shear rate. In our parametric study, R F (Flory radius) and τ f (relaxation time of free chains) are taken to be the unit of length and time, respectively. The activation length δ is assumed to be on the order of ∼0.5R F and the dimensionless constant A is set equal to 10-4 . We assume that each chain is comprised from 5000 monomers and the concentration of dilute polymer solution is ∼0.1(R −3 F ). The energetic affinity between the polymer R ad and ρ = R Ff , chains and fillers and size of the fillers are characterized by dimensionless quantities ε = E kB T respectively. Figure 2.2(a) represents the dependence of low shear rate steady state (LSRSS) viscosity of the reinforced polymer on the polymer–particle interaction through affinity parameter ε. The values are normalized to the steady state viscosity of the unfilled solutions (ηm ). Here, it is assumed that = 10% and χ = 10. The results clearly show the strong effect of monomer-filler energetic affinity on the overall shear viscosity of the mixture, especially within the range of small particle size. Figure 2.2(b) shows the size effect of spherical fillers on LSRSS viscosity of the composites at = 10% and χ = 10. Even within the range of weak monomer-particle attraction considered in this chapter, the concentration of adsorbed chains at the vicinity of particles could be substantially higher than that in the bulk. This indicates an indirect dependence of the

(b)

(a) 60

Normalized LSRSS viscosity

ρ=5 ρ=1

50

100

Φ = 0.1 χ = 10

Normalized LSRSS viscosity

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40 30 20 10 0 0

0.1

0.2

0.3

0.4

ε = Δ Ead / kBT

0.5

0.6

0.7

Φ = 0.1 χ = 10

ε = 0.1 ε = 0.5

10

1 1

10

100

ρ = R f /R F

Figure 2.2 The variation of LSRSS viscosity with (a) monomer–particle interaction energy (ε = E ad /kB T ) and (b) the filler size (ρ = R f /R F ). Volume fraction of dispersed particles was assumed to be 10% ( = 0.1) and relative monomer–particle friction is taken to be χ = ξ1 /ξ0 = 10 in all cases. The viscosity values are normalized to the viscosity of unfilled polymer solution.


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Handbook of Multiphase Polymer Systems (a)

10 Normalized LSRSS viscosity

9 8 7 6

(b) 9

Φ = 0.01 Φ = 0.05 Φ = 0.10

χ = 10 ρ = 5 ε = 0 3

Φ = 0.15 Φ = 0.20

5 4 3 2 1 0 10

ρ = 50

8 Normalized LSRSS viscosity

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ε = 0.3 χ = 10

ρ=5

7 6 5 4 3 2 1 0

γτf

100

0

0.05

0.1

0.15

0.2

0.25

Filler volume fraction (Φ)

Figure 2.3 (a) The variation of LSRSS viscosity with shear rate at different filler concentrations (). (b) Effect of filler size (ρ = R f /R F ) on the variation of viscosity with filler concentration.

overall properties on filler size. At the limit of small particles, this high density interfacial layer achieves a relatively high volume fraction and can significantly affect the steady state shear viscosity of the mixture. By increasing the particle size, the overall viscosity of the composite converges to h()ηm , implying that the hydrodynamic interaction is the dominant reinforcing mechanism in composites made with larger particles. The variation of steady state viscosity of the suspensions with applied shear rate is depicted in Figure 2.3(a) for different concentration of filler particles. All other parameters are kept constant. The neat dilute polymer solution is considered to be shear rate independent and, hence, the observed nonlinearity is totally due to the thixotropic effect of chain-filler detachment. The detachment process is favored by the resultant entropic tension exerted by the chains which, according to Eq. (2.14), leads to a reduction of the number density of adsorbed chains at higher deformation rates, and a significant decrease in shear viscosity of the filled systems. Figure 2.3(b) shows the values of LSRSS viscosities as a function of filler volume fraction, , at two different filler sizes. In the case of ρ = 5, the shear viscosities are almost equal to the predictions of Eq. (2.1). However, when the particle size is comparable with molecular dimensions (i.e. ρ = 1) the magnitudes of shear viscosity are significantly higher than the predictions of Eq. (2.1). This phenomenon is experimentally observed for suspensions of well-dispersed nanoparticles at low concentrations [14]. In the present model, this behavior is solely due to the energetic interaction between polymer chains and fillers. Figure 2.4(a) demonstrates the effect of χ on the storage modulus G (ω) in a frequency sweep, where the values of , ρ, and ε are fixed. When the monomer-surface friction is weak, the typical slope of the neat polymer (2:1 in log-log scale) is obtained for the filled system at low frequencies. As the ratio ξ1 /ξ0 increases, a secondary plateau in the low frequency region forms which is associated with the relaxation of adsorbed chains. This transition, from liquid to solid-like behavior, indicates that the stress relaxation can be effectively hindered by the presence of nanoparticles, when the relaxation time and number density of the adsorbed chains are sufficiently high. This deceleration in relaxation process at low frequencies is reported frequently for a variety of nanofilled polymer suspensions. Figure 2.4(b,c) shows a similar behavior corresponding with the increase in polymer–particle energetic affinity and reduction of particle size, both of which result in enhancement of the concentration of adsorbed chains in the system. The results of frequency sweeps, for suspensions at different filler concentrations, are also shown in Figure 2.4(d), where all other parameters are

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Macro, Micro and Nano Mechanics of Multiphase Polymer Systems (c)

(a) 1

1

χ = 10

ρ=5

χ = 50 0.1

G′ k BT

ρ = 20

χ = 1000

0.1

G′ k BT

0.01

ρ = 50

0.01

0.001

0.001

Φ = 0.01 χ = 100

Φ = 0.01 ε = 0.5 0.0001 0.01

0.1

ωτf

1

0.0001 0.01

10

0.1

1

ωτf

10

(d)

(b) 1

0.1

21

1

ε = 0.1

Φ = 0.01

ε = 0.3

Φ = 0.05

ε = 0.5

Φ = 0.10 0.1

G′ k BT

G′ k BT

0.01

0.01

0.001

ε = 0.5

Φ = 0.01

χ = 100

χ = 100 0.0001 0.01

0.1

ωτf

1

10

0.001 0.01

0.1

ωτf

1

10

Figure 2.4 The variation of frequency response of the filled suspensions with (a) friction between the monomers and particle (χ = ξ1 /ξ0 ), (b) monomer–particle interaction energy (ε = E ad /kB T ), (c) filler size (ρ = R f /R F ), and (d) filler volume fraction ().

kept constant. This figure shows that at higher concentration of dispersed particles, the system could show a relatively independent frequency behavior. This rubber-like behavior, which is also reported experimentally [15, 20, 21], is due to the increase in volume fraction of the polymer–filler interfacial zone (especially for smaller filler particles) and hence the increase in fraction of adsorbed chains in filled systems.

2.3 Entangled Systems In polymer melt of sufficiently long chains or in polymer solution of sufficiently high concentration (above the overlap concentration), the dynamics of flexible polymer chains is controlled by the effect of topological

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constraints imposed by interchain entanglements [43]. Since the covalently bonded molecular chains cannot cross each other, the large-scale mobility of an entangled chain is restricted to its one-dimensional Brownian motion along the backbone. Hence, the overall viscoelastic behavior of entangled systems appears to depart significantly from the predictions of general bead-spring models. Instead, the well-known reptation model of de Gennes, Doi and Edwards [43] can more realistically represent the influence of topological constraints on diffusional motion of entangled polymers. Obviously, adhesive particles dispersed in an entangled suspending medium reduce chain mobility. However, due to the reversible nature of energetic interaction between fillers and chains, the reptative motion of the chains is not fully suppressed; adsorbed chains have the possibility to exercise their reptative motion within the encompassing tube during those time intervals which are simultaneously detached from adsorbing particles. Full relaxation is reached when the representative chain completely renews its original tube. The effective reptation time of the reversibly adsorbing chain can be determined using the sticky reptation model [42].

2.3.1

Microscopic Structure

Consider an ensemble of entangled polymer molecules with Ne entanglement segments and a random distribution of non-aggregated rigid spherical nanoparticles. The monodispersity of both species is assumed. The schematic configuration of an adsorbed chain is shown in Figure 2.1. The chains can reversibly adsorb on the colloidal surface and form a polydisperse succession of loops, tails and sequences of bound monomers. It is also assumed that the dispersed particles are sufficiently small such that even at low volume fractions, the average particle wall-to-wall distance could be on the order of the average size of a polymer coil. Hence, at equilibrium, chains may simultaneously attach to more than one nanoparticle. The effect of topological constraints on an adsorbed chain can be modeled by a confining tube with average diameter a, equal to the size of an entanglement segment (Figure 2.1). It has been suggested that the effective entanglement length may change close to an impenetrable wall [57]. In that case, this is a local effect which decays on a length scale comparable to the bulk value of a. In the present analysis, this possible effect is neglected and the tube diameter of adsorbed chains is taken to be similar to that of free chains. The probability for an entangled chain to make contact with nanoparticles at Nc entanglement segments can be approximated by p(Nc ) =

Ne Nc

q Nc (1 − q) Ne −N c

(2.18)

where q is the probability for one entanglement segment to bind to the surface of a nanoparticle. The underlying assumption for Eq. (2.18) is that the number of adsorbed entanglement segments of a single macromolecule is a random quantity and all entanglement segments have equal probability to attach to a nanoparticle. Obviously, the accuracy of this approximation for the probability distribution of Nc depends on the spatial distribution of nanoparticles and the relative values of Nc and Ne . Given the fact that q is usually a very small number, the distribution function shown by Eq. (2.18) can be approximated by

p(Nc ) =

Ne ! q Nc exp[−q(Ne − Nc )] Nc !(Ne − Nc )!

(2.19)

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23

A B

C a

D Figure 2.5 Schematic configuration of a free and an adsorbed chain in an entangled melt. The effect of molecular entanglement of surrounding chains is represented by encompassing tubes with diameter a. The adsorbed chain is attached on the surface of two nanoparticles at points B and C.

For a long polymer chain, well above the entanglement threshold, Ne is usually very large. At the limit of Ne Nc , we have 1 (N c ) Nc exp[−N c ] p(Nc ) ∼ = Nc !

(2.20)

where N c = Ne q is the average number of adsorbed entanglement segments per chain. In addition to the adsorbed chains, there is a population of free chains, which are not interacting with any particle (during the characteristic time of their relaxation), and hence simply exercise their regular reptative motion (Figure 2.5). The free reptation of the adsorbed chains, however, is suppressed because of their association with nanoparticles. Since the polymer–particle interaction is considered to be a reversible process, the adsorbed entanglement segment desorbs from the particle surface after a finite residence time τ , due to thermal fluctuation. This quantity can be utilized to express another definition for N c in Eq. (2.3) as N c = ρτ , where ρ represents the average number of attached entanglement segments in a chain per unit time. If each nanoparticle is considered as a temporary crosslink point between adsorbing entangled polymers, then the diffusion of adsorbed chains can be considered to be somewhat similar to the diffusion of associating polymers [58, 59]. Within this framework, the relaxation of the entire adsorbed chain is the result of partial relaxation of bridged segments (e.g. BC in Figure 2.5) and dangling ends (e.g. AB or AC after detachment of the chain from the particle at B) due to longitudinal Rouse type motion of these segments along the encompassing tube, in addition to the reptative motion of the entire chain during the time intervals in which the chain is not attached to any nanoparticle. Here, as a first order approximation, we neglect the partial relaxation of bridged segments and dangling ends and assume that an adsorbed chain can only make a reptation step when it is simultaneously detached from all temporary junctions. This relaxation process is called sticky reptation. Implementation of the relaxation of bridges and dangling ends in the model is possible; however, this is not examined here in order to reduce the number of adjustable parameters when the model predictions are compared with experiments.

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Let us assume that tad is the average time in which a chain stays in contact with nanoparticle(s). Each tad is followed by a time interval equal to tfr that chain is free and carries no temporal crosslink. According to Eq. (2.20), the fraction of time that a chain spends free can be evaluated as tfr = p(0) ∼ = exp[−Ne q] tad + tfr

(2.21)

We are interested in estimating the average reptation time for adsorbed chains. This represents the characteristic time that the chain needs to completely renew its encompassing tube. Since it is assumed that the adsorbed chains can only diffuse during tfr , then their effective reptation time, τa , can be estimated by [42] τf ∼ τa ∼ = = τ f exp[Ne q] p(0)

(2.22)

where τ f represents the characteristic reptation time of free chains. Parameter q implicitly depends on the volume fraction of dispersed nanoparticles, their size, and their interaction with surrounding polymer molecules.

2.3.2

Macroscopic Properties

Equation (2.22) shows that by increasing q, the relaxation of adsorbed chains could be considerably slower than the free chains. Therefore, for a system composed of populations of macromolecules with fast and slow relaxation dynamics, the mechanism of stress relaxation seems to be similar to double reptation of binary blends, due to the effect of constraint release [33–36]. This possibility is examined in the next section of the chapter. Here, we express the stress relaxation function in the following form to include chain reputation as well as tube renewal due to constraint release of the surrounding chains [60]: G(t) = G 0 [(1 − ψ)μ f (t) + ψμa (t)]α

(2.23)

where φ is volume fraction of the adsorbed chains and G 0 is plateau modulus. μ f and μa represent the fraction of surviving tube segments for free and adsorbed chains, respectively, which are defined by [61]: 8 1 −tn 2 exp π 2 n n2 τf

(2.24a)

8 1 −tn 2 μa (t) = 2 exp π n n2 τa

(2.24b)

μ f (t) =

odd

odd

Parameter α in Eq. (2.6) represents the contribution of constraint release in stress relaxation. If it is assumed that each chain moves inside its tube, independent of the motion of other chains (i.e. no constraint release), then α = 1, following the Doi-Edwards version of the reptation model. If the diffusional motion of surrounding chains is taken into account, then it can be shown that α = 2, following the theory of double

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Macro, Micro and Nano Mechanics of Multiphase Polymer Systems (a)

25

(b)

1

1

0.1

0.1

G′/G 0

G ′/G 0 ψ=0

0.01

ψ = 0.1

0.01

c = 10 c = 50

ψ = 0.2

c = 100

ψ = 0.5 0.001 0.01

0.1

1

10

c = 500 0.001 0.01

ωτ f

0.1

1

10

ωτ f

Figure 2.6 The variation of normalized storage modulus with (a) concentration of adsorbed chains (ψ) and (b) relative relaxation times of adsorbed and free chains (c = τa /τ f ) in a frequency sweep.

reptation [60, 62, 63]. The storage and loss moduli as functions of frequency ω can be calculated from Eq. (2.23), using the following expressions:

∞

G (ω) = ω

G(t) sin(ωt)dt

(2.25a)

G(t) cos(ωt)dt

(2.25b)

0

G (ω) = ω

∞ 0

2.3.3

Results

We start with a parametric study on the effect of fraction and relaxation time of the adsorbed chains on the linear (steady state) viscoelastic response of a filled melt. Here, only the principal mode of relaxation is taken into account (i.e. n = 1 in Eq. (2.24)). The relative relaxation time of the adsorbed and free chains is characterized by c = τa /τ f and the parameter α is set equal to unity (i.e. no constraint release). Figure 2.6(a) represents the effect of concentration of adsorbed chains (ψ) on the normalized shear modulus G in a frequency sweep. The parameter c is taken to be constant and equal to 100. Unfilled systems (ψ = 0) show the typical slope of 2:1 (in log-log scale) at low frequencies, providing a comparison for the effect of chain adsorption. As the concentration of adsorbed chains increases, the magnitude of storage modulus increases and a secondary plateau in low frequency region forms. The emergence of the plateau region is manifested when only 10% of the polymer chains are assumed to be adsorbed. Figure 2.6(b) shows that increasing the relaxation time of the adsorbed chains (at constant concentration of 20%) leads to a similar trend in viscoelastic behavior. These results indicate that the model is able to predict the enhancement in elasticity and formation of secondary plateau at low frequency domains, similar to the frequency sweep results of nanofilled polymer melts. According to the model, this behavior is associated with the increase in fraction and relaxation time

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of the adsorbed chains. The magnitude of these quantities is presumably controlled by the volume fraction and size of the dispersed particles, as well as the interaction between particles and surrounding chains. The surface area of well-dispersed colloidal particles increases by raising their volume fraction and more effectively by reducing their diameter. These effects increase the probability of polymer adsorption (i.e. q) and the fraction of adsorbed chains (i.e. ψ). Moreover, in the presence of strong polymer–particle interactions, the average residence time τ of the entanglement segments near the particle surface increases, which results in a significant reduction in mobility of the adsorbed chain. In the presented theory of sticky reptation, this effect is shown by the exponential dependence of relaxation time τa with quantity N c = ρτ , as indicated by Eq. (2.22). This implies that the stress relaxation at low frequencies (comparable with τa−1 ) can be hindered by the presence of nanoparticles, leading to a transition from liquid to solid-like behavior in those frequency regions. This deceleration in relaxation process at low frequencies is reported frequently for a variety of nanofilled polymer melts. Zhong and Archer [21] have reported on the linear viscoelastic response of poly(ethylene oxide) (PEO) with nearly monodisperse molecular weight of 189 kDa (i.e. N = 86Ne ), reinforced with isotropic silica nanospheres with average diameter of 12 nm. They have conducted relaxation and frequency sweep tests on PEO/silica nanocomposites in the melt state with different concentrations of dispersed nanoparticles. In what follows, the predictions of the proposed model in this chapter are quantitatively compared with their experimental data. The fitting parameters of the study include G 0 , τ f , q, and ψ. The values of the first three parameters extracted from fitting with experiments are 1300 kPa, 0.7 sec, and 0.084. ψ is the only fitting parameter whose value changes with the concentration of silica nanoparticles; we found that ψ = 0.15 at 2 vol% silica whereas ψ = 0.45 at 4 vol% of silica. Figure 2.7 compares the model predictions and experimental data for G in frequency sweeps. The best fits to the data of unfilled melts were obtained when the effect of constraint release was taken into account (i.e. α = 2). In contrast, predictions are in much better agreement

1.0E+07

1.0E+06

1.0E+05

1.0E+04

G ′ (Pa)

no silica 2 vol% silica 4 vol% silica

1.0E+03

1.0E+02

1.0E+01

0.01

0.1

1

10

100

ω(1/s) Figure 2.7 Comparison of the predicted storage modulus by model (solid lines) with experimental results of Zhong and Archer. Reprinted from [21]. (12/24/2002) Copyright (2002) American Chemical Society.

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27

with experimental data of filled melts when α = 1 and the effect of constraint release is neglected. This can be attributed to the relatively frozen dynamics of adsorbed chains in filled systems. In this way, a fraction of the tube forming chains exhibits an extremely slow relaxation and, as a result, the survival time of the encompassing tube increases. By increasing the concentration of nanoparticles and consequently the fraction of adsorbed chains, an entrapped chain exercises its reptation motion in a practically time-independent network and without constraint release. Further investigation, however, is necessary to validate the generality of this observation for different nanocomposites melts.

2.4 Conclusion Growing applications of polymer nanocomposites require a comprehensive understanding of their equilibrium and kinetic structure at a wide spectrum of time and length scales. Different techniques at various time and length scales from molecular scale (e.g. atoms), to mesoscale (e.g. coarse-grains, particles, monomers) and to macroscale, have shown success in addressing various aspects of polymer nanocomposite properties. In this chapter, we reviewed two classes of mesoscale models for linear viscolesatic behavior of unentangled polymer solutions and entangled polymer melts reinforced with non-agglomerated nanoparticles at low volume fraction and with short range energetic affinity with polymer chains. In filled dilute polymer solutions, the structure of the adsorbed polymer layer was determined using the scaling theory of adsorption in good-solvent conditions. The dynamics of the system was modeled by the classical Maxwell constitutive relations. The relaxation of entangled chains was described by the sticky reptation model [42]. The system relaxation was analyzed by the combination of stress relaxation of free and adsorbed chains. The proposed models were able to rationalize the specific properties of reinforced polymer liquids based on the role of energetic affinity towards particles at sub-colloidal sizes. This effect intensifies by increasing volume fraction and reducing size of the particles which leads to higher fraction of adsorbed chains in the system. It was shown that behaviors such as high shear viscosity at low filler concentrations or solid-like properties in low frequency regimes could be attributed to the slowdown of the relaxation process in polymer chains. This process is controlled by the frictional interaction between monomers and particles, the density of the adsorbed polymer chains on the particle surface (controlled by monomer-particle adsorption energy), and volume fraction of the interfacial layer which can be enhanced by reduction of filler size or increasing filler concentration. Although the proposed models show the ability to predict different characteristics of viscoelastic properties of nanofilled polymers, they have a few notable limitations. The models hold true only for systems with homogeneously dispersed nanoparticles. This is an idealization, since the formation of particle aggregates is always possible in experimental conditions (due to either incomplete exfoliation during synthesis or deformation-induced flocculation). The porosity in the structure of individual aggregates increases the effective volume fraction of the rigid phase, which can lead to a significant contribution of hydrodynamic effect. In addition, the clusters of aggregated particles follow a different relaxation pattern which is rooted in the stored elastic energy of the strained clusters and the failure/restoration properties of filler–filler bonds. Moreover, in the proposed model for entangled systems, the effects of filler size and concentration are only implicitly accounted for. Understanding how the values of q and φ change with size and concentration of nanoparticles is possible by performing complementary molecular simulation studies on relevant systems. Finally, it should be emphasized that in the proposed models, the adsorption of molecules on the surface of nanoparticles is regarded as a reversible phenomenon. In reality, this assumption is not necessarily valid for any filled system. For a polymer melt, a subtle interplay between compressibility, monomer energetic affinity with the surface, and surface/chemical heterogeneities is expected to determine reversibility or irreversibility of polymer–particle interaction [64].

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Acknowledgements A.S. Sarvestani gratefully acknowledges the financial support provided by the Department of Mechanical Engineering and Office of Vice President for Research at the University of Maine. This work was supported by a grant to E. Jabbari from the National Science Foundation under Grant No. CBET-0756394.

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3 Theory and Simulation of Multiphase Polymer Systems Friederike Schmid Institute of Physics, Johannes-Gutenberg Universität Mainz, Germany

3.1 Introduction The theory of multiphase polymer systems has a venerable tradition. The ‘classical’ theory of polymer demixing, the Flory-Huggins theory, was developed in the 1940s [1, 2]. It is still the starting point for most current approaches – be they improved theories for polymer (im)miscibility that take into account the microscopic structure of blends more accurately, or sophisticated field theories that allow to study inhomogeneous multicomponent systems of polymers with arbitrary architectures in arbitrary geometries. In contrast, simulations of multiphase polymer systems are quite young. They are still limited by the fact that one must simulate a large number of large molecules in order to obtain meaningful results. Both powerful computers and smart modeling and simulation approaches are necessary to overcome this problem. In the limited space of this chapter, we can only give a taste of the state-of-the-art in both areas, theory and simulation. Since the theory has reached a fairly mature stage by now, many aspects of it are covered in textbooks on polymer physics [3–9]. The information on the state-of-the art of simulations is much more scattered. This is why we have put some effort into putting together a representative list of references in this area – which is of course still far from complete. The chapter is organized as follows. In Section II, we briefly introduce some basic concepts of polymer theory. The purpose of this part is to make the chapter accessible to readers who are not very familiar with polymer physics; it can safely be skipped by the others. Section III is devoted to the theory of multiphase polymer systems. We recapitulate the Flory-Huggins theory and introduce in particular the concept of the Flory interaction parameter (the χ parameter), which is a central ingredient in most theoretical descriptions of multicomponent polymer systems. Then we focus on one of the most successful mean-field theories for inhomogeneous (co)polymer blends, the self-consistent field theory. We sketch the main idea, discuss various


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aspects of the theory and finally derive popular analytical approximations for weakly and strongly segregated blends (the random phase approximation and the strong-segregation theory). In Section IV, we turn to discussing simulations of multiphase polymer systems. A central concept in this research area is ‘multiscale modeling’: Polymers cannot be treated at all levels of detail simultaneously, hence coarse-grained models are used in order to study different aspects of the materials in different simulations. This allows one to push the simulation limits to larger length and time scales. We describe some of the most popular coarse-grained structural and dynamical models and give an overview over the state-of-the-art of simulations of polymer blends and copolymer melts.

3.2

Basic Concepts of Polymer Theory

For the sake of readers who are not familiar with polymer theory, we begin with recapitulating very briefly some basic concepts. Polymers are macromolecules containing up to hundreds of thousands of atoms. At first sight, one would not expect such molecules to be easily amenable to theoretical modeling; however, it turns out that the large size of the molecules and their highly repetitive structure in fact simplifies things considerably. Since polymer molecules interact with many others, details of local interactions average out and polymers can often be characterized by a few effective quantities, such as their topology, the local stiffness along the backbone, the bulkiness, the compatibility/incompatibility of the building blocks, etc. Decades ago, pioneers like Flory [3], Edwards [5] and de Gennes [4] have established theoretical polymer science as a highly successful field of research, which brings together scientists from theoretical chemistry, statistical physics, materials science, and even the biosciences, has created a wealth of new beautiful theoretical concepts, and has not lost any of its fascination for theorists up to date.

3.2.1

Fundamental Properties of Polymer Molecules

The characterizing property of polymers is their highly modular structure. They are composed of a large number of small building blocks (monomers), which are often all alike, but may also be combined to arbitrary sequences (in the case copolymers and biopolymers). The monomers are arranged in chains, which are usually flexible on the nanometer length scale, i.e., they can form kinks at little energetic expense, they curve around and may assume a large number of conformations at room temperature. The properties of such flexible polymers are largely determined by the entropy of the chain conformations. For example, the number of available conformations is reduced if molecules are stretched, which leads to a purely entropic restoring spring force [10] (rubber elasticity). Exposed to stress, polymeric systems respond by molecular rearrangements, which takes time and results in time-dependent strain (viscoelasticity). The fundamental processes that govern the behavior of polymeric materials do not depend on the chemical details of the monomer structure. For qualitative purposes, polymer molecules can be characterized by a few properties such as:

r r r r

The architecture of the molecules (linear chains, rings, stars, etc.); Physical properties (local chain stiffness, chain size, monomer volume); Physiochemical properties (monomer sequence, compatibility, charges); Special properties (e.g., a propensity to develop crystalline or liquid crystalline order).

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Theory and Simulation of Multiphase Polymer Systems

3.2.2

33

Coarse-Graining, Part I

The notion of ‘coarse-graining’ has lately become a buzzword in materials science, but the underlying concept is actually quite old in polymer science. The need for coarse-graining results from the fact that polymeric materials exhibit structure on very different length scales, ranging from Angstrom (the monomeric scale) to hundreds of nanometers (typical molecule extensions) or micrometers (supramolecular aggregates). It is not possible to treat all of them within one common theoretical framework. Therefore, different theoretical descriptions have been developed that deal with phenomena on different length and time scales. On the microscale, chemical details are taken into account and the polymers are treated at an atomistic level. This is the realm of theoretical chemistry. On the mesoscale, simplified molecule models come into play (string models, lattice models, bead-spring models, see below), whose behavior can be understood with concepts from statistical physics. Finally, on the macroscale, polymeric materials are described by continuous fields (composition, strain, stress, etc.) with certain mechanical properties, and their behavior can be calculated with methods borrowed from the engineers. In the following, we shall mainly focus on the mesoscale level, where polymers are described by extended molecules made of simplified ‘monomeric’ units, each representing several real monomers. Even within that level, one still has some freedom regarding the choice of the coarse-grained units. This is illustrated in Figure 3.1, where different coarse-grained representations of a polymer are superimposed onto each other. Polymers have remarkable universality properties, which allow one to link different coarse-grained representations in a rather well-defined manner, as long as the length scales under consideration are much larger than the (chemical) monomer length scale. For example, starting from one (atomistic or coarse-grained) model, we can construct a coarse-grained model by combining m ‘old’ units to one ‘new’ unit. If m is sufficiently large, the average squared distance d2 between two adjacent new units will depend on m according to a

Figure 3.1 Mesoscopic coarse-grained representations of a polymer molecule. Light pearl necklace in the background: Bead chain with bonds of fixed length. Solid and dashed lines: chain of coarse-grained units linked by bonds of variable length.

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characteristic power law d 2 ∼ m 2ν

(3.1)

where the exponent ν depends on the environment of a polymer, but not on chemical details [4, 5]. In a dense polymer melt, one has ν = 1/2 (see below). Similar scaling laws can be established for other chain parameters. 3.2.3

Ideal Chains

In mesoscopic polymer theories, one often uses as a starting point a virtual polymer chain where monomers that are well separated along the polymer backbone do not interact with each other, even if their spatial distance is small. Such polymers are called ‘ideal chains’. Even though they are mere theoretical constructions, they provide a good approximative description of polymers in melts and in certain solvents (‘Theta’-solvents, see below). 3.2.3.1

A Paradigm of Polymer Theory: The Gaussian Chain

Let us now consider a flexible ideal chain with N monomers, which we coarse-grain several times as sketched in Figure 3.1, until one coarse-grained monomer unites m ‘real’ monomers. For large m, the resulting chain is a random walk in space consisting of uncorrelated random steps di of varying length. According to the central limit theorem of probability theory [11], the steps are approximately Gaussian distributed, P(d) ∼ exp(−d 2 /2mσ 2 ), where σ does not depend on m. The same random walk statistics can be reproduced by a Boltzmann distribution with an effective coarsegrained Hamiltonian N /m 1 kB T 2 Hm = d 2 mσ 2 i=1 i

(3.2)

The Hamiltonian Hm describes the energy of a chain of springs with spring constant k B T m/σ 2 . The coarsegraining procedure has thus eliminated the information on chemical details (they are now incorporated in the single parameter σ ), and instead unearthed the entropically induced elastic behavior of the chain which lies at the heart of rubber elasticity. Eq. (3.2) is also an example for universal behavior in a polymer system (see section 3.2.2). The coarse-grained chain is self-similar. Every choice of m produces an equivalent model, provided the spring constant is rescaled accordingly. The distance between two coarse-grained units exhibits a scaling law of the form (3.1) as a function of m, d 2 = 3σ 2 m 2ν with ν = 1/2. Based on these considerations, it seems natural to define a ‘generic’ ideal chain model based on Eq. (3.2) with m = 1, N −1 1 HG [ri ] = (ri+1 − ri )2 kB T 2σ 2 i=1

(3.3)

the so-called ‘discrete Gaussian chain’ model. For theoretical purposes, it is often convenient to take the continuum limit: The index i in Eq. (3.2), which counts the monomers along the chain backbone, is replaced by a continuous variable s, the chain is parametrized by a continuous path R(s), and the steps d correspond

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to the local derivatives dR/ds of this path. The effective Hamiltonian then reads HG [R] 3 = 2 kB T 2b

N

ds 0

dR ds

2 (3.4)

This defines the continuous Gaussian chain. The only material parameters in Eq. (3.4) are the chain length N and the ‘statistical segment length’ or ‘Kuhn length’ b. Even those two are not independent, since they both depend on the definition of the monomer unit. An equivalent chain model can be obtained by rescaling N → N/λ and b2 → b2 λ. Hence the only true independent parameter is the extension of the chain, which can be characterized by the squared gyration radius Rg2

1 = N

N

ds (R(s) − R)2 = b2 N /6

(3.5)

0

where R = 1/N ∫ dsR(s) is the center of mass of the chain. The quantity Rg sets the (only) characteristic length scale of the Gaussian chain, and all length-dependent quantities scale with Rg . For example, the structure factor is given by 2

1 N ds eik R(s) = N g D k 2 Rg2 S(k) = N 0

(3.6)

with the Debye function g D (x) =

2 −x (e − 1 + x) x2

(3.7)

The Gaussian chain is not only a prototype model for ideal chains, it also provides a general framework for mesoscopic theories of polymer systems. The Hamiltonian HG (Eq. (3.4)) is then supplemented by additional terms that account for interactions, external fields, constraints (e.g., chemical crosslinks) etc. In this more general context, the Hamiltonian (3.4) is often referred to as ‘Edwards Hamiltonian’. Finally in this section, we note that from a mathematical point of view, the probability distribution of chain conformations defined by Eq. (3.4) is a Wiener measure [11, 12]. The continuum limit leading to Eq. (3.4) is far from trivial, but well-defined. We shall not delve further into this matter. 3.2.3.2

Other Chain Models

The Gaussian chain model is a common starting point for analytical theories of long flexible polymers on sufficiently large length scales. On smaller length scales, or for stiffer polymers, or for computer simulation purposes, other types of coarse-grained models have proven useful. We briefly summarize some popular examples. The wormlike chain model is a continuous model designed to describe stiff polymers. They are represented by smooth paths R(s) with fixed contour length N, where the parameter s runs over the arc length of the curve, i.e., the derivative vector u = dR/ds has length unity, |u| ≡ 1. The paths have a bending stiffness η, such that they are distributed according to the effective Hamiltonian η HW LC [R] = kB T 2

N

ds 0

d2 R ds 2

2 (3.8)

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The wormlike chain model is particularly useful if local orientational degrees of freedom are important. The freely jointed chain is a discrete chain model where the chain is composed of N links of fixed length. It is often used to study general properties of ideal chains. The spring-bead chain is a chain of beads connected with springs. It has some resemblance with the discrete Gaussian chain model, except that the springs have a finite equilibrium length. Spring-bead models are popular in computer simulations. In lattice models, the monomer positions are confined to the sites of a lattice. This simplifies both theoretical considerations and computer simulations. 3.2.4

Interacting Chains

The statistical properties of chains change fundamentally if monomers interact with each other. Such interactions are readily introduced in the coarse-grained models presented above. In the discrete models, one simply adds explicit interactions between monomers. In the continuous path models, one supplements the energy contribution for individual ideal chains, Eq. (3.4) or (3.8), by an interaction term, such as v ˆ = H I [ρ] 2

w dr ρˆ + 6 2

dr ρˆ 3 + · · ·

(3.9)

(for weak interactions), where the ‘monomer density’ ρ(r ˆ ) is defined as ρ(r) ˆ =

α

N

ds δ(r − Rα (s))

(3.10)

0

and the sum α runs over the polymers Rα (s) in the system. Eq. (3.9) corresponds to a virial expansion of the local interaction energy in powers of the density. In many cases, only the quadratic term (v) needs to be taken into account (‘two-parameter Edwards model’). The Ansatz (3.9) is suitable for dilute polymer systems – dense systems are discussed below (Section 3.2.4.2). The generalization to multiphase systems where monomers may have different type A, B, . . . is straightforward. One simply operates with different densities ρˆ A , ρˆ B , . . . and interaction parameters v A A , v AB , . . .. Interactions complicate the theoretical treatment considerably and in general, exact analytical solutions are no longer available. The properties of interacting polymer systems have been explored theoretically within mean-field approximations, renormalization-group calculations, scaling arguments, and computer simulations. To set the stage for the discussion of multiphase systems in Sections 3.3 and 3.4, we will now briefly sketch the most important scenarios for monophase polymer systems. 3.2.4.1

Polymers in Solution and Blobs

We first consider single, isolated polymer chains in solution. Their properties depend on the quality of the solvent, which is incorporated in the second virial parameter v in Eq. (3.9) (the three-body parameter w is typically positive [13]). In good solvent (v > 0), monomers effectively repel each other, and the chain swells. Extensive theoretical work [4] has shown that the scaling behavior (Eq. (3.1)) remains valid, but the exponent ν increases from ν = 1/2 for ideal chains to ν ≈ 3/(d + 2), where d is the spatial dimension (more precisely, ν = 0.588 in three dimensions). This is the famous ‘Flory exponent’, which characterizes the scaling behavior of so-called ‘self-avoiding chains’. Accordingly, the gyration radius of the chain scales

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with the chain length like Rg2 ∼ N 2ν

(3.11)

In bad solvent (v < 0), monomers effectively attract each other and the chain collapses. At the transition between the two regimes, the ‘Theta point’ (v θ ≈ 0), the scaling behavior basically corresponds to that of ideal chains (ν = 1/2), except for subtle corrections due to the three-body w-term [4]. Eq. (3.11) describes the behavior of single, unperturbed chains. Even in good solvent, the self-avoiding scaling is often disturbed. For example, the chains cannot swell freely if they are confined, or if they are subject to external forces. Another important factor is the concentration of chains in the solution: If many chains overlap, the intrachain interactions are screened on large length scales. Loosely speaking, monomers cannot distinguish between interactions with monomers from the same chain and from other chains. As a result, chains no longer swell and ideal chain behavior is recovered. This mechanism applies in three or more spatial dimensions. Two dimensional chains segregate [4, 14, 15]. All of these situations can be analyzed within one single ingenious framework, the ‘blob’ picture introduced Daoud et al. in 1975 [17]. It is based on the assumption that there exists a crossover length scale ξ below which the chain is unperturbed. Blobs are volume elements of size ξ within which the polymers behave like self-avoiding chains. On larger scales, the polymer behaves like an ideal chain consisting of a string of blobs. Every blob contains m ∼ ξ 1/ν monomers and carries a free energy of the order kB T. These simple rules are the whole essence of the blob model. We shall illustrate their use by applying them to a number of prototype situations depicted in Figure 3.2. Concentrated polymer solution (Figure 3.2 a). For polymer concentrations , we calculate the crossover length scale ξ from self-avoiding to ideal behavior. Since ξ is the blob size, we can simply equate = m/ξ 3 , i.e., ξ ∼ φ −ν/(3ν−1) . Polymer confined in a slit (Figure 3.2 b). We consider the free energy penalty F on the confinement. Here, the blob size is set by the width R of the slit. Each blob contains m ∼ R1/ν monomers, hence the total free energy scales like F ∼ N/m ∼ NR−1/ν . Polymer confined in a cavity (Figure 3.2 c) The result b) also holds for chains confined in a tube. In closed cavities, however, the situation is different due to the fact that the cavity constrains the monomer concentration. The resulting blob size is ξ ∼ (N/R3 )ν /(1−3ν) , and the free energy of confinement scales as (a)

(b)

(c)

(d)

Figure 3.2 Illustration of the blob model in different situations: (a) concentrated polymer solution; (b) Polymer confined in a slit; (c) chain confined in a spherical cavity; (d) structure formation in solutions of miktoarm star copolymers (after Ref. [16]). See text for explanation.

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F ∼ (R/ξ )3 ∼ (R/N ν )3/(1−3ν) . This has been discussed controversially, but was recently confirmed by careful computer simulations [18]. ABC miktoarm star copolymers in selective solvent (Figure 3.2d). Last, we cite a recent application to a multiphase polymer system. Zhulina and Borisov [16] have studied ABC star copolymers by means of scaling arguments. They derived a rich state diagram, according to which ABC star copolymers may assemble to several types of nanostructures, among other spherical micelles, dumbbell micelles, and striped cylindrical micelles. This is only one of numerous examples where scaling arguments have been used to analyze complex multicomponent systems.

3.2.4.2

Dense Melts

Dense melts can be considered as extreme cases of a very concentrated polymer solution, hence it is not surprising that the chains effectively exhibit ideal chain behavior. In fact, the situation is more complicated than this simple argument suggests. The quasi-ideal behavior results from a cancellation of two effects: On the one hand, the intrachain interactions promote chain swelling, but on the other hand, the chain pushes other chains aside (‘correlation hole’), which in turn exert pressure and squeeze it. Deviations from true ideal behavior can be observed, e.g., at the level of chain orientational correlations [19, 20]. Nevertheless, the ideality assumption is a good working hypothesis in dense melts and shall also be used here in the following.

3.2.5

Chain Dynamics

In this chapter we focus on equilibrium and static properties of polymer systems. We can only touch on the possible dynamical behavior, which is even more diverse. In the time scales of interest, the motion of polymers is diffusive, i.e., the inertia of the macromolecules is not important. Three prominent types of dynamical behavior have been established. In the Rouse regime, the chain dynamics is mainly driven by direct intrachain interactions. This regime is encountered for short chains. The dynamical properties of ideal chains can be calculated exactly, and the results can be generalized to self-avoiding chains using scaling arguments. One of the important properties of Rouse chains is that their sedimentation mobility does not depend on the chain length N. Hence the diffusion constant scales like D ∼ 1/N , and the longest internal relaxation time, which can be estimated as the time in which the chain diffuses a distance Rg , scales like τ ∼ N 2ν+1 . In the Zimm regime, the dynamics is governed by long-range hydrodynamic interactions between monomers. This regime develops for sufficiently long chains in dilute solution. They diffuse like Stokes spheres with the diffusion constant D ∼ 1/Rg , and the longest relaxation time scales like τ ∼ Rg3 . In concentrated solutions, the hydrodynamic interactions are screened [5] and Rouse behavior is recovered after an initial Zimm period [21]. The reptation regime is encountered in dense systems of chains with very high molecular weight. In this case, the chain motion is topologically constrained by the surrounding polymer network, and they are effectively confined to move along a tube in a a snake-like fashion [4, 22]. The diffusion constant of linear chains scales like D ∼ 1/N 2 and the longest relaxation time like τ ∼ N 3 . This description is very schematic and oversimplifies the situation even for fluids of linear polymers. Moreover, most polymer materials are not in a pure fluid state. They are often cooled down below the glass transition, or they partly crystallize – in both cases, the dynamics is frozen. Chemical or physical crosslinks constrain the motion of the chains and impart solid-like behavior. In multiphase polymer systems, the situation is further complicated by the fact that the glass point or the crystallization temperature of the

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Spheres (S)

Cylinders (C)

Gyroid (G)

39

Lamellae (L)

Figure 3.3 Self-assembled copolymer mesophases.

different components may differ, such that one component freezes where the other still remains fluid [23–32]. In the following discussion, we shall limit ourselves to fluid multiphase polymer systems.

3.3 Theory of Multiphase Polymer Mixtures After this general overview, we turn to the discussion of polymer blends. We consider dense mixtures, where the polymers are in the melt regime (Section 3.2.4.2). Moreover, we assume incompressibility – the characteristic length scales of density fluctuations are taken to be much smaller than the length scales of interest here. Monomers of different type are usually slightly incompatible (see Section 3.3.1.3). In polymers, the incompatibilities are amplified, such that macromolecules of different type tend to be immiscible: Blended together, they demix and develop an inhomogeneous multiphase structure where microdroplets of one phase are finely dispersed in another phase. In order to overcome or at least control this situation, copolymer molecules can be added in which the two incompatible components are chemically linked to each other. They act as compatibilizer, i.e., they shift the demixing transition and reduce the interfacial tension between different phases in the demixed region. At high concentrations, they are found to self-organize into a variety of ordered mesophases (microphase separation; see, e.g., structures shown in Figure 3.3). Hence copolymers can also be used to manufacture nanostructured materials in a controlled way. Nowadays, the theory of structure formation in polymer blends has reached a highly advanced level and theoretical calculations have predictive power, e.g., with respect to structures that can be expected in new polymeric materials. In this section, we shall present some of the most successful theoretical approaches. 3.3.1

Flory-Huggins Theory

We begin with sketching the Flory-Huggins theory, which is the classical theory of phase separation in polymer blends, and which in some sense lays the foundations for all later, more sophisticated theories of polymer mixtures. 3.3.1.1

Basic Model for Binary Blends

We consider a binary blend of homopolymers A and B with length NA and NB , and volume fractions A and B . According to Flory [1] and Huggins [2], the free energy per monomer is approximately given by fF H B A ln( A ) + ln( B ) + χ A B = kB T NA NB

(3.12)

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Handbook of Multiphase Polymer Systems 4

4 N =NA/2 B

3

χNA

3

demixed

χNA

2

N =NA B

2 N =2NA

1

B

N =2N B

0 0

A

1

mixed

ΦA

1

0

ΦA

0 1

Figure 3.4 Left: Phase diagram for a binary AB polymer blend with B-chains twice as long as A-chains according to the Flory-Huggins theory. Thick solid line shows the binodal line (i.e., the demixing line), thin dashed line the spinodal line (i.e., the line where the homogeneous blend becomes unstable). Right: Binodals for binary AB blends with different chain length ratios as indicated.

with A + B = 1. The first two terms account for the mixing entropy of the two components, and the last term for the (in)compatibility of the monomers. The parameter χ is the famous ‘Flory Huggins parameter’, which will be discussed in more detail below. The generalization of this expression to ternary ABC homopolymer blends, etc. is straighforward, one only needs to introduce several χ -parameters χ AB , χ BC , and χ AC . Here we will only discuss binary systems. By minimizing the free energy, Eq. (3.12), one easily identifies the region in phase space where the mixture phase separates into an A-rich phase and a B-rich phase. At low of χ , the blend remains √ √ values 2 = (1/ N + 1/ N ) for the critical composition homogeneous. Demixing sets in at a critical value 2χ c A B √ A,c = 1/(1 + N A /N B ). The region of stability of the homogeneous (mixed) blend is delimited by the ‘binodal’ line (see Figure 3.4). Beyond the binodal, the homogeneous blend may still remain metastable. It becomes unstable at the ‘spinodal’, which is defined as the line where the second derivative of fFH in Eq. (3.12) with respect to A vanishes. An example of a phase diagram with a binodal and a spinodal is shown in Figure 3.4 (left). Figure 3.4 (right) demonstrates the shift of the binodal with varying chain length ratio NA /NB . The Flory-Huggins free energy, Eq. (3.12), was originally derived based on a lattice model, but it can also be applied to off-lattice systems. It does, however, rely on three critical assumptions:

r r r

The polymer conformations are taken to be those of ideal chains, independent of the composition (ideality assumption, cf. Section 3.2.4.2). The melt is taken to be incompressible, and monomers A and B occupy equal volumes. Local composition fluctuations are neglected (mean-field assumption).

In reality, none of these assumptions is strictly valid. The polymer conformations do depend on the composition, most notably for chains of the minority component. The incompressibility assumption is reasonable, but the volumes per monomer are not equal. As a consequence, the χ -parameter is not a fixed parameter (at fixed temperature), but depends on the composition of the blend (see Section 3.3.1.3). Finally, the composition fluctuations shift phase boundaries and may even fundamentally change the phase behavior. (see Section 3.3.2.5).

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3.3.1.2

41

Inhomogeneous Systems: Flory-Huggins-de Gennes Theory

Equation (3.12), only describes homogeneous systems. The simplest approach to generalizing the FloryHuggins theory to inhomogeneous systems, e.g., polymer blends containing interfaces, consists in adding a penalty on composition variations (∇ A )2 = (∇ B )2 . The coefficient of the square gradient term can be derived within a more advanced mean-field treatment, the random phase approximation, which will be described further below (Section 3.3.3.1). One obtains the Flory-Huggins-de Gennes free energy functional for polymer blends,

FFHdG [ A (r)] = ρ0

dr

f F H ( A (r)) +

kB T 36

b2A b2 + B A B

(∇ A )2

(3.13)

(with B = 1 − A ), where bA and bB are the Kuhn lengths of the homopolymers A and B, and fFH ( A ) is given by Eq. (3.12). The functional (3.13) can be applied if composition variations are weak, and have characteristic length scales of the order of the gyration radius of the chains (‘weak segregation regime’, see Section 3.3.3.1). A very similar functional can be derived in the opposite case, where A- and B-polymers are fully demixed and separated by narrow interfaces. In this ‘strong segregation’ regime, the blend can be described by the functional (see Section 3.3.3.2) FSSL [ A (r)] = ρ0

kB T dr χ A B + 24

b2A b2 + B A B

(∇ A )

2

(3.14)

We note that at strong segregation, the mixing entropy terms in in fFH , Eq. (3.12), can be neglected, hence the functionals (3.13) and (3.14) are identical except for the numerical prefactor of the square gradient term. In the strong segregation limit, the square gradient penalty results from an entropic penalty on A and B segments due to the presence of the interface, whereas in the weak segregation limit, it is caused by the deformation of whole chains. 3.3.1.3

Connection to Reality: The Flory-Huggins Parameter

In the Flory-Huggins theory, the microscopic features of the blend are incorporated in the single Flory-Huggins parameter χ . Not surprisingly, this parameter is very hard to access from first principles. In the original Flory-Huggins lattice model, χ is derived from the energetic interactions between monomers that are neighbors on the lattice. The interaction energy between monomers i and j is taken to be characterized by energy parameters ij . The χ -parameter is then given by χ=

z−2 (2 AB − A A − B B ) 2k B T

(3.15)

where z is the coordination number of the lattice. It is reduced by two (z − 2) in order to account for the fact that interactions between neighbor monomers on the chain are fixed and have no influence on the demixing behavior. In reality, the situation is not as simple. The miscibility patterns in real blends tend to deviate dramatically from that predicted by the Flory-Huggins model. Several blends exhibit a lower critical point instead of an upper critical point, indicating that the demixing is driven by entropy rather than enthalpy. The critical temperature Tc of demixing often does not scale linearly with the chain length N, as one would expect from

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Eq. (3.15) (see Figure 3.4). In blends of polystyrene and poly(vinyl methyl ether) (PS/PVME), for example, Tc is nearly independent of N, and the critical concentration c is highly asymmetric even for roughly equal chain lengths, in apparent contrast with Figure 3.4 (NB = NA ) [33]. Formally, this problem can be resolved by arguing that the Flory-Huggins expression for χ , Eq. (3.15), is oversimplified and that χ kB T is not a constant. Several factors contribute to the χ -parameter, leading to a complex dependence on the temperature, the blend composition, and even the chain length. Monomer incompatibility: Monomers may be incompatible both for enthalpic and entropic reasons. For example, consider two nonpolar monomers i and j. The van-der Waals attraction between them is proportional to the product of their polarizabilities α, hence ij ∝ α i α j and χ ∝ (α A − α B )2 . More generally, the enthalpic incompatibility of monomers i and j can be estimated by χ H ∝ (δ A − δ B )2 , where the δ i are the Hildebrand solubility parameters of the components [34, 35]. In addition, entropic factors may contribute to the monomer incompatibility, which are, e.g., related to shape or stiffness disparities [36–40]. Since the enthalpic and entropic contributions evolve differently as a function of the temperature, the χ -parameter will in general exhibit a complicated temperature dependence. Equation-of-state effect: In general, the volume per polymer depends on the composition of the blend. Already the volume per monomer is usually different for different monomer species; at constant pressure, χ therefore varies roughly linearly with A [41]. In blends of monomers with very similar monomer structure, e.g., isotopic blends, the linear contribution vanishes and a weak parabolic dependence remains, which can partly [42] (but not fully [43]) be explained by an ‘excess volume of mixing’. Chain correlations: Since the demixing is driven by intermolecular contacts, intramolecular contacts only contribute indirectly to the χ -parameter. The estimate for χ can be improved if one replaces the factor z − 2 by an ‘effective coordination number’ zeff which is given by the mean number of interchain contacts per monomer [44]. Moreover, the ideality assumption (see Section 3.2.4.2) is not strictly valid. Chains in the minority phase (e.g., A chains in a B-rich phase) tend to shrink in order to reduce unfavorable contacts [45]. Since the segregation is effectively driven by χ N, χ slightly depends on the chain length N as a result. The situation becomes even more complex if copolymers are involved, which assume dumbbell shapes even in a disordered environment [46, 47]. This also may affect the effective χ -parameter [48]. Composition correlations: The effective interactions between monomers change if the local environment is not random. We have already noted earlier that the composition may fluctuate. Large-scale fluctuations can be incorporated in the Flory-Huggins framework in terms of a fluctuating field theory (see Sections 3.3.2.5 and 3.3.3.1). Fluctuations (correlations) on the monomeric scale renormalize the χ -parameter (nonrandom mixing). Moreover, the local fluid structure may depend on the local composition (nonrandom packing). In view of these complications, establishing an exhaustive theory of the χ -parameter remains a formidable task [37, 49–54]. Even the reverse problem of designing simplified particle-based polymer models with a well-defined χ -parameters turns out to be highly non-trivial [55]. The very concept of a χ -parameter has been challenged repeatedly, e.g., by Tambasco et al. [56], who analyzed experimental data for a series of blends and found that their thermodynamic behavior can be related to a single ‘g − 1-parameter’, which is independent of composition, temperature and pressure. They suggest that this parameter may be more appropriate to characterize blends than the χ -parameter. However, it has to be used in conjunction with an integral equation theory, the BGY lattice theory by Lipson [57–59], which is much more involved than the Flory-Huggins theory especially when applied to inhomogeneous systems. Freed and coworkers [36, 37, 40, 60] have proposed a generalized Flory-Huggins theory, the ‘lattice cluster theory’, which provides a consistent microscopic theory for macroscopic thermodynamic behavior. In a certain limit (high pressure, high molecular weight, fully flexible chain), this theory reproduces a Flory-Huggins type

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free energy with an effective χ -parameter [49] χeff

2( p A A + p B B ) (r A − r B )2 = + χ0 1 − z2 z(z − 2)

(3.16)

where χ 0 is given by Eq. (3.15) and the parameters rj , pj depend on the structure of the monomers j. Dudowicz et al. have recently pointed out that this model can account for a wide range of experimentally observed miscibility behavior [38, 39]. Another pragmatic and reasonably successful approach consists in using χ as a heuristic parameter. Assuming that it is at least independent of the chain length, it can be determined experimentally, e.g., from fitting small-angle scattering data to theoretically predicted structure factors [61–63]. Alternatively, χ can be estimated from atomistic simulations [55, 64]. The results from experiment and simulation tend to compare favorably even for systems of complex polymers [65]. 3.3.2

Self-consistent Field Theory

We have taken some care to discuss the Flory-Huggins theory because it establishes a framework for more general theories of polymer blends. In particular, it provides the concept of the Flory-Huggins parameter χ , which we will take for granted from now on, and take to be independent of the composition, despite the question marks raised in the previous section (Section 3.3.1.3). In this section, we present a more sophisticated mean-field approach, the self-consistent field (SCF) theory. It was first proposed by Helfand and coworkers [66–70] and has since evolved to be one of the most powerful tools in polymer theory. Reviews on the SCF approach can be found, e.g., in [8, 71, 72]. 3.3.2.1

How It Works in Principle

For simplicity, we will first present the SCF formalism for binary blends, and discuss possible extensions later. Our starting point is the Edwards Hamiltonian for Gaussian chains, Eq. (3.4), with a Flory-Huggins interaction term, H I [ρˆ A , ρˆ B ]/k B T = ρ0 χ

ˆ A ˆB dr

(3.17)

ˆ j = ρˆ j /ρ0 and incompressibility is requested, i.e., where we have defined the ‘monomer volume fractions’ ˆ B ≡ 1 everywhere. The quantity ρˆ j depends on the paths Rα of chains of type j and has been defined ˆ A+ in Eq. (3.10). We consider a mixture of nA homopolymers A of length NA and nB homopolymers B of length NB in the volume V. The canonical partition function is given by

1 ˆ ˆ −HG [Rα ]/k BT ˆ A+ ˆ B − 1) Z= ρ0 DRα e e−ρ0 χ dr A B δ( n A !n B ! α

(3.18)

The product α runs over all chains in the system, ρ 0 DRα denotes the path integral over all paths Rα (s), and we have introduced the factor ρ 0 in order to make Z dimensionless. The path integrals can be decoupled by inserting delta-functions ∫ Dρ j δ(ρ j − ρˆ j ) (with j = A,B), 1 and using the Fourier representations of the delta-functions δ(ρ j − ρˆ j ) = ∫i∞ DW j e N W j (ρ j −ρˆ j ) and

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δ(ρ A + ρ B − ρ 0 ) = ∫i∞ Dξ e N ξ (ρ A +ρ B −ρ0 ) . Here N is some reference chain length, and the factor 1/N is introduced for convenience. This allows one to rewrite the partition function in the following form: 1

DW A DW B Dξ

Z∝ i∞

∞

Dρ A Dρ B e−F/k B T

(3.19)

with ⎧ F[WA , WB , ξ, ρA , ρB ] ρ0 ⎨ dr W j j = χ N dr A B − kB T N ⎩ j=A,B

−

⎫ ⎬ Qj N dr ξ ( A + B − 1) − Vj ln ρ0 N nj ⎭ j j=A,B

(3.20)

( j = ρ j /ρ 0 ), where Vj denotes the partial volume occupied by all polymers of type j in the system, and the functional Nj 1 (3.21) Q j = DR e−HG [R]/k B T e− N 0 ds W j (R(s)) is the partition function of a single, noninteracting chain j in the external field Wj . Thus the path integrals are decoupled as intended, and the coupling is transferred to the integral over fluctuating fields Wj and ξ . Now the self-consistent field approximation consists in replacing the integral (3.19) by its saddle point, i.e., minimizing the effective Hamiltonian H with respect to the variables ρ j (r) and Wj (r). The minimization procedure results in a set of equations, ˆ j = j ˆ i − ξ W j = χ N

with with

j = A, B i, j = A, B

and i = j

(3.22)

ˆ j denotes the average of ˆ j in a system of noninteracting chains subject to the external fields where Wj (r). We note that the latter are real, according to Eq. (3.22), even though the original integral (3.19) is carried out over the imaginary axis. Intuitively, the Wj (r) can be interpreted as the effective mean fields acting on monomers due to the interactions with the surrounding monomers. Together with the incompressibility constraint, the equations (3.22) form a closed cycle which can be solved self-consistently. For future reference, we note that it is sometimes convenient to carry out the saddle point integral only with respect to the variables Wj (r). This defines a free energy functional FSCF [ A ], which has essentially the same form as F (Eq. 3.20), except that the variables W α (r) and ξ (r) are now real Lagrange parameter fields ˆ B = B , and B = 1 − A and depend on A . The same functional can also ˆ A = A , that enforce be derived by standard density functional approaches, using as the reference system a gas of non-interacting Gaussian chains [73]. In some cases, one would prefer to operate in the grand canonical ensemble, i.e., at variable polymer numbers nj . The resulting SCF theory is very similar. The last term in Eq. (3.25) is replaced by (− j z j Q j ), where zj is proportional to the fugacity of the polymers j. The formalism can easily be generalized to other inhomogeneous polymer systems. The application of the theory to multicomponent A/B/C/. . . homopolymer blends with a more general interaction Hamiltonian H I [ρˆ A , ρˆ B , ρˆC , . . .] replacing Eq. (3.17) is straightforward. The self-consistent field equations (3.22) are

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45

simply replaced by

Wj =

N δH I [{ρi }] −ξ kB T δρ j

(3.23)

If copolymers are involved, the single-chain partition function Qc for the corresponding molecules must be adjusted accordingly. For example, the single-chain partition function for an A:B diblock copolymer of length Nc with A-fraction f reads Qc =

1

DR e−HG [R]/k B T e N

f Nc 0

ds W A (R(s))+ N1

Nc f Nc

ds W B (R(s))

(3.24)

The general SCF free energy functional for incompressible multicomponent systems is given by FSCF [{ρ j }] H I [{ρi }] 1 1 = − ρ j − ρ0 dr W j ρ j − dr ξ kB T kB T N N j j Qα − n α ln ρ0 nα α

(3.25)

where the sum j runs over monomer species and the sum α over polymer types. The SCF theory has been extended in various ways to treat more complex systems, e.g., compressible melts and solutions [74, 75], macromolecules with complex architecture [76], semiflexible polymers [77] with orientational interactions [78, 79], charged polymers [80], polydisperse systems [81, 82], polymer systems subject to stresses [83, 84], systems of polymers undergoing reversible bonds [85, 86], or polymer/colloid composites [87–89].

3.3.2.2

How It Works in Practice

In the previous section, we have derived the basic equations of the SCF theory. Now we describe how to solve them in practice. The first task is to evaluate the single-chain partition functions Q j and the corresponding density averages ρˆ j for noninteracting chains in an external field. We consider a single ideal chain of length N in an external field W(r, s), which may not only vary in space r, but also depend on the monomer position s in the chain in the case of copolymers. It is convenient to introduce partial partition functions q(r, s) = q † (r, s) =

DR e−HG [R]/k B T e N 1

DR e−HG [R]/k B T e N 1

s 0

s 0

ds W (R(s ),s )

δ(R(s) − r)

ds W (R(s ),N −s )

δ(R(s) − r)

(3.26) (3.27)

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where the path integrals DR are carried out over paths of length s. According to the Feynman-Kac formula [90], the functions q(r, s) and q† (r, s) satisfy a diffusion equation 2 b W (r, s) ∂ q(r, s) = − q(r, s) ∂s 6 N 2 W (r, N − s) ∂ † b q (r, s) = − q † (r, s) ∂s 6 N

(3.28) (3.29)

with the initial condition q(r, 0) = q† (r, 0) = 1. Numerical methods to solve diffusion equations are available [91], hence q and q† are accessible quantities. The single-chain partition function can then be calculated via Q = dr q(r, N ) = dr q † (r, N ) (3.30) and the distribution of the sth monomer in space is q(r, s) q† (r, N − s)/Q. More specifically, to study binary homopolymer blends, one must solve the diffusion equations for the † partial partition functions q j = q j in an external field W = Wj (with j = A, B). The averaged volume fractions of monomers j are then given by ˆ j (r) =

1 nj ρ0 Q j

0

Nj

†

ds q j (r, s) q j (r, N − s)

(3.31)

in the canonical ensemble, and ˆ j (r) = z j N

Nj 0

†

ds q j (r, s) q j (r, N − s)

(3.32)

in the grand canonical ensemble. If AB diblock copolymers with fraction f of Amonomers are present, one † must calculate the partial partition functions qc and qc in the external field W(r, s) = WA for s < f Nc and ˆ A is n C /ρ0 QC f A (r) W(r) = WB for s ≥ f Nc . The contribution of the copolymers to the volume fraction fN † with f A (r) = 0 c ds qC (r, s)qC (r, N − s) in the canonical ensemble, and z c /N fA (r) in the grandcanonical ensemble. With this recipe at hand, one can calculate the different terms in Eqs. (3.22). The next problem is to solve these equations simultaneously, taking account of the incompressibility constraint. This is usually done iteratively. We refer the reader to Section 3.4. in [91] for a discussion of different iteration methods. 3.3.2.3

Application: Diblock Copolymer Blends, Part I

To illustrate the power of the SCF approach, we cite one of its most spectacular successes: The reproduction of arbitrarily complex copolymer mesophases. In a series of seminal papers, Matsen and coworkers have calculated phase diagrams for diblock copolymer melts [95, 96]. Figure 3.5 compares an experimental phase diagram due to Bates and coworkers [92–94] with the SCF phase diagram of Matsen and Bates [95]. The SCF theory reproduces the experimentally observed structures. At high values of χ N (‘strong segregation’) the SCF phase diagram features the correct sequence of mesophases at almost the correct value of the fraction of A-monomers f . At low values of χ N (‘weak segregation’), the two phase diagrams are distinctly different.

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Theory and Simulation of Multiphase Polymer Systems 40

47

40

C

S

C

L

30

30

G

χ N 20

χN 20 10

S

L

C

Scp

C

S Scp

G

10 disordered

0 0.0

0.2

0.4

0.6

disordered 0.8

f

1.0

0 0.0

0.2

0.4

0.6

0.8

1.0

f

Figure 3.5 Experimental phase diagram for polystyrene-polyisoprene diblock copolymer melts (left, after Refs. [92–94]) compared with phase diagram obtained with the SCF theory by Matsen and coworkers (right, after Ref. [95]) in the coordinates χ N and A block volume fraction f. The labels S,C,L, and G correspond to the structures shown in Figure 3.3. In addition, the SCF phase diagram features a close-packed sphere phase Scp . All phase transitions are first order except for the disordered/lamellar-transition at f = 1/2 in the SCF phase diagram. Courtesy of Mark W. Matsen, adapted from M.W. Matsen, J. Phys.: Cond. Matter 14, R21 (2002).

This can be explained by the effect of fluctuations and will be discussed further below (Sections 3.3.2.5 and 3.4.4.2, see also Figure 3.8). Tyler and Morse have recently reconsidered the SCF phase diagram and predicted the existence of yet another mesophase, which has an orthorombic unit cell and an Fddd structure and intrudes in a narrow regime at the low χ N-end of the gyroid phase [97]. This phase was later indeed found in a polystyrene-polyisoprene diblock copolymer melt by Takenaka and coworkers [98]. 3.3.2.4

Related Mean Field Approaches

So far, we have focussed on sketching a variant of the SCF theory which was originally developed by Helfand and coworkers [70]. A number of similar approaches have been proposed in the literature. Scheutjens and Fleer have developed a SCF theory for lattice models [99], which is applied very widely [6]. Scheutjens-Fleer calculations are very efficient and incorporate in a natural way the finite (nonzero) range of monomer interactions. To account for this in the Helfand theory, one must introduce additional terms in Eq. (3.9) [69, 70], which indeed turn out to become important in the vicinity of surfaces [75]. Carignano and Szleifer [100] have proposed an SCF theory where chains are sampled as a whole in a surrounding mean field. Hence intramolecular interactions are accounted for exactly and the chain statistics correspond to that of self-avoiding walks (Section 3.2.4.1). This approach is more suitable than the standard SCF theory to study polymers in solution, or melts of molecules with low-molecular weight, where the ideality assumption (see Section 3.2.4.2) becomes questionable [101]. In this chapter, we have chosen a field-theoretic way to present the SCF theory. Freed and coworkers [73, 102] have derived the same type of theory from a density functional approach, using a reference system of non-interacting Gaussian chains. Compared to the density functional approach, the field-theoretic approach has the advantage that the effect of fluctuations can be treated in a more transparent way (see Section 3.3.2.5). On the other hand, information on the local liquid structure of the melt (i.e., monomer correlation functions, packing effects. etc.), can be incorporated more easily in density functional approaches [103, 104]. Density functionals have also served as a starting point for the development of dynamical theories which allow to study the evolution of multiphase polymer blends in time [105–108] (see Section 3.4.3.2).

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Handbook of Multiphase Polymer Systems 180 I

10

Disordered

140 Disordered

III 11

T (°C)

100 2φ & 3φ

60

20

0

0.2

0.4

II

2φ

DμΕ

12

G μΕ IV

Lamellar

13

BμE –20

χΝ

0.6

φHomopolymer

0.8

1

0.4

Lamellar 0.5

0.6

0.7

0.8

0.9

1

φHomopolymer

Figure 3.6 Experimental phase diagram for symmetric ternary blends of PDMS+PEE+PDMS-PEE (NC ∼ 5NH ) featuring lamellar phase, phase-separated region 2 , and microemulsion channel BμE (left, from Ref. [109]), compared with theoretical phase diagrams (right) from SCF theory (solid lines) and Monte Carlo simulations at C = 50 (dashed lines/circles). The dotted lines separate disordered regions with different local structure: DμE, defect driven disorder, and GμE, genuine microemulsion morphology. The regions I–IV are discussed in the text. Left Figure: Reprinted from [454]. Copright (1999) with permission from Royal Society of Chemistry.

3.3.2.5

Fluctuation Effects

Mean-field approaches for polymer systems like the SCF theory tend to be quite successful, because polymers overlap strongly and have many interaction partners. However, there are several instances where composition fluctuations become important and may affect the phase behavior qualitatively. To illustrate some of them, we show the phase diagram of ternary mixtures containing A and B homopolymers and AB diblock copolymers in Figure 3.6. The left graph shows the experimental phase diagram [109], the right graph theoretical phase diagrams obtained by Döuchs et al. [110, 111] from the SCF theory (solid lines) and from field-based computer simulations (dashed line, see Section 3.4.3.2 for details on the simulation method). Regions where different types of fluctuations come into play are marked by I–IV.

I) Fluctuations are important in the close vicinity of critical points, i.e., continuous phase transitions. They affect the values of the critical exponents, which characterize, e.g., the behavior of the specific heat at the transition [90]. In Figure 3.6, such critical transitions are encountered at high homopolymer concentration, where the system essentially behaves like a binary A/B mixture with a critical demixing point. This point belongs to the Ising universality class, hence the system should exhibit Ising critical behavior. It has to be noted that in polymer blends, critical exponents typically remain mean-field like until very close to the critical point [112–114]. II) The effect of fluctuations is more dramatic in the vicinity of order-disorder transitions (ODT), e.g., the transition between the disordered phase and the lamellar phase at low homopolymer concentrations. Fluctuations destroy the long-range order in weakly segregated periodic structures, they shift the ODT and change the order of the transition from continuous to first order (Brazovskii mechanism [115, 116]).

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49

This effect accounts for the differences between the experimental and the SCF phase diagram in Figure 3.5. III) The SCF phase diagram features a three-phase (Lamellar + A + B) coexistence region reaching up to a Lifshitz point. Lifshitz points are generally believed to be destroyed by fluctuations in three dimensions. IV) In strongly segregated mixtures, fluctuations affect the large-scale structure of interfaces. Whereas mean-field interfaces are flat, real interfaces undulate. The so-called ‘capillary waves’ may destroy the orientational order in highly swollen lamellar phases. A locally segregated, but globally disordered ‘microemulsion’ state intrudes between the homopolymer-poor lamellar phase and the homopolymerrich two-phase region in Figure 3.6.

Both in the cases of III) and IV), the effect of fluctuations is to destroy lamellar order in favor of a disordered state. However, the mechanisms are different. This is found to leave a signature in the structure of the disordered phase, which is still locally structured with a characteristic wavevector q* [111]. In the Brazovskii regime, the wavevector q* corresponds to that calculated from the SCF theory (defect driven disorder regime, DμE). In the capillary wave regime, the characteristic length scale increases, compared to that calculated from the SCF theory (genuine microemulsion regime, GμE). Formally, the effect of fluctuations is hidden in the overall prefactor ρ 0 /N in the SCF Hamiltonian H (Eq. 3.25). The larger this factor, the more accurate is the saddle point integration that lies at the heart of the SCF approximation. We can thus define a ‘Ginzburg parameter’ C = Rgd ρ0 /N , which characterizes the strength of the fluctuations. Here the factor Rgd must be introduced to make C dimensionless (d is the spatial dimension), and Rg = Nb2 is the natural length scale of the system, the radius of gyration of an ideal chain of length N. The Ginzburg parameter roughly corresponds to the ratio of the volume spanned by a chain, Rgd , and the volume actually occupied by a chain, N/ρ 0 , and thus measures the degree of interdigitation of chains. At C → ∞, the SCF approximation becomes exact. The numerical simulations shown in Figure 3.6 were carried out at C = 50 (using the length of the copolymers as the reference length), which still seems large. Nevertheless, the effect of fluctuations is already quite dramatic. √ In three dimensions (d = 3), the Ginzburg parameter is proportional to the square root of the chain length, N . This is why the mean-field theory becomes very good for systems of polymers √ with high molecular weight, and only fails at selected points in the phase diagram. The relation C ∝ N has motivated the definition of an ‘invariant polymerization index’ N¯ = N b6 ρ02 ∝ C 2 , which is also often used to quantify fluctuation effects [72, 91]. In two dimensional systems, C is independent of the chain length and fluctuation effects are much stronger. Furthermore, topological constraints become important (i.e., the fact that chains cannot cross each other) which are not included in the Helfand model. In three dimensional systems of linear polymers, they only affect the dynamics (leading to reptation), but in two dimensions, they also change the static properties qualitatively [15]. Finally, we note that fluctuations can also be treated to some extent within the SCF theory, by looking at Gaussian fluctuations about the SCF solution [117]. This is useful for calculating structure factors and carrying out stability analyses. However, Gaussian fluctuations alone cannot bring about the qualitative changes in the phase behavior and the critical exponents which have been described above.

3.3.3

Analytical Theories

The SCF equations have to be solved numerically, which can be quite challenging from a computational point of view. In addition, they also serve as a starting point for the derivation of simpler approximate theories, which may even have analytical solutions in certain limits.

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Two main regimes have to be distinguished here. In the weak segregation limit, χ N is small, and the A and B homopolymers or copolymer blocks are barely demixed. This is the realm of the ‘random phase approximation’ (RPA), which can be derived systematically from the SCF theory. In the strong segregation limit, χ N is large, the polymers or copolymer blocks are strongly demixed and the system can basically be characterized in terms of its internal interfaces. 3.3.3.1

Weak Segregation and Random Phase Approximation

We first consider the situation at low χ N. In this case, the composition varies smoothly, A-rich domains still contain sizeable fractions of B-monomers and vice versa, and the interfaces between domains are broad, i.e., their width is comparable to the radius of gyration of the chains. The idea of the RPA is to perform a systematic expansion about a homogeneous reference state. More precisely, we use the SCF free density functional, Eq. (3.25), as a starting point, and then expand F about the homogeneous state. Defining A = ρ A /ρ 0 as usual, using A + B ≡ 1, and introducing the Fourier representation (k) = dr eikr (r), we obtain a functional of the form ⎧ ρ0 ⎨ 1 FRPA [ A (r)] = Vρ0 f homo + k B T | A (k)|2 2 (k) N ⎩ 2V k =0 ⎫ ⎬ 1 × (k) (k ) (−k − k ) (k, k ) + · · · (3.33) A A A 3 ⎭ 6V 2 k,k =0 where f homo is the SCF free energy per chain in the reference system, and the coefficient n depend on the direct monomer interactions and on the intrachain correlations of free ideal Gaussian chains. We focus on the leading coefficient. To calculate 2 for a given blend, we define the pair correlators K i j [71, 117] K i j (k) = ρî (k)ρˆ j (k)

1 ρ0 N

(3.34)

which give the density-density correlations in an identical blend of noninteracting, ideal Gaussian chains and can thus be expressed in terms of Debye functions gD (x) (Eq. (3.7)). For example, for binary blends of ¯ j ( j = A, B), the pair homopolymers with chain length Nj , gyration radii Rg, j , and mean volume fractions correlators are given by

N ¯ N ¯ 2 2 KBB = K AB = K B A = 0 (3.35) A g D k 2 Rg,A B g D k 2 Rg,B K AA = NA NB For pure diblock copolymer blends with A-fraction f , one gets

K B B = (1 − f )2 g D (1 − f ) k 2 Rg2 K A A = f 2 g D f k 2 Rg2

1 2 2

g D k Rg − f 2 g D f k 2 Rg2 − (1 − f )2 g D (1 − f )k 2 Rg2 K AB = 2

(3.36)

Having calculated K i j , one can evaluate 2 according to [71, 117] 2 =

K A A + K B B + K AB + K B A − 2χ N K A A K B B − K AB K B A

(3.37)

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The function 2 is particularly interesting, because it is directly related to the structure factor of the homogeneous phase, S(k) ∝ 2 (k)−1. Hence the RPA provides expressions for structure factors which can be compared to small angle scattering experiments, e.g., to determine effective interaction parameters. This is probably its most important application. We will now discuss specifically the application of the RPA to binary homopolymer blends and to diblock copolymer blends. (i) Binary Homopolymer Blends and Flory-Huggins-de Gennes Functional According to Eqs. (3.35) and (3.37), the RPA coefficient 2 for binary blends is given by N /N A N /N B

+

− 2χ N 2 (k) = (3.38) 2 2 ¯ A g D k 2 Rg,A ¯ B g D k 2 Rg,B We assume that the composition varies only slowly in the system (on length scales not much shorter than 2 2 Rg ), and expand 2 for small wave vectors. Using gD (x) ≈ 1 − x/3 and Rg, j = b j N j /6, and inserting our result in the RPA expansion (3.33), we obtain the free energy functional ⎧ ⎨ 1 1 1 FRPA [ A ] ≈ Vρ0 f homo + k B Tρ0 + − 2χ | A (k)|2 ¯ A NA ¯ B NB ⎩ 2V k =0 ⎫ ⎬ 1 1 b2A b2B | A (k)|2 k 2 + + (3.39) ¯A ¯B ⎭ 2V k =0 18 The first two terms in (3.39) correspond to the second order expansion of the integral ρ 0 dr f SCF ( A ), where f SCF ( A ) is the SCF free energy per chain in a homogeneous system with A-volume fraction A . It thus seems reasonable to replace them by the full integral. The last term is a square gradient term in real space. Together, one recovers the Flory-Huggins-de Gennes functional of Section 3.3.1.2, Eq. (3.13). (ii) Copolymer Melts, Leibler Theory, and Ohta-Kawasaki Functional In diblock copolymer blends, Eqs. (3.36) and (3.37) yield the RPA coefficient G(1) − 2χ N (3.40) 2 (k) = G( f )G(1 − f ) − (G(1) − G( f ) − G(1 − f ))2 /4 with the short hand notation G( f ) = 2g D ( f k 2 Rg2 ). At low χ N, 2 (k) is positive. Upon increasing χ N, one encounters a spinodal line where 2 (k) becomes zero for some nonzero k = q* , and the disordered state becomes unstable with respect to an ordered microphase separated state. Since the function 2 (k) is spherically symmetric in k, it does not favor a specific type of order. The information on possible ordered states is contained in the higher order coefficients n , most notably, in the structure of the cubic term, 3 . In a seminal paper of 1979, Leibler has carried out a fourth order RPA expansion and deduced a phase diagram which already included the three copolymer phases L, C, and S (Figure 3.3) [118]. Milner and Olmsted later showed that the Leibler theory is also capable of reproducing the gyroid phase [119]. The RPA phase diagram roughly coincides with the full SCF phase diagram, as established 1996 by Matsen and Bates [95], up to χ N < 12. Unfortunately, fluctuations have a massive effect on the phase diagram at these small χ N (see Figure 3.5), therefore the predictive power of the Leibler theory must be questioned. Nevertheless, it is useful for identifying potential ordered phases and phase transitions in

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copolymer systems. Generalized Leibler theories still prove to be efficient tools to analyze phase transitions in complex copolymer blends by analytical considerations [120]. Next we attempt to construct a simplified free energy functional for diblock copolymer melts, in the spirit of the Flory-Huggins-de Gennes functional. To this end, we again expand 2 in powers of k, as in (i). Compared to homopolymer blends, however, there is an important difference: 2 (k) has a singularity at k → 0 and diverges according to 2 (k) ≈

2

1 3 2 2 − f ) k Rg2

at

f 2 (1

k→0

(3.41)

The singularity accounts for the fact that large-scale composition fluctuations are not possible in copolymer blends, since the A- and B-blocks are permanently linked to each other. It ensures that the structure factor S(k), vanishes at k → 0, suppresses macrophase separation and is thus ultimately responsible for the onset of microphase separation in the RPA theory. A 1/k2 term like (3.41) in a density functional corresponds to a long-range Coulomb type interaction. This observation motivated Ohta and Kawasaki [121] in 1986 to propose a free energy functional for copolymer melts, which combines a regular squaregradient functional accounting for direct short-range interactions with a long-range Coulomb term accounting for the connectivity of the copolymers. In real space, the Ohta-Kawasaki functional has the form ρ0 B ρ0 A 2 (3.42) dr W( A ) + (∇ A ) + dr dr G(r, r ) δ A (r) δ A (r ) FOK [ A ] = N 2 N 2 ¯ A . The last term introduces the long-range interactions, with G(r, r ) defined such with δ A (r) = A (r) − that G(r, r ) = −δ(r − r )

(3.43)

which corresponds to G(r, r ) ∼ 1/|r − r | in infinitely extended systems. Given Eq. (3.41), it seems natural to identify A = 3/(2 f 2 (1 − f )2 Rg2 ). The choice of W and B is somewhat more arbitrary. The function W( A ) is a free energy density with two degenerate minima and can be approximated by a fourth order polynomial in A . As for the coefficient of the square gradient term, B, Ohta and Kawasaki originally estimated it from the asymptotic behavior of 2 at k → ∞, 2 (k) ≈

1 k 2 Rg2 2 f (1 − f )

at

k→∞

(3.44)

which yields B = Rg2 /(2 f (1 − f )). Later, they noted that this choice of B gives the wrong interfacial width at stronger segregation, which has implications for the elastic constants and the equilibrium period of the ordered phases, and suggested to replace B by a constant in the strong segregation limit [122]. The Ohta-Kawasaki functional reproduces microphase separation and complex copolymer phases such as the gyroid phase [123, 124] and even the Fddd phase [124]. It can be handled much more easily than the Leibler theory or the full SCF theory (see Section 3.4.3.2), therefore it is particularly popular in largescale dynamical simulations of copolymer melts (see Section 3.4.3.2). Different authors have generalized it to ternary blends containing copolymers [123, 125, 126]. In particular, Uneyama and Doi have recently proposed a general density functional for polymer/copolymer blends that reduces to the Flory-Huggins-de Gennes functional in the homopolymer case and to the Ohta-Kawasaki functional in the diblock case.

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3.3.3.2

53

Strong Segregation

We turn to discussing the situation at high χ N. The A-rich and B-rich (micro)phases are then well-separated by sharp interfaces. The free energy contribution from the interfacial regions (i) and the chain conformations inside the A- or B-domains (ii) can be treated separately. (i) Interfacial Profiles and Ground State Dominance In the interfacial region, the free energy is dominated by the contribution of the direct A-B interactions and the local stretching of segments. Chain end effects can be neglected. This simplifies the situation considerably. We first note that the diffusion equation ((3.28) or (3.29)) for a j-chain or a j-block of a chain has the same structure than the time-dependent Schrödinger equation, if one identifies s ↔ it. As is well known from quantum mechanics, the general solution can formally be expressed as [127] q j (r, s) = n cn ψn, j (r)e−n, js , where {ψ n,j (r), n,j } are Eigenfunctions and Eigenvalues of the operator (b2j /6 − W j (r)/N ). At large s, the smallest Eigenvalue 0,j dominates, i.e., q j (r, s) ∝ ψ0, j (r)e−0, j s , and the resulting density in the large Nj limit is ρ j ∝ |ψ0, j |2 e−0, j N j . This type of approximation is called ‘ground state dominance’. It is commonly used to study polymers at interfaces and surfaces. In the case of blends, we have the freedom to shift the fields Wj (r) by a constant value, hence we can set 0,j = 0. The self-consistent field equations can thus be written as ρ j = ρ0 |ψ j (r)|

2

with

b2j

Wj − 6 N

ψj = 0

and

Wj =

N δH I −ξ k B T δρ j

(3.45)

where ψ j is normalized such that Q j = dr|ψ j (r|2 = V j is the partial volume occupied by the polymers j, and ξ (r) ensures j |ψ j |2 ≡ 1. In order to derive an epression for the free energy, we first note that Eqs. (3.45) minimize a Lagrange action,

kB T ρ0 L = HI + (3.46) dr b2A (∇ψ A )2 + b2B (∇ψ B )2 6 with respect to ψ j under the constraint |ψ A |2 + |ψ B |2 ≡ 1. One easily checks that L vanishes for homogeneous bulk states, and that the minimized L is equal to the extremized SCF Hamiltonian FSCF , Eq. (3.25) up to a constant. Hence L can be identified with the interfacial free energy. Rewriting it in terms of the volume fractions φ j and using (∇ A )2 = (∇ B )2 , one obtains the free energy functional Fint [ A (r)] = H I + ρ0

dr

kB T 24

b2A b2 + B A B

(∇ A )2

(3.47)

which reproduces Eq. (3.14) for Flory-Huggins interactions (3.17). For bA = bB = b and Flory-Huggins interactions, the self-consistent field equations √ (3.45) are solved by a tanh profile, (ρ A − ρ B√) ∼ ρ0 tanh(z/wSSL ) with the interfacial width wSSL = b/ 6χ and the interfacial tension σSSL = k B Tρ0 b χ /6 [66–68]. (ii) Copolymer Conformations and Strong Stretching Theory The free energy functional (3.47) is sufficient to describe strongly segregated homopolymer blends. In copolymer blends, additional contributions come into play due to the fact that the copolymer junctions are confined to the interfaces and the copolymer blocks stretch away from them into their respective A or B domains. The associated costs of configurational

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free energy can be estimated within a second approximation scheme, the ‘strong stretching’ theory (SST) [128–131]. The main idea of the SST was put forward in 1985 by Semenov [128], who noted that for strongly stretched copolymer blocks, the paths fluctuate around a set of ‘most probable paths’. This motivates to approximate the single-chain partition function Q, Eq. (3.21), by its saddle point, i.e., the path integral in Q is replaced by an integral over ‘classical’ paths Rc that extremize the integrand and thus satisfy the differential equation [132] 3 d2 Rc 1 = ∇W (Rc ) 2 2 b ds N

(3.48)

We will treat the copolymer blocks as independent chains of length M. The classical paths corresponding to one block are then characterized by their boundary conditions, Rc (0) = r j and Rc (M) = re , where the junction rj is confined to an interface and the free end re is distributed everywhere in its domain. Next, we note that for infinitely long blocks, the classical paths must satisfy dRc /ds|s=M = 0

M →∞

for

(3.49)

at the free end. Mathematically speaking, they would not have a well-defined end position otherwise. Physically speaking, the ‘average’ chain representing the classical path does not sustain tension at the free end, which seems reasonable. In the following, Eq. (3.49) is also imposed for finite (large) blocks as an additional boundary condition. Eq. (3.48) is then overdetermined and can, in general, no longer be solved for arbitrary end positions re . To ensure that chain ends are indeed free to move throughout the domain, the field W(r) must have a special shape. Specifically, near flat interfaces it must be parabolic as a function of the distance z from the interface [129, 130], 3 π2 2 1 W (z) = − z N 8 b2 M 2

(3.50)

This is one of the main results of the SST. It generally applies to situations where strongly stretched polymers are attached to an interface, e.g., strongly segregated copolymer blocks, [72, 133] or polymer brushes in solvents of arbitrary quality [131]. The SST field must always have the form (3.50), and the remaining task is to realize this by a suitable choice of the chain end distribution P(re ). In the incompressible blend case, P(re ) must be chosen such that the density in the domains is constant, ρ 0 . Luckily, we do not have to evaluate P(re ) explicitly to calculate the free energy. The SST field has another convenient property: One can show that the stretching energy of classical paths of fixed length N in a field satisfying Eqs. (3.48) and (3.49) is exactly equal to the negative field energy,

N 0

3 ds 2 2b

dRc ds

2

1 =− N

N

ds W (Rc (s))

(3.51)

0

Summing over all blocks in a domain, the total stretching energy is thus given by 1 Fstretch =− kB T N

dr ρ(r) W (r) ≈ ρ0

3 π2 8 b2 M 2

dr z 2

(3.52)

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where the integral is over the volume of the domain, and z denotes the closest distance to an interface. The total energy of the system can be estimated as the sum over the stretching energies in the different domains, Eq. (3.52), and the interfacial energy, Eq. (3.47), and then used to evaluate the relative stability of different phases. In the strong segregation limit, only the C, L, and S phase are found to be stable [72], in agreement with SCF calculations at high χ N. The validity of the strong stretching theory seems to be restricted to very large chains [134]. This is presumably to a large extent due to the requirement (3.49), which does not necessarily hold for classical paths of finite length. Netz and Schick [135, 136] have shown that an unrestricted ‘classical theory’, which just builds on the saddle point integration of Q and avoids using (3.49), gives results that agree better with the SCF theory. However, the classical theory has to be solved numerically, and the computational advantage over the full SCF theory is not evident. The SST has found numerous applications [72] and has been extended and improved in various respect. It provides an analytical approach to analyzing multicomponent polymer blends in a segregation regime where the SCF theory becomes increasingly cumbersome, due to the necessity of handling narrow interfaces. 3.3.4

An Application: Interfaces in Binary Blends

To close the theory section, we discuss the simplest possible examples of an inhomogeneous polymer system: An interface in a symmetrical binary homopolymer blend. This system has been studied intensely in experiments [137–143]. By mixing random copolymers of ethylene and ethyl-ethylene with two different, but very well defined copolymer ratios, Carelli et al. [137, 143] were able to tune the Flory-Huggins parameter very finely and study interfacial properties in a wide range of χ N between the weak segregation limit and the strong segregation limit. Figure 3.7 compares the results for the interfacial width and compares them with the mean-field prediction for the weak segregation limit, the strong segregation limit, and the full numerical result.

500

Interfacial width 2w (Å)

400

300

200

100

0 2

8

14

20

26

32

38

χN

Figure 3.7 Intrinsic width of interfaces between A- and B-phases in binary polyolefin blends as a function of χ N, compared with the predictions of the weak segregation theory, the strong segregation theory, and the full SCF theory. From Ref. [137].

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We should note that there is a complication here. As we have mentioned earlier, fluid–fluid interfaces are never flat, they exhibit capillary waves [144, 145]. This leads to an apparent broadening of the interfacial width w [146]. The apparent width depends on the lateral length scale of the observation L and can be calculated according to [147, 148] w = 2

w02

L kB T ln + 4σ a0

or

w = 2

w02

L 2k B T ln + πσ a0

(3.53)

depending how it is measured. Here w0 is the ‘intrinsic’ width, σ the interfacial tension, and a0 a ‘coarsegraining length’, which is roughly given by the interfacial width [146, 148]. Both the quantities L and a0 are not very well defined in an actual experiment. Fortunately, they only enter logarithmically, therefore the result is not very sensitive to their values. The theoretical curves shown in Figure 3.7 include the capillary wave broadening, calculated using the interfacial tension from the respective theory. Comparing the curves in Figure 3.7, one finds that the weak segregation theory consistently overestimates the width, and the strong segregation theory consistently underestimates it. The numerical SCF values interpolate between the two regimes and are in excellent quantitative agreement with the experimental data over almost the whole range of χ N. The SCF theory is also found to perform very well compared to computer simulations [148, 149]. It reproduces many features of the interfacial structure, such as chain end distributions, local segment orientations, etc. at a quantitative level, if capillary waves are accounted for [148]. This illustrates the power of the SCF theory to describe the local structure of inhomogeneous polymer systems, even if the global structure is affected by large-scale composition fluctuations.

3.4

Simulations of Multiphase Polymer Systems

Whereas theoretical work on multiphase polymer systems has a long-standing tradition, the field of simulations in this area is much younger. This is because polymer simulations are computationally very expensive, which has essentially rendered them unfeasible until roughly 20 years ago. In this section, we will attempt to give an overview over the current state-of-the-art of simulations of inhomogeneous multicomponent polymer systems.

3.4.1

Coarse-Graining, Part II

One of the obvious challenges in multiphase polymer simulations is that polymers are such big molecules, which moreover self-organize into even larger supramolecular structures. Polymeric materials exhibit structure on a wide range of length scales, from the atomic scale up to micrometers. Their specific material properties are to a great extent determined by local inhomogeneities and internal interfaces, and depend strongly on the interplay between these mesostructures in space and time. In order to understand the materials and make useful predictions for new substances, one must analyze their properties on all time and length scales of interest. Therefore, multiscale modeling has become one of the big topics in computational polymer science. The central element of multiscale modeling is coarse-graining. By successively eliminating degrees of freedom (electronic structure, atomic structure, molecular structure, etc.), a hierarchy of models is constructed (see Section 3.2.2). For each type of model, optimized simulation methods are developed, which allow one to investigate specific aspects of the materials. Having identified suitable classes of coarse-grained models, one can proceed in two different manners.

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(i) Generic Modeling This approach has been favored historically, and up to date, the overwhelming majority of simulations of multicomponent polymer systems is still based on it (see Section 3.4.4). Generic models are simple and computationally efficient. They are not designed to represent specific materials; rather, they are the simulation counterpart of the theoretical models discussed in the previous section. They are suited to study generic properties of polymer and copolymer systems, i.e., to identify the behavior that can be expected from their stringlike structure, their chemical incompatibility etc. Simulations of generic models are also particularly suited to test theories. Generic models are used in all areas of materials science, and in most cases, they only give qualitative insights into the behavior of a material. This is different for polymers, because of their universal properties (see Section 3.2). For example, we have already seen in Figure 3.7 that a generic theoretical model (the Edwards model) quantitatively predicts important aspects of the interfacial structure in real polymer melts. Nevertheless, the predictive power of generic models is restricted, and relies on the knowledge of ‘heuristic’ parameters such as the χ -parameter. Therefore, a second approach is attracting growing interest.

(ii) Systematic Bottom-Up Modeling The idea of systematic coarse-graining is to establish a hierarchy of models for the same specific material, starting from an ab-initio description, with well-defined quantitative links between the different levels. Ideally, the goal is to replace many degrees of freedom by a selection of fewer ‘effective’ degrees of freedom. If one is only interested in equilibrium properties, the problem is at least well defined. For each possible coarse-grained configuration, one must evaluate a partial partition function of the full system under the constraints imposed by the values of the coarse-grained degrees of freedom. This procedure results in an effective potential in the coarse-grained space, which is, in general, a true multibody potential – it cannot be separated into contributions of pair potentials. If one is interested in dynamical properties, the situation is even more complicated. One must replace a dynamical system for all variables by a lower dimensional system for a subset of effective variables. This can be done approximately, e.g., using Mori Zwanzig projector operator techniques [150]. The new dynamical system is inevitably a stochastic process with memory, i.e., the future time evolution not only depends on the current state of the system, but also on its entire history. Obviously, such ‘ideal coarse-graining’ is not feasible for polymer systems. Instead, researchers adopt a heuristic approach, where they first define a coarse-grained model, which typically has no memory and only pair potentials, and match the properties of the coarse-grained model with those of the fine-grain model as best they can: The model parameters are chosen such that the coarse-grained model reproduces physical properties of interest, such as correlation functions or diffusion constants [151–154]. Already at an early stage, researchers have started to develop schemes for mapping real polymers on lattice models [155–157]. Nowadays, off-lattice models are more common. Early approaches focussed on the task of reproducing the correct intrachain correlations by optimizing the bond potentials in the chains [155]. Later, the interchain correlations were considered as well, which can be matched by adjusting the non-bonded, intermolecular potentials in the coarse-grained model. It is important to note that the resulting effective potentials depend on the concentration and the temperature [158] (much like the χ -parameter itself). Different methods to determine effective potentials have been devised and even automated packages are available [153, 159–162]. The reverse problem – how to reconstruct a fine-scale model from a given coarsescale configuration – has also been addressed [163, 164]. Nowadays, the available techniques for mapping static properties are relatively advanced. In contrast, the field of mapping dynamical properties is still in its infancy [165]. The standard multiscale approach is sequential, i.e., numerical simulations are carried out separately for different levels. Currently, increasing effort is devoted to developing hybrid schemes where several coarsegraining levels are considered simultaneously within one single simulation [166, 167].

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Despite the large amount of work that has already been devoted to systematic coarse-graining, coarsegrained simulation studies of realistic multicomponent polymer blends are still scarce. Mattice and coworkers have carried out lattice simulations of blends containing polyethylene and polypropylene homopolymers and copolymers [168–170]. Faller et al. have developed and studied a coarse-grained model for blends of polyisoprene and polystyrene [171–173]. 3.4.2

Overview of Structural Models

After these general remarks, we shall give a brief overview over the different models that are currently used in multicomponent polymer simulations. 3.4.2.1

Atomistic Models

Atomistic simulations are computationally intensive, and rely very much on the quality of the force fields. (Force fields are a separate issue in multiscale modeling, which shall not be discussed here.) Therefore, atomistic simulations of blends are still relatively scarce. So far, most studies have focussed on miscibility aspects [174–184]. Already early on, atomistic and mesoscopic simulations were combined in multiscale studies: Atomistic simulations were used to determine the Flory-Huggins χ -parameter, coarse-grained methods were then applied to study large-scale aspects of phase separation [185–191] or mesophase formation [192]. Only few fully atomistic studies deal with aspects beyond miscibility, e.g., the formation of lamellar structures in diblock copolymers [193], or the diffusion of small molecules in blends [194]. 3.4.2.2

Coarse-Grained Particle Models

The coarse-grained models for polymers can be divided into two main classes: Coarse-grained particle models operate with descriptions of the polymers that are considerably simplified, compared to atomistic models, but still treat them as explicit individual objects. Field models describe polymer systems in terms of spatially varying continuous fields. We begin with discussing some of the most common particle models. Lattice Chain Models Lattice models have the oldest tradition among the coarse-grained particle models for polymer simulations, and are still very popular. The first molecular simulations of multiphase polymer systems – studies of binary homopolymer blends by Sariban and Binder in 1987 [195, 196] and by Cifra and coworkers in 1988 [197] – were based on lattice models. They are particularly suited to be studied with Monte Carlo methods, and several smart Monte Carlo algorithms have been designed especially for lattice polymer simulations [198, 199]. In molecular lattice models, the polymers are represented as strings of monomers confined to a lattice. A natural approach consists in placing the ‘monomers’ on lattice sites and linking them by bonds that connect nearest-neighbor sites. For many applications, it has proven useful to apply less rigid constraints on the links and allow for bonds of variable length, which may also connect second-nearest neighbors [200] or stretch over even longer distances [201]. Moreover, the lattice is usually not entirely filled with monomers, but also contains a small fraction of voids. This is because most Monte Carlo algorithms for polymers do not work at full filling, and special algorithms have to be devised for that case [202]. One particularly popular lattice model is the ‘bondfluctuation model’, devised in 1988 by Carmesin and Kremer [203]. It is based on the cubic lattice; monomers do not occupy single sites, but√entire cubes in a cubic lattice. They are connected by √ √ ‘fluctuating bonds’ of varying length, (2, 5, 6, 3 or 10 lattice constants). In the bond-fluctuation model, a polymer system behaves like a dense polymer melt already at the volume fraction 0.5. Therefore, it can be simulated very efficiently.

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Despite their intrinsically anisotropic character, lattice models are able to reproduce most known selfassembled mesophases in copolymer melts, even the gyroid phase in diblock copolymers [204]. Nowadays, they are used to study such complex systems as ABC triblock copolymer melts confined in cylindrical nanotubes [205], which feature a rich spectrum of novel morphologies, e.g., stacked disks, curved lamellar structures, and various types of helices. Off-Lattice Chain Models For many years, only lattice simulations were sufficiently efficient that they could be used to study polymer blends at a molecular level. With computers becoming more and more powerful, off-lattice chain models have become increasingly popular. Compared to lattice models, they have the advantage that they provide easy access to forces and can also be used in Molecular Dynamics or Brownian Dynamics simulations. They do not impose restrictions on the size and shape of the simulation box (in lattice models, the box dimensions have to be multiple integers of the lattice constant). The structure of space is not anisotropic as in lattice models. Whereas the inherent anisotropy of lattice models does not seem to cause problems if the lattice model is sufficiently flexible and if the chains are sufficiently long, simulations of shorter chains can be hampered by lattice artifacts. In bead-spring models, polymers are represented by chains of spherically symmetric force centers connected by springs. Numerous variants have been proposed, which differ in the choice of the spring potentials (the bonded interactions) and the choice of the pairwise interactions between the beads (the non-bonded potentials). The simplest choice of spring potential is a harmonic potential. In order to prevent chains from crossing each other in dynamical simulations, an anharmonic cutoff on the spring length is often imposed. A popular choice is the ‘Finitely Extensible Nonlinear Elastic’ (FENE) potential k 2 (b − b0 )2 VFENE (b) = d ln 1 − 2 d2

(3.54)

which reduces to a harmonic spring potential with equilibrium spring length b0 at b ≈ b0 , and diverges at |b − b0 | → d. In some applications, the springs are constrained to have fixed lengths – however, this requires the use of special algorithms and changes slightly the distribution of bond angles [198]. In addition, some bead-spring models also include bending potentials that allow one to tune the chain stiffness, or even torsional potentials. The non-bonded interactions drive the segregation of the monomers. As we have discussed in Section 3.3.1.3, both energetic and entropic factors can contribute to making monomers incompatible. Many models operate with energetic monomer (in)compatibilities, but models with entropically driven segregation are also common. As an example, we consider one commonly used type of potential, the truncated Lennard-Jones potentials for r < rc VLJ (r ) = (σ/r )12 − (σ/r )6 + C

(3.55)

(V LJ = 0 otherwise), where the parameter C is chosen such that V LJ (r) is continuous at r = rc . If the cutoff parameter rc is larger than 21/ 6 σ (a common choice is r = 2.5σ ), the potential has a repulsive core surrounded by an attractive well. In this case, energetic incompatibility can be imposed by using species dependent interaction parameters ij with 2 AB < AA + BB . If the cutoff parameter is rc ≤ 21/6 , the potential is purely repulsive. In this case, one can still enforce monomer segregation by choosing species dependent and non-additive interaction radii σ ij with 2σ AB >σ AA + σ BB . The mechanism driving the segregation is the Equation-of-State effect discussed in Section 3.3.1.3. A simulation model that is based on this idea has been

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proposed by Grest, Lacasse and Kremer in 1996 [206]. It is more efficient than conventional models with energetically driven segregation, because the interaction range is shorter. We note that high pressures have to be applied to keep the chains together and to drive the demixing. In the context of ‘Dissipative particle dynamics’ (DPD) simulations, (see below) it has also become popular to use soft non-bonded potentials without a hard core. A typical DPD-potential has the form [207] VDPD (r ) =

v (1 − r/rc )2 2

for r < rc

(3.56)

(V DPD = 0 otherwise). Demixing is induced by using species dependent parameters vi j > 0 with 2v AB > (v A A + v B B ). The mechanism driving the segregation is again related to the Equation-of-State effect – like particles overlap more strongly than unlike particles. The DPD simulation method was originally proposed in the context of fluid simulations, where every DPD particle supposedly represents a lump of true particles. This motivates the absence of hard, impenetrable cores and even a roughly linear shape [208]. Early on, DPD potentials were also used in polymer simulations [207]. As long as topological constraints are not important, the monomer potentials do not need to have a hard core (see also the discussion at the end of the next section). ‘Edwards’ Models A special class of chain models are the Edwards models, which shall be treated separately. The idea is to implement directly an Edwards-type interaction H I (Eq. (3.9)) in a particle-based simulation. The molecules are modeled as off-lattice chains as before, and the non-bonded interactions are given by a potential V = H I [{ρi }] that depends on local monomer densities ρi (r). To complete the definition of the model, one must prescribe how to evaluate the local densities. This is most easily done by simply counting the monomers in each cell of a grid. Other prescriptions that do not impose a grid are also conceivable. Edwards models are not yet very common in simulations of multiphase polymer systems, but they will very likely gain importance in the future. The basic idea was put forward by Zuckermann and coworkers [209, 210] in 1994 in the context of polymer brush simulations. It was first applied to studying microphase separation in block copolymers by Besold et al. [211], who also showed that the model produces correct single-chain behavior in solution [212] (i.e., the chains behave like regular self-avoiding chains). In the following, the power of the approach has been demonstrated in a series of impressive work by Müller and coworkers [213] (see also Section 3.4.3.1). Two notes are in order here. First, it is difficult to impose strict incompressibility in particle-based simulations. Instead, dense melt simulations usually operate at finite compressibility: One introduces an additional term [66] H I,comp. =

κ (ρ A + ρ B − ρ0 )2 2

(3.57)

with a high modulus κ in the interaction Hamiltonian. Second, chains may overlap in the Edwards models. They share this ‘problem’ with the DPD models introduced in the previous section. In three dimensions, the lack of hard core interactions has no effect on the static properties of linear polymers [212, 213]. Topological constraints become important in two dimensions [15], or in melts of cyclic polymers [214], or, most notably, when looking at dynamical properties. In Edwards models, chains do not entangle, and reptation dynamics has to be put in ‘by hand’ [215]. Ellipsoid Model In 1998, Murat and Kremer proposed a model that allows to study weakly segregated polymer blends on the scale of the gyration radius Rg and beyond [216]. If details of the conformations are not of interest, the polymers can be replaced by single, soft particles with ellipsoidal shape. The idea was

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then pursued mainly by Eurich, Maass, and coworkers [217–219], who also suggested to extend the model to diblock copolymers by modeling them as dimers to account for their dumbbell shape [219]. Sambrisky and Guenza have recently worked out a microscopic foundation for coarse-graining diblock copolymers into such dumbbells, which is based on liquid-state integral equations [220]. 3.4.2.3

Field-Based Models

Field-based models for polymer systems no longer treat the molecules as individual objects, but describe them in terms of locally varying fluctuating fields. Among these, the molecular field theories still incorporate some information on the conformations of molecules, and the Ginzburg-Landau models focus on the large-scale structure only. Field-Theoretical Models Field-theoretical models are the molecular field equivalent of the particle-based ‘Edwards’ models discussed above. The idea is to use the starting point of the selfconsistent field theory, i.e., a field-theoretic expression for the partition function (e.g., Eq. (3.19)), and to evaluate the functional integrals over fluctuating fields by simulation methods. Since some fields are complex, one is confronted with a sign problem (the integrand oscillates between positive and negative). Fredrickson and coworkers have demonstrated that the integrals can nevertheless be evaluated in many cases using a method borrowed from elementary particle physics, the ‘Complex Langevin’ simulation method [8, 221, 222]. We refer to Ref. [8] for a general presentation of the method, and Ref. [223] for technical details. To be more specific, let us consider the system discussed in Section 3.3.2.1, a blend of polymers/copolymers containing two types of monomers A and B, with Flory-Huggins interactions (3.17). Rather than evaluating the integral over all five fluctuating fields WA,B , ξ , ρ A,B in Eq. (3.19), we reconsider the original partition function, Eq. (3.18), and decouple the integrals over different paths (polymer conformations) by means of a Hubbard-Stratonovich transformation [90]. This is possible, because the Flory-Huggins interaction H I [{ρî }] is quadratic in the densities ρî . The result is a functional integral over ‘only’ two fields – one which is conjugate to the total density ρˆ = ρˆ A + ρˆ B and imaginary, and one which is conjugate to the composition mˆ = ρˆ B − ρˆ A and real:

DW+

Z∝ i∞

∞

DW− e−FFTS /k B T

(3.58)

with ρ0 FFTS [W+ , W− ] = N

1 χN

dr W−2 −

dr W+ −

α

N Vα ln ρ0 Qα /n α Nα

(3.59)

This partition function is the starting incompressible AB (co)polymer systems. The idea to study multiphase polymer systems by direct evaluation of fluctuating field integrals like Eq. (3.58) was first put forward by Ganesan and Fredrickson in 2001 [224]. They used Complex-Langevin simulations to look at fluctuation effects in pure symmetric diblocks. Since then, Fredrickson and coworkers have shown that the method can be extended in various ways, e.g., it can deal with external stresses [225]), and it can be applied very naturally to charged polymers [226, 227]. Nevertheless, there are still problems with it. The theoretical foundations of the Complex Langevin method are not fully established. One can show that under certain conditions, it produces the correct statistical averages if the system reaches equilibrium [8]. However, the stability of the simulations cannot be ensured. Even if the Langevin step size chosen is small, a

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fraction of simulation runs still crashes [223]. Such problems are unknown from other simulation methods. Because it is generally not widely used and very young in the polymer community, the Complex Langevin method still suffers from teething troubles. An alternative approach to carrying out field-theoretical simulations in melts has been suggested by Doüchs et al. [110] (see also Ref. [91]). They suggested to treat only the real integral over W − with computer simulation and approximate the problematic imaginary integral over W + by its saddle-point. The integral over W − can be evaluated by standard Monte Carlo methods. The results from this partial saddle point approach were shown to agree quantitatively with the full Complex Langevin simulation [110]. Possible strategies to improve upon the saddle point integral have been pointed out by Bäurle and coworkers [228–230]. As already pointed out, field-theoretic models are in some sense equivalent to particle-based Edwards models (if the latter use discrete Gaussian chain models). In field-theoretical simulations, the mean-field limit can be reached very naturally and at low computational cost. Hence they are more efficient than particle-based simulations at high polymer densities, close to the SCF limit, whereas particle-based simulations are superior at lower polymer densities. An application of field-theoretic simulations has already been shown in Section 3.3.2.5. The simulation data shown in Figure 3.6 were obtained with field-theoretic Monte Carlo simulations. Molecular Density Functionals Molecular density functionals are free energy functionals of the type (3.25). They are used in dynamical density functional simulations (see Section 3.4.3.2). Ginzburg-Landau Models Ginzburg-Landau models no longer incorporate specific information on chain conformations and thus have a much simpler structure than molecular field models. Examples are the FloryHuggins-de Gennes functional for homopolymer blends (Eq. (3.13)), or the Ohta-Kawasaki functional for diblock copolymer melts (Eq. (3.42)). This is the highest level of coarse-graining discussed in the present article. 3.4.3

Overview of Dynamical Models

In many cases, only static equilibrium properties are of interest, and then most dynamical models are equivalent. When looking at dynamical properties such as dynamical response functions, or at nonequilibrium situations, the choice of the dynamical model becomes relevant. In the following, we summarize some important models that have been used to study multiphase polymer systems. 3.4.3.1

Particle-Based Dynamics

Kinetic Monte Carlo (MC) A priori, the Monte Carlo (MC) method has been invented as a method to evaluate high dimensional integrals (i.e., thermal averages), and is designed for studying dynamics. Nevertheless, MC simulations are used for dynamical studies, based on the fact that like many static properties, dynamical phenomena are also often governed by universal principles. In kinetic MC simulations, one analyzes the artificial Monte Carlo evolution of a system in order to gain insight into real dynamical processes in the system. The main requirement is that the Monte Carlo moves are only local and reasonably ‘realistic’, i.e., chain crossings are not allowed. Kinetic Monte Carlo simulations have been used, e.g., to study the early stages of demixing in binary blends or the ordering dynamics in block copolymer melts. Molecular Dynamics (MD) In Molecular Dynamics simulations, one solves directly Newton’s or Hamilton’s equations of motions. This is the most straightforward approach to studying dynamical processes in a system. As usual, of course, the devil is in the detail [198].

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Brownian Dynamics In Brownian Dynamics simulations, one also solves equations of motion, but the underlying dynamics is not Hamiltonian. As we have noted earlier (Section 3.4.1), systematic coarse-graining necessarily turns a dynamical system into a stochastic process. Brownian Dynamics simulations account for this fact. They include dissipative and stochastic forces, which supposedly represent the interaction of the coarse-grained degrees of freedom with those that have been integrated out. In general, however, these forces are not derived systematically from the original model, but postulated heuristically. More specifically, the particles j experience three types of forces: m j r¨ j = FCj + F Dj + F Rj

(3.60)

The first term FCj correspondes to the conservative forces, which are derived from the interparticle potentials of the structural model under consideration. These forces also enter the standard Molecular Dynamics simulations. The second F Dj term is a dissipative force that couples to the velocity of the particles. The last term F Rj is a Gaussian stochastic force with mean zero whose amplitude is related to the dissipative force by a ‘fluctuation-dissipation theorem’ [231]. In a canonical simulation, the last two terms constitute a thermostat, i.e., they maintain the system at a given temperature T. Microcanonical models where energy is randomly shifted between ‘internal’ and ‘external’ degrees of freedom have been designed as well. In the simplest (canonical) Ansatz, the dissipative force on a particle j at time t is proportional to its velocity v j (t). According to the fluctuation-dissipation theorem, the stochastic force then fulfills F Dj = −γ j v j

←→

R R F j,α (t) Fk,β (t ) = 2γ j k B T δαβ δ jk δ(t − t )

(3.61)

Other choices are possible. For example, the dissipative force on j can depend on the velocities of other particles k as well, and/or on the history of the system. The fluctuationdissipation relation for the stochastic force has to be adjusted accordingly [231]. The dynamics defined by Eq. (3.61) is commonly used because it is so simple, but it has the disadvantage that it is not Galilean-invariant – a system moving at constant speed is treated differently than a system at rest. Therefore, it does not incorporate hydrodynamic effects correctly, and it is not suited to study nonequilibrium systems such as polymer melts in shear flow. Dissipative Particle Dynamics (DPD) The problems of Eq. (3.61) are avoided in the recently developed ‘Dissipative Particle Dynamics’ (DPD) method [232, 233]. DPD is a special type of Brownian Dynamics which is Galilean-invariant and has become very popular in simulations of complex fluids in recent years. The dissipative and stochastic forces are constructed such that they conserve the total momentum and angular momentum in the system. Consequently, they couple to relative velocities of particles rather than absolute velocities, and they act as central forces (to preserve angular momentum). Specifically, the forces acting on a particle j are given by FCj −

γ ω(r jk ) (ˆr jk v jk ) rˆ jk +

!

2γ k B T ω(r jk ) rˆ jk ζ jk

" (3.62)

k = j

where r jk = r j − rk is the vector separating two particles j and k, rjk its length, rˆ jk the unit vector in the same direction, ω(r) an arbitrary function with a cutoff, and ζ jk are symmetric and uncorrelated Gaussian random numbers with mean zero and variance one. In the literature, the term ‘DPD simulations’ often refers to simulations that use DPD dynamics in combination with soft ‘DPD potentials’ (see Section 3.4.2.2, (Eq. (3.56)). However, DPD can of course be used as a dynamical model [233] or simply as a thermostat [234] in combination with any type of potential.

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In contrast to kinetic Monte Carlo simulations or simple Brownian dynamics simulations (using Eq. (3.61)), DPD simulations take full account of hydrodynamic interactions. Studies of microphase separation in copolymer melts have shown that this makes a difference. The dynamics is strongly affected by hydrodynamic effects in certain regions of phase space – in particular, hydrodynamic interactions play a critical part in helping the system to escape from metastable transient states [235]. Single Chain in Mean Field (SCMF) In 2005, Müller proposed an efficient method to study the ordering dynamics of polymer blends within Edwards models (see Section 3.4.2.2) [236, 237]. The idea is to take snapshots of the density configurations in regular intervals, and to let the chains move in the fields created by these snapshots, e.g., by kinetic Monte Carlo simulations. If the fields were updated after every Monte Carlo move, this would correspond to a regular simulation of the Edwards model. Daoulas et al. have shown that it is possible to update much less often [237] without changing the results. This makes the method very efficient and especially suited for the use on parallel computers. A similar idea had been put forward already in 2003 by Ganesan and Pryamitsin [238], in a less transparent formulation that involves self-consistent fields, to study stationary inhomogeneous polymer systems in an externally imposed flow field. In the absence of flow, the model of Ganesan and Pryamitsin is equivalent to that of Müller et al. A hybrid method that combines kinetic Monte Carlo and the self-consistent field formalism has also recently been proposed by Bäurle and Usami [239]. SCMF simulations are hampered by the general limitations of the Edwards models – chains can cross each other. However, there are ways to introduce the dynamical effect of entanglements at least in cases where the equilibrium configuration space isnot affected by topological constraints [215]. 3.4.3.2

Field-Based Dynamics

Dynamic Density Functional Theory (DDFT) Dynamic density functional theories (DDFT) are based on density functionals for polymer systems, such as, e.g., Eq. (3.25). They supplement them by a model for their dynamical evolution at nonequilibrium. For diffusive dynamics, the dynamical equations in an imposed flow profile v have the general form. [105–108] ∂ρi + ∇(vρi ) = ∂t

dr

ij

∇r i j (r, r )∇r

δF + ηi (r, t) δρ j (r )

(3.63)

where i j (r, r ) is a generalized mobility, and ηi (r, t) a Gaussian white noise with mean zero. If the amplitude of the latter is very small or zero, one has ‘mean-field dynamics’ and the system evolves towards a minimum of the free energy functional (although not necessarily the global minimum). If the noise is larger and satisfies the fluctuation dissipation theorem, ηi (r, t)η j (r , t ) = −2k B T δ(t − t ) ∇r i j (r, r )∇r

(3.64)

the density configurations {ρ i (r)} in an equilibrium simulation (no flow, sufficiently long runs) will be distributed according to P[ρ] ∝ exp(−F[ρ]/k B T ). The kinetic Onsager coefficient i j (r, r ) depends on the microscopic dynamics in the system. Since it characterizes the current of component i in response to an external force acting on component j, it is reasonable to assume that it is proportional to the local density ρ i (r). An efficient choice which however disregards the connectivity of the chains is thus i j (r, r ) = Mi ρi (r) δ(r − r )δij for compressible systems [240]. or (r, r ) = Mρ Aρ B δ(r − r ) for binary incompressible systems (local dynamics) [106].

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65

To account for the chain connectivity, one must include information on the chain correlations. For example, for Rouse chains, i j (r,r ) should be proportional to the pair correlators K i j defined in Eq. (3.34) [108, 241]. At first sight, using such complex Onsager coefficients in a simulation seems forbidding, but thanks to a clever trick due from Maurits and Fraaije [241], it becomes feasible [91, 242–245]. The DDFT method has been extended, e.g., to account for viscoelastic [246] or hydrodynamic effects [83, 247, 248]. A lattice version has also been proposed [249]. Time-Dependent Ginzburg-Landau (TDGL) Theories and Cell Dynamics Like dynamic density functional theories, time-dependent Ginzburg-Landau (TDGL) theories supplement a free energy functional F[ ] of an ‘order parameter’ field (r) by a model for the dynamical evolution of . The TDGL theories of interest in multiphase polymer systems mostly operate with locally conserved order parameters and have the same general structure as Eq. (3.63). For example, time-dependent Flory-Huggins de Gennes theories are used to study the demixing dynamics in polymer blends, and time-dependent Ohta-Kawasaki theories to study the ordering kinetics in copolymer systems. Discrete lattice versions of TDGL theories are often referred to as ‘cell dynamics’ models. We discuss specifically the time-dependent Ohta-Kawasaki theory. Starting from the free energy functional (3.42) for melts of diblock AB copolymers and choosing an Onsager coefficient that describes local diffusive dynamics, (r, r ) = M δ(r − r ), we obtain the dynamical equations [250] ∂ A δFFA + ∇(v A ) = M + η(r, t) ∂t δ A (r) $ ρ0 # ∂W ¯ A ) + η(r, t) = − B A − A( A − N ∂ A

(3.65)

¯ A is the total volume fraction of monomers A in the system. Hence the some-what awkward long-range where ‘Coulomb’ term in Eq. (3.42) becomes short range, and the dynamical equations only depend on local terms. Because of the appealingly simple structure of the final theory, Eq. (3.65), it is widely used for simulations of copolymer systems at equilibrium and under shear (see below). Bahiana and Oono have formulated a discrete version on a lattice [251] which is equally popular in cell dynamics simulations. TDGL simulations are much faster than molecular field simulations, but of course, the underlying model is less accurate. Honda and Kawakatsu have recently proposed a multiscale hybrid method that combines the two approaches, using dynamic density functional input to improve on the accuracy of the TDGL model [167]. Such hybrid approaches will presumably gain importance in the future. 3.4.4

Applications

After this overview of the main models used for simulations of multiphase polymer systems, we will now illustrate them by reviewing simulation work that has been done on homopolymer blends and copolymer melts. We focus on simulations of generic models. Atomistic studies or studies of bottom-up models are scarce and have already been discussed earlier (Sections 3.4.1 and 3.4.2.1). 3.4.4.1

Homopolymer Blends

Bulk Properties We have already mentioned the pioneering simulations of binary blends by Sariban and Binder and by Cifra et al. [195–197] Following up on this work, a number of studies, mainly by Binder and coworkers, have considered the critical behavior of binary blends [44, 252–256]. As discussed in Section

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3.3.2.5, fluctuations shift the critical demixing point and change the critical exponents from mean-field-like to Ising-like. However, the region where the critical behavior deviates from the mean-field prediction shrinks with increasing chain length, and effective mean-field behavior could be observed already for relatively moderate chain lengths [252]. Coming from the other end, field-based Monte Carlo simulations of the FloryHuggins-de Gennes functional (3.13) have confirmed that noise shifts the coexistence curve and changes the critical behavior from mean-field to Ising [257]. Along with these studies of static blend properties, extensive work has also been dedicated to the dynamics of demixing. On the particle-based side, this was mainly investigated using kinetic Monte Carlo [242, 258, 259]. Reister et al. have compared results from kinetic Monte Carlo simulations and different versions of the dynamic density functional theory in a study of spinodal decomposition in symmetric blends [242]. Other field-based simulations have mainly relied on time-dependent Ginzburg Landau models, which have the advantage that one can reach much later stages of demixing. They were used to study demixing processes at equilibrium [126, 260–263] and under shear [264, 265]. Particularly interesting morphologies can be obtained if the dynamics in the two phases is distinctly different, i.e., one component becomes glassy during the demixing process [31] or crystallizes [29, 30, 266]. The particle-based studies mentioned so far have used coarse-grained models of blends that demix explicitly for energetic reasons. A number of authors have explored other factors that are believed to affect the chain miscibility with generic models [267], e.g., the effect of nonrandom mixing [268, 269], shape disparity [270], stiffness disparity [271], different architectures [272], and a different propensity towards crystallization [23, 27].

Internal Interfaces In the miscibility gap, polymer blends have highly inhomogeneous structures with droplets of one phase dispersed in the other phase. Their material properties are largely determined by the properties of the interfaces separating the two phases. While the distribution, size, and shape of the droplets depend on how the blend has been processed, the interfaces separating them often have time to reach local equilibrium and can be studied by means of equilibrium simulations. The first study of an interface in a binary blend was carried out by Reiter et al. [273] in 1990. Since then, several authors have investigated interfaces in symmetric [147, 148, 274–277] or asymmetric [278–280] binary blends (e.g., blends with stiffness and/or monomer size disparities) by means of generic particle-based simulations. The local interfacial structure is of interest because it determines the mechanical stability of an interface. For example, the local interfacial width gives the volume in which chains belonging to different phases can entangle. On the other hand, we have already discussed in Section 3.3.4 that interfaces exhibit capillary waves, which are significant on all length scales. This becomes apparent from the fact that the capillarywave contribution to the total width, as given by Eq. (3.53), diverges both if the system size L becomes very large and if the microscopic coarse-graining length a0 becomes very small. Therefore the length scales of the capillary waves and those of the local interfacial profiles cannot be separated clearly, and it is not clear, a priori, whether the concept of a ‘local interfacial structure’ is at all meaningful. This question has been investigated by Werner et al. [148] by simulations of the bond-fluctuation model (see Section 3.4.2.2). They demonstrated that it is indeed possible to describe homopolymer interfaces consistently in terms of a convolution of ‘intrinsic’ profiles with capillary wave undulations. They also studied the influence of confinement both on the capillary waves [281, 282] and on the intrinsic interfacial width [283]. Equilibrium simulations give information on the stability of interfaces under mechanical stress, but in a rather indirect way. A few authors have probed directly the rheological properties of interfaces with nonequilibrium particle- and field-based simulation methods, looking, e.g., at shear thinning and interfacial slip [284–287]. Detailed simulations of nonequilibrium interfaces are expensive, but with the development of new efficient simulation algorithms and modern fast computers, they become feasible.

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Surfaces and Films Another topic discussed intensely in the literature is the behavior of blends in the vicinity of surfaces, and that of blends confined to thin films. From a simulation point of view, these two situations are identical, because surfaces are typically studied in slab geometries (i.e., with periodic boundaries in two directions and free boundaries in the third). A large amount of work has been dedicated to the somewhat artificial situation of surfaces to which both types of monomers have exactly equal affinity. Even though these surfaces are perfectly neutral, one component will typically segregate to them: In incompatible blends, the minority component aggregates at the surface in order to reduce unfavorable contacts [288–290] (in contrast, the minority chains are removed from the surface in miscible blends [291]). In blends of chains with different stiffness, the stiffer chains are pushed towards the surface, because they loose less entropy there [292, 293]. For the same reason, linear chains aggregate at surfaces in blends of linear and star polymers [294]. Cavallo et al. have systematically investigated the phase behavior of films confined between neutral walls as a function of the film thickness. If the film is made thinner, one observes a crossover from three-dimensional to two-dimensional Ising behavior [295, 296]. Fluctuation effects in thin films are observed to be much stronger than in the bulk, consistent with our discussion in Section 3.3.2.5: In two dimensions, the Ginzburg parameter no longer scales with the chain length and stays finite for all chain lengths. A second transition occurs when the film becomes so thin that polymers are effectively two-dimensional, i.e., they can no longer pass each other. This reflects the theoretically expected fundamental difference between the demixing behavior of overlapping and non-overlapping two-dimensional polymers [15]. The more realistic situation of selective walls, to which one component adsorbs preferentially, has been addressed as well by different authors. In this case, the phase behavior is governed by wetting phenomena and capillary condensation [297–299]. The studies discussed so far were based on particle simulations. A few authors have used field-based simulations to explore dynamical aspects of phase separation in thin polymer blends. Morita et al. have studied the interplay of spinodal decomposition and interfacial roughening due to droplet formation with dynamic density functional simulations [300]. Shang et al. have used a time-dependent Ginzburg-Landau approach to study the spinodal phase separation of a thin film on a heterogeneous substrate [301].

3.4.4.2

Copolymer Systems

Copolymers as Compatibilizers Copolymers were originally designed as natural surfactant molecules that increase the miscibility of incompatible homopolymers and enhance their interfacial properties. They are usually much more expensive than their respective homopolymers, but adding a small amount of copolymer can already improve the properties of the homopolymer blend significantly. A number of researchers have considered the effect of copolymers on the demixing transition for different copolymer architectures [256, 302, 303]. Dadmun and Waldow [256] have pointed out that copolymers not only shift the transition point towards higher values of the Flory-Huggins parameter χ , but also change the critical exponents via a Fisher renormalization mechanism. In the phase-segregated regime, copolymers aggregate to the interface, reduce the interfacial tension, and enlarge the interfacial width. The interfacial structure of homopolymer interfaces with adsorbed copolymers has been explored in detail by several authors [277, 302, 304–307]. Milner and Xi noted in 1996 that the main compatibilizing effect of copolymers probably has a kinetic origin: [308] Copolymers reduce the rate of droplet coalescence during the processing of the blend via a Marangoni effect: If the copolymer concentration drops somewhere at the surface of a droplet, the surface tension increases locally. This induces flow in the direction of the weak point. Hence the copolymer film stabilizes itself kinetically, much like a soap film. In addition, the copolymer blocks stretching into the bulk

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form repulsive coronae. Experimental studies suggest that the resulting steric repulsion between droplets may be even more prohibitive for droplet coalescence than the Marangoni effect [309]. Kim and Jo have studied the influence of copolymers on the dynamics of demixing in a series of works [310–314], but they used kinetic Monte Carlo, hence they did not include hydrodynamic effects and could not study the effect of the Marangoni flow. If one increases the copolymer concentration beyond a certain threshold, the macrophase separated phase at high χ N eventually gives way to a microphase separated phase (see, e.g., the phase diagram shown in Figure 3.6). A few authors have explored the full phase behavior of ternary systems [110, 111, 315, 316]. In ternary systems containing A and C homopolymers and ABC triblock copolymers, a whole zoo of new tricontinuous gyroid phases can be observed [317]. Pure Bulk Copolymer Melts The propensity of copolymers to self-assemble into complex mesostructures makes them attractive for various micro- and nanotechnological applications, which is why pure copolymer melts have become interesting in their own right. Many simulation studies are now concerned with the properties of pure copolymer melts. Early studies of fluctuations and chain correlations near the order/disorder transition (ODT), coming from the disordered side [46, 47, 318, 319] have reproduced the ODT singularity in the structure factor and revealed the dumbbell structure of the chains mentioned earlier [47]. Later, intense work has been devoted to studying ordered lamellar structures below the ODT in melts of symmetrical diblock copolymers, and analyzing them with respect to their structure, their dynamics, and their fluctuations [320–331]. Simulation studies of asymmetric diblock copolymer melts have also reproduced most of the other mesophases – the cylindrical phase, the bcc sphere phase, and even the gyroid phase [204, 211, 235, 332–335]. Locating the actual position of the ODT accurately is difficult, especially in lattice models [336], since the natural periodicity of the structures is in general incompatible with the box size. This results in complex finite size artefacts. Nevertheless, phase diagrams have been obtained in recent years [337–339]. Particle-based and field-based simulations have provided evidence that for symmetrical diblock copolymers ( f = 1/2), the transition to the lamellar phase is shifted [224, 340], compared to the mean-field phase diagram, and becomes first order [340–342], in agreement with the theoretical expectation [116] (see Section 3.3.2.5). Two examples of phase diagrams obtained with different simulation methods are shown in Figure 3.8: One was calculated by Matsen et al. [338] using Monte Carlo simulations of copolymers with length N = 30 in a simple lattice model (left), and one by Lennon et al. [342] using field-theoretical calculations (right) at Ginzburg parameter C = 50. Both phase diagrams significantly improve on the SCF phase diagram of Figure 3.5 (right) in the weak segregation regime, and reproduce the main qualitative features of the experimental phase diagram (Figure 3.5, left): The transitions are first order everywhere. The ODT is shifted to higher χ N. Direct phase transition between the disordered phase and the complex mesophase (PL or G, respectively) or the lamellar phase are possible for a range of copolymer block fractions f . The phase diagrams even reproduce a small hump of the ODT transition line at the boundary to the complex mesophase (PL or G), which is also observed experimentally. The only ‘problem’ with the Monte Carlo phase diagram is that it features a perforated lamellar (PL) phase instead of a gyroid (G) phase. This may be a finite size artefact, as suggested by the authors, or a property of the lattice model under consideration. (Gyroid phases have been found in lattice models [204], but it should be noted that the free energies of the PL and the G phase are very close according to SCF calculations.) Nevertheless, the simulations demonstrate convincingly that the discrepancies between the experimental phase diagram and the SCF phase diagram shown in Figure 3.5 can to a large extent be attributed to the effect of fluctuations. In recent years, researchers have also begun to simulate melts of more complex copolymers, e.g., starblock copolymers [343–347], rod-coil copolymers [348, 349], diblock copolymers with one crystallizing component

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69

18

75

C

C

L

50

16

PL

χeff N 14

χ eff N 25

12

disordered 0 0.0

0.2

0.4

0.6

f

0.8

1.0

10 0.3

0.4

0.5

f

Figure 3.8 Phase diagrams for diblock copolymer melts as obtained from Monte Carlo simulations of a lattice model (chain length N = 30) by Matsen et al. (left) [338] and from field-theoretical Complex-Langevin simulations (Ginzburg parameter C = 50) by Lennon et al. (right). [342] Symbols L,C,G correspond to diblock phases of Figure 3.3; in addition, the Monte Carlo phase diagram features a perforated lamellar (PL) phase. Left figure: Courtesy of reference [342]. Right figure: Reprinted from reference [342], Copyright (2008) with permission from American Physical Society.

[24–26, 350–352], triblock copolymers [353–356], or random copolymers [357, 358]. An interesting study on diblock copolymers with one amphiphilic block has recently been presented by Khokholov and Khalatur [359]. Since the amphiphilic block favors interfaces, the morphology of the mesophases changes completely and is characterized by thin channels and slits. Lay et al. and Palmer et al. have studied the computationally challenging problem of microphase separation in randomly crosslinked binary blends [360–362], and also the inverse problem, to which extent ordered copolymer structures can be stabilized by crosslinking [363]. A great deal of simulation work on copolymer melts has been done with timedependent Ginzburg-Landau approaches [364–378]. These studies have mostly addressed dynamical questions, i.e., the kinetics of ordering, disordering processes in pure melts [366–374] and in mixtures containing copolymers [375–378]. Already in pure diblock copolymer melts, ordering/disordering processes were found to proceed via intricate pathways that involve nontrivial intermediate states (e.g., the perforated lamellar state which is only metastable at equilibrium). In mixtures, the interplay of macrophase and microphase separation leads to a wealth of new transient morphologies [375–378] (see also Ref. 379). Confined Copolymer Melts In recent years, there has been growing interest in confined copolymer systems. The first studies have explored the ordering of copolymers melts in thin films between neutral walls [380–383] or general (selective) walls [384–394], both with particle-based models and dynamic density field theories. Triblock copolymers [395–400] and copolymer blends [382] have also been considered. The dynamics of copolymer ordering in confinement was studied with time-dependent Ginzburg-Landau methods [402]. A series of papers have dealt with the technologically relevant question, whether and how surface patterns can be transferred into copolymer films [389, 403–408]. The current focus shifts to nanocylindrically or spherically confined blends [205, 409–418]. A variety of new structures can already be observed in systems of diblock copolymers confined to nanocylinders, e.g., mesh structures, single and double helices [410]. For triblock copolymers, the spectrum is even more diverse [205]. The confined structures depend on the bulk structure and on the shape of the confining channels. Li and coworkers have shown that possible morphologies can be screened efficiently with the method of simulated annealing [419–424].

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Copolymers Under Shear Finally in this section, we briefly mention the large amount of work that has addressed the effect of shear on the microstructure of copolymer systems, with both field-based and particlebased methods [425–441]. Special attention has been given to the phenomenon that lamellae reorient under shear [431, 442–445]. Experimentally, it is observed that lamellae orient parallel to the shear flow at low shear rates, and perpendicular to the shear flow at high shear rates. Simulations give conflicting results. The parallel state consistently becomes unstable at high shear rates, but its relative stability at low shear rates (or results on the indicators of relative stability such as the entropy production) seems to depend on the type of model [443, 445].

3.5

Future Challenges

The equilibrium theory of fluid polymer mixtures is fairly advanced. Thanks to the universal properties of polymers, it requires relatively little input information (χ parameter etc.) to be predictive at a quantitative level. However, if one goes beyond equilibrium and beyond the fluid state, the situation is much less satisfying. For example, very little work has been done on crosslinked polymer blends [360–363], even though they are common in applications. Physical crosslinks can be established if domains crystallize or become glassy. As we have seen above, research on blends with one crystallizing or glassy component is also rather scarce. Another important issue is the influence of the blend processing on the properties of the resulting materials, i.e., the structure of phase separating blends under shear. The mechanical properties of immiscible blends depend crucially on their microscopic morphologies, i.e., the sizes and shapes of droplets, which in turn depend on the manufacturing process. Theoretical nonequilibrium state diagrams that relate the processing conditions (shear rates, geometry, copolymer concentration, etc.) with final morphologies are still missing. Since the relevant length scales are relatively large and hydrodynamics are important, the simulation method of choice should be a cell dynamics method that incorporates hydrodynamics, e.g., a Lattice-Boltzmann method [446]. Methods that combine Ginzburg-Landau functionals for immiscible fluids with Lattice-Boltzmann models for Newtonian fluids have been developed [447] and used to study demixing processes at rest [448] and under shear [449–451]. Giraud and coworkers have proposed a Lattice-Boltzmann method for viscoelastic fluids, which is more suitable to describe polymers [452], and carried out first simulations of viscoelastic liquid mixtures [453]. Nevertheless, the whole field is still in its infancy. As the field of polymer simulations reaches maturity, the bottom-up modeling approach (Section 3.4.1) will gain importance. So far, the vast majority of theoretical and simulation studies of (co)polymer blends was based on generic model systems. One or two decades from now, realistic simulations of specific polymer blends will probably be equally, if not more, common. One of the major challenges in this context is to develop hybrid multiscale methods that combine different levels of coarse-graining, i.e., use a relatively coarse basic model and fine-grain selectively in interesting regions of the materials (e.g., interfaces) [166].

Acknowledgements The author thanks Mark W. Matsen for introducing her to the self-consistent field theory a long time ago and for providing Figures 3.5 and 3.8 (left), and Glenn H. Fredrickson for discussions and for the permission to show Figure 3.8 (right). She has benefitted from collaborations and/or discussions with Jörg Baschnagel, Kurt Binder, Dominik Düchs, Burkhard Dünweg, Avi Halperin, Venkat Ganesan, Kurt Kremer, Marcus Müller, Wolfgang Paul, Ulf Schiller, Jens Smiatek, Jens-Uwe Sommer, Andreas Werner, and many others. Financial support from the German Science Foundation (DFG) is gratefully acknowledged.

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4 Interfaces in Multiphase Polymer Systems György J. Marosi Budapest University of Technology and Economics, Budapest, Hungary

4.1 Introduction Interfaces, in their widest meaning, divide the space into distinct compartments, thus the interface is one of the basic phenomena of the organization of the world. The segregation of certain organic molecules into multiphase systems, assisted by phase borders, was an essential condition for the origin of life. Concerning multiphase polymeric materials, their behavior is determined, beyond the bulk characteristics of the phases and their ratio, by the intermediate phase (interphase) of altered structure and molecular dynamics. The role of material located in the interfacial region is far larger in the determination of the macroscopic characteristics of multiphase systems than its share in volume or mass. Refining the phase structure increases the share and importance of the interfacial region and this growth accelerates significantly in the submicron range. In nanostructured multiphase polymer systems, in which the specific surface area of phase borders can be several orders of magnitude larger than between microphase structures, almost all the polymer chains belong to the interphase (the bulk disappears). The extent of the influence of the phase contact on their structure and mobility depends on the distance and type of interaction. The interfacial interactions determine the formation of the structure at the phase borders and consequently its physical characteristics, chemical stability and biological effects. The interface-related physical effects include alteration of strength, transport and electrical characteristics. The stability of (nano)composites is influenced not only by physical effects (such as interfacial heat conductivity) but also by chemical effects of catalytic ions and molecular mobility at the interfaces. The structure and mobility of biomolecules at phase borders influence the bioavailability of drug delivery systems, the immune response around implants, the strength of biomaterials and the controllability of all biological effects. As the interface-related fields of application include the use of (nano)composites, electrical units, packaging materials, stabilizing agents, catalysts, tribological agents, and biomaterials the relevant market shear is huge. The importance of nanocomposites and nanomedicine is based basically on their enhanced interfacial effects. Interface design is relevant even in the waste management, therefore a wide scope has to be considered when the interfacial phenomena of the multiphase systems are discussed.


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In the past, the separate discussions about the different interface-related fields, including the artificial distinction between flat and particulate surfaces/interfaces, limited the synergistic common initiatives between the representatives of the relevant disciplines. The aim of this chapter is to show the common fundamentals, similar approaches and convergent activities in the different fields of multiphase polymer and biopolymer systems and to try to promote the common thinking of these research communities. The first step for achieving this aim was to establish common classifications concerning characteristics, modification and analyses of the interfaces (including both flat and particulate ones). The evaluation of thermodynamic and kinetic factors establishes a common basis for all types of modifications. The relationship between the macroscopic properties and the interfacial structure has been most widely studied in the case of mechanical response, which is also reviewed briefly. The most challenging task is to categorize the different types of interfacial modifications performed for various purposes and industrial segments. Modifications of morphology, segmental mobility, compatibility, reactivity and responsiveness has been considered, but still some important areas such as catalysis and tribology are not involved in the scope of this study.

4.2

Basic Considerations

Some relevant terms have to be summarized before discussing the main interfacial characteristics. The term surface means a border between phases of different state matter, while the term interface is a sharp boundary between phases of the same (e.g. solid–solid) state (the difference is artificial as both of them can be considered as interface). If a layer exists between the phases, differing from the bulk materials, it is called the interphase (IP), interlayer, or interfacial zone. The interphases form boundary layers of different width usually in the micro- or nanometer range. The use of the terms ‘interphase’ or ‘interface’ depends in most cases on the point of view (micro- or nanometer scale). The extent and nature of the IP hinge upon how the interphase formation, governed by both thermodynamic and kinetic effects, takes place. On the micro-scale, the IP can be considered a distinct 3D continuum of average mechanical and other properties. On the nano-scale the relaxation processes of polymer segments in the region of interfacial influence has to be considered. Increasing deviation from Gaussian chain statistics occurs as the effect of interface becomes dominant. A ‘train-loop-tail’ structure of restricted mobility is characteristic of the chains near to a solid surface (excepting the particular case of polymer brushes grown perpendicularly to the surface) in contrast to the random distribution of the coil of mobile chains in a melt (see in Figure 4.1).

Figure 4.1 Interfacial structures.

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INTERFACE characteristics

structural charge

mobility

morphological geometrical

conducting amorphous semicond. crystalline insulating oriented

volume thickness hierarchy

interaction

migration segmental sorption interfacial bonds crosslinking degradation barrier gateway

restricted enhanced varied

ad~ de~

secondary acid-base covalent

activation inhibition

activation inhibition

Figure 4.2 Classification of main characteristics related to interfaces.

As a good interfacial contact restricts the number of possible conformations available for the chains, their entropy is reduced. The internal energy of chains can be altered too (depending on the energy of the surface–polymer interaction). In nanocomposites the average interparticle distance can be reduced to the magnitude of the radii of gyration (Rg ) of the chains (at a few % nano-particle content) and all the chains of the system become more or less immobilized. In these systems the IP means the entropic difference among the entangled segments at different level of immobilization depending on their distance from the phase border. The thickness of IP cannot be defined exactly as the level of immobilization changes continuously. The IP zone is formed in spontaneous process or engineered consciously when phases of different character (such as hard–soft, charged–neutral, polar–apolar, living–lifeless) are attached to each other. The rules governing the phenomena (and determining the design of IPs) are basically the same in all kinds of materials. Apart from certain differences, various similarities can be found in the surface-related aspects of biocompatibility, tribology, packaging, blending, fire protection, catalysis, formation of (nano-)composites, optoelectronical devices, and other multiphase systems.

4.3 Characteristics of Interfacial Layers The characteristics of interfaces and interfacial interactions are summarized in Figure 4.2. Structural characteristics include the molecular/segmental and charge arrangement in material units taking part in the formation of micro- or nanoscale interlayers. The morphological features of the interfacial region are influenced by the interaction between the associated groups of the phases, mobility of adjacent segments and local shear forces. Strong interaction, that restricts the molecular or segmental mobility, leads to stabile amorphous structure in the interfacial region (requested in many pharmaceuticals). The interactions, however, can also transmit a template for certain crystalline organization, while local shear forces initiate orientation (favored in many polymer composites). The volume of the interlayers within the multiphase system increases by several orders of magnitude when the size of inclusions is reduced from micro- to nanoscale. Good wetting of a rough surface is also accompanied with the increase of the internal contact area between the chains and the inclusions. The thickness of the IP, falling in the range of thin Langmuir−Blodgett (LB) monomolecular films and macroscopic layer of several hundred μm, is determined by the balance between entropy (segmental arrangement) and free energy (adhesion forces) effects. Hierarchic structure of layers of different thickness is characteristic to natural multiphase structures and to some advanced composites [1]. An interface separates and/or integrates the phases and a separating interlayer (containing for example inorganic nanoparticles) can act also as a joining site between incompatible phases.

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Figure 4.3 Fibers embedded in an interpenetrating network of A () polymer (adhering to the fiber) and B (≡) polymer (non-adhering to the fiber) alternating well and poorly adhered sites.

Mobility is also influenced by the interfacial structure. Well adhered inorganic inclusion barriers hinder the migration within the material, while the voids around detached interfaces provide channels for the molecular transport. The segmental mobility is mostly restricted by the interfacial interaction, but penetration of plasticizer molecules (e.g. water from moisture environment) in a surface layer can increase the local mobility. This causes a confusing increase or decrease of glass transition temperature (Tg ), depending on the character of the substrate and the polymer, when the thickness of a studied interfacial layer is decreased. These issues are of great importance because they contribute to the understanding of the mechanical, rheological, diffusion and electrical characteristics of multiphase polymers systems in which the interfacial ratio is dominant. For example a recently proposed structure with varied sites of high and low adhesion level along the surface of reinforcing fibers, as shown in Figure 4.3, is a promising concept for increasing the toughness of composites [2, 3]. The adsorption–desorption process is in close relation with the adhesion. The driving force for the adsorption is the lowering of the system’s free energy. The surplus energy of the surface of materials attracts the mobile molecules available in the surrounding medium (vapor or liquid phase). Surfaces of high surface energy (surface tension ∼500–5000 mJ/m2 , covalent. ionic, metallic bonds) are mostly covered, in normal environments, with layers of adsorbed water (already at

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