Liquid Crystal Elastomers (The International Series of Monographs on Physics, 120) - Revised edition

International Series of Monographs on Physics Series Editors

J. Birman, City University of New York S. F. Edwards, University of Cambridge R. Friend, University of Cambridge M. Rees, University of Cambridge D. Sherrington, University of Oxford G. Veneziano, CERN, Geneva

International Series of Monographs on Physics

138. 137. 136. 135. 134. 133. 132. 131. 130. 129. 128. 127. 126. 125. 123. 122. 121. 120. 119. 118. 117. 116. 115. 114. 113. 112. 111. 110. 109. 108. 107. 106. 105. 104. 103. 102. 101. 100. 99. 98. 97. 96. 95. 94. 91. 90. 88. 87. 86. 83. 82. 73. 71. 70. 69. 51. 46. 32. 27. 23.

I. M. Vardavas, F. W. Taylor: Radiation and climate A. F. Borghesani: Ions and electrons in liquid helium C. Kiefer: Quantum gravity, Second edition V. Fortov, I. Iakubov, A. Khrapak: Physics of strongly coupled plasma G. Fredrickson: The equilibrium theory of inhomogeneous polymers H. Suhl: Relaxation processes in micromagnetics J. Terning: Modern supersymmetry M. Mariño: Chern-Simons theory, matrix models, and topological strings V. Gantmakher: Electrons and disorder in solids W. Barford: Electronic and optical properties of conjugated polymers R. E. Raab, O. L. de Lange: Multipole theory in electromagnetism A. Larkin, A. Varlamov: Theory of fluctuations in superconductors P. Goldbart, N. Goldenfeld, D. Sherrington: Stealing the gold S. Atzeni, J. Meyer-ter-Vehn: The physics of inertial fusion T. Fujimoto: Plasma spectroscopy K. Fujikawa, H. Suzuki: Path integrals and quantum anomalies T. Giamarchi: Quantum physics in one dimension M. Warner, E. Terentjev: Liquid crystal elastomers L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems J. Wesson: Tokamaks, Third edition G. Volovik: The Universe in a helium droplet L. Pitaevskii, S. Stringari: Bose-Einstein condensation G. Dissertori, I.G. Knowles, M. Schmelling: Quantum chromodynamics B. DeWitt: The global approach to quantum field theory J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition R. M. Mazo: Brownian motion - fluctuations, dynamics, and applications H. Nishimori: Statistical physics of spin glasses and information processing - an introduction N. B. Kopnin: Theory of nonequilibrium superconductivity A. Aharoni: Introduction to the theory of ferromagnetism, Second edition R. Dobbs: Helium three R. Wigmans: Calorimetry J. Kübler: Theory of itinerant electron magnetism Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons D. Bardin, G. Passarino: The Standard Model in the making G. C. Branco, L. Lavoura, J.P. Silva: CP Violation T. C. Choy: Effective medium theory H. Araki: Mathematical theory of quantum fields L. M. Pismen: Vortices in nonlinear fields L. Mestel: Stellar magnetism K. H. Bennemann: Nonlinear optics in metals D. Salzmann: Atomic physics in hot plasmas M. Brambilla: Kinetic theory of plasma waves M. Wakatani: Stellarator and heliotron devices S. Chikazumi: Physics of ferromagnetism R. A. Bertlmann: Anomalies in quantum field theory P. K. Gosh: Ion traps S. L. Adler: Quaternionic quantum mechanics and quantum fields P. S. Joshi: Global aspects in gravitation and cosmology E. R. Pike, S. Sarkar: The quantum theory of radiation P. G. de Gennes, J. Prost: The physics of liquid crystals B. H. Bransden, M. R. C. McDowell: Charge exchange and the theory of ion-atom collision M. Doi, S. F. Edwards: The theory of polymer dynamics E. L. Wolf: Principles of electron tunneling spectroscopy H. K. Henisch: Semiconductor contacts S. Chandrasekhar: The mathematical theory of black holes C. Møller: The theory of relativity H. E. Stanley: Introduction to phase transitions and critical phenomena A. Abragam: Principles of nuclear magnetism P. A. M. Dirac: Principles of quantum mechanics R. E. Peierls: Quantum theory of solids

Liquid Crystal Elastomers

M. Warner and E. M. Terentjev Cavendish Laboratory, University of Cambridge

CLARENDON PRESS

•

OXFORD

3

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press, 2003 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2003 Reprinted 2005 First published in paper back (revised edition) 2007 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by the author using LaTex Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–852767–1 (Hbk.)

978–0–19–921486–0 (Pbk.)

1 3 5 7 9 10 8 6 4 2

PREFACE Liquid crystals are unusual materials. As their name suggests, they inhabit the grey area between liquids and solids. They have long range orientational order, typically of the unique axes of their component rod-like or plate molecules. Spatial variations of this average direction of molecular orientation are resisted by so-called curvature (Frank) elasticity. On the other hand liquid crystals can flow, albeit as anisotropic liquids. Polymers too are unusual materials. Above the glass transition, the physics is mostly dominated by the high entropy inherent in the disorder of their component long chain molecules. Resistance to molecular shape change arises mostly from the imperative to maintain high entropy. Viscoelastic flow and rubber elasticity are macroscopic manifestations of this principle. Thus rubber, where the long molecules are linked together, also inhabits the grey region between liquids and solids. Though nominally a solid, rubber is capable of very high deformations, greater than any other type of solid. Its internal molecular motion is rapid, as in a liquid, with the resulting amorphous solid being highly extensible rather than glassy. If it were not for the few crosslinks holding the chains into a percolating network, rubber would flow under stress, as ordinary polymers and other liquids do. The bulk (compression) modulus of typical rubber is of the same order as that of all liquids, and solids, but the shear modulus is about 10−4 − 10−5 times smaller. Thus rubber essentially deforms as a liquid, that is by shearing at constant volume. It is a weak solid and therein lies its enormous technological importance. This book is concerned about the phenomena arising when these two marginal materials, liquid crystals and polymers, are combined into one even more mysterious material – polymer liquid crystals. For two compelling reasons we shall concentrate on such polymers crosslinked into networks, that is, on elastomers and gels made from polymer liquid crystals: 1. Liquid crystal elastomers exhibit many entirely new effects that are not simply enhancements of native liquid crystals or polymers. We shall see their thermal phase transformations giving rise to spontaneous shape changes of many hundreds of per cents, transitions and instabilities induced by applied mechanical stress or strain, and some unusual dynamical effects. Strangest of all, we shall see elastomers under some conditions behaving entirely softly, deforming as true liquids do without the application of stress. All these new forms of elasticity have their genesis in the ambiguities between liquid and solid that are present in liquid crystals and polymers, but are only brought to light in a crosslinked rubbery network. 2. A molecular picture of rubber elasticity is now well established. Since the late 1930s its entropic basis has been understood and turns out to be as universal as, say, the ideal gas laws. The rubber shear modulus, µ , is simply ns kB T where ns counts the number of network strands per unit volume, and temperature T enters for the same entropic reason it does in the gas laws. There is no mention of the v

vi

PREFACE

chemistry of chains or other molecular details and the picture is thus of great generality. We call this the classical theory, to which various complexities such as crosslink fluctuations, entanglements and nematic interactions have later been added. By contrast to simple polymers, which change shape only in response to external forces, liquid crystal polymers do so spontaneously when they orientationally order their monomer segments. Can one nevertheless create a picture of their rubber elasticity of the same generality as that of classical rubber? It turns out that one can, with the sole extra ingredient of chain shape anisotropy (a single number directly measurable by experiment). We shall treat this anisotropy phenomenologically and find we can explore it at great length. One could go into many theoretical complexities, taking into account effects of finite chain extensibility, entanglements and fluctuations – however, in all cases, the underlying symmetry of spontaneously anisotropic network strands enters these approaches in the same way and the new physical phenomena are not thereby radically influenced. Alternatively, one could try to calculate the polymer chain anisotropy that appears in the molecular picture of rubber elasticity. There is, however, no universal agreement about which way to do this. A further complication is that polymer liquid crystals can be either main chain or side chain variants, where the rod-like elements are found respectively in, or pendant to, the polymer backbone. Nematic and smectic phases of considerable complexity and differing symmetry arise according to the molecular geometry. For instance side chain fluids can exist in 3 possible uniaxial nematic phases, NI , NII and NIII , with still further biaxial possibilities. In this book, by concentrating on Liquid Crystal Elastomers, rather than polymer liquid crystals per se, we relegate these theoretical uncertainties in the understanding of polymer liquid crystals to a subsidiary role. Key physical properties of crosslinked elastomers and gels are established without any detailed knowledge of how chains become spontaneously elongated or flattened. When more molecular knowledge is required, an adequate qualitative understanding of nematic and smectic networks can be obtained by adopting the simplest molecular models of polymer liquid crystals. In contrast, a treatise on polymer liquid crystals would have to address these issues rather more directly. These two reasons, the existence of novel physical phenomena and their relative independence from the details of molecular interactions and ordering, explain the sequence of arguments followed by this book. We introduce liquid crystals, polymers and rubber elasticity at the rather basic level required for the universal description of the main topic – Liquid Crystal Elastomers. Then we look at the new phenomena displayed by these materials and, finally, concentrate on the analysis of key features of nematic, cholesteric and then smectic rubbery networks. Rubber is capable of very large extensions. Many important new phenomena of nematic origin only occur at extensions of many tens of percents and are themselves highly non-linear. Linear continuum theory is utterly incapable of describing such a regime and this inadequacy is a motivation for our molecular picture of nematic rubber

PREFACE

vii

elasticity. However, it is clear that in liquid crystal elastomers we have not only the Lam´e elasticity of ordinary solids and the Frank curvature elasticity of liquid crystals, but also novel contributions arising from the coupling of the two. The richness and complexity of this new elasticity are such that it is worthwhile also analysing it using the powerful and general methods of continuum theory. There is a second motivation for studying continuum theory – for smectic elastomers there is not yet any underlying molecular theory and phenomenological theory is the best we can do. Because of their important technological applications, for instance in piezo- and ferroelectricity, an understanding of smectic elastomers is a vital priority. The latter chapters of our book are devoted to this, addressing the linear continuum approaches to elastomers with more complicated structure than simple uniaxial nematics. We also build a bridge between the elasticity methods of rubber and the application of continuum theory into the non-linear regime. At this point we revisit the symmetry arguments which explain why ‘soft elasticity’ is possible and why it cannot be found in classical elastic systems. We were tempted to take ‘Solid Liquid Crystals’ as our title. This would have been apt but obscure. We hope that this book will illuminate the peculiar materials that merit this description. Mark Warner and Eugene Terentjev February 2003 P REFACE TO THE PAPERBACK E DITION Since our book first appeared, there has been considerable progress in many aspects of chemistry, physics and applications of liquid crystalline elastomers. We could not cover all the new developments but focused in particular on nematic photo-elastomers and on smectic elastomers (but sadly had to leave out other very interesting work, for instance aspects of swelling and liquid crystalline gels). We have revised our book generally, but in particular added a new chapter (12). There, we describe recent advances in making smectic monodomains, the new mechanical effects found in smectic networks, and a molecular theory of smectic elastomers, directed as in nematics towards potentially large deformations. A rich range of phenomena emerge. Photo-elastomers also offer new possibilities – their non-uniform contractions lead to bend, and they can be steered mechanically by the polarisation of the light stimulating their elastic response. Our Appendices are now presented on-line. They address in each case the highly technical needs of a specialist slice of our readership that will be able to find them readily by searching on “Liquid Crystal Elastomers – On-line Appendices” [currently they are at www.lcelastomer.org.uk]. To retain their utility as a technical aid in reading this book, they are referred to at the appropriate place, and are listed in our Table of Contents. Material beyond that of the printed medium, such as films, will also be accessible. MW and EMT November 2006

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PREFACE

Figure Acknowledgments We are grateful to the many authors who have allowed us to reproduce figures from their work. The source of the figure is made clear in the caption or in the text where it is referred to. In each case the relevant publisher has also kindly given permission for us to reproduce the figure: Reproduced by permission of Wiley-VCH: Figures 1.7, 5.6, 5.10, 5.23, 6.5(a), 6.7(a), 6.7(b), 6.8, 12.5, 1.8, 12.6, 12.8, 12.9, 13.6(a). Reproduced by permission of The American Physical Society: Figures 3.6, 5.15, 5.19, 6.1(a), 8.1, 8.2, 8.15(b), 8.19(a), 8.19(b), 9.5(a), 9.5(b), 9.6(a), 9.6(b), 9.7, 9.8(a), 9.8(b), 9.12, 9.13, 9.14, 9.15, 9.16, 9.17, 9.19, 9.20, 11.5, 11.8(a), 11.8(b), 13.1, 13.2, 13.3, B.1, B.2, B.3. Reproduced by permission of Elsevier: Figure 5.7 (Anwer and Windle, 1991) and Figs. 8.11, 8.12, 8.14(a), 8.14(b) (Conti et al., 2002). Reproduced by permission of EDP Sciences: Figures 5.16, 5.20, 6.1(b), 6.5(b), 6.13, 7.10, 7.12(a), 7.12(b), 8.18(b), 10.10(b), 11.7, 11.9(a), 11.9(b), 11.10(a), 11.10(b), 11.11(a), 11.11(b), 11.12(a), 11.12(b), 11.13(b), 11.14, 6.9. Reproduced by permission of The Institute of Physics: Figure 8.21(a). Reproduced by permission of the American Chemical Society: Figures 8.21(b), 9.3, 10.10(a). Reproduced by permission of The Royal Society of Chemistry: Figure 9.2. Reproduced by permission of Nature: Figure 13.4(b).

ACKNOWLEDGEMENTS From the initial speculations of the late 60s to the ideas of artificial muscles in the late 1990s, the presence of Pierre-Gilles de Gennes can be felt in the field and in this book. We are grateful for his personal encouragement over many years in many different areas. In 1981 Heino Finkelmann discovered all three elastomer phases and subsequently in the 1990s found routes to nematic, cholesteric and smectic monodomains. We have been inspired by both his pioneering chemistry and deep physical insights into the processes underlying liquid crystal elastomers, and also benefited enormously from his advice and help with techniques of sample preparation. Sam Edwards, one of the founders of modern polymer physics, introduced us to polymer networks. He has been a constant source of advice and motivation in our work on liquid crystal elastomers and in the writing of this book. His philosophy of the necessity of molecular models of polymers when one is confronted with non-linear effects, as is the case throughout this book, will be evident to the reader. Over the years, we have enjoyed fruitful scientific collaboration and exchange of ideas with F. Kremer, G.R. Mitchell, P. Palffy-Muhoray, W. Stille, R.V. Talroze, R. Zentel and many others. Geoffrey Allen and Ed Samulski first introduced us to the idea of nematic effects in elastomers generally and have encouraged our work over many years. Several people have influenced us in a particularly crucial way: Robert Meyer shared his penetrating insight into many new effects in elastomers, including but not limited to Freedericks transitions, stripes and field deformations of cholesterics. We drew much from contact with Tom Lubensky on soft elasticity and non-linear continuum theory. Our work on the continuum description of smectics was in part in collaboration with him. Peter Olmsted’s clear and concise formulation of soft elasticity in molecular theory, and his symmetry arguments for its existence, is an approach adopted in large measure in our book and helped us formulate our understanding of the subject. We also thank Stuart Clarke and Yong Mao for guidance, advice and criticism over their years of collaboration with us. Richard James exposed us to the concept of quasi-convexification and the mechanisms of macroscopic, inhomogeneous soft deformation. He and Antonio DeSimone convinced us that nematic elastomers were an outstanding example of where this process can occur. The first chapter of Tom Faber’s remarkable book on fluid mechanics inspired our Birds Eye View of the material of this book. A. DeSimone and S. Conti, generously gave advice and also material for the relevant sections of Chapter 8. Samuel Kutter provided many figures from his own work. He, Paolo Teixeira, James Adams and Daniel Corbett commented critically on the book, much improving it, though its shortcomings have to remain ours. David Green helped us overcome technical difficulties of LATEX. Finally, we thank our wives Adele and Helen for their endurance and patience over the period when this book was written, which was far too long by any measure.

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CONTENTS

1

A bird’s eye view of liquid crystal elastomers

2

Liquid crystals 2.1 Ordering of rod and disc fluids 2.2 Nematic order 2.3 Free energy and phase transitions of nematics 2.4 Molecular theory of nematics 2.5 Distortions of nematic order 2.6 Transitions driven by external fields 2.7 Anisotropic viscosity and dissipation 2.8 Cholesteric liquid crystals 2.9 Smectic liquid crystals

9 9 11 15 20 22 25 29 33 38

3

Polymers, elastomers and rubber elasticity 3.1 Configurations of polymers 3.2 Liquid crystalline polymers 3.2.1 Shape of liquid crystalline polymers 3.2.2 Frank elasticity of nematic polymers 3.3 Classical rubber elasticity 3.4 Manipulating the elastic response of rubber 3.5 Finite extensibility and entanglements in elastomers

47 48 52 54 61 62 67 70

4

Classical elasticity 4.1 Deformation tensor and Cauchy–Green strain 4.2 Non-linear and linear elasticity 4.3 Geometry of deformations and rotations 4.3.1 Rotations 4.3.2 Shears and their decomposition 4.3.3 Square roots and polar decomposition of tensors 4.4 Compressibility of rubbery networks

75 75 78 83 83 84 90 91

5

Nematic elastomers 5.1 Structure and examples of nematic elastomers 5.2 Stress-optical coupling 5.3 Polydomain textures and alignment by stress 5.4 Monodomain ‘single-crystal’ nematic elastomers 5.4.1 Spontaneous shape changes 5.4.2 Nematic photoelastomers 5.5 Field-induced director rotation 5.6 Applications of liquid crystalline elastomers xi

1

95 96 99 101 104 106 108 111 115

xii

CONTENTS

6

Nematic rubber elasticity 6.1 Neo-classical theory 6.2 Spontaneous distortions 6.3 Equilibrium shape of nematic elastomers‡ 6.4 Photo-mechanical effects 6.5 Thermal phase transitions 6.6 Effect of strain on nematic order 6.7 Mechanical and nematic instabilities 6.7.1 Mechanical Freedericks transition 6.7.2 The elastic low road 6.8 Finite extensibility and entanglements

120 120 123 129 131 136 139 145 146 148 150

7

Soft elasticity 7.1 Director anchoring to the bulk 7.1.1 Director rotation without strain 7.1.2 Coupling of rotations to pure shear 7.2 Soft elasticity 7.2.1 Soft modes of deformation 7.2.2 Principal symmetric strains and body rotations 7.2.3 Forms of the free energy allowing softness 7.3 Optimal deformations 7.3.1 A practical method of calculating deformations 7.3.2 Stretching perpendicular to the director 7.4 Semi-soft elasticity 7.4.1 Example: random copolymer networks 7.4.2 A practical geometry of semi-soft deformation 7.4.3 Experiments on long, semi-soft strips 7.4.4 Unconstrained elastomers in external fields 7.5 Semi-soft free energy and stress 7.6 Thermomechanical history and general semi-softness 7.6.1 Thermomechanical history dependence 7.6.2 Forms of the free energy violating softness

154 155 155 158 159 160 164 166 167 167 169 173 174 175 177 178 179 183 184 185

8

Distortions of nematic elastomers 8.1 Freedericks transitions in nematic elastomers 8.2 Strain-induced microstructure: stripe domains 8.3 General distortions of nematic elastomers 8.3.1 One-dimensional quasi-convexification 8.3.2 Full quasi-convexification 8.3.3 Numerical and experimental studies 8.4 Random disorder in nematic networks 8.4.1 Nematic ordering with quenched disorder 8.4.2 Characteristic domain size 8.4.3 Polydomain-monodomain transition

187 188 194 201 202 205 207 210 212 213 216

CONTENTS

9

Cholesteric elastomers 9.1 Cholesteric networks 9.1.1 Intrinsically chiral networks 9.1.2 Chirally imprinted networks 9.2 Mechanical deformations 9.2.1 Uniaxial transverse elongation 9.2.2 Stretching along the pitch axis 9.3 Piezoelectricity of cholesteric elastomers 9.4 Imprinted cholesteric elastomers 9.5 Photonics of cholesteric elastomers 9.5.1 Photonics of liquid cholesterics 9.5.2 Photonics of elastomers 9.5.3 Experimental observations 9.5.4 Lasing in cholesterics

xiii

220 221 221 222 227 228 233 236 242 245 246 249 251 253

10 Continuum description of nematic elastomers 10.1 From molecular theory to continuum elasticity 10.1.1 Compressibility effects 10.1.2 The limit of linear elasticity 10.1.3 The role of nematic anisotropy 10.2 Phenomenological theory for small deformations 10.3 Strain-induced rotation 10.4 Soft elasticity 10.4.1 Symmetry arguments 10.4.2 The mechanism of soft deformation 10.5 Continuum representation of semi-softness 10.6 Unconstrained director fluctuations 10.7 Unconstrained phonons 10.8 Light scattering from director fluctuations

256 257 257 258 260 262 265 268 269 271 273 276 279 282

11 Dynamics of liquid crystal elastomers 11.1 Classical rubber dynamics 11.1.1 Rouse model and entanglements 11.1.2 Dynamical response of entangled networks 11.1.3 Long time stress relaxation 11.2 Nematohydrodynamics of elastic solids 11.2.1 Viscous coefficients and relaxation times 11.2.2 Balance of forces and torques 11.2.3 Symmetries and order parameter 11.3 Response to oscillating strains 11.4 Experimental observations 11.4.1 Oscillating shear 11.4.2 Steady stress relaxation

288 289 291 293 296 298 300 301 303 304 308 309 313

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CONTENTS

12 Smectic elastomers 12.1 Materials and preparation 12.1.1 Smectic A elastomers 12.1.2 Smectic C and ferroelectric C∗ elastomers 12.2 Physical properties of smectic elastomers 12.2.1 Smectic-A elastomers 12.2.2 Smectic-C elastomers 12.3 A molecular model of Smectic-A rubber elasticity 12.3.1 The geometry of affine layer deformations 12.3.2 Response to principal deformations 12.3.3 General deformations of a SmA elastomer 12.4 Instability and CMHH microstructure 12.5 Comparison with experiment 12.6 Smectic-C rubber elasticity 12.6.1 SmC soft deformations 12.6.2 SmC deformations with microstructure

317 317 319 321 322 322 326 328 330 331 339 340 342 344 345 348

13 Continuum description of smectic elastomers 13.1 Continuum description of smectic A elastomers 13.1.1 Relative translations in smectic networks revisited 13.1.2 Nematic -strain, -rotation and -smectic couplings 13.2 Effective smectic elasticity of elastomers 13.3 Effective rubber elasticity of smectic elastomers 13.4 Layer elasticity and fluctuations in smectic A elastomers 13.5 Layer buckling instabilities: the CMHH effect 13.6 Quenched layer disorder and the N-A phase transition 13.7 Smectic C and ferroelectric C∗ elastomers

350 350 351 353 355 360 364 371 374 378

References

382

Index

393

Author Index

400

O NLINE A PPENDICES

(www.lcelastomer.org.uk)

A

Nematic order in elastomers under strain

B

Biaxial soft elasticity

C

Stripe microstructure

D

Couple-stress and Cosserat elasticity

E

Expansion at small deformations and rotations

F

Smectic C soft elasticity

1 A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS Liquid crystal elastomers bring together, as nowhere else, three important ideas: orientational order in amorphous soft materials, responsive molecular shape and quenched topological constraints. Acting together, they create many new physical phenomena that are the subject of this book. This bird’s eye view sketches how these themes come together and thereby explains the approach of our book. In the early chapters we introduce the reader to liquid crystals and to polymers since they are our building blocks. One could regard the first part of our book as a primer for an undergraduate or graduate student embarking on a study of polymer or liquid crystal physics, or on complex fluids and solids. Then elastomers are discussed both from the molecular point of view, and within continuum elasticity. We need to understand how materials respond at very large deformations for which only a molecular approach is suitable. Also one needs to understand the resolution of strains into their component pure shears and rotations, the latter also being important in these unusual solids. We also provide a primer for the basics of these two areas that are otherwise only found in difficult and advanced texts. Classical liquid crystals are typically fluids of relatively stiff rod molecules with long range orientational order. The simplest case is nematic – where the average ordering direction of the rods, the director n , is uniform. Long polymer chains, with incorporated rigid anisotropic units can also order nematically and thus form liquid crystalline polymers. By contrast with rigid rods, these flexible chains elongate when their component rods align. This results in a change of average molecular shape, from spherical to spheroidal as the isotropic polymers become nematic. In the prolate anisotropy case, the long axis of the spheroid points along the nematic director n , Fig. 1.1.

F IG . 1.1. Polymers are on average spherical in the isotropic (I) state and elongate when they are cooled to the nematic (N) state. The director n points along the principal axis of the shape spheroid. (The mesogenic rods incorporated into the polymer chain are not shown in this sketch, only the backbone is traced.) 1

2

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

So far we have no more than a sophisticated liquid crystal. Changes in average molecular shape induced by changes in orientational order do little to modify the properties of this new liquid crystal. Linking the polymer chains together into a gel network fixes their topology, and the melt becomes an elastic solid – a rubber. Radically new properties can now arise from this ability to change molecular shape while in the solid state. To understand this we have to consider rubber elasticity. In rubber, monomers remain highly mobile and thus liquid-like. Thermal fluctuations move the chains as rapidly as in the melt, but only as far as their topological crosslinking constraints allow. These loose constraints make the polymeric liquid into a weak, highly extensible material. Nevertheless, rubber is a solid in that an energy input is required to change its macroscopic shape (in contrast to a liquid, which would flow in response). Equivalently, a rubber recovers its original state when external influences are removed. Systems where fluctuations are limited by constraints are known in statistical mechanics as ‘quenched’ - rigidity and memory of shape stem directly from this. It is a form of imprinting found in classical elastomers and also in chiral solids, as we shall see when thinking about cholesteric elastomers. Can topology, frozen into a mobile fluid by constraints, act to imprint liquid crystalline order into the system? The expectation based on simple networks would be ‘yes’. This question was posed, and qualitatively answered, by P-G. de Gennes in 1969. He actually asked a slightly more sophisticated question: Crosslink conventional polymers (not liquid crystalline polymers) into a network in the presence of a liquid crystalline solvent. On removal of the solvent, do the intrinsically isotropic chains remember the anisotropy pertaining at the moment of genesis of their topology? The answer for ideal chains linked in a nematic solvent is ‘no’! Intrinsically nematic polymers, linked in a nematic phase of their own making, can also elude their topological memory on heating. How this is done (and failure in the non-ideal case) is a major theme of this book. Second, what effects follow from changing nematic order and thus molecular shape? The answer is new types of thermal- and light-induced shape changes. The third question one can ask is: While in the liquid-crystal state, what connection between mechanical properties and nematic order does the crosslinking topology induce? The answer to this question is also remarkable and is discussed below. It leads to entirely new effects – shape change without energy cost, extreme mechanical effects and rotatory-mechanical coupling. We give a preview below of these effects in the form of a sketch – details have to await the later chapters of the book. Rubber resists mechanical deformation because the network chains have maximal entropy in their natural, undeformed state. Crosslinking creates a topological relation between chains that in effect tethers them to the solid matrix they collectively make up. Macroscopic deformation then inflicts a change away from the naturally spherical average shape of each network strand, and the entropy, S, falls. The free energy then rises, ∆F = −T ∆S > 0. This free energy, dependent only on an entropy change itself driven by molecular shape change, explains why polymers are sometimes thought of as ‘entropic springs’. Macroscopic changes in shape are coupled to molecular changes. In conventional rubber it is always the macroscopic that drives the molecular; the induced conformational entropy of macromolecules offers the elastic resistance.

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

3

n

lm

heat

1

cool

I

N

F IG . 1.2. A unit cube of rubber in the isotropic (I) state. Embedded in it is shown the average of the chain distribution (spherical). The block elongates by a factor λm on cooling to the nematic (N) state, accommodating the now elongated chains. Nematic polymers suffer spontaneous shape changes associated with changing levels of nematic (orientational) order, Fig. 1.1. One now sees a reversal of influence: changes at the molecular level induce a corresponding change at the macroscopic level, that is induce mechanical strains, Fig. 1.2: a block of rubber elongates by a factor of λm > 1 on cooling or 1/λm < 1 on heating. This process is perfectly reversible. Starting in the nematic state, chains become spherical on heating. But mechanical strain must now accompany the molecular readjustment. Very large deformations are not hard to achieve, see Fig. 1.3. Provided chains are in a broad sense ideal, it turns out that chain shape can reach isotropy both for the imprinted case of de Gennes (on removal of nematic solvent) and for the more common case of elastomers formed from liquid crystalline polymers (on heating). Chains experiencing entanglement between their crosslinking points also evade any permanent record of their genesis. Many real nematic elastomers and gels in practice closely conform to these ideal models. Others are non-ideal – they retain some nematic order at high temperatures as a result of their order and topology combining with other factors such as random pinning fields and compositional fluctuations. They still show the elongations of Fig. 1.3, but residues of non-ideality are seen in the elastic effects we review below. This extreme thermomechanical effect, and the phenomena of Figs. 1.5 and 1.7, can only be seen in monodomain, well aligned samples. Without very special precautions during fabrication, liquid crystal elastomers are always found in polydomain form, with very fine texture of director orientations. The great breakthrough in this field, developing a first method of obtaining large, perfect monodomain nematic elastomers, was made

F IG . 1.3. A strip of nematic rubber extends and contracts according to its temperature. Note the scale behind the strip and the weight that is lifted!

4

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

nO nnOo W

W

n’ n

Rotation by 90O

wq

(a)

(b)

n

(c)

F IG . 1.4. (a) Rotations of the director and matrix by angles θ and Ω, respectively. From (b) to (c) the director, and thus chain shape distribution, is rotated by 90o from n o to n . The rubber is mechanically clamped and hence the chains in (c) that would be naturally elongated along n must be compressed: the dotted spheroid in (c) is compressed to the actual solid spheroid. by K¨upfer and Finkelmann in 1991. Nematic-elastic coupling was the third question we posed and gives rise to new rotational phenomena ubiquitous in liquid crystal elastomers. It is possible to rotate the director and the rubber matrix independently, see Fig. 1.4 (a). Such relative rotations of the body and of its internal anisotropy axis show that nematic elastomers are not simply exotic, highly-extensible, uniaxial crystals. Such materials belong to a class displaying so-called Cosserat elasticity, but with the distinction that deformations and rotations can be large in elastomers. Imagine now rotating the director while clamping the body so its shape does not change, Figs. 1.4(b) and (c). The natural, prolate spheroidal distribution, when rotated by 90o to be along n , has a problem. Chains do not naturally fit, since the clamped body to which they are tethered is not correspondingly elongated along n to accommodate their long dimensions. Chains in fact must have been compressed to fit, at considerable entropy loss if they were very anisotropic. A rotation of 180o recovers the initial state, so the free energy must be periodic, and turns out to be F = 21 D1 sin2 (θ − Ω). The rotational modulus, D1 , was first given by de Gennes in the infinitesimal form 12 D1 (θ − Ω)2 . A rotation of the director in Fig. 1.4(b) would lead to a ‘virtual’ intermediate state depicted by dotted lines in Fig. 1.4(c). Subsequent squeezing to get back the actual body shape demanded by the clamp condition (full lines) of Fig. 1.4(c) costs an energy proportional to the rubber modulus, µ , and to the square of the order, Q, (since Q determines the average chain shape anisotropy). Thus D1 ∼ µ Q2 . In contrast to ordinary nematics, it costs energy to uniformly rotate the director independently of the matrix. In liquid nematics it is director gradients that suffer Frank elastic penalties, and thus long-wavelength spatial variations of the rotation angle cost vanishingly small energy. Thermal excitation of these rotations causes even monodomain nematic liquids to scatter light and to be turbid. Not so monodomain nematic elastomers which are optically clear because even long wavelength director rotations cost a finite rubber-elastic energy 1 2 2 D1 θ and cannot be excited, see Fig. 1.5. The excitations have acquired a mass, in the language of field theory.

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

5

F IG . 1.5. A strip of monodomain ‘single-crystal’ nematic rubber. It is completely transparent and highly birefringent (image: H. Finkelmann). Local rotations, so central to nematic elastomers, yield a subtle and spectacular new elastic phenomenon which we call ‘soft elasticity’. Imagine rotating the director but now not clamping the embedding body, in contrast to Figs. 1.4(b) and (c). One simple response would be to rotate the body by the same angle as the director, and this would clearly cost no energy. However, contrary to intuition, there is an infinity of other ways by mechanical deformation to accommodate the anisotropic distribution of chains without its distortion as it rotates. Thus the entropy of the chains does not change, in spite of macroscopic deformations. Figure 1.6 illustrates the initial and final states of a 90o director rotation. They are separated by a path of states, characterised by an intermediate rotation angle θ and by a corresponding shape of the body, one of which is shown. This θ -state is shown in the sketch (b) accommodating the spheroid without distorting it. A special combination of shears and elongations/compressions is required, but it turns out not very difficult to achieve in experiment! One of the traditional ways to rotate the director in liquid crystals is by applying an electric (or magnetic) field and generating a local torque due to the dielectric anisotropy. Due to the nematic-elastic coupling, the director rotation is very difficult if an elastomer sample is mechanically constrained. Apart from a few exceptions (all characterised by a very low rubber-elastic modulus, such as in highly swollen gels) no electrooptical response can occur. However, if the elastomer is mechanically unconstrained, the situation changes remarkably. In a beautiful series of experiments, Urayama (2005,2006) has confirmed the prediction of soft elasticity: that the field-induced director rotation has no energy cost, can easily reach 90o rotation angles and has associated mechanical strains that almost exactly follow the sketch in Fig. 1.6. Practically, when dealing with rubbers, one might instead impose a mechanical distortion (say an elongation, λ , perpendicular to the original director) and have the other

F IG . 1.6. Rotation of chain shape distribution, from n o to n , with an intermediate state θ shown. The unconstrained rubber deforms to accommodate the rotating director without distorting the chain distribution.

6

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

components of strain, and the director orientation, follow it. The result is the same – extension of a rubber costs no elastic energy and is accompanied by a characteristic director rotation. The mechanical confirmation of the cartoon is shown in stress-strain curves in Fig. 1.7(a) and the director rotation in Fig. 1.7(b). We have made liquid crystals into solids, albeit rather weak solids, by crosslinking them. Like all rubbers, they remain locally fluid-like in their molecular freedom and mobility. Paradoxically, their liquid crystallinity allows these solid liquid crystals to change shape without energy cost, that is to behave for some deformations like a liquid. Non-ideality gives a response we call ‘semi-soft’. There is now a small threshold before director rotation (seen in the electrooptical/mechanical experiments of Urayama (2005,2006), and to varying degrees in Fig. 1.7); thereafter deformation proceeds at little additional resistance until the internal rotation is complete. This stress plateau, the same singular form of the director rotation, and the relaxation of the other mechanical degrees of freedom are still qualitatively soft, in spite of a threshold. There is a deep symmetry reason for this apparently mysterious softness that Fig. 1.6 rationalises in terms of the model of an egg-shaped chain distribution rotating in a solid that adopts new shapes to accommodate it. Ideally, nematic elastomers are rotationally invariant under separate rotations of both the reference state and of the target state into which it is deformed. If under some conditions, not necessarily the current ones, an isotropic state can be attained, then a theorem of Golubovi´c and Lubensky shows that in consequence soft elasticity must exist. It is a question of care with the fundamental tenet of elasticity theory, the principle of material frame indifference. We shall examine this theorem and its consequences many times in this book, including what happens when the conditions for it to hold are violated, that is when semi-softness prevails. 100

se (MPa)

q

(deg)

80

0.09

60

0.06 40

0.03

20 0

1

1.15 (a)

1.3

1.45

l

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

l

(b)

F IG . 1.7. (a) Stress-deformation data of K¨upfer and Finkelmann (1994), for a series of rubbers with the same composition and crosslinking density, but differing in preparation history: some show a normal elastic response while others are remarkably soft. (b) The angle of director rotation on stretching nematic elastomer perpendicular to the director for a variety of different materials, from Finkelmann et al. (1997). The solid line from, theoretical modeling, accurately reproduces singular points and characteristic shape of data.

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

7

Elastic softness, or attempts to achieve it, pervade much of the elasticity of nematic elastomers. If clamps or boundary conditions frustrate uniform soft deformation trajectories, microstructures will evolve to allow softness with the cost of interfaces being a relatively smaller price to pay. There are similarities between this so-called ‘quasiconvexification’ and that seen in martensite and other shape-memory alloys. Cholesteric liquid crystals have a helical director distribution. Locally they are very nearly conventional nematics since their director twist occurs typically over microns, a much longer length scale than that associated with nematic molecular ordering. They can be crosslinked to form elastomers which retain the cholesteric director distribution. Several phenomena unique to cholesterics emerge: Being locally nematic, cholesteric elastomers would like on heating and cooling to lose and recover orientational order as nematic elastomers do. However, they cannot resolve the requirement at neighbouring points to spontaneously distort by λm , but in different directions. Accordingly, their chains cannot forget their topologically imprinted past when they attempt to reach a totally isotropic reference state (the second de Gennes’ prediction of 1969). Thus cholesteric rubbers also cannot deform softly in response to imposed strains. Their optical and mechanical responses to imposed stress are exceedingly rich as a result. They are brightly coloured due to selective reflection and change colour as they are stretched – their photonic band structure changes with strain. They can emit laser radiation with a colour shifted by mechanical effects. Further, the effect of topological imprinting can select and extract molecules of specific handedness from a mixed solvent. Such rubbers can act as a mechanical separator of chirality – a new slant on a problem that goes back to Pasteur. We have sketched the essentials of nematic (and cholesteric) rubber elasticity. This survey leaves out many new phenomena dealt with in later chapters, for instance electromechanical Freedericks effects, photo-elastomers that drastically change shape on illumination, rheology and viscoelasticity that crosses between soft and conventional depending upon frequency and geometry, and so on. Smectics are the other class of liquid crystal order. They have plane-like, lamellar modulation of density in one direction (SmA), or additionally a tilt of the director away from the layer normal (SmC). Many other more complex smectic phases exist and could also be made into elastomers. In many smectic elastomers, layers are constrained not to move relative to the rubber matrix. Deformations of a rubber along the layer normal are thus resisted by a layer spacing modulus, B, of the order of 102 times greater than the shear modulus of the matrix. Distortions in plane, either extensions or appropriate shears, are simply resisted by the rubber matrix. Thus SmA elastomers are rubbery in the two dimensions of their layer planes, but respond as hard conventional solids in their third dimension. Fig. 1.8 shows this behaviour. Such extreme mechanical anisotropy promises interesting applications. The director tilt associated with the transition from SmA to SmC induces distortion in the polymer chain shape distribution. Since chain shape is coupled to mechanical shape for an elastomer, one expects, and sees in Fig. 1.9, spontaneous distortion. This response to order change is analogous to the elongations associated with orientational order of chains on entering the nematic state, but here we instead have shear. The amp-

8

A BIRD’S EYE VIEW OF LIQUID CRYSTAL ELASTOMERS

Stress s (kPa)

80

60

40

z

x

20

s

s 0 0

0.05

0.1

0.15

0.2

0.25

Strain DL/L O

F IG . 1.8. In-plane fluidity and parallel rigidity in a smectic A elastomer (Nishikawa et al., 1997). The Young modulus parallel and perpendicular to the layer normals differ very greatly - the rubber elasticity is two-dimensional.

k0

(a)

n0

k0

n

(b)

F IG . 1.9. (a) A SmA elastomer (Hiraoka et al., 2005). (b) Spontaneous shear λxz in achieving the SmC state. litude is also large, of the order of 0.4 in the figure. As in the nematic case, the broken symmetry suggests a mechanism for SmC solids richer still than that of SmA elastomers, including SmC soft elasticity equivalent to that of Fig. 1.6. The tilted, SmC, liquids also exist in chiral forms which must, on symmetry grounds be ferroelectric. Their elastomers are too. Ferroelectric rubber is very special: mechanically it is soft, about 104 times lower in modulus than ferro- and piezoelectrics because, as sketched above, its molecules are spatially localised by topological rather than energetic constraints. Distortions give polarisation changes comparable to those in ordinary ferroelectrics. But the response in terms of stress must necessarily be 104 times larger than in conventional materials. We end our preview as we started – solids created by topological constraints are soft and highly extensible. Liquid crystal elastomers share this character with their important cousins, the conventional elastomers. But their additional liquid crystalline order gives them entirely new kinds of elasticity and other unexpected phenomena.

2 LIQUID CRYSTALS Liquid crystalline rubbery solids are polymer networks with nematic or smectic order. They display most of the complexities of conventional liquid crystals: directional but not translation long range order, optical birefringence and phase transitions. In fact they are liquid crystals with the exception that they cannot flow. Liquid crystal networks have many properties in addition to simple nematics and smectics, but to start understanding them, one must review conventional liquid crystals. Our review will be very selective since we require only the basics of what is a large and subtle subject. Moreover, excellent monographs exist, such as (de Gennes and Prost, 1994) and (Chandrasekhar, 1977). More specialist reviews explain corners of the field, for instance Landau theory or polarisational effects. Readers familiar with liquid crystals can skip most sections of this chapter – these provide and set up the style of discussion and the notation essential for the rest of this book. In discussing Freedericks transitions, eqn (2.29) onwards, we briefly look forward to a major difference between conventional nematics and liquid crystal elastomers. What are the essential differences when nematic liquid crystals are ‘solidified’ to form elastomers or gels? We shall see that rubber has all the mobility of liquids locally but not in a bulk sense – they cannot flow. The ordering thus remains mobile, albeit with some tethering to the solid matrix. All liquid crystal properties other than flow are manifested. We shall dwell in this chapter on properties of simple nematics, cholesterics and smectics that will be radically changed in networks. Detailed molecular models play little role in nematic elastomers, apart from describing phase transitions and behaviour close to them, and for details of photoelastomers. Otherwise nematic, cholesteric and smectic elastomers, like conventional rubbers, are remarkably universal. We shall see that the properties of conventional elastomers depend essentially on the density of crosslinks and on temperature, much like an ideal gas. Liquid crystal elastomers depend upon these two factors, but also upon the shape anisotropy of their constituent polymer chains. This anisotropy is liquid crystalline (molecular) in origin, but can be measured directly or derived from macroscopic shape changes, a path we shall mostly follow. For this reason we do not dwell on detailed microscopic models of liquid crystals, and also not on models of polymers in the next chapter. 2.1 Ordering of rod and disc fluids Nematics are anisotropic fluids. They derive their name from the thread-like defects in their anisotropy, i.e. disclinations that are observed under the microscope. The Greek word νη µα for thread was taken by G. Friedel for the name of this phase. Molecular asymmetry is a precondition for macroscopic anisotropy. Weak asymmetry, for instance in the N2 molecule is insufficient to lead to spontaneous ordering. Increasing either the 9

10

LIQUID CRYSTALS (PAA)

CH3 O

N N

O CH3

O

(MBBA)

(5CB)

CH3 O

N C

CH N

CH 2 CH C C 2 CH3 C 2 C CH CH 2 CH C C 2 C CH3 C 2 C 2 CH CH

F IG . 2.1. The chemical structure of para-azoxyanizole (PAA). This, and many other mesogenic (liquid crystal phase-forming) molecules are characterised by the same general pattern of two para-substituted aromatic rings rigidly linked into a rod-like structure. The terminal groups often vary, from a simple CH3 in PAA, to longer flexible chains in MBBA, or dipolar units, e.g. a CN group in cyanobiphenyls (5CB). shape anisotropy (a steric influence), or the anisotropy of polarisability (a thermotropic influence), results in anisotropic liquids with long range directional ordering. The archetypical mesogenic molecule that forms such a fluid is para-azoxyanizole (PAA), see Fig. 2.1. Its shape is rod-like and its conjugated chemical bonds render it more polarisable along its long axis. S TERIC EFFECTS Anisotropy in shape encourages molecules to be parallel, when in dense solution or in the melt, since then they can translate more freely without overlapping. They thereby maximise the disorder of translation, increase the entropy and thus lower the free energy. It is the same entropic driving force that persuades a conventional system at a high enough temperature to be a liquid rather than a solid. However, in a rod-fluid rotational freedom must be compromised in order to gain this entropy. Shape exclusion is clearly much more effective at higher rod concentrations in solution since the possibility of rods overlapping occurs more often. On dilution rods can also randomise directions while not compromising translational entropy. Such nematics where concentration and its entropic consequences dominate are known as lyotropics, see Fig. 2.2. A spectacular case of ordering by the steric effect of anisotropic excluded volume alone is in water solutions of tobacco mosaic virus (TMV). Here rod-like mo˚ and width D ∼ 200 A ˚ form a nematic lecular aggregates of length up to L ∼ 3000 A phase when their concentration in solution exceeds a critical value of about 5-10(D/L), arguably, with no attractive forces present at all. More recently, the nematic ordering of carbon nanotubes or long DNA fragments in dilute solutions have been reported. At the melt densities of pure rod fluids, the critical axial ratio L/D below which steric forces fail to cause ordering at all temperatures, is thought to be in the region of L/D ∼ 6. T HERMOTROPIC EFFECTS There is another ordering mechanism, provided by the soft, long-range potential forces. Greater molecular polarisability along the rod axis creates a greater van der Waals attraction between two rods when they are parallel (a) than perpendicular (b), see Fig. 2.2. A fluctuation of electron density on one molecule gives an instantaneous dipole moment p, say. Its electric field at a neighbouring molecule

NEMATIC ORDER

p

p’ (a)

p

11

p’

(b)

F IG . 2.2. A sketch of virial ideas, showing the reasons for orientational ordering of rod-like molecules due to the entropic (steric) effect of anisotropic excluded volume, shown by dotted shape, and to long-range van der Waals interaction of induced molecular dipoles p and p  . Scenario (a) is preferable to (b) for both steric (at high densities) and van der Waals effects. induces a dipole p  , much greater in the aligned configuration (a) than in case (b), where the other molecule presents a much less polarisable aspect to the field. The dipoles attract each other and lower the energy of the pair of molecules, more if they are parallel than if they are perpendicular. When long-range van der Waals forces of anisotropic attraction are the dominant ordering influence, a reduction in temperature will lead to nematic ordering. Such systems are known as thermotropic. PAA is a good example of these. At a temperature T > 135oC, even at the highest densities (in the melt, with no solvent at all), shape effects are insufficient to produce the nematic phase which can only result from cooling. We shall be interested in directionally ordered molecules, irrespective of the mechanism by which they order (generally it is both). Rod-like molecules, similar to PAA, continue to order when incorporated into polymer chains and thereby create the essential alignment we require to obtain nematic (and later) smectic elastomers. For all the reasons given above, anisotropic disc-like molecules will generate nematic (and other) phases too. In some cases liquid crystal polymers have been created from incorporating discs into polymer chains. 2.2 Nematic order The sketch Fig. 2.3(a) of a nematic fluid shows rods correlated with a direction n , the nematic director. The director is a unit vector, only showing the principal axis of alignment. In a fluid of rods, as in Figs. 2.2 and 2.3, the direction ‘up’ is not distinguished from ‘down’; indeed it could not be since the rods drawn are not themselves capable of making the distinction. For this reason n is drawn as a double headed vector. In practice rods do have an internal direction, for instance a dipole moment along their long axis, but the up-down (quadrupolar) symmetry of nematics is not broken. If it were, we would have ferroelectric nematics with a spontaneous polarisation from the predominance of, say, ‘up’ molecular dipoles over those ‘down’. In nature uniaxial nematics are not polar but quadrupolar, with the symmetry described by the point group D∞h (a symmetry of a simple cylinder). The orientational order can now be defined. In Fig. 2.3(a) a test rod’s spine is drawn with an angle θ to n . The nematic order parameter is defined via the average of second

12

LIQUID CRYSTALS

n

1

P2 (cos q)

q 0.5

0

-0.5

(a)

0

45

90 Angle q (deg)

135

180

(b)

F IG . 2.3. (a) The distribution of molecular axes around the average alignment direction n . (b) The Legendre polynomial P2 as a function of angle θ in the range 0 to 180o . Note that it varies between 1 and -0.5, that angles θ and π − θ are equivalent, and that positive and negative values of P2 refer to geometrically very different states. Legendre polynomial1 , as Q = P2 (cos θ ) =  32 cos2 θ − 12 

(2.1)

where . . . denotes an average over rod directions θ . From Fig. 2.3 one can see how Q = 1 corresponds to perfect nematic order with rods directed up (θ = 0) or down (θ = π ). Q = 0 is when rods are randomly oriented, that is the phase is isotropic and cos2 θ  = 1/3. Moderate nematic order of Q = 1/2 sees rods with an average angle of θ ∼ 35o , whereas Q = −1/2 has all rods confined to the plane perpendicular to n , that is all rods have (θ = π /2). Nematic order can be measured directly by nuclear magnetic resonance (de Gennes and Prost, 1994). A more macroscopic measure of nematic ordering can be adopted. The fluid of aligned rods in Fig. 2.3(a) has a refractive index m along n typically greater than that, m⊥ , in all the perpendicular directions. This is because rods (see Fig. 2.1) are mostly more polarisable along their lengths and their long axes in a nematic are correlated with the director. Again ‘up’ and ‘down’ are not distinguished and the difference ∆m = m − m⊥ depends on the nematic order Q as: ∆m = ∆mo Q. The intrinsic anisotropy, ∆mo , depends on molecular factors and can be calculated, or estimated by extrapolation to low temperatures where the nematic order becomes high, Q → 1. We shall later describe nematically induced shape changes of large, floppy polymers and will calculate the equivalent of ∆m for molecular shapes. This type of macroscopic tracing of nematic order is powerful and very general. As we shall see, it absolves us from adopting any particular microscopic model. We shall often deal with nematic order viewed from a general coordinate frame, not simply along the director as in the above example of refractive index. In fact, nematic 1 The Legendre polynomials P (cos θ ) naturally describe orientations since they are the eigenfunctions of n the angular momentum operator (actually, its square) which is the generator of rotations. Dipolar order is described by P1 (cos θ ) = cos θ . Since equal numbers of rods in a nematic have an angle θ as π − θ , and since cos(π − θ ) = − cos θ , the dipolar order of a nematic vanishes, P1  = 0. P2 is the next function to try.

NEMATIC ORDER

13

z

q

n

x

f

u

y

F IG . 2.4. The coordinates of a rod used to define the order parameter tensor. order is tensorial in character and we have viewed in a principal frame where the director is along the z axis:   0 m⊥ 0 m =  0 m⊥ 0  ≡ Diag (m⊥ , m⊥ , mz ) . (2.2) 0 0 m (For brevity we shall often denote diagonal tensors by the ‘Diag’ form). For a general orientation of n we have for the refractive index tensor: mi j = m⊥ δi j + (m − m⊥ ) ni n j

(2.3)

The reader can confirm that if n points along one of the coordinate axes, z say, and hence ni = 0 unless i = z, then mzz = m , mxx = myy = m⊥ . If n is not along a coordinate axis, then ni = 0 for more than one i and there are off-diagonal elements to mi j . We have in effect rotated coordinate frames and the second rank matrix m has developed off-diagonal elements. The microscopic definition of the order parameter tensor is the analogous extension from the scalar Q. Let u be the unit vector describing the axis of the test rod in Fig. 2.2. Using the θ , φ coordinates of Fig. 2.4, the projections of the rod are uz = cos θ , ux = sin θ cos φ and uy = sin θ sin φ . The mean square projections are uz uz  = cos2 θ , ux ux  = sin2 θ cos2 φ , uy uy  = sin2 θ sin2 φ  and all other ui u j  with i = j vanish. Since we are interested in angular distributions rather than in the physical extent of an extended object, we have taken a unit vector, u , for which one has 1 = (uu)2 = (uu)2  = ux ux  + uy uy  + uz uz  = Tr (uuu ). The above averages certainly satisfy this identity. In fact the identity adds nothing to the content of the tensor ui u j , so we subtract out the spherical part 13 Tr (uuu ) δ ≡ 13 δ from the tensor uuu . Then the equivalent of eqn (2.3) is: (2.4) Qi j =  32 ui u j − 12 δi j  (see Fig. 2.4). One can check that Qzz is indeed the average P2 (cos θ ) = Q we defined before, if the coordinate axis z is chosen along n . The other elements ux ux  and uy uy  are related to uz uz  since the average over the free angle φ is trivial in the case of uniaxial order: cos2 φ  = sin2 φ  = 1/2. They can

14

LIQUID CRYSTALS

thus be written as ux ux  = uy uy  = (1 − cos2 θ )/2 = (1 − uz uz )/2 = (1 − Q)/3. For this orientation of n we have for the matrix representing the order parameter:   −Q/2 0 0 Q= 0 −Q/2 0  , (2.5) 0 0 Q while in general the order parameter is Qi j = Q

3

1 2 ni n j − 2 δi j



.

(2.6)

The order parameter tensor is, by construction, traceless and agrees with any of the macroscopic definitions of the ordering (for instance m ) if they too are made traceless.   Let the average value of the refractive index be m¯ = 13 Tr m = 13 (m + 2m⊥ ). Then the ˜ becomes new traceless m    1  0 0 −2 0 0 m⊥ − m¯ m ˜ = 0 0  ≡ 23 ∆m  0 − 12 0  ≡ 23 ∆mo Q . (2.7) m⊥ − m¯ 0 0 m − m¯ 0 0 1 The phase we have described is uniaxial. All angles φ in Fig. 2.4 are equivalent and, as we have seen, for such nematics macroscopic average quantities such as the refractive index take the same values in all directions perpendicular to n . Here this means mxx = myy = m⊥ . In optics, the distinguished direction (along the director n ) is called extraordinary (e) and the others ordinary (o). The refractive index tensor described by m governs the passage of the variously polarised light beams through the liquid crystal. It is known as the refractive index indicatrix and precisely mirrors the local nematic order parameter. When there is no symmetry about n , that is where all the perpendicular directions φ are not equivalent, then we have a biaxial fluid with a more complex order parameter (Stephen and Straley, 1974), Q . We shall discuss induced biaxial order in strained nematic elastomers in Sect. 6.6. Biaxial phases are not generally thermodynamically stable. They require rods that are far from circular, for instance with the symmetry of strips, or that the rods are so connected to each other so that their azimuthal freedom is restricted. Thus biaxial nematics have been found in molecules with complex structure (Chandrasekhar et al., 1988) and more directly in polymers (Leube and Finkelmann, 1991) for this reason. The combination of main chain and side chain ordering, with rodlike elements perpendicular to the main ordering direction possibly creating a secondary ordering direction in the perpendicular plane, has been suggested (Bladon et al., 1992). We shall not be interested in mechanisms of how spontaneously biaxial states arise, but intrinsically biaxial elastomers are a real possibility. We describe their soft elasticity in Appendix B. Whether induced or spontaneous, a more detailed description than that leading to eqn (2.4) is required to define the biaxial state. The elements ux ux  and uy uy  of the tensor uuu  can no longer be simplified by averaging directly over a free azimuthal rotation φ . In ux ux  = sin2 θ cos2 φ  we rearrange sin2 θ cos2 φ  = 12 sin2 θ (1+cos 2φ ) =

FREE ENERGY AND PHASE TRANSITIONS OF NEMATICS

15

2 2 2 1 1 1 3 (1 − Q) + 2 sin θ cos 2φ . Likewise uy uy  = sin θ sin φ  becomes 3 (1 − Q) − 2 1 Q 2 sin θ cos 2φ . Thus the x and y principal values of , that is Qxx and Qyy , are now distinguished. The vital quantity is the biaxial order b = 32 sin2 θ cos 2φ . It is clearly

zero if φ is free since cos 2φ  = 0. For perfect uniaxial order Q = 1 and there is no projection onto the perpendicular plane: sin θ = 0. Then the φ variation cannot be sensed and b again vanishes. Putting these values of uuu  into (2.4) for Q one obtains 

 −(Q − b)/2 0 0 Q= 0 −(Q + b)/2 0  . 0 0 Q 2.3

(2.8)

Free energy and phase transitions of nematics

In Fig. 2.3 we saw that an order parameter of Q = 1/2 represented a nematic with a moderate degree of typical alignment of rods. By contrast a state with Q = −1/2 is geometrically very different and physically very implausible in conventional nematics. The value P2 (cos θ ) = − 12 implies that all the rods would then be confined to the plane perpendicular to n . Both the van der Waals and the excluded volume contributions to the free energy would be most unfavourable. Thus a system free energy depending on the equilibrium order parameter Q must distinguish between states of ±Q, in contrast to magnetic (polar) systems where there is no distinction between positive and negative states. The general, Landau-de Gennes expansion of the free energy in powers of the full tensor order parameter, Q , is       Fnem = 13 ATr Q · Q − 49 BTr Q · Q · Q + 29 CTr Q · Q · Q · Q + . . .

(2.9)

where Fnem can thus extend to being a function of the biaxiality. Inserting the ordinary uniaxial Q , from eqn (2.5) or (2.6), into eqn (2.9) yields the usual free energy density expressed as a function of the scalar order parameter, Q. As an important consequence of nematic symmetry, the Landau expansion of the nematic free energy density contains odd powers of Q: (2.10) Fnem = 12 AQ2 − 13 BQ3 + 14 CQ4 + . . . − f Q The linear term − f Q represents the effect of an external field, for instance f = 12 δ

o E 2 where E is an applied electric field (which also has the effect of defining the direction of alignment, that is the director n) and where δ
∝ Q is the anisotropic part of the relative dielectric constant. Without an applied field, the free energy density expansion Fnem is schematically shown in Fig. 2.5. As the temperature is lowered, a metastable minimum with Q > 0 first appears at Tu . At Tni the absolute minimum at Q = 0 jumps discontinuously to an order Qm > 0. The transition is thus of the first order. The existence of coexisting isotropic and nematic states around this transition creates the possibility of thermal hysteresis. The nominal transition point Tni is defined where the two minima have equal depth. The second minimum at Qm = 0 exists in addition to the first at Q = 0 because of the − 13 BQ3 term, that is because of the need to distinguish between states

16

LIQUID CRYSTALS

Fnem

a

b

c

d

Q -0.2

0

0.2

0.4

0.6

0.8

1

F IG . 2.5. The Landau free energy density Fnem of a nematic liquid plotted against the scalar magnitude of order parameter Q. The plots correspond to characteristic points on the temperature scale: (a) the first appearance of Qm on cooling at T = Tu , (b) the transition point Tni , (c) at some temperature below Tni the order parameter Qm (shown by arrow) increases and the ordered phase has a lower free energy; (d) at the supercooling point T = T ∗ the disordered phase at Q = 0 is no longer a metastable state. of ±Q. This deep connection between the requirements of alignment geometry and the first order character of the phase transition was first recognised by Landau (Landau and Lifshitz, 1986). In fact nematics are only weakly first order. Their latent entropy at the transition is very small. The sign of this is the smallness of the coefficient B. Normally the vanishing of B yields a so-called critical point where a second order phase transition occurs, with the attendant critical divergence of many physical properties, for instance the specific heat and the correlation length of fluctuations. In nematics there is an incipient critical behaviour associated with a hidden second order transition at a temperature T ∗ just below Tni , see curve (d) of Fig. 2.5. The most sensitive temperature behaviour in the problem can encapsulated by writing A = Ao · (T − T ∗ ). The free energy density in the biaxial case is F = 12 AQ2 − 13 BQ3 + 14 CQ4 + . . .   + 12 A/3 + 2BQ/3 +CQ2 /3 + . . . b2 + . . .

(2.11)

The biaxial part is only expanded as far as b2 since the only explicit use of the Landau energy we will make is in evaluating the penalty for inducing b in an otherwise uniaxial phase, b = 0. Exercise 2.1: Analyse the characteristics of the first order nematic phase transition implicit in the Landau free energy density Fnem eqn (2.10). Solution: Thermodynamic equilibrium is achieved by adopting a minimum of Fnem (Q). One solution of equation ∂ Fnem /∂ Q = 0 always exists at Q = 0, representing an extremum, but not always a minimum. Another minimum appears as a solution of (A − BQ +CQ2 ) = 0, a simple quadratic for Qm (T ), that is

FREE ENERGY AND PHASE TRANSITIONS OF NEMATICS Qm (T ) =

1 2C

B±

17



B2 − 4A(T )C

when the expression under the square root is greater than zero. At the transition point Tni we also have a condition Fnem (Qm ) = 0 in addition to ∂ Fnem /∂ Q = 0. Combining these two conditions is a considerable simplification to a quadratic relation for Q:

1

4 Qm ∂ Fnem /∂ Q − Fnem (Qm ) Qm =Qni

=0.

The equilibrium order at the transition point is Qni = 3Ani /B, making a discontinuous jump from the state Q = 0. Inserted into Fnem (Qni ) = 0 this yields Ani = 2B2 /9C. Thus Tni = T ∗ +

2B2 and Qni = 2B/3C. 9CAo

(2.12)

The enthalpy of this transition is determined by the jump in system entropy between the states with Q = 0 and Q = Qni : ∆H = Tni ∆S = −Tni

2Ao B2 = − 21 Tni Ao Q2ni 9C2

(2.13)

per unit volume, on cooling to the nematic state. The entropy is negative in the nematic state since it is measured with respect to an arbitrary origin, S = 0, in the isotropic state. This first order phase transition is regarded as ‘weak’, when the coefficient B is small and both jumps in the order parameter and in the entropy are accordingly also small.

A full discussion of Landau theories applied to nematics, including their foundations, can be found in (Gramsbergen et al., 1986) and in (Hornreich, 1985). Landau descriptions of (albeit weakly) first order systems are of qualitative rather than quantitative significance. Strictly, the Landau free energy is an expansion of F(Q) for small Q, valid for second order systems close enough to the transition where the order parameter becomes indefinitely small. First order systems, such as nematics, have however a discontinuous jump to a finite order parameter. Qualitatively, however, the Landau energy does give the right behaviour. The exercise shows that Ao , T ∗ , B and C determine the transition and can be fixed from measurements of Qni , of Tni − T ∗ (from observing critical properties), of Tni and of the latent entropy. It is better to take this phenomenological approach to F(Q) here than to attribute any deeper significance to the coefficients. The minimum in Fnem at Q = 0 becomes a maximum at T = T ∗ (the quadratic coefficient A reverses sign). Thus T ∗ is the limit to supercooling of the isotropic state. The other minimum at Qm > 0 is lost when A(T ) > B2 /4C or equivalently when Tu = T ∗ + 14 (B2 /AoC), the limit to superheating of the nematic state, see Fig. 2.5. Superheating and supercooling are of course characteristic of a first order transition. Below the limit of supercooling T = T ∗ there is also a solution Qm < 0 shown by a dashed line in Fig. 2.6. It is only metastable since it is of higher energy than the main, Qm > 0, state. We shall see nematic objects with negative order parameters when we discuss

LIQUID CRYSTALS

Nematic order

18 1.0 0.8 0.6

Q ni

0.4 0.2 0.0

T*

-0.2 -0.4

T ni T

u

Temperature -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

F IG . 2.6. The order parameter Qm (T ) against temperature T . The dashed line shows the metastable solution Qm < 0, dual to the principal order parameter branch Qm corresponding to the deep minimum in Fnem in Fig. 2.5. The upper and lower limits of the transition Tu and T ∗ and the direction of hysteresis are shown by arrows; the zero-field transition point is at T = Tni . The subsequent plots show the evolution of Qm when an external field − f is applied: the paranematic phase at high temperatures becomes more pronounced as f increases, while the discontinuous jump of Qm becomes smaller and disappears at the critical point. liquid crystal polymers and certain elastomers. This oblate character is not entirely unphysical. The structural component we are concentrating on in this book on rubbers and elastomers is the polymer chain backbone, since it carries the elastic function. The backbone conformation is a slave to the main rod-like nematic component that does have a positive order parameter. The Q(T ) plot in Fig. 2.6 with no external field summarises the situation. A large number of experiments have been performed to investigate the nematicisotropic phase transition, which is often called the ‘clearing point’. The reason for this name will become clear in later sections of this book. There is a degeneracy and the fluctuations of nematic director n are very large at long wave lengths. Because n is also the axis of optical birefringence, the light is strongly scattered by its fluctuations in the nematic phase – and so the material appears turbid. In the isotropic phase there is no director, no birefringence, no significant scattering of light – and so the liquid is clear, transparent. The mentioned degeneracy, when long wavelength director fluctuations are not penalised by elastic energy, is removed in nematic elastomers. What can one say about the values of the three phenomenological parameters describing the transition, coefficients Ao , B and C? Obviously, different materials will have these parameters slightly different. Nevertheless, it is instructive to examine the characteristic orders of magnitude. In order to determine three parameters one needs three independent measurements: • The first measurement could be the jump of order parameter at the transition Qni . There might be some error in its determination, which should depend (among

FREE ENERGY AND PHASE TRANSITIONS OF NEMATICS

19

other factors) on the rate of cooling through the transition. Q can vary between 0 and 1. A number of molecular theories predict this jump to be in the range Qni  0.4, an illustrative value we adopt here. It gives B ≈ 0.6C from eqn (2.12). • The second measurement could then be the width of temperature hysteresis, or the interval between the transition and the limit to supercooling, T ∗ . In usual thermotropic nematic liquid crystals this interval is rather small. Let us, again, take qualitatively Tni −T ∗  1o K. This gives B2 ≈ (4.5CAo )×1K or B ≈ (7.5Ao )×1K. Note that the dimensions of B and C is energy density, J/m3 , while Ao has dimensions of J/(m3 K). • A third measurement would then be the enthalpy of transition ∆H. It is typically obtained from the differential scanning calorimetry by integrating the characteristic peak of the first order transition. Again, a large uncertainty may accompany such a measurement because at any non-zero cooling rate the conditions are not exactly equilibrium. However, keeping a qualitative, order of magnitude approach, the value of transition enthalpy per unit mass is usually around ∼ 1 − 2 J/g. The classical nematic material 4 -pentyl-4-cyanobiphenyl (5CB) has ∆H ∼ 1.5 J/g. [Compare this value with, for instance the enthalpy released during the typical (strong first order) melting transition in zinc, ∆H ∼ 106 J/g]. Taking density ρ = 1.5 g/cm3 and Tni  350K, we obtain (2Ao B2 /9C2 ) ∼ 104 J/(m3 K), or Ao ∼ 1.2 × 105

J J J . Then B ∼ 9.3 × 106 3 and C ∼ 5.6 × 106 3 . 3 m K m m

An external field, the − f Q contribution in eqn (2.10), can induce order at high temperatures (the paranematic state, in analogy to the effect of magnetic fields on spins at high temperatures). A linear term added to the curves of Fig. 2.5 shifts to higher Q both the minimum at Q = 0 and those at finite Q. The transition is also shifted to higher temperatures since the mimimum initially at finite Q is deepened (stabilised). The transition eventually disappears at a critical point ( fc , Tc ). The critical order parameter, Qc , is half the order parameter jump, Qni , at the zero-field transition. See the sequence of f = 0 curves in Fig. 2.6. Usually magnetic fields have a weak effect on the nematic transition. Electric fields are stronger, but seeing the critical point is still difficult (Hornreich, 1985). In contrast, mechanical fields can exert a powerful influence on the nematic behaviour of nematic elastomers, as is seen in Sect. 6.6. Indeed many nematic elastomers appear to be in a supercritical state from the internal stresses they suffer. They do not have discontinuities in their order parameter at any temperature. Exercise 2.2: Find the critical temperature and field for a nematic liquid crystal. Determine also how the transition temperature and order at the transition vary with applied field. Solution: At the critical point, the energy Fnem (2.10) must have the merging of all its maxima and minima (see the graphs of Fig. 2.5). This requires:  (Q) = AQ − BQ2 +CQ3 − f = 0 Fnem

 (Q) = A − 2BQ + 3CQ2 = 0 Fnem

20

LIQUID CRYSTALS  (Q) = −2B + 6CQ = 0 . Fnem

From the last of these we have Qcp = B/(3C). Putting Qcp into the second equation yields Acp = B2 /(3C) which, since A depends on temperature, gives us the critical temperature Tcp − T ∗ = B2 /(3CAo ). Returning Qcp and Acp to the first equation gives the critical field f cp = B3 /(27C2 ). The dependence of the transition order Qni with field comes from the first equa (Q) = 0, which is now a cubic rather than the simple equation we solved tion, Fnem in the field free case. We scale values of the order parameter at the transition √ by the critical value and try a solution to the cubic of the form Q = Qcp (1 ± 3 − τ ) where τ is to be determined but must contain the temperature dependence. Insert√ ing this trial into the first equation we have as a condition for the surds . . . to vanish: τ = 9AC/B2 . This is indeed a temperature shift relative to T ∗ of the transition in the presence of the field, reduced by the shift at zero field Tni (0) − T ∗ = 2B2 /(9CAo ), see eqn (2.12). With this combination we can express τ as twice the relative shift in Tni with f :

τ = 2(Tni ( f ) − T ∗ )/(Tni (0) − T ∗ ) . The remaining terms in the first equation vanish if τ B3 /(27C2 ) − 2B3 /(27C2 ) = f which can be rearranged into τ = 2 + f / f cp . (One must test that these solutions give the correct condition for the transition, namely that the free energies at the minimal values Q± are equal: Fnem (Q+ ) = Fnem (Q− ).) With no external field one has τ = 2 and hence the order at the transition with no field, cf. eqn (2.12), is either Qni ( f = 0) = 2Qcp or Qni = 0. Thus in the Landau picture the critical order parameter is 1/2 the zero field jump, Qc = 21 Qni .

Nematic elastomers are much more complex than simple nematics. However, they possess the same uniaxial quadrupolar symmetry. ‘Up’ and ‘down’ are not distinguished for its director (nn → −nn). Thus Landau theory, which is based on symmetry considerations, tells us that their thermal properties will be qualitatively the same as those of simple nematics. We shall find that the free energy of nematic elastomers have a critical temperature T ∗ and the coefficient C modified from the values taken by the corresponding nematic polymer (uncrosslinked) melt. The modification depends on the network’s thermal and mechanical history. Networks are solids and thus the director can resist aligning along external fields, most drastically a mechanical (stress) field, even in the absence of anchoring at the boundaries. In simple nematics, where the orientation of the director n is readjusted without resistance, ignoring anchoring at surfaces and other boundary effects for the moment, the external field (generally E or B ) sets this orientation. We shall see later, in Sect. 6.6, the collapse and enhancement of the magnitude of the nematic order by the application of external fields coaxially with the director, or perpendicularly but without director rotation. 2.4

Molecular theory of nematics

We have seen that the form of the Landau-de Gennes free energy established the character of the nematic’s thermal behaviour: the Q3 gives a first order phase transition and T ∗ basically sets the transition temperature. Molecular theory gives some insight into

MOLECULAR THEORY OF NEMATICS

21

what sets T ∗ and into how the other Q terms arise. We shall need a molecular theory to model elastomers near their transition and to model their photo response. The simplest, mean field, nematic model of thermotropic nematics is due to Maier and Saupe (MS) and is the quadrupolar equivalent of Curie theory for magnetic (dipolar) systems (Maier and Saupe, 1959). By approximating the P2 (cos θi j ) interaction between a pair (i, j) of rods as that of a test rod interacting with a molecular field of all the others represented as a mean field, one obtains for the mean field potential U(θ ): U(θi ) = −∑ Jo P2 (cos θi j ) j ≈ −JQP2 (cos θi ) . j

Jo is the strength of the anisotropic part of the van der Waals interaction and J is that considering the total, average effect of all the partners j, see also the discussion of eqn. (2.19) below. The label i will be dropped since all the rods are assumed equivalent to the test rod when they are in the mean field. The negative sign ensures that rods want to be parallel or anti parallel to each other, see Figs. 2.2 and 2.3. The distribution of rod orientations, p(θ ), is   JQP2 (cos θ ) 1 p(θ ) = exp (2.14) Z kB T where Z is the partition function, Z = ises p(θ ). The free energy is

π 0

d θ sin θ exp(JQP2 (cos θ )/kB T ), and normal-

FMS = −kB T ln Z + 12 JQ2 .

(2.15)

The second term, 12 JQ2 , corrects for the double counting [(i, j) and ( j, i)] arising in replacing the pair interaction Ui j with the single particle, mean field U. As yet Q is undetermined. Taking ∂ FMS /∂ Q = 0 yields 1 ∂Z or Z ∂Q  π kB T 1 ∂ Z Q= ≡ d θ sin θ p(θ )P2 (cos θ ) . J Z ∂Q 0 0 = JQ − kB T

(2.16) (2.17)

The latter result is what one expects as a definition of Q, giving confidence in the mean field correction in FMS . Alternatively one can view the minimisation as a self consistency problem: adopting eqn (2.17) as a calculation of Q, one finds a consistency problem since p(θ ), while determining Q in eqn (2.17), is itself a function of Q, see eqn (2.14). The solution of eqn (2.17) yields an equilibrium Q(T ) qualitatively like the zero field curves in Fig. 2.6. The transition is at a temperature Tni = J/(4.541kB ) where the Q = 0 and the Q = 0 solutions are of the same free energy. This is close to the T ∗ for this model. To make contact with Landau theory, one can instead expand FMS in powers of Q up to the fourth order and find qualitatively the same form as eqn (2.10). Explicitly, the Q2 term gives

22

LIQUID CRYSTALS

T∗ =

J . 5kB

(2.18)

Thus T ∗ in the Maier-Saupe model is set essentially by the interactions between the rods. In writing this form, one has taken Ao = J/T which has a weak T dependence that we ignore, as we also do in B and C. We shall need eqn (2.18), relating T ∗ with J, and in turn how J is modified by, say, dilution of the rods by solvent or by their being photoisomerised. The simplest model possible is offered below: recall that J derives from a mean field correction to a representation of pair interactions and would thus be expected to depend on the volume fraction, φ , of rods presumably like J ∼ zφ 2 Jo

(2.19)

where z is the number of neighbours interacting, with strength Jo , with the test rod. Another example of such a dilution is the depletion of rods when they bend on absorbing a photon and do not contribute to the nematic interaction any more. Such photoelastomers are discussed in sections 5.4.2 and 6.4. 2.5

Distortions of nematic order

Ignoring the effect of boundaries, the free energy of a nematic fluid is degenerate with respect to the direction of n , which can be swung around by an infinitesimal guiding field, either E or B . However, since nematic fluids have long range directional order, there is a penalty associated with spatially varying the director, n (rr ). We stress both aspects in this section. In the next section we show that the loss of this degeneracy is central in nematic elastomers. Non-uniform directors will figure in a novel way later when we discuss instabilities in nematic elastomers. The director can be splayed, twisted or bent, the penalty for such distortions being the celebrated Frank elastic free energy density: FFr = 12 K1 (div n )2 + 12 K2 (nn · curl n )2 + 12 K3 (nn × curl n )2

(2.20)

with the Ki being the corresponding splay, twist and bend curvature elastic constants respectively. Deriving this expression (de Gennes and Prost, 1994) requires care that it obeys all symmetry requirements, the most obvious being invariance with respect to n → −nn. With the simplification K2 = K3 = K, the Frank free energy density FFr reduces to 2 2 1 1 2 K1 (div n ) + 2 K(curl n ) , apparently only quadratic in n , but even then this free energy is not at all innocent. The requirement that n (rr ) remains a unit vector, n 2 (rr ) = 1 at all points r , means that FFr yields highly non-linear problems. Only few problems can be solved exactly; these special cases are catalogued in a series of beautiful and instructive illustrations (de Gennes and Prost, 1994).

DISTORTIONS OF NEMATIC ORDER

23

F IG . 2.7. Three principal distortion modes in Frank elasticity: splay (a), twist (b), and bend (c) of the director n .

F IG . 2.8. Illustration of twist and bend distortions in nematic liquid crystals. Vectors show orientations of the director at one point in space, n(rr ), and a distance δ r away, the small change δ n and the direction of local curl n. For twist (a) deformation one obtains n  curl n, for bend (b) n ⊥ curl n. Since n is a unit vector, the small change δ n (rr ) in its orientation on going to a different point in space r + δ r , must be perpendicular to n at linear order. That is, if n (rr + δ r ) = n (rr ) + δ n (rr ), then 1 = (nn + δ n )2 = 1 + 2(nn · δ n ) + (δ n )2

(2.21)

(nn · δ n ) = − 12 (δ n )2 → 0 .

(2.22)

and hence Splay is a rather obvious distortion, created by moving glass sheets apart as indicated in Fig. 2.7(a) with the director anchored to the surface. Twist can be created by rotating anchoring surfaces relatively about a common axis, Fig. 2.7(b), and bend by splaying anchoring surfaces, but with a director anchored perpendicular to the surfaces. Twist and bend can be identified in Figs. 2.7(b) and (c), and in greater detail in Figs. 2.8(a) and (b) which explain the vectorial character of the corresponding terms in the elastic energy density FFr (2.20). The cross product ∇ × δ n , between the gradient (that is in the direction of spatial variation δ r ) and the variation δ n , is parallel to n for twist and perpendicular to n for bend deformation, as illustrated Figs. 2.8(a) and (b) respectively. This identifies the respective twist and bend terms in (2.20) since curl n = ∇ × δ n is either dotted or crossed with n , respectively.

24

LIQUID CRYSTALS

It is often important to know, at least approximately, the magnitude of elastic constants K1 , K2 and K3 without performing a special experiment (see Sect. 2.6 for examples of such experiments). A crude estimate of the order of magnitude recognises that these constants have dimensions of energy per unit length (Joules per meter, or Newtons, in SI). A characteristic energy scale of a thermodynamically equilibrium liquid phase is ∼ kB T per molecule and the only relevant length scale is the length of rod-like molecules. Hence, at room temperature (kB T ∼ 4 × 10−21 J) a typical nematic liquid crystal (with length a ∼ 1 − 2 nm) will have Ki ∼ 2 − 4 × 10−12 N, or perhaps a few times greater if stronger attractive van der Waals forces between molecules are taken into account. There are several successful molecular theories of curvature elasticity in nematic liquid crystals, obtaining the three Frank constants in terms of molecular parameters (length a and thickness t of the rod, strength of intermolecular pair interaction and the degree of nematic order Q). A purely entropic model of Straley (Straley, 1973), considering rigid rods aligned due to the anisotropic excluded volume effect, gives K1 ≈ 0.06ρ 2 (a4t)kB T, K2 ≈ 0.02ρ 2 (a4t)kB T and K3 ≈ 0.2ρ 2 (a4t)kB T. Here ρ is the number density of the liquid [note that the maximal density at close packing is ρ ∼ 1/(at 2 )]. A comprehensive theory for a liquid of spherocylinders includes both anisotropic steric repulsion at contact and long-range van der Waals attraction pair interactions (Gelbart and Ben-Shaul, 1982). Among a rich set of results and conclusions, theory shows the Frank constants to be proportional to the square of the nematic order parameter Q. In fact, the proportionality Ki ∝ Q2 can be deduced directly from the phenomenological Landau expansion of the free energy, an extension of eqn (2.10) to the case when the director n is non-uniform. In general, when the variable describing the ordered phase is fluctuating in space, the free energy density should also contain the gradient penalty of the form 12 g(∇Q)2 . This is the lowest order of such a correction in the series expansion, both in powers of Q and ∇: the linear gradient term cannot exist because it would lead to a non-uniform (modulated) phase in equilibrium. A few special cases when such phases occur do indeed have a linear-gradient term in their free energy density – for example cholesteric liquid crystals, see Chapter 9. Because the tensor order parameter Qi j given by eqn (2.6) is the proper field variable in the nematic phase, the Landau free energy density of the fluctuating nematic liquid crystal should be written as   2    Fnem = 13 A Tr Q · Q − 49 B Tr Q · Q · Q + 19 C Tr Q · Q + 29 κ1 (∇ j Qi j )2 + 19 κ3 (∇k Qi j )2 .

(2.23)

The two gradient terms, written here in the matrix form using the convention of summation over repeated pairs of indices, are the only possible scalar invariants that can appear in the expression for the local free energy. Much attention has been put in recent years into investigations of possible extensions and generalisations of eqn (2.23). We only wish to illustrate the following point. Suppose the scalar magnitude of order parameter

TRANSITIONS DRIVEN BY EXTERNAL FIELDS

25

Q is constant, i.e. it is an equilibrium function of temperature as shown in Fig. 2.6, but not a function of coordinates in the sample. Then, for instance, ∇ j Qi j = 32 Q ni div n . Performing the corresponding matrix manipulations and integrating by parts to eliminate surface terms, one obtains from eqn (2.23): Fnem = 12 A Q2 − 13 B Q3 + 14 C Q4 + 12 (κ1 + κ3 )Q2 (div n )2 + 12 κ3 Q2 (curl n )2

(2.24) .

In terms of the Frank elasticity (2.20), one has K1 ≡ (κ1 + κ3 )Q2 , K2 = K3 ≡ κ3 Q2 . Two conclusions follow: at leading, quadratic order in the order parameter Q, the bend and twist constants are not distinguishable (i.e. the difference K3 − K2 should be proportional to Q4 or other small corrections). The correlation length of fluctuations of order parameter could be interpreted as the size of the volume within which the molecules fluctuate coherently and one can define the local macroscopic value of order parameter and the average direction of n . This important length scale is often called the nematic ξN and is given by the square root of the ratio of the uniform and gradicoherence length ent terms, ξN = κ /A. Taking the estimated values, one obtains ξN ∼ 5 − 10 nm. This length is the so-called cutoff of the macroscopic continuum description (2.24): the concepts of thermodynamic phase, equilibrium order of rod like molecules, their average director, are not applicable at distances below ξN . 2.6

Transitions driven by external fields

The Freedericks transitions are the classical examples of director distortion resulting from the competition of surface anchoring with bulk rotation induced by a field in an incompatible direction. A full discussion of the various types of this transition (de Gennes and Prost, 1994; Chandrasekhar, 1977) shows their utility in determining the values of Frank constants Ki and their exploitation in display devices. We sketch the simplest transition here because we shall later see that nematic elastomers are profoundly different. They display electrical and mechanical ‘anti-Freedericks transitions’ of a diametrically opposite character to those in liquid nematics. Without boundary constraints a nematic’s alignment direction will rotate freely since its energy is degenerate with n. For instance, n follows the imposed electric field E as it rotates in the yz or xy-plane of Figs. 2.9(a) and (b) respectively. However, if the surfaces of cell boundaries at z = 0 and z = L anchor the director in the y direction, n(z = 0) = n(z = L) = y, then in the absence of a field the director is everywhere aligned along the y-axis. To be otherwise would require a director distortion and its attendant Frank elastic energy. If an electric field is applied perpendicularly to this preferred direction, the nematic has to compromise between two effects. On the one hand, the dielectric energy can be reduced by having n parallel to E over the greatest possible volume (including near the surfaces). On the other hand, the Frank elastic energy is reduced by deforming n (rr ) as slowly as possible (from along y at the boundaries to a perpendicular direction in the bulk). Magnetic fields have an analogous effect and are

26

LIQUID CRYSTALS

F IG . 2.9. The geometry of the splay-bend and the twist Freedericks transitions. In the splay-bend case (a) the director components are given by n = {0, cos θ , sin θ } with the electric field along z ; the twist case (b) is described by n = {− sin φ , cos φ , 0} with the electric field along −xx easier to consider because, in fact, the electric polarisation generates an internal field , which may be different from that applied. We neglect this modification in the illustrations of director distortion that follow. Using a magnetic field instead of E would render the analysis exact. Depending on the orientation of the electric field, Fig. 2.9, the director deforms in a splay-bend or in a twist fashion and, therefore, different Frank constants control its response. In the splay-bend geometry n = {0, cos θ , sin θ }, the contributors to the Frank energy (2.20) are (div n )2 (splay) and (nn × curl n )2 (bend). In the case of twist deformation, the y-component of curl n survives and the Frank free energy density has only the (nn · curl n )2 contribution. The total free energies for the two cells are: Fsplay−bend = A Ftwist = A

 L

dz

1

dz

1

0

 L 0

2 2 2 1 1 n E 2 K1 (div n ) + 2 K3 (n × curl n ) − 2 δ

o (E · n ) 2 2 1 n E 2 K2 (n · curl n ) − 2 δ

o (E · n )



,



(2.25) (2.26)

where A is the area of the plates and the dielectric anisotropy δ
is defined below eqn (2.10). Minimising FFr over variation of the director angle with z is an EulerLagrange problem with a constraint that n 2 = 1. Keeping n a unit vector is easily ensured by writing the director components in terms of sin and cos, see Fig. 2.9. We have already seen that electric fields can (i) alter the magnitude Q of the nematic order (exercise 2.2) as well as (ii) redirect it, that is rotate n . A third, more subtle, possibility is flexoelectricity (Meyer, 1969) where E can induce a non-uniform director field, or conversely a locally non-uniform director field can induce a polarisation P = e1 n div n + e2 [nn × curl n ]. These effects are discussed at length in monographs (de Gennes and Prost, 1994; Pikin, 1991). In this book, non-uniform directors are mostly encountered in cholesterics, see Chapter 9. Exercise 2.5 shows how polarisation arises via flexoelectricity when a cholesteric elastomer is distorted. Twist Freedericks transition Cholesteric elastomers are sometimes influenced by Frank twist energy so it is useful to explore this for liquids. In the twist geometry we have for the curl of the director, the

TRANSITIONS DRIVEN BY EXTERNAL FIELDS

27

F IG . 2.10. Splay-bend director distortions in a half-space anchored on the boundary at z = 0. A external field, E , competing with the Frank elastic penalty near the anchoring wall defines a length scale – the penetration depth ζ . twist and the total free energy respectively: curl n = φ  (sin φ , − cos φ , 0) n · curl n = −φ  (sin2 φ + cos2 φ ) = −φ   L   Ftwist = A dz 12 K2 φ 2 − 12 δ

o E 2 sin2 φ .

(2.27)

0

Below a critical field Ec = (1/L)(π K2 /
o δ
)1/2 , the director remains anchored perpendicular to E everywhere (de Gennes and Prost, 1994). For larger fields, a distortion away from n o parallel to y grows continuously, the deviations initially being δ n x ≈ sin(π z/L). Since the potential is V = EL, clearly the potential for beginning distortion is  Vc = Ec L = π K2 /
o δ
. (2.28) This critical voltage depends only on material parameters and is independent of sample thicknesses. The transition is characterised by a voltage since the thicker the sample, the weaker the Frank distortion is, thereby requiring a weaker field. A weaker field over a thicker sample yields the same potential difference. We will see that Freedericks effects in solid nematics are entirely different - this cancelling length scaling is lost. The analogous construction for the splay-bend geometry produces a threshold depending on a weighted average of K1 and K3 instead of on K2 . To summarise, a bulk field E , resisted by the rigid anchoring on cell boundaries, eventually yields a non-uniform state with an energy balance (Frank vs. dielectric) Ki (π /L)2 
o δ
E 2 . Splay-bend in a half space A simple situation illustrating this balance between free energy contributions of different origin and the emerging characteristic length scale, is that of a half-space bounded by an anchoring surface – essentially half of the Freedericks set-up, see Fig. 2.10. In a half space it is convenient to choose a different origin of rotations of n . We define the angle between the director and the electric field to be ω ; in most of space well away from z = 0 one has ω ∼ 0. There is rigid anchoring along y at z = 0, that is ω (z = 0) = π /2. Now we have n = (0, sin ω , cos ω ), with splay, curl and bend, and the overall Frank free energy density given respectively by: divnn = −ω  sin ω

28

LIQUID CRYSTALS

curl n = −ω  (cos ω , 0, 0) n × curl n = ω  (0, − cos2 ω , sin ω cos ω ) FFr = 12 K1 ω 2 sin2 ω − 12 K3 ω 2 cos2 ω → 12 K ω 2 where we have set K1 = K3 = K, a single constant approximation, to simplify algebra. One now sees why the splay-bend Freedericks transition rests upon a mixture of the splay and bend constants – near the wall sin ω ∼ 1 and splay dominates, in the bulk cos ω ∼ 1 and bend dominates, although far from the wall the variation of ω is slow (ω  → 0) and neither are significant. The Euler-Lagrange equation for the minimum free energy, eqn (2.25) for this splay geometry, reads: ζ 2 ω  = cos ω sin ω , (2.29) where the characteristic length is ζ = (K1 /
o δ
E 2 )1/2 . This length scale is where the electrical and Frank energy densities match: K1 (∇nn)2 ∼ K1 /ζ 2 =
o δ
E 2 .

(2.30)

It is a measure of how deeply the direction of order in the surface penetrates the bulk; ζ is a “penetration depth”. The reader can confirm that a solution to the non-linear eqn (2.29), satisfying ω (0) = π /2 and ω (∞) = 0 is tan

1

2 ω (z)

= e−z/ζ ,

with ω ≈ π /2 − 2z/ζ at z  ζ .

How will these non-uniform phenomena be found in elastomeric nematics? Director rotation with respect to the solid matrix of an elastomer is argued in Chapter 1, page 4, to cost an energy density ∼ µ Q2 . Distorting the director in a polymeric nematic costs the usual Frank energy and this effect is also present in crosslinked systems. It influences the response to distortions in solid nematics if the director variation is fast enough. Analogously to eqn (2.30) for the length ζ , a new length, ξ the nematic penetration depth, emerges for nematic elastomers: variation over this length scale, that is ∇ ∼ 1/ξ , gives a Frank energy density comparable to the nematic rubber elastic cost of rotations. Thus K(∇nn)2 ∼ K/ξ 2 ∼ µ Q2 and hence ξ = K/µ Q2 . Because mechanical fields are so much stronger than typical electrical fields (µ 
o δ
E 2 ), this penetration depth is much smaller than ζ , in fact ξ ∼ 10−8 m. With strain fields and a memory of the original director, nematic elastomers turn out to have many more than the three Frank elastic distortion energies governing director variation (Terentjev et al., 1996). Exercise 2.3: Estimate the electrical and Frank energies of the anchored nematic half-space problem. Solution: These energies are equal in the optimal case given above (by minimising the free energy the system adopts a configuration when the two competing

ANISOTROPIC VISCOSITY AND DISSIPATION

29

contributions are in balance). We can evaluate either one, for instance the dielectric energy: F =A

 ∞ 0

 dz

2 2 1 2 δ

o E sin ω

= A 21 δ

o E 2

 ∞ 0

dz



4t 2 (1 + t 2 )2

(2.31) 

= 12 A

δ

o E 2 K2 

(the integral being done trivially by the substitution z = −ζ lnt and yielding dz(. . .) = ζ ). The resultant energy is as expected in such problems – the geometric mean of the two separate energy scales.

2.7

Anisotropic viscosity and dissipation

The dynamics of anisotropic fluids, such as nematic liquid crystals, displays macroscopic phenomena on a continuum level analogous to that of the Frank elastic free energy (2.20). By movements of a ‘liquid particle’ one understands the translations and changes of local director orientation of the physically infinitesimal volume, generally of the correlation size ξN ∼ 10nm, including many molecules in thermal equilibrium with the reservoir, and characterised by the local nematic tensor order parameter Qi j (T ) = Q(ni n j − 13 δi j ). If one assumes the constant magnitude of Q, two physical fields describe the state of motion of nematic liquid, the fluid velocity v(r,t) and the variation of director orientation δ n (r,t) = n − n 0 with respect to the equilibrium n 0 . The original derivation of hydrodynamic equations for the nematic liquid is due to Ericksen (Ericksen, 1961) and Leslie (Leslie, 1968) and the theory has their names. The modern understanding of Leslie-Ericksen nematohydrodynamics is presented in some detail in key monographs on liquid crystals (de Gennes and Prost, 1994; Chandrasekhar, 1977). In this review chapter we only discuss the matters relevant for the further examination of nematic elastomers, omitting many fine and subtle points of this complicated subject. The equation of fluid motion, usually called the Navier-Stokes equation, can be written in the usual form of local balance of forces – in vector components

∂  ∂ Πki ρ vi = − , ∂t ∂ xk

(2.32)

where ρ is the density of liquid and Πki the tensor of momentum flux density. In a viscous fluid Πki = ρ vk vi − σki + P δki with P the local pressure and σki the so-called viscous stress tensor. In an isotropic liquid σki is proportional to the symmetric strain rate, Aki :  ∂ vi ∂ vk , σki = 2η Aki ≡ η + ∂ xk ∂ xi with η the viscosity coefficient . In nematic liquid crystals, with their intrinsic uniaxial anisotropy, the stress tensor must depend not only on the fluid velocity gradients, but also on the components of local nematic order Qki , cf. eqn (2.6), and its gradients. Rotations and corresponding torques may arise if the director changes orientation at a

30

LIQUID CRYSTALS

˙ = 1 curl v . The stress different rate from the rate of local fluid rotation, denoted by Ω 2 ˙ × n] that tensor should also depend on the antisymmetric combinations N = dtd n − [Ω describe this difference. As a result of all such considerations, one obtains the viscous stress tensor of the uniaxial nematic liquid as

σi j = α1 ni n j Akm nk nm + α2 ni N j + α3 n j Ni +α4 Ai j + α5 ni nk Ak j + α6 n j nk Aki

(2.33)

(assuming an incompressible fluid, Akk = div v = 0). Here the 6 viscosity coefficients α1 , ...α6 (called the Leslie coefficients) all depend on the magnitude of nematic order parameter Q(T ). A relation we discuss below reduces their effective number to 5. Chapter 11 ascribes a geometrical meaning to the rather impenetrable terms of eqn (2.33). They can be made analogous to the expansion of the elastic free energy in terms of strains (rather than strain rates) that concern us in nematic elastomers. This will make natural why there are only 5 effective coefficients. One can show that near the weak first order transition, as Q → 0, they behave as (Imura and Okano, 1972)

α1 ∝ Q2 , (α2 , α3 , α5 , α6 ) ∝ Q. Thus, in the isotropic phase, only one of the Leslie coefficients survives: α4 → 2η . Molecular theory (Kuzuu and Doi, 1983; Osipov and Terentjev, 1989) also shows that in a typical nematic liquid of rod-like molecules these coefficients may have very different magnitudes and even signs: α2 , α3 , α6 are negative, α4 , α5 are positive. The magnitudes are typically |α2 | ∼ α5  |α3 | ∼ |α6 |. The constant α1 is generally small and may be positive or negative in different materials. Far from the nematic transition as Q → 1, the ‘isotropic coefficient’ α4 ∼ |α2 | ∼ α5 . The experimental values for Leslie coefficients, and for the γ coefficients defined below, for the classical nematics MBBA (de Gennes and Prost, 1994) and PAA (de Jeu, 1980) are representative: Viscosity coefficients (10−3 Pa · s) α1 α2 α3 α4 α5 α6 γ1 γ2 MBBA at 25oC 6.5 −77.5 −1.2 83.2 46.3 −34.4 76.3 −78.7 −2 6.8 −7.1 PAA at 122oC 4 −6.9 −0.2 6.8 5 In a nematic liquid crystal one also needs a dynamic equation describing the evolution of the director. By analogy with eqn (2.32) for the balance of forces, one writes the balance of torques   dnn d n× = [nn × h ] − [nn × R], (2.34) I dt dt where the left hand side is the variation of angular momentum, with I the inertia moment density. Such inertial effects are often neglected in laminar flow problems. The dissipative driving force that induces the director rotation in the liquid flow is R: Ri = γ1 Ni + γ2 n j A ji . The corresponding mechanical torque density is Γ = [nn × R].

(2.35)

ANISOTROPIC VISCOSITY AND DISSIPATION

31

The nematic molecular field, h , causes the director to have the orientation n 0 in equilibrium. At equilibrium this vector is obviously parallel to n 0 , h = ψ (rr )nn0 . With distortions, it is determined by varying the total Frank free energy to obtain a minimal energy to the cost of director non-uniformity. This variational problem reduces to the Euler-Lagrange equation in terms of the Frank free energy density:  ∂ ∂ FFr ∂ FFr − ∼ K∇2 n . hi = ∂ xk ∂ [∇k ni ] ∂ ni The balance of torques in the absence of inertial effects is then expressed by the vanishing of the right hand side of eqn (2.34), in effect h = R. The neglect of complicated inertial effects in the classical fluid dynamics is controlled by the small magnitude of Reynolds number, expressing the relative strength of inertial and viscous forces, Re = ρ v /η  1, with  the characteristic length of spatial variation of v (rr ) and η the typical viscosity. In nematic liquids one also needs to examine the moment of inertia term I n¨ ∼ I ω 2 , where here ω is a frequency characterising the motion of the fluid. We must compare this torque density with the elastic restoring torque density ∼ K/2 . The moment of inertia density is of order ρ a2 , where a is the molecular dimension. The ratio of the two torques is, therefore, ρ a2 ω 2 2 /K. There are two characteristic frequencies in the dynamics of nematic liquid (Stephen and Straley, 1974), the fast mode (corresponding to shear acoustic waves in a fluid, ωF ∼ η /ρ 2 ) and the slow hydrodynamic mode (describing the dynamics of nematic director, ωS ∼ K/η 2 ). In the hydrodynamic regime, the ratio of torques is estimated as  2 ρ K a2 −4 a ∼ 10 η 2 2 2 and is thus always negligible. Another important dimensional number is introduced to characterise the relative magnitude of viscous and Frank elastic torques contributing to eqn (2.35), the Ericksen number Er = η v /K. The small Ericksen number Er  1 means that the director orientation is mostly controlled by equilibrium free energy effects, while at Er  1 the director generally follows the local orientation provided by the viscous flow: the right hand side of eqn (2.35). Two differential equations, (2.32) and (2.34), with the viscous stress and torque given by eqns (2.33) and (2.35), form the complete set describing the linear viscous effects in the nematic fluid. Neglecting the effects of heat convection, the total energy dissipation (often called the entropy production, the rate of doing work against viscous effects) in such a fluid is expressed by the volume integral of the quantities defined above:     ˙ ≡ dV (σi j Ai j + h · N ) . (2.36) T S˙ = dV σi j Ai j + h · n˙ − Γ · Ω Thermodynamic equilibrium and the reciprocity of linear response functions demand certain exact relations between the coefficients:

γ1 = α3 − α2 , γ2 = α6 − α5 = α3 + α2 , the last equality being called the Parodi relation.

32

LIQUID CRYSTALS Exercise 2.4: Estimate the relaxation times for the Freedericks effect. Solution: Let us return to the splay geometry of the Freedericks effect, Fig. 2.9(a) and the equilibrium free energy eqn (2.25) describing the director deformations caused by external field. Let us assume that no fluid flow occurs, that is the fluid velocity v (rr ) = 0 at all times. The director, with components {0, cos θ (z,t), sin θ (z,t)}, is the only dynamical variable in this problem. It satisfies the boundary condition θ = 0 at z = 0, L. The local free energy density is F = 12 K1 cos2 θ (θ  )2 − 1 2 2 2 δ

o E sin θ . Neglecting all inertial effects and setting v = 0, we find that the equations for the force and torque balance reduce to:

dP d d = σzz = (α2 + α3 )θ˙ sin θ cos θ dz dz dz  d ∂ FFr ∂ FFr d − γ1 ni = hi = . dt dz ∂ [∇z ni ] ∂ ni

(2.37)

The first of these equations allows to determine the local pressure, which we are not interested in at present, while the second determines the relaxation process in the electrooptic cell, Fig. 2.9. It has a generic form of the linear kinetic equation describing an overdamped system approaching its equilibrium (Landau and Lifshitz, 1981), in our case:

γ1

d d sin θ = δ

o E 2 sin θ + K1 dt dz γ1 θ˙ ≈ δ

o E 2 θ + K1 θ  ,



d sin θ dz

 (2.38) (2.39)

the last equation being the limit of small deformations θ  1, near the Freedericks threshold. Crudely assuming uniform director rotation (essentially, the limit of a very thick cell), the kinetic equation (2.39) reduces to γ1 θ˙ ≈ δ

o E 2 θ and one identifies the characteristic ‘switching-on’ (or ‘field-on’) relaxation time τE . When the field is switched off, it is the remaining Frank elastic term in eqn (2.39) that drives the director back to its equilibrium orientation with θ = 0 provided by the boundary conditions, with the corresponding ‘field-off’ relaxation time τK . Thus

τE =

γ1 ; δ

o E 2

τK =

γ1 L2 . π 2 K1

(2.40)

Performing the analysis near the Freedericks threshold more accurately, i.e. assuming the director profile θ = θ0 (t) sin(π z/L), we obtain the condition for small amplitude of modulation

τK θ˙0 =

  δ

o E 2 L 2 V − 1 θ ≡ − 1 θ0 , 0 Vc π 2 K1

(2.41)

with Vc the Freedericks threshold voltage, cf. eqn (2.27). Thus the small amplitude of director rotation above the threshold evolves with time according to the exponential

CHOLESTERIC LIQUID CRYSTALS

θ0 (t) ≈ exp

33

      π 2K t 1 1 ≡ exp t , δ

o E 2 sin θ − 2 1 − γ1 τE τK L

(2.42)

giving another, dynamic, way of estimating the Freedericks threshold: at τE ≤ τK the director starts rotating, initially with an exponential increase of θ0 (t).

2.8 Cholesteric liquid crystals Cholesteric phases arise when a nematic system of aligned rods is modified due to the molecules being chiral. The rods possess a handedness and the fluid is now noncentrosymmetric at the local level. It typically displays macroscopic chiral structures – the director is induced to twist spatially and adopts a helical distribution (Straley, 1976; van der Meer et al., 1976). By contrast non-chiral nematic liquids pay an energy penalty for distortions of their director away from the uniform state, the Frank elasticity of eqn (2.20); see in particular the twist in Fig. 2.7. For chiral nematics however, there is another, lower-order invariant in the gradient expansion (2.23) of the free energy, a non-centrosymmetric scalar combination: 4 9λ

Qi j
jkl ∇k Qli = λ Q2 (nn · curl n )

(2.43)

2 1 n 2 K2 (n · curl n + qo )

(2.44)

Ftwist →

This chiral linear gradient addition to the Frank free energy density is the cross term in the modified twist contribution to the free energy density, Ftwist . By identification, qo = λ Q2 /K2 and there is also an arbitrary and irrelevant additive constant 12 K2 q2o to the free energy density. The linear term induces (de Gennes and Prost, 1994) the cholesteric liquid crystal’s deformed director field: the twist of Figs. 2.7(b) and 2.8(a) is the state with the lowest elastic energy, at a value −qo for the twist (nn · curl n ). Such a twisted state is shown in Fig. 2.11 where the cholesteric helix is along the z axis. No splay or bend of the director is involved. The chiral coefficient λ is a pseudo scalar, that is, it changes sign on inversion. (Thus in a non-chiral material, where an inversion leads to an equivalent state, λ = −λ = 0.) Further remarks about the importance of pseudo scalars and vectors are made below eqn (2.52) on page 36. The azimuthal angle the local director within the xy-plane makes with x in equilibrium is, before any possible distortions of the director,

φo (z) = qo z

(2.45)

y f

z

O

x

p/q

O

F IG . 2.11. The helical distribution of director in a classical cholesteric liquid crystal.

34

LIQUID CRYSTALS

and thus qo is the chiral pitch wavenumber. The corresponding helical pitch is po = 2π /qo . However, recall that n ≡ −nn and so the true repeat distance is only po /2. One can denote the pitch direction by the unit vector pˆ . Note that pˆ ≡ − pˆ since the helical screw is unchanged by its being bodily rotated by 180o about any axis perpendicular to pˆ . Thus the pitch direction pˆ can be reversed, but as with any truly chiral object, the helix itself is only inverted by reflection – an important fact when we consider piezoelectric response. Cholesterics are essentially twisted nematics and are known from high precision optical studies (Berreman and Scheffer, 1970) to be locally practically uniaxial about the local director, n(rr ). In principle though they must be biaxial since three unique directions can be defined: n, pˆ and n × pˆ . But deviations from local cylindrical symmetry are small: the pitch, po , is typically ∼ 500-1000 times greater than the local length scale determining rod interactions and packing. Twist on this scale only weakly perturbs the molecular packing. Cholesteric liquid crystals reflect light strongly when the wavelength is equal or commensurate to the helical pitch length. There is a Bragg-like reflection from the onedimensional periodicity in the optical density that creates a (single) photonic band gap at normal incidence, in which propagation of light of the corresponding wavelength (colour) is forbidden. A solution of this problem, based on the original work of de Vries (1951), is reviewed in detail in (Belyakov et al., 1979). Since the pitch can be comparable in length to the wavelength of visible light, this accounts for the vivid colours that cholesteric produce. We discuss this later in describing the photonic behaviour and lasing of cholesteric elastomers. Electric and magnetic fields applied transverse to or along the pitch axis distort the helical director pattern of a liquid cholesteric. 1. Transverse fields cause the director to linger in parts of the helix when it is parallel to the field; in other parts it twists more rapidly to minimise the volume in which it is electrically unfavourable. The helix is now non-uniform and the ideal uniform rate of twist, qo , no longer achieved. An elastic penalty is paid as the electrical energy is lowered. The competition between these two influences results in a coarsening and lengthening of the helix. 2. Longitudinal fields can swing the director toward the pˆ axis to give a conical director pattern, see Fig. 2.12. As one advances along the helix, the rotating director appears to act as the generator of a cone. Twist is reduced (and bend created) as the electrical energy is reduced. Again a competition gives an equilibrium distorted helix. These are classical problems first posed and solved by R.B. Meyer and P.-G. de Gennes (Meyer, 1968; de Gennes, 1968), to which we return in an analogous form for cholesteric elastomers subjected to a mechanical strain. We adopt the classical term ‘conical’ for such cholesterics with a longitudinal component of n following the original authors. However, ‘hour glass’ might be more appropriate: In Fig. 2.12 one can take n → −nn and thus invert the cone with no change of properties, in particular, the helical pitch pˆ . Equally the inversion pˆ → − pˆ reverses the helix axis but the sense (e.g. right handed-

CHOLESTERIC LIQUID CRYSTALS

35

ness) is preserved. Thus one cannot use a cone to define a preferred direction, for example the vector connecting the centre of a cone base with its apex. Only under very special conditions, where the locally nematic order is biaxial and where the local order axes are not in or perpendicular to the plane defined by z and n , could an effect arise (Pleiner and Brand, 1993). Cholesterics are weakly biaxial in practice, but the local axes are defined by this special plane and therefore polar order by this indirect route is very unlikely or very weak in the conventional cholesterics we consider. Exercise 2.5: Describe the distribution of twist and bend in classical and conical cholesterics. Does any flexoelectric polarisation arise from distorting from a classical to a conical cholesteric? Solution: Taking the coordinate system of Fig. 2.12, the director and its curl are: n = (sin θ cos φ , sin θ sin φ , cos θ ); curl n = − sin θ φ  (cos φ , sin φ , 0) ,

(2.46)

where φ  = d φ /dz. The direction of the vector curl n determines how much twist and bend are present. For instance for the classical cholesteric of Fig. 2.11 we have θ = π /2 and hence sin θ = 1 whence curl n = −φ  (cos φ , sin φ , 0) = −φ  n . Thus the twist is n · curl n = −φ  and the bend is n × curl n = −φ  n × n = 0. The twist energy density is F = 12 K2 (φ  − qo )2

(2.47)

and is minimised for the uniform twist φ  = qo . For a general conical state, θ = π /2, the twist is n · curl n = − sin2 θ φ  and is reduced by the inclination of the director towards the helix axis, pˆ . Bend is now present and is n × curl n = sin θ cos θ φ  (sin φ , − cos φ , 0). It is clear that splay is absent in a conical cholesteric: div n = dnz /dz = d cos θ /dz = 0. The energy density is now F = 12 K2 (sin2 θ φ  − qo )2 + 12 K3 sin2 θ cos2 θ φ 2 .

(2.48)

Flexoelectric polarisation arises (Meyer, 1969; Patel and Meyer, 1987) from a nonuniform director distribution involving splay and bend:

x

m

p

f

n q

z

y

F IG . 2.12. The cholesteric conical (or ‘hour-glass’) state. The polar angle θ of the director n is measured from the pitch axis pˆ . The projection of n to the plane perpendicular to pˆ , labelled m here and in eqn (2.50), makes the azimuthal angle φ with the x axis.

36

LIQUID CRYSTALS P = e1 n div n + e2 [nn × curl n ]

(2.49)

(We follow the sign convention of (de Gennes and Prost, 1994) of +e2 rather than −e2 of the original author, Meyer, 1969). Flexoelectricity is present even in centrosymmetric systems, that is, it is not a consequence of chirality. In distorting from the classical helix to the conical case, we have created bend and thus a flexoelectric polarisation that follows it: m × pˆ ] . P = e2 sin θ cos θ φ  (sin φ , − cos φ , 0) ≡ e2 sin θ cos θ φ  [m

(2.50)

The polarisation is in the plane perpendicular to pˆ and rotates with z, always perpendicular to the projection of n in the perpendicular plane, cf. Fig. 2.12. There is thus no macroscopic P ; polarisation is averaged to zero over all directions perpendicular to pˆ .

Symmetry of cholesterics In the results of exercise 9.1, polarisation P being perpendicular to the helix axis pˆ illustrates a general principle: one does not obtain a polar vector response, such as electric polarisation P , in a direction defined by a unit pseudovector pˆ , for which pˆ ≡ − pˆ , see the remark on cones and hour-glasses above. With pˆ and n o defining a local reference frame along the helix, the third orthogonal direction pˆ × n o can be identified. The helical director distribution can be simply written in a coordinate-free form (in parallel with Fig. 2.11): n (rr ) = n o cos φ (rr ) + pˆ × n o sin φ (rr )

(2.51)

where the phase angle φ (rr ) is

φ (rr ) = q ( pˆ · r )

(2.52)

with q = 2π /p being the wavenumber of the helix of pitch p. We now pause to discuss the nature of pseudoscalars, -vectors and -tensors, most important entities in chiral phases. The intrinsic chiral symmetry of an object modifies its properties from those of non-chiral objects. Any normal, polar vector object, P, changes sign on the operation of space inversion, that is, on the simultaneous changes {x → −x, y → −y, z → −z} of coordinates. A chiral, pseudo-vector by contrast does not change sign under inversion of coordinates. It has the same relation to the new coordinates as it had to the old, but now of course the coordinate system is left instead of righthanded. Thus rotations associated with the vector are now left- instead of righthanded. The simplest example of a chiral vector construction is the vector product of two ordinary polar vectors: Ω = [vv × u], see Fig. 2.13. Sometimes pseudovectors are known as axial vectors. The effect of chirality is even more spectacular in scalars. An ordinary scalar is totally invariant with respect to any space transformation. A chiral (pseudo) scalar changes sign on the inversion of space or on reflection in any mirror plane. A good example of a pseudoscalar is the mixed product of three polar vectors: s = [vv × u ] · w . Of

CHOLESTERIC LIQUID CRYSTALS

37

F IG . 2.13. The cross product of the two polar vectors v and u is a chiral (pseudo) vector Ω = v × u . Think of the right hand rule for the direction of Ω which reverses on use of the left hand. course, a product of two pseudo-scalars is a true scalar (just as the product of a pseudoscalar and a polar vector returns a pseudo-vector, and so on). An intrinsically chiral physical system can have pseudoscalar material parameters (coefficients), say, Γ; as a result, there are contributions ∼ Γ [vv × u ] · w to the free energy density which, of course, have to be true, invariant scalars. Since in this case we have taken the product of two pseudoscalars, this contribution to the free energy density is clearly a scalar. These questions of chiral symmetry are addressed in standard books (Arfken, 1985) and can be summarised by examining the transformation rules governing pseudoscalars (s), pseudovectors (vv) and pseudotensors (tt ), ... If the transformation of coordinate system is described by a matrix U , these objects transform as:   s = Det U s

  vi = Det U Ui j v j

  ti j = Det U UikU jl tkl . . .

(2.53)

This is in with the transformation of non-chiral objects which  contrast   not require  do the Det U factor. If the transformation U is a rotation, it has Det U = 1. If it is   an inversion, then U = −δ and, accordingly, Det U = −1. Applied to eqn (2.53), the results we discuss in the above emerge.   Reflection in one plane, say the  1 0 paragraphs 0 0 . Again Det U = −1, whereupon the changes xy-plane, corresponds to U = 00 10 −1 in the pseudo-tensors can again be determined from eqn (2.53). Returning to cholesterics, the pitch axis pˆ can be taken to be a pseudovector and the azimuthal angle φ must then be an ordinary scalar. Alternatively pˆ can be regarded as a vector but then φ must be a pseudoscalar. In either case the helix wavenumber q is a pseudoscalar. Turning a screw or a helix around in space does not change its sense, that is changing pˆ → − pˆ must have no effect on the director distribution. In eqn (2.51) sin φ is odd and changes sign with pˆ which appears in its argument, but so also does its coefficient and there is no sign change overall. The cos φ term is trivially invariant. Reflection changes the sign of pˆ but also the pseudoscalar q in eqn (2.52) so that φ is unchanged. The second term in eqn (2.51) then changes sign because of the pˆ × n o part and thus the sense of the helix reverses. The remarks produce two vital constraints on any free energy – it must involve even powers of pˆ , and it must be a scalar and not a pseudoscalar.

38

2.9

LIQUID CRYSTALS

Smectic liquid crystals

Smectic liquid crystals have orientational order as nematics do, but additionally have a limited degree of spatial ordering in the form of layering. From the symmetry point of view there is a great variety of possible phases, combining one-dimensional layered order with various degrees of structure and various types of alignment of mesogenic groups. The word smectic derives from the Greek for soapy, an attribute that the phases have from the flow properties associated with layers sliding largely intact over each other. We shall first consider the most simple smectic order, appropriately called ‘smectic A’ or lamellar Lα phase, where the molecular anisotropy is coaxial with the layer normal. In other words, the nematic director (which is the principal axis of uniaxial optical birefringence) is locked perpendicular to the smectic layers, Fig. 2.14. The layering smectics arises often from the drive of the aliphatic tails segregating from their conjugated, rod-like, mesogenic cores by forming the layered morphology seen in Fig. 2.14(a). We shall be interested in the polymer forms of smectics where, since both the spacers and the main chain often tend to be immiscible with the conjugated side chains rods Fig. 2.14(b) (Shibaev et al., 1982) microphase separation is even more likely. These complex molecules are crosslinked to form smectic elastomers and rodlike crosslinkers can also be important, Fig. 2.14(c). Another huge class of materials display the same layer structure for analogous reasons – amphiphilic surfactants in solutions and block copolymers in solutions and melts, see Fig. 2.15(a). Their morphology can be quite complex and a large literature exists on this subject (Bates, 1990; Seddon, 1996). When the composition of block copolymers, or the proportion of hydrophilic side-groups and hydrophobic long polymer chains, is near 50% such systems phase separate into the simple lyotropic Lα lamellar phase. Similarly, layered phases formed from the aggregation of amphiphilic molecules as shown in Fig. 2.15(b) can be polymerised by linking the heads or tails together and will thereby be of potential interest as smectic elastomers, Chapter 12. In this section we review the symmetry and elasticity of SmA liquids. Important

do

(a)

(b)

(c)

F IG . 2.14. Schematic of mesogenic group arrangements in phases of smectic A symmetry: (a) a classical smectic liquid, (b) a smectic side-chain polymer with the backbone confined between the layers and (c) a smectic elastomer with crosslinking groups incorporated into the smectic layers.

SMECTIC LIQUID CRYSTALS

A

39

do

B (a)

(b)

F IG . 2.15. Examples of “lyotropic” lamellar Lα phase: (a) diblock AB copolymer melts undergo microphase separation in a broad region of compositions, roughly around 50%, see (Bates, 1990) for details; (b) solutions of surfactant molecules form bilayer lamellar phases within a broad range of molecular structure and solution concentrations. Crosslinking the chains in (a) or hydrophobic tails of surfactants in (b) would result in a “smectic elastomer or gel” (Fischer et al., 1995). differences emerge when the analogue elastomers are created. Finally we review the properties of SmC liquids, where the nematic director is tilted with respect to the layer normal. An additional degree of freedom then exists – rotation of the director on a cone about the layer normal. This will have important mechanical ramifications when we consider the solid (elastomeric) SmC analogues. The role of chirality is also important since it leads to ferro-electricity in SmC liquids. The corresponding solids are soft, non-crystalline ferroelectrics that can be mechanically manipulated like no other ferroelastic materials. Smectic A liquids Smectic or lamellar order is characterised by a one-dimensional wave of density or composition, ρ (z) ≈ ρo + |ψ | cos(qo z + Φ). The amplitude of the density modulation is |ψ |. The layer normal is along z and Φ(rr ) is an arbitrary phase which thus describes the distortion of the layers. Far from the phase transition point this modulation may be significantly coarsened, that is higher harmonics could enter the expression for ρ (z). By symmetry, the phase transition directly from the isotropic to smectic A phase (a change between full rotational symmetry and the point group D∞h representing a cylinder) is the same as that between the isotropic and nematic phases. The uniaxial orientational order underlying the smectic phases has to be gained at the transition which thus has to be first order. This is the case in many systems that are candidates for smectic elastomers, in particular all lamellar phases. The transition between an established nematic phase and a smectic A as depicted in Fig. 12.3(c) can, by symmetry, be second order. A comprehensive, yet accessible, description of symmetries during phase transformations in liquid crystals is given in (Pikin, 1991). However, in the majority of liquid crystals with the I-N-A phase sequence, the N-A transition appears to be first order. There are several reasons, involving either an effect of insufficient magnitude of nematic order Q (de Gennes and Prost, 1994) or thermal fluctuations δ n (Halperin et al., 1974), that change the nature of critical behaviour at TNA . Later in this chapter we shall return to this question in more detail and examine the additional effect of the elastic network, and in particular the effect of its random crosslinks, on this phase transition.

40

LIQUID CRYSTALS

F IG . 2.16. Rotation of a system of layers about the y-axis gives tan θ = −∂ v/∂ x. The layer normal k then develops an x-component δ kx ≈ −∂ v/∂ x or more generally the ∇⊥ v. component of the normal in the original perpendicular plane is δ k = −∇ When the smectic order is well established, the phase of layer modulation can be written as Φ = −qo v(rr ) with v(rr ) describing the displacement of layers relative to their original position.2 Note that v is not a vector but only the component of such along the layer normal (z or n o for smectic A): displacements in the layer plane have no meaning for smectic liquid crystals. The continuum description of smectic and lamellar phases uses the gradients of layer displacement v(rr ) as effective distortion fields. The elastic free energy density must be invariant under the symmetry transformations of the phase, given here by the point group D∞h (the symmetry of a simple cylinder), or the point group D∞ (a cylinder with a screw thread) if non-centrosymmetric molecules form the smectic phase. Rotations of the layers in the fluid state cannot be relevant to the free energy. For instance rotation of θ about the y-axis relates to a gradient of layer displacement as tan θ = −∂ v/∂ x, see Fig. 2.16. Thus the corresponding harmonic form (∂ v/∂ x)2 + ∇⊥ v)2 does not enter FSmA (we introduce the in-plane gradient ∇ ⊥ to give (∂ v/∂ y)2 ≡ (∇ a coordinate-free form). Rotations will no longer be irrelevant in smectic elastomers. Layers are formed at a certain equilibrium distance from each other, called the layer spacing do = 2π /qo . Altering this spacing, causes a layer strain (d − do )/do = ∂ v/∂ z, see Fig. 2.17(a). This costs a free energy measured by the layer compression constant B, of dimensions of energy density, in eqn (2.54). Well-established layers, as do flat membranes or interfaces, have their curvature penalised. See Fig. 2.17(b) for a sketch of curvature in one of the two perpendicular directions. The energy is expressed by the second term in eqn (2.54), with the constant K having the dimensions of energy per length. At leading order, the free energy density then takes the form FSmA =

1 2B

∂v ∂z

2

+

1 2K

∂ 2v ∂ 2v + ∂ x2 ∂ y2

2 .

(2.54)

2 In most of the liquid crystal literature one usually finds this displacement called u(rr ). We need to use a different notation here, v(rr ), because in smectic elastomers there are two relevant displacement fields, of the layers and of the rubbery network, and we have already used the notation u(rr ) for the latter. Let us hope this does not cause confusion.

SMECTIC LIQUID CRYSTALS

d0

v(z+d0 ) d0+v(z+d0)-v(z)

41

R

v(z) z x (a)

(b)

F IG . 2.17. Distortions of smectic layers: (a) neighbouring layers at z and z + do suffer differing displacements v and hence acquire a new separation d ≈ do (1 + ∂ v/∂ z); (b) layers are bent and acquire a local radius of curvature R = (∂ 2 v/∂ x2 )−1 .

F IG . 2.18. Bending and compressing smectic layers: (a) The local curvature of smectic layers causes splay deformation of the nematic director, provided n represents the layer normal: δ n x = −∂ v/∂ x, etc. (b) The relation between bending, compression and the layer spacing. Membrane fluctuations are controlled by its curvature modulus K and exert an effective pressure on neighbouring layers, thus determining the compression modulus B. Too much fluctuation would result in layer expansion, too little in layer compaction. Nematic order in the planes underpins the smectic. It is clear that layer bending causes nematic splay deformation, see Fig. 2.18(a), with the Frank splay elastic penalty 1 2 2 K1 (div n ) . Thus one can identify the layer bending modulus with the Frank splay constant, K ≡ K1 . The definitive book by de Gennes and Prost on liquid crystals contains much detail on continuum properties of smectic A systems, including a comprehensive analysis of their elasticity. We remind the reader of another important tool for estimating the character and size of moduli. There is a relation between the two elastic moduli of a lamellar phase: the ratio K/B has a dimensionality of length squared and has been denoted by λ 2 (de Gennes and Prost, 1994). There are many compelling arguments, ranging from the qualitative statement, that ‘no other length scale is available’, to a detailed evaluation of layer-compression modulus B from the effective pressure exerted by layer fluctuations, (Helfrich, 1978) and Fig. 2.18(b), which indicates that this characteristic length scale is, in fact, the equilibrium layer spacing do . In other words, one may regard the size of a fluctuating lamellar stack as a balance of competition between the layer bending and compression effects: K/B = ξS2  do2 .

(2.55)

This relation can be useful in estimating the values of Frank and compression constants in different phases. For instance, in a usual smectic A liquid crystal, where the

42

LIQUID CRYSTALS

layer curvature is given by the Frank constant K1 ∼ 10−11 J/m and the layer spacing is, crudely, the length of a rod-like molecule do ∼ 2.5 nm, the layer compression constant should take the value B  K1 /do2 ∼ 1.7 × 106 J/m3 . This value is indeed very close to the B found in smectic liquids, but we shall find higher values of B in elastomers. There are several other possible invariant contributions to the free energy proportional to the squares of (∂ 2 v/∂ z2 ) and {(∂ 2 v/∂ x∂ z), (∂ 2 v/∂ y∂ z)}, which are nominally of the same order as the last term in eqn (2.54). However, the first of these derivatives must be discarded at this stage as it represents non-uniformity in the compression of layers and is small (this statement may have to be reconsidered in highly swollen gels where the layer separation is not so rigid and additional related effects could be anticipated). Smectic layer connectivity makes each of the derivatives in the second group exactly equal to zero (Orsay Group on Liquid Crystals, 1971), or at least penalised by a modulus identifiable with B. Consider a uniform system of flat layers with their equilibrium (unit) z axis. Let these layers be arbitrarily distorted, v = v(rr ), but at normal k o along the  constant spacing and without dislocations, see the region (C ) of Fig. 2.19(a). The new local layer normal is k (rr ). If one draws an arbitrary closed contour C across  this system, the sum of projections k (rr ) · dll of the normal onto this contour is zero: C k (rr ) · dll = 0. Such an integral simply counts the number of layers entering and leaving the area encircled by C , provided the layer spacing is constant. When a dislocation is present and C is drawn around the dislocation line, this integral is equal to its Burgers vector. Via Stokes theorem this contour integral transforms into the integral over area A surrounded by C : A curl k (rr ) · dSS , which tells us that curl k ≡ curl δ k ≡ 0. Recall from Fig. 2.16 ∇⊥ v, then that δ k = −∇ curl δ k = (∂ 2 v/∂ z∂ y, −∂ 2 v/∂ z∂ x, 0) = 0 whence

∂ 2 v/∂ z∂ x = ∂ 2 v/∂ z∂ y = 0. If the layer spacing varies, then so will these derivatives vary from 0, at a cost given by the compression modulus B. Hence these terms add nothing new to the free energy.

F IG . 2.19. (a) Integrating over a closed contour drawn across smectic layers. When the contour does not include discontinuities (as the edge dislocations to the left of C ), this amounts to summing equal and opposite contributions from layers entering and leaving the contour area. (b) An example of layer splay (bend deformation of layer normals) achieved by an array of edge dislocations.

SMECTIC LIQUID CRYSTALS

43

do/cosq do

Lz

q (a)

(b)

F IG . 2.20. (a) When a system of layers is rotated it presents an effectively larger laying spacing with respect to the original (stretch) direction, z. (b) To avoid collision with the bounding plates, layers must adopt undulatory rotations. The no displacement boundary condition at the plates suppresses undulations in their vicinity. One may note that if the nematic director is associated with the layer normal, n =kk (rr ), a dislocation-free smectic A can only deform with curl n = 0, if it preserves a constant equilibrium layer spacing. The appealing analogy between smectics A and superconductors (de Gennes and Prost, 1994) establishes the parallel between twist and bend deformations of nematic director (layer normal) and electron-pair correlation length, divergent in the superconducting phase. Twist and bend of the director (which both depend on curl n , see Sect. 2.5) translate to twist and splay of the layers. Thus curl n = 0 means that layer twist and splay can only exist mediated by defects: arrays of dislocations, as in Fig. 2.19(b), or twist-grain boundaries in strongly chiral smectics (Chaikin and Lubensky, 1995). The layer tilt instability of smectics Stretch deformation along the layer normal is resisted by the cost B of increasing layer separation. Smectic A liquids find a cheaper way of elongating, by rotating their layers. Simple geometry shows that layers, of intrinsic separation d0 , rotated by an angle θ then present a longer distance d0 / cos θ along the stretch direction, as indicated in the marker bars of Fig. 2.20(a) or more graphically where the two depictions meet. This geometrical mechanism for avoiding strain is associated with many authors including Helfrich and Hurault who discovered in it cholesterics and with Clark and Meyer who discovered it in smectics (Clark and Meyer, 1973). We shall refer to it as the CMHH effect in this book. The rotation angle is related to the layer displacement by tan θ = −∂ ν /∂ x when rotation is about the Thus the apparent spacing in the stretch direction √ y axis, see Fig. 2.16. 1 2 is d = d0 1 + tan θ  d0 (1 + 2 (∂ ν /∂ x)2 ) for small angles. However in the presence of boundaries at z = 0 and z = Lz (for instance the glass plates that contain the liquid and which are being separated), the layers cannot simply rotate and avoid colliding with the plates, ν = −x tan θ . Thus the rotation must be periodic to avoid the walls, see Fig. 2.20(b). Such a solution involves bending the layers which, as we saw in eqn (2.54), incurs essentially a splay cost. The outcome is a compromise between the advantage to rotating layers and the concomitant cost of bending them. Exercise 2.6: Find the threshold zz-strain for the CMHH undulation instability of SmA liquids.

44

LIQUID CRYSTALS Solution: Consider the uniform zz-strain ε0 to be supplemented by that generated by undulatory displacements ν1 (x, z) so that ν = ε0 z + ν1 (x, z). Then the effective strain is that from ∂ ν /∂ z, but reduced by that generated by rotations:

ν = ν0 + ∂ ν1 /∂ z − 12 (∂ ν1 (x, z)/∂ x)2 . For the small undulations just as the threshold, we can take the undulatory form for displacement and hence angle to be:

ν1 (x, z) = ν1 sin(kz z) cos(kx x) θ (x, z) = −ν1 kx sin(kz z) sin(kx x). The displacement and hence rotation vanishes at z = 0 and also at z = Lz if the wavevector kz = π /Lz . The strain energy is 12 Bε 2 = 12 Bε02 +Bε0 ∂ ν1 /∂ z+ 12 B(∂ ν1 /∂ z)2 − 1 2 2 Bε0 (∂ ν1 /∂ x) + . . . (ignoring terms higher order in ν1 ). Averaging over the volume of the sample, the second term vanishes since it is linear in sin and cos, and the second two have their quadratic sin and cos dependence replaced by the average value 1/4. Along with the layer bend energy density 12 K(∂ 2 ν /∂ x2 )2 which is likewise averaged, one obtains for the total energy density:   2 2 2 2 1 1 2 (2.56) 2 Bε0 + 8 θ1 Kkx + Bkz /kx − Bε0 , where we have replaced the combination ν1 kx by θ1 , the physically meaningful combination rather than the displacement which, without its gradient,  is somewhat arrbitrary. The choice of kx minimising the energy density is kx = π /(Lz ξS ), where ξS is the smectic penetration depth defined in eqn (2.55). The minimal energy is now (2.57) Fmin = 12 Bε02 + 18 θ12 [2ξS π /Lz − ε0 ] , whence it pays to develop undulations (θ1 = 0) and lower the energy when the strain reaches the threshold value ε0 = 2πξ /Lz .

As the amplitude of undulations increases one must consider non-linear corrections to this analysis (Singer, 1993). The threshold is small in liquids because the Frank costs are small compared with layer deformations, but in elastomers the strain thresholds will turn out to be much larger. CMHH undulations can be relieved by inserting replacement layers by dislocation propagation. In elastomers this apparently does not happen since the undulatory state is long-lived. Smectic C liquids Smectic C is a phase special among all the varieties of liquid crystals because of the discovery of spontaneous ferroelectricity in the chiral C∗ -smectics (Meyer et al., 1975). This class of materials has attracted great industrial and scientific interest. See the key monographs on the subject of ferroelectric liquid crystals (de Gennes and Prost, 1994; Pikin, 1991), which give detailed accounts of symmetries and the resulting physical effects. The basic feature of smectic C structure is the tilt of the average molecular axis (the director n ) with respect to the layer normal, while maintaining a disordered (liquid) placement of rods within layers. The tilt angle θ is often regarded as the order parameter of A-C phase transition, see Fig. 2.21(a). It is a good scalar measure,

SMECTIC LIQUID CRYSTALS

45

F IG . 2.21. (a) A sketch of the smectic A and C phases, indicating the tilt θ and the associated layer compression. (b) The helically twisted structure of chiral smectic C∗ ; the coordinates show the relative orientation of n , its projection to the layer plane c and spontaneous polarisation P s for right-handed chirality. although strictly the order parameter is more complicated – an in-plane vector ξ with components (−nz ny , nz nx ) = 12 sin 2θ (− sin φ , cos φ ) (Pikin, 1991). When the material is non-centrosymmetric (chiral), the resulting smectic C∗ structure becomes helically twisted in equilibrium, Fig. 2.21(b), in such a way that the director preserves the constant tilt angle θ but rotates around a cone, in the azimuthal plane, as one travels along the layer normal, φ = q z. The low symmetry of such a phase, with the layer normal k and the director n at an increasing angle to each other, allows spontaneous electric polarisation to develop in the direction perpendicular to both, that is P is along k × n . Note that in centrosymmetric, non-chiral smectic C such polarisation cannot develop P cannot be distinguished. since the directions of P and −P Ferroelectricity in chiral smectic C∗ liquid crystals is called ‘improper’. This term q (deg)

P (10-5C/m2)

30

q

P E

8 20

P

6

k c

n 4 10 2 0

30

(a)

20

DT=Tc -T

10

0

0

(b)

F IG . 2.22. (a) Simultaneous plots of θ (T ) and P(T ) on cooling below the A-C phase transition, illustrating the direct proportionality between the order and the polarisation in improper ferroelectrics (Ostrovskii et al., 1978); (b) A sketch of bookshelf geometry of smectic C alignment. Switching the external field would demand the optical axis n to switch to the opposite side of the θ -cone of orientations.

46

LIQUID CRYSTALS

is used to distinguish them from classical (‘proper’) ferroelectrics where the vector of spontaneous polarisation is the genuine order parameter and where there is generally a discrete manifold of preferred axes. The rise of polarisation in the SmC∗ phase is only a secondary effect, arising due to a coupling of P to the actual order parameter. In a chiral system, the free energy density has the invariant contribution, cf. Fig. 2.21(b), m˜ p P · ξ = m˜ p (Px nz ny − Py nz nx ) = − 12 m˜p P sin 2θ ,

(2.58)

with m˜ p a corresponding piezoelectric coefficient. Its chiral essence is like that of the pseudoscalar constant λ in cholesterics which leads to the helical pitch discussed below eqn (2.44). When one adds the standard dielectric susceptibility term, 21 χ −1 P2 , minimisation with respect to the amplitude of polarisation gives P = 12 m˜ p χ sin 2θ .

(2.59)

Figure 2.22(a) shows the experimental data on simultaneous measurements of the tilt angle and spontaneous polarisation in a ferroelectric smectic C∗ liquid crystal with the abbreviated name DOBAMBC (Ostrovskii et al., 1978). The coupling (2.58) is often called piezoelectric in the liquid crystal literature, referring to the fact that polarisation can be induced (or altered) by mechanically distorting the structure. Any deformation that alters the tilt angle θ would thus affect P . However, in a liquid, this is not piezoelectricity in the classical sense as no elastic strain is directly involved. In elastomers true piezoelectricity arises. The other aspect of this important coupling is seen in the electroclinic effect (Garoff and Meyer, 1977), when the director tilt θ is induced in the chiral smectic A∗ phase by applying an external electric field in the layer plane. In equilibrium, the C∗ -smectic is helically twisted and so its polarisation rotates in the layer plane. In order to obtain a uniformly polarised material one needs to unwind the helix – the classical way of doing so is forming the so-called bookshelf geometry of smectic C∗ confined between two boundary layers, Fig. 2.22(b). By applying an external electric field across such cell one can switch the polarisation from ‘up’ to ‘down’, by rotating the director azimuthally around the cone to its 180o -opposite position. Since this is the basis of some advanced liquid crystal displays (Clark and Lagerwall, 1980), a huge literature exists on this subject.

3 POLYMERS, ELASTOMERS AND RUBBER ELASTICITY The ability to identify the structure of macromolecules is a relatively recent occurrence in history, in spite of our reliance on such materials since ancient times. Some significant events in this advance have been the development of a rubber industry based on coagulated rubber latex, procedures for vulcanising rubber with sulphur and heat, discovered by Goodyear in 1844, the development of cellulose nitrate and xanthane and the development of gutta percha. One of the founders of polymer chemistry was Hermann Staudinger, who first proposed the now familiar ‘chain’ idea for these large molecules (Staudinger, 1920). The DuPont Company began synthesising polyesthers and polyamides in the 1930s which provided an impetus for polymer industry in the United States (Carothers, 1929). Parallel efforts, largely stimulated by the two World Wars, were focussed on phenolformaldehyde resins (Baekeland, 1909) and at Farbenfabriken Bayer (Weil, 1926; Hofmann, 1936). Several essential notions from basic polymer physics are required to describe liquid crystalline elastomers. In the melt, chain conformations are non-excluding random walks and thus are Gaussian. The problem of self-avoidance which makes polymer solutions so subtle does not arise here because of the excluded volume screening. This pertains to elastomers too and thus their statistical mechanics is relatively straightforward. Although the chains are ideal, they may be entangled and these constraints will be felt when chains are extended. In the review of this chapter, we shall therefore again take a most partial and incomplete view: it will emerge that polymers in general, and liquid crystal elastomers in particular, are universal in their physical properties, that is these properties depend only weakly on their detailed structure. To the joy of physicists, their complex chemistry can be at first neglected. For instance the shear modulus of a rubber is well described by µ = ns kB T where ns is the number of network strands per unit volume. A problem, at first sight of great complexity, has been reduced to counting (ns ) and an energy scale set by temperature, kB T . The relationship of ideal, equilibrium rubber elasticity has the same status and simplicity as the perfect gas law P = n kB T and it is to this level of simplicity that we shall aspire in discussing the molecular basis of liquid crystal elastomers. We shall find that for many properties this is possible and indeed adequate. As with liquid crystals, there are many fine books that cover all aspects of polymers that we shall require, (Flory, 1953; Flory, 1969; de Gennes, 1979; Doi and Edwards, 1986), the latter two being directed toward more advanced topics in polymers such as entanglements, dynamics and solutions. We touch upon the problem of entanglements in this chapter and, later, in reviewing the theory of nematic elastomers and the dynamic-mechanical response. After reviewing classical, isotropic polymers in the above spirit, we shall turn to nematic polymers. These materials combine both the properties of liquids composed 47

48

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

of long polymers, and the properties of liquid crystals. They are the building blocks of liquid crystal elastomers, the new class of elastic solids this book is concerned with. We conclude by developing the molecular basis of classical rubber elasticity. Due to their low shear moduli, rubbers deform at essentially constant volume. Rubber is also capable of very large deformations, as any child playing with a slingshot would confirm. In liquid crystal elastomers many new phenomena emerge at large strains which is why we shall require a molecular theory. 3.1 Configurations of polymers The classic example of a polymer is polyethylene, a long chain of segments shown in Fig. 3.1. The degree of polymerisation, N, may be quite large (N = 102 - 104 ). The CC bonds are nearly tetrahedral (109o ), but there is a significant degree of crank motion generated in exploring the three possible positions of the next −CH2 − group. This generates an enormous number of equivalent configurations, 3N in total, for an ideal single chain of −(CH2 − CH2 )n −. For any chain, especially those with a complex chemical structure, the effective step length  over which the chain can essentially bend may be equivalent to many monomers. However, the principle of polymer chains possessing a vast number of conformations is preserved so long as the total number of monomers, N, is large compared with the number of monomers per effective step length. The rubbery response of networks (and indeed the characteristic response of polymers in general) depends on this separation of scales (the total length of a chain, often called the arc length L , being much greater than the effective step length ). The opposite limiting case, of L  , corresponds to an almost completely rigid rod molecule of length L (something that we have discussed in relation to ordinary nematic liquid crystals). We shall confine ourselves to sufficiently long chains where the entropic properties of polymers are pronounced. Figure 3.2 shows three schematic snapshots of such a chain with R), of considerable internal flexibility in the joints. By considering the distribution, p(R the chain’s end-to-end vector R , one can make the idea of an effective jointed unit and the irrelevancy of local structure more precise. Let us take a chain composed of N rods of length a freely jointed together as in Fig. 3.2. The whole chain conformation traces a path of a random walk with a fixed step

F IG . 3.1. The molecular unit (monomer) of a polyethylene chain. Covalent bonds of carbon make a tetrahedron – arrows on the two outgoing bonds show where this unit is connected to other identical monomers, thereby specifying the position of two more C atoms. (a) The trans conformation with the −C − C− links in one plane. (b) One of the gauche conformations where the first or last C atom is out of plane.

CONFIGURATIONS OF POLYMERS

49

F IG . 3.2. A random walk composed of freely jointed segments with N = 100 such rods or ‘steps’. Three different trajectories in space are illustrated. The end-to-end vector, R , is the sum of the steps u of the component rods. length a (in this simple model, evidently, a = ). Equivalently, this is a trajectory of a Brownian particle diffusing in space under the influence of a fixed-magnitude stochastic force. The mean square end-to-end vector for such a random walk of N steps is, in each direction, R2y  = R R2z  = 13 R R2  = 13 a2 N ≡ 13 aL R2x  = R (3.1) R where L = Na is the actual arc length of the chain and corresponds to the total time of the analogous Brownian diffusion. In terms of the joint vectors u i of length a, the end-to-end distance R is given by R = ∑i u i . Since vectors u i are uncorrelated with R2  follows each other in their direction, the average uui u j  = 31 δi j a2 and the result for R immediately. Let the total number of possible conformations of such a chain, or the number of possible random walks with no restrictions on their starting and ending points, be ZN (this is 3N in our simplistic 3-state model for polyethylene). Since energy plays no role in this idealised chain model, this number of conformations is also the partition function for the chain: ZN = ∑configs exp(−H /kB T ) with the energy H of each configuration equal to zero or an irrelevant constant. The number of configurations with the ends fixed, R), is a great deal smaller: ZN (R R) = pN (R R) ZN ZN (R

(3.2)

R) expressing the probability a given conformation will have an end-to-end vecthe pN (R R) is a Gaussian distritor R . It is easy to show from the central limit theorem that pN (R bution: 3/2 2 2 3 R) = e−3RR /2Ro (3.3) pN (R 2 2π Ro characterised by its variance Ro . The product of two parameters expressing the detail of chemical structure of a polymer, its step length a, and the arc length L, appears R), namely as aL = R2o , simply as the single parameter of probability distribution p(R reproduce eqn (3.1). This combination Ro is the only significant quantity associated with an idealised chain. It is directly measurable by neutron scattering in the melt and by light and neutron scattering in solution, as the average radius of chain gyration.

50

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

For a non freely-jointed chain, the effective step length will be increased beyond the physical length of a monomer a and, given a fixed overall arc length L, the number of effective steps in a chain will decrease from the full number of monomers N to a lower value. We now more precisely define an effective step length, denoted by o , from the measurable quantities Ro and L: o = R2o /L (by analogy with R2o = aL).

(3.4)

Flory’s coefficient C∞ = o /a (Flory, 1953) is a direct measure of just how much local chemical structure can stiffen and extend a chain beyond what it would be, if freely jointed. Whatever the stiffening, Ro remains the single measure of the chain size distribution. This is true if L is long enough compared with o so that the distribution is Gaussian. The free energy of the single polymer chain we have described above is F = R), where we use eqn (3.2) and (3.3) for Z to obtain: −kB T ln ZN (R   2 R) = Fo + kB T 3R R /2R2o +C . (3.5) F (R Fo = −kB T ln ZN is the free energy of an unconstrained chain and is an additive constant. C is another additive constant arising from the normalisation of probability distribution pN . Fo and C simply make a reference point of free energy and we neglect it, since it does not depend on the chain end-to-end distance R . R) by simply counting configurations, assuming that they all We have obtained F (R have equal internal energy. The free energy (3.5) is purely entropic, the prefactor of kB T being a signal of this. However, energy is involved in the distortion of chemical bonds. R), then we If the internal energy per molecule associated with bond distortion were U (R would instead have: R) = U (R R) − T S (R R) F (R with S the entropy per molecule. The classical freely jointed model evidently has U = 0 and an entropy  2  R /2R2o . R) = −kB 3R (3.6) S (R In fact the free energy (3.5), quadratic as it is in R, represents Hooke’s law for the extension of a single chain. One indeed thinks of polymers as entropic springs with Hooke’s constant 3kB T /R2o . The stored (free) energy is entropic, because it measures a change (reduction) in the number of possible conformations (and thus – the entropy) when the ends of such a chain are pulled apart (R increases). Ultimately one would reach a state of a fully extended chain with R = L and, thus, only one possible configuration. This very unfavourable situation is, of course, well beyond the limit of applicability of the Gaussian law (3.3) for a truly random walk. In addition to this very basic argument, there is some residual temperature dependence in Ro in eqn (3.5) and (3.6) since thermal energy determines the effective stiffness of chemical bonds and hence the effective step length . The dependence is weak compared with the dramatic effects of nematic ordering leading, for instance, to spontaneous shape changes of between 10 and 400% in elastomers.

CONFIGURATIONS OF POLYMERS

51

Moreover, for most of the book, we only require that chains have some anisotropy. As usual in polymers, most effects are universal and do not depend on specific chain properties. We accordingly mostly discard stiffness variation effects, the exceptions being classical and nematic photoelastomers. The free energy for an isolated polymer chain with free ends extended by a distance R is a paradigm for a polymer network where the macroscopic deformation ultimately leads to the extension of constituent chains. Is the internal energy contribution to F actually zero, that is, can we consider network chains as purely entropic springs? Studies on networks show that the internal energy U forms a small part of F , typically less than 5%. There is also a possibility that a small part of U is actually due to distortion of bonds. A much larger part of internal energy is determined by nematic effects, that is anisotropic short-range steric and long-range van der Waals forces discussed in Sect. 2.1. Such anisotropic contributions to the internal energy U are only residual when we are dealing with conventional isotropic networks, but will form an important contribution to the nematic equivalent of the free energy (3.5). In any case, the fact remains that entropy and molecular disorder dominate most aspects of polymer physics. Chains are driven by the need to maximise their disorder – the number of their possible conformations. Another complication is that, of course, chains are not phantoms – they interact with other chains and themselves. In dilute solution the interaction with other chains can be discarded, but self-avoidance of a chain cuts down the number of configuraR) of (3.3). In addition tions in (3.2) more drastically than does the Gaussian factor pN (R to the reduction of the ZN factor, the result R2o ∼ N is modified to the celebrated law R2o ∼ N 6/5 for the self-avoiding random walk. Such chains are more extended than their ideal analogues. As the concentration of chains is increased the interactions with other chains become more important and eventually dominate the overall behaviour to the point where self-avoidance is effectively screened out. This problem is one of the principal concerns of de Gennes’ book (de Gennes, 1979) to which the reader should turn for further details. We shall ignore this subtle problem and exploit the simplicity of the ideal form of chains in the melt and in concentrated solution. Their configurations are those of the phantom, single Gaussian chains we analysed above. There is solid experimental evidence from neutron scattering for this assumption (Arrighi et al., 1992). Even with their ideal conformations in the concentrated state, chains are not really phantoms - they are entangled with each other. Entanglements slow down dynamics in the melt, but do not have a thermodynamic effect in equilibrium (Edwards and Vilgis, 1988; Kholodenko and Vilgis, 1998). In a network as chains are extended, their configurations are restricted more powerfully than simply by their ends being fixed. Unlike in the corresponding melt, knots cannot be untied and their frozen-in topology must be respected. One can examine (Deam and Edwards, 1976; Ball et al., 1981) topological effects in networks in order to explain the experimental deviation from classical predictions. It turns out that for sufficiently long chains crosslinked in the melt such deviations are important. However, in this book we shall find that liquid crystal order leads to deviations from classical behaviour that are much more significant than the role of entanglements, even in short solution-crosslinked chains. In our enthusiasm to abstract the essential polymer physics in its simplest and most

52

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

universal form, we must not forget the vital role of chemistry in modifying and designing molecules of great delicacy. Polymers for non-linear optics require stability against a variety of physical and chemical factors, specific 3-dimensional structures and high hyperpolarisability (that is a dipolar response depending on the square of the field). Polymer solutions and blends are dominated by questions of the compatibility of the components. Polymers of specific structure (handedness, stereo-regularity or irregularity, topology, . . . ) are required in a myriad of applications and novel science, for instance in suppression or enhancement of crystallinity, optical properties, viscoelasticity, etc. The classic case, centrally related to the systems of this book, is the first synthetic rubber, polyisoprene, where the control of its stereochemistry is vital to the mimicry of natural rubber properties. These remarks are also true of liquid crystalline, nematic and smectic, elastomers where molecular architecture has to be controlled in a sophisticated manner. The orientational order, induced by the incorporation of rigid rod-like segments into chains, is what drives these new elastomers to be so unusual. The residual flexibility is what preserves the high entropy and the resulting rubbery response. To an extent these requirements are contradictory and their correct balance requires considerable synthetic insight. We shall review this briefly, discussing polymer liquid crystals. 3.2

Liquid crystalline polymers

Nematic and smectic elastomers are networks of polymer chains that have intrinsic liquid crystalline ordering in addition to their conventional high polymer properties. In such polymer liquid crystals (PLCs) one combines the spontaneous orientation of liquid crystals with the entropically driven behaviour we have seen in polymers. Creating a PLC is a delicate balance; too much chain stiffness eliminates the large number of configurations of a chain making it an entropic spring. The stiffness turns the chain to

F IG . 3.3. Typical main chain polymer liquid crystals. Note the classic rod-like, nematic-forming sections in the middle of each monomer, but also the flexible −(CH2 )n − spacers between the rods which allow the chain the choice of many conformations. Polymer (a) with the n=10 spacer is known as DDA-9, with the shorter n=7 spacer it is AZA-9; their monomers are not unlike the nematic PAA, Fig. 2.1. Polymer (b), with different rigid rod structure, has been used in several main chain nematic elastomers mentioned in the text.

LIQUID CRYSTALLINE POLYMERS

53

F IG . 3.4. Typical side chain polymer liquid crystals. The rod-like, nematic-forming elements are now pendant to the backbone chain via flexible −(CH2 )n − link. (a) The polyacrylate backbone with cyanobiphenyl side group, similar to another classical nematic liquid crystal 5CB. (b) The benzoic ester mesogenic group attached to the polysiloxane chain (these two classes, acrylate and siloxane polymers, represent the vast majority of all SC PLC made so far). (c) The different type of SC PLC with the mesogenic group connected ‘side-on’ to the backbone, siloxane in this case. being a simple rod, albeit a long one. Too little stiffness or too few nematic-forming rods leads to an insufficient drive towards orientational order and results in an ordinary isotropic melt of chains. Two strategies of synthesis can be followed. Rigid rod-like elements can be linked together in a head-to-tail fashion to form a main chain (MC) polymer. Judicious choice of linkages between the rods gives sufficient flexibility that ensures the chain is Gaussian, and thus a random walk as in Fig. 3.2. This is the opposite of for instance the almost completely rigid rod-like viruses and synthetic polypeptides mentioned in Sect. 2.1. Examples of the chemical structure of main-chain PLCs are shown in Figs. 3.3(a) (d’Allest et al., 1988) and (b) (Percec and Kawasumi, 1991). Rods can otherwise be pendant to a flexible backbone. Such polymer has the topology of a comb and is called a side-chain (SC) PLC, pioneered in the mid-70s (Plate and Shibaev, 1987). Again, the linkages to the backbone and the volume fraction of rods in the polymer must be carefully chosen to allow the chain conformational freedom while not over-diluting or over-constraining the nematic phase (Shibaev et al., 1982). Examples are shown in Figs. 3.4(a) (Talroze et al., 1990), (b) (Finkelmann et al., 1981) and (c) (Li et al., 1993). When chiral centres are added, the nematic order is induced to twist and the melts become cholesteric (Finkelmann et al., 1981; Shibaev et al., 1981). A large body of literature exists for liquid crystalline polymers, see for example, a collection of reviews (McArdle, 1989) or the monograph by Donald and Windle (Donald and Windle, 1992). Much is known and understood about the ordering and the physical properties of such systems. For instance the variation of spacer length [(CH2 )10

54

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

in Fig. 3.4] has been long known to change thermal properties such as phase stability (Shibaev et al., 1982). Many questions remain a matter of argument. However, progress can be made within the spirit of the early part of this chapter, where a statistical description of polymer chains in the melt rested on only one key parameter, R2o , the mean square size. 3.2.1

Shape of liquid crystalline polymers

It is the average shape of the nematic polymer backbone that is important, since it is the backbone that generates the equilibrium elastic response of a network into which it is linked. Ordinary polymers are naturally spherical (i.e. their average gyration volume is a sphere) and hence only one dimension (Ro ) is sufficient to characterise them. They deform to a shape that mirrors the network deformation. By contrast, nematic polymers, whether side- or main-chain, have backbones with an average shape distorted by the nematic ordering of the associated rods. Therefore, more than one dimension is needed to describe their anisotropic shape. The aligned rod-like segments of a main-chain polymer will clearly elongate the average shape of gyration, essentially stretching the backbone along their principal axis, the director n . When the nematic order is high, the whole chain is forced into a directed shape: short chains unfold and become rods themselves, albeit slightly flexible. In sufficiently long chains entropy is in part recovered by the creation of the celebrated hairpin defects (de Gennes, 1982; Li et al., 1993), rapid reversals of chain direction, two of which are seen in the sketch of Fig. 3.5(a). The hairpins recover some entropy in the form of their random placements along the chain contour length. In contrast, the side-chain polymer may have the backbone in different conformations for the same degree of nematic ordering of pendant rods. This depends on the type of linking the mesogenic groups to the backbone, often called the spacer: in

(a)

(b)

(c)

(d)

F IG . 3.5. The shapes of nematic polymer backbones. The MC polymer (a) shows very high backbone anisotropy, the intermediate case of side-on PLC (b) shows weaker, but still substantial backbone alignment, while in the two end-on SC PLCs (c,d) the mesogenic groups may be only weakly coupled to the backbone; here the choice between the oblate (c) and the prolate (d) backbone arrangement is made by the spacer selection.

LIQUID CRYSTALLINE POLYMERS O

35

O

100

R (A)

R (A) 80

30

60

25

40

55

Iso

Nem

20 15

20

Nem

Smec 0 80

Iso

10

100

120

140

Temperature ( C)

160

180

20

(a)

30

40

50

60

Temperature ( C)

70

80

(b)

F IG . 3.6. The radii of gyration of nematic polymers as measured by small-angle neutron scattering. (a) The MC nematic polymers DDA-9 (filled circles) and AZA-9 (open diamonds) in the isotropic  and in the nematic state (d’Allest et al., 1988). The radius of gyration R ∼  L along the director n becomes much larger than √ that perpendicular to the director, R⊥ ∼ ⊥ L. (b) The side-on polysiloxane in the isotropic, nematic and smectic states (Lecommandoux et al., 1997). The longer dimension of the backbone in the nematic state, R (filled circles), flattens to become shorter than R⊥ (open circles) in the smectic phase, see Sect. 12.1. the sketch 3.5(b) the ‘side-on’ linking promotes the backbone extension along n , similarly to the main-chain case, but evidently to a much less extent. The ‘end-on’ type of coupling 3.5(c) often makes the backbone flatten in the plane perpendicular to n . However, there are many cases when a more rigid and bent spacer promotes the extended, prolate backbone even in the ‘end-on’ situation, for example, Fig. 3.5(d). In any case, the local uniaxial symmetry of a nematic must be preserved. The mean square end-to-end vector remains sufficient to characterise the shape of a chain and its probability distribution, if it is long enough to be Gaussian. In a principal frame there are now three such mean square quantities and in general we have: Ri R j  = 13 i j L

(3.7)

where now the effective step lengths form a tensor i j [compare with the analogous isoR), uniaxial tropic form, eqn (3.1)] and define an anisotropic Gaussian distribution pN (R if the mesogenic units form an ordinary nematic phase. A standard experimental procedure to directly measure the radius of gyration of polymer chains, including molecular shape changes induced by gradients of flow or by strain if in a network, is neutron scattering. Test chains are deuterated and neutrons incident in a beam respond to the triplet wave function of the D-nuclei, providing a strong contrast against the background of protonated chains (Higgins and Benoit, 1994). Deuterium is substituted for hydrogen in certain positions on each monomer. In some simple cases, such as polyethylene −(CH2 − CH2 )n − or polystyrene, the substitution H → D can be complete. In other cases, only a small fraction of the protons on each monomer is substituted, which should be the case for complex mesogenic molecules as in Figs. 3.3 and 3.4. Although there might be a residual doubt that the resulting deuterated polymer

56

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

behaves physically the same as the hydrogenated one, it is outweighed by the advantage of being able to directly measure molecular features. Such a test chain, exploring all its allowed conformations in thermal motion, appears in a neutron scattering experiment as an average object of the size and shape given by the radii of gyration. Knowing the average number of monomers on each chain, in effect the arc length L, one can easily measure the parameters of the step length tensor i j . Figure 3.6(a) shows the shape anisotropy of the polymer DDA-9, Fig. 3.3(a). The main-chain PLC melt anisotropy jumps from zero to a finite value on cooling through the transition temperature Tni . With ever increasing nematic order at low temperatures, the anisotropy can increase to very high values as the main chain polymer stretches out its backbone, possibly via hairpin reversals of chain along or against n, see Fig. 3.5(a). For instance at T = 108o C, the ratio of radii of gyration is R /R⊥ ∼ 8 giving a ratio of effective step lengths  /⊥ ∼ 60. Still larger values obtain below this temperature. The most extreme mechanical effects will be found in elastomers made from such main chain polymers. Analysis of hairpin defects show they become exponentially unlikely and their increasing separation gives a parallel effective step length that grows exponentially with the nematic order (de Gennes, 1982; Wang and Warner, 1986; Li et al., 1993). At the same time the transverse extent of chains continues to diminish. Side chain polymers of both the extended and flattened backbone varieties have been studied as well (Ohm et al., 1988; Cotton and Hardouin, 1997; Lecommandoux et al., 1997). In general their anisotropy is less extreme because the backbones are less strongly coupled to the ordering rods – see Fig. 3.6(b). Nevertheless, substantial anisotropy of polymer backbones have been recorded, for instance a poly-cyanobiphenyl acrylate chain of Fig. 3.4(a) with N ∼ 50 units (Mitchell et al., 1991) gives R2  + 2R2⊥ 

≈ (3.5 nm)2 with R /R⊥  ≈ 1.14 . 3 We shall see that intrinsic backbone chain anisotropies can at times be large even in SC systems. In uniaxial polymers, mean square sizes in all directions in the plane perpendicular to n are identical, Rx = Ry = R⊥ . For such nematic polymer melts with the director n along z we have the accordingly uniaxial tensor of step lengths:   ⊥ 0 0 (3.8)  o =  0 ⊥ 0  0 0  where  and ⊥ are the effective lengths of steps in the directions parallel and perpendicular to n and depend on Q. Thus R2z  = 13  L and R2x  = R2y  = 13 ⊥ L (cf. Figs. 3.5 and 3.6). In a general coordinate frame with n not necessarily oriented along z, x or y, the matrix  is not diagonal, as in eqn (3.8), but is given by:  o = ⊥ δ + [ − ⊥ ] n n .

(3.9)

An analogous approach is used in ordinary nematic liquid crystals to identify the order parameter via some relevant physical property. Section 2.2 makes the connection

LIQUID CRYSTALLINE POLYMERS

57

between the order Q and the refractive index tensor. The diamagnetic susceptibility, χi j , is another example. It is uniaxial with its unique principal axis along n (de Gennes and Prost, 1994), and is analogous to expressions (3.8) or (3.9):

χi j = χ⊥ δi j + [χ − χ⊥ ] ni n j .

(3.10)

The magnetic anisotropy is χ − χ⊥ and is a measure of Q. The most important parameter in (3.9) is the shape anisotropy of the chain distribution, δ  =  − ⊥ . We shall see that the fractional anisotropy δ /⊥ = [ /⊥ − 1] is proportional to Q and determines all principal features of nematic rubber elasticity. Just as in ordinary nematic liquid crystals where one can

directly measure the diamagnetic or dielectric anisotropy, we have seen that the ratio  /⊥ is measured by neutron scattering from the deuterated backbone. An additional, mechanical, characterisation is available in elastomeric networks. As polymer chains spontaneously deform, following their mesogenic rod-like groups when they become nematic, so does the whole network that the chains are crosslinked into. In this way one may deduce the characteristic molecular property  /⊥ from the overall sample changing its dimensions. There is, however, a big difference with ordinary nematic liquids: in a liquid of diamagnetic or dielectric rods the anisotropy of the corresponding susceptibility is a measure of rod alignment, given by the order parameter Q, as in eqns (2.5) and (2.7). In contrast, the anisotropy δ  = [ − ⊥ ] measures the shape anisotropy of the polymer backbone which although coupled to the nematic rods, might not have an identical degree of alignment – see Fig. 3.5. Therefore, in liquid crystal polymers we should distinguish between the anisotropy of polymer backbones expressed via  and the orientational (nematic) order parameter Q of mesogenic rod-like units, which is also a directly measurable quantity, by optical birefringence methods testing the average dielectric anisotropy of aligned rods, eqn (2.7), or by X-ray scattering testing the coherence in the rod packing direction. The two tensors are always coaxial. Since Q is traceless it expresses the deviations from isotropy δ . Thus  and Q are intimately related, see exercise 3.1 and the discussion below it. For small order the anisotropy is proportional to Q,   −1  α Q , ⊥ but the magnitude and even the sign of the proportionality coefficient α may be difficult to predict. A side-chain polymer with ‘end-on’ linking, Fig. 3.5(c) and (d), may have α → 0 or take negative values up to ∼ −0.5 (Mitchell et al., 1992; Guo et al., 1994). In main chain polymers order is directly that of the backbone; exercise 3.1 shows for freely jointed main chain polymers that α = 3. In Chapter 6 we shall see that when nematic order establishes itself in crosslinked networks of liquid crystalline polymers, macroscopic mechanical deformation will result. The sample will spontaneously stretch, in a uniaxial fashion along n , by the ratio 1/3  λm . This spontaneous deformation will turn out to be λm =  /⊥ , see Sect. 6.2.

58

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

n

R R F IG . 3.7. The gyration tensor spheroids. The arrow indicates the nematic director. The rods are suppressed in the diagram. Their coupling to the backbone may produce prolate (elongated along n ) or oblate chain conformations. In some cases the coupling between the aligning rods and the backbone may be so weak that the chain remains spherical, on average. Thus, there will turn out to be two independent measures of this vital chain parameter, by direct neutron scattering and by macroscopic mechanical means, and these will have 2  to be consistent. We denote the ratio ( /⊥ ) ≡ R /R⊥ by r. This characteristic of the polymer backbone will be of great significance later. Other, related mechanical methods have also been used to distinguish between prolate and oblate anisotropy induced in the backbones of nematic side chain melts. One can draw fibres and examine with X-rays whether rods align parallel (prolate) or perpendicular (oblate) to the stretch direction (which is clearly also the direction of natural polymer elongation). It is safer to do this with networks where one can draw and wait to equilibrate above Tg (Zentel et al., 1987). Prolate is most common, but oblate has been found by these and other authors. The Freiburg group (Hammerschmidt and Finkelmann, 1989) stretched chains and then looked at the circular dichroism induced by the ordering of selected bonds in the backbone. They thereby directly determined the sign of δ , that is whether chains naturally extend or flatten. The tensor  defines the spheroid of gyration Ri R j , eqn (3.7). Figure 3.7 extracts only the backbone from the sketches in Fig. 3.5, in particular ignoring the rods in sidechain polymers. We then have the appropriate uniaxial prolate and oblate spheroids, with δ  > 0 and δ  < 0, respectively. The isotropic phase, or a side-chain polymer with a ‘null’ coupling of rods to the backbone, has its gyration tensor in the shape of a sphere R), must be and hence has δ  = 0. The Gaussian distribution of chain conformations, p(R generalised for the anisotropic case:  R) = p(R

3 2π L

3

1 Det[  ]

1/2

 3 −1 exp − Ri i j R j . 2L

(3.11)

  −1 −1 The inverse step length tensor is  −1 = Diag −1 in its diagonal frame (see ,  ,  ⊥ ⊥  eqn (2.2) for a reminder of this notation), with the distribution in this frame being:

LIQUID CRYSTALLINE POLYMERS



3R2y 3R2z 3R2x R) ∼ exp − − − p(R 2⊥ L 2⊥ L 2 L

59

 .

(3.12)

The nematic polymer fluids are qualitatively identical with their simple nematic counterparts we reviewed in Chapter 2. There is a first order transition in such melts from the isotropic to the nematic phase. The order of rod-like mesogenic units of these polymers is much the same as that of conventional nematics. See Fig. 5.6(a) for a direct NMR determination of order with temperature, and Fig. 3.6(b) for its manifestation via chain anisotropy. Both show a jump. The resulting anisotropy of the backbone in mainchain melts is generally very high. The anisotropy r =  /⊥ induced in the backbones in SC melts is in general much smaller and often is oblate r < 1. Although many macroscopic properties, for instance the optical birefringence, follow the order of the rods, for rubbery networks we shall concentrate on this anisotropy of the backbone, since it is there that the elastic function rests. However, in some cases it would be misleading to identify the nematic order of the network with the nematic order parameter of the mesogens. Oblate chains would have an apparent prolate refractive index tensor since the side-chain rods invariably have a higher molecular polarisability, see Fig. 3.4. They are aligned along the director n with positive order while that of the backbones would be negative. Exercise 3.1: Give an explicit expression for the step length tensor  of a freely jointed nematic main chain polymer (cf. Sect. 3.1). Solution: Let the links be rods of length a, the α th link forming an angle θα with the direction of average alignment n (chosen along z), with azimuthal angle φα , see Fig. 2.4. Let the unit vector along the axis of the α th rod be u (α ) . The end-to(α ) , see Fig. 3.2. The end vector of a chain is the sum of its steps, R = a ∑N α =1 u matrix of step lengths  identified in (3.7), derives from the average =

  3a 3 R R = u(α ) u(β ) . R ∑ Na N α ,β

(3.13)

Since  reduces to the average uuu , just as Q did in eqn. 2.4 in Sect. 2.2, it is clear that  and hence the shape of chains will closely relate to Q itself. We now follow steps similar to those used to evaluate Q . Being freely jointed, there is no correlation between u (α ) and   the directions ( α ) ( β ) only survives of different links α and β . Therefore, the average u u when α = β , i.e. for the same link, and gives uuu , the same for all N links. The coordinate projections, in the frame where the director is along z, are uz = cos θ , ux = sin θ cos φ and uy = sin θ sin φ . Only diagonal terms survive in ui u j , since in any nematic phase cos θ sin θ  = 0, because θ and π − θ are equivalent states for a rod. The average sin φ cos φ  vanishes for similar reasons. In a uniaxial nematic all φ values  likely  in the plane perpendicular to the ordering  are equally axis n , therefore cos2 φ = sin2 φ = 1/2 (in a biaxial nematic this equivalence

u (β )

60

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY is broken, see the discussion leading to eqn (2.8) and see Appendix B). Thus in a uniaxial phase 1 2   0 0 2 sin θ   1 2 . (3.14)  = 3a  0 0 2 sin θ  2  0 0 cos θ   The nematic order parameter of the aligning rods is Q = 32 cos2 θ − 12 and hence there is a direct relationship between it and the components of  : 

1−Q  = a 0 0

0 1−Q 0

 0 0 , 1 + 2Q

(3.15)

or, comparing with the definitions of Q , eqn (2.5), and of  , eqn (3.9):     Q = a [1 − Q]δ + 3Q n n  = a δ + 2Q  = a(1 + 2Q)

and ⊥ = a(1 − Q) .

(3.16) (3.17)

Note that the chain of freely jointed rods is a model for main-chain nematic polymer. For a side chain polymer the angular relationship between chain steps and the rod alignment would be different.

The anisotropy for a nematic main chain freely jointed polymer is r=

 1 + 2Q = ⊥ 1−Q

∼ 1 + 3Q for Q small.

For small nematic order Q, different molecular models of nematic polymer chains will behave qualitatively like the freely jointed rod model here. As order gets stronger, stiffer polymers (for instance, worm-like chains with increasing bending rigidity) will have more correlations between monomers induced. A semiflexible worm, in the limit RR  ∼ L2 , of very strong order, can stretch out to be a large rod of length L, reaching R if its persistence length  becomes comparable to its chemical length L. In contrast, the average trajectory of a freely jointed chain, in the limit of perfect  nematic order Q → 1, becomes narrow in its transverse directions, Rx Rx  = Ry Ry → 0. It still remains a random walk along the director, that is Rz Rz  = a2 N, because complete reversals of direction, θ = 0 → θ = π , are possible in going from one monomer to the next. In this limit it would be important to be more discerning about which molecular model of chains one adopts for a particular physical system. The mean square size of a chain, R2o , determines the properties of isotropic polymer melts and networks - for instance the scale of its entropy and thus its response as a Hookean spring, see Sect. 3.1 and  (3.6) for example. For nematic polymers the  eqn R2  = 13 L Tr  . For freely jointed chains, the step length tensor mean square size is R Q), see eqn (3.16). Then R R2  = depends on the order parameter tensor as  = a(δ + 2Q aL, independently of nematic order in this model. Expressions (3.17) for  (Q) and ⊥ (Q) confirm this.

LIQUID CRYSTALLINE POLYMERS

61

We can connect the step length tensor to the nematic order more generally than in the freely jointed chain model. We can expand  in terms of the only tensor in the problem, Q :  = a0 δ + a1 Q + a2 Q · Q + . . . .

(3.18)

However the tensor product Q · Q is given by 12 Q Q + 12 Q2 δ and likewise all higher Q)n are reducible to powers (Q    n   n  Q)n = 13 Qn 1 + 2 − 12 Q δ + 23 Qn−1 1 − − 12 (Q

(3.19)

Q)n must be a uniaxial tensor with the same principal axes as Q , This is expected since (Q the most general such tensor being the sum of a multiple of the spherical tensor and a multiple of the traceless, uniaxial tensor Q . Since only one independent number defines a traceless, uniaxial tensor this decomposition can always be done. (When Q is biaxial two independent numbers are required.) The order tensor Q represents the deviations away from a spherical shape with the same mean as the original tensor. Eliminating all the higher terms in  in favour of δ and Q one must have Q  = bδ + cQ where the coefficients b and c depend on Q and are of the form b = b0 + b2 Q2 + . . .

and

c = c0 + c1 Q + . . .

Use of eqn (3.19) in eqn (3.18) shows for instance that b0 = a0 , b2 = 12 a2 , . . . and c0 = a1 , c1 = 12 a2 , . . . and so on. Differing models of nematic chains will give different coefficients from the freely jointed model. In general and in terms of ⊥ (Q) and δ (Q) = Q.  − ⊥ , one can easily show that  = 13 ( + 2⊥ )δ + 23 (δ /Q)Q 3.2.2

Frank elasticity of nematic polymers

As in any nematic system, one also finds the Frank elastic energy (2.20) – the penalty on curvature deformations of the average direction of alignment n . The Frank elastic constants for an SC melt are expected to be similar to those of a fluid of the corresponding monomers. The estimate of order of magnitude of K1 , K2 and K3 made by dividing the characteristic energy scale of aligning rods, ∼ kB T , by the characteristic length relevant for their alignment, their length a, gives as in ordinary thermotropic nematic liquid crystals around room temperature Ki ∼

4 × 10−21 J  4 × 10−12 N . 10−9 m

This is fairly close to the typically observed values, considering the crudeness of the estimate. The dependence Ki ∝ Q2 near the phase transition is also a universal feature, a direct consequence of the Landau expansion (2.10) with gradient terms added.

62

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

However, in main-chain nematic polymers one may expect deviations from these values because the long range correlations of monomers along the chain interfere with the definition of the ‘length scale’ as the length of a rod-like monomer. There may also be deviations because the number of monomers coherently participating in forming the excluded volume masks the concept of the ‘energy scale’ being simply kB T . Qualitative scaling arguments (de Gennes, 1980; Meyer, 1982) suggest that the Frank bend and twist elastic constants for main-chain polymers do not deviate much from conventional values. In contrast, the splay constant K1 becomes very large as the average step length  increases dramatically,  ∝ exp[1/(1 − Q)] (cf. Fig. 3.6) or, equivalently, the number of chain reversals (hairpins) along the director diminishes. More elaborate molecular theories have been developed (Grosberg and Zhestkov, 1986; Petschek and Terentjev, 1992). If only entropic, excluded volume effects are considered, K1 ∼ kB T ( /a2 ). Including the long-range van der Waals pair interactions makes a significant enthalpic contribution to Ki but does not alter the fact that K1 ∝ ( /a)  1. Frank elasticity underlies effects in nematic elastomers as well, although, as we have seen in Chapter 1, page 4, there are stronger restraints even on the uniform variations of the director. One can calculate (Terentjev et al., 1996) the modification to Frank elasticity in elastomers due to rubber elasticity and find that this modification may even lead to some Frank constants being negative! There is in these cases a drive to spontaneous spatial variation of the director in nematic solids under strain when this modification overwhelms the underlying rubber elasticity. Effects arise however only on very short length scales. 3.3

Classical rubber elasticity

Let us now return to the classical picture of simple, isotropic and long polymer chains. Most of the unusual, characteristically polymeric properties we associate with polymers of high molecular weight derive from their resistance to distortion of their average shape. The entropy of a single chain, eqn (3.6), is lowered as the distance between its ends is extended. Fewer conformations implies that the free energy rises. This stored elastic free energy is at the root of visco-elasticity of high polymers. Consider a chain in an extensional flow, for example in a constriction (entering a mould); Fig. 3.8. In the region of velocity gradients, ∼ ∇ v , the material is under the action of local stress (think of a viscous stress in an ordinary Newtonian fluid, σ  η ∇ v). At the point X, in the reference frame moving with the fluid, this gradient flow extends the molecule, increasing its end-to-end dimension along the flow axis and storing elastic energy. On later moving to a region of uniform flow, where no external stress is acting upon the system, the chain retracts to its equilibrium average conformation and releases its stored free energy; there is an internal stress acting. This delayed elastic reaction to the viscous flow is called visco-elasticity. The surrounding fluid may be a simple solvent or indeed other polymer chains in the case of a flowing melt. A simpler illustration of entropic-mechanical effects is rubber elasticity. We present the classical picture of rubber here, because this book aims to develop an analogous simple view for nematic elastomers – a straightforward extension of the classical approach describing isotropic polymer networks. Consider a network of crosslinked chains

CLASSICAL RUBBER ELASTICITY

63

F IG . 3.8. The extension of a polymer chain by a convergent flow field. (a) Solvent flowing into a constriction. The flow is extensional since the fluid velocity is greater further down the constriction. (b) A suspended polymer at the point X of (a). In the frame of the polymer the velocity is in opposite directions at opposite ends of the polymer and in this frame the chain suffers an extension. sketched in Fig. 3.9. A question may be asked – how many crosslinks are sufficient to transform a visco-elastic polymer melt or solution into an elastomer or gel, which is nominally solid? (i.e. responds with an elastic stress, and not a flow, to external deformations). The answer for such a threshold crosslinking density is not trivial by any means (Stauffer, 1985; Broderix et al., 1997). However, we only need to consider crosslinks well above this concentration, when their sufficient number ensures a percolating path of elastically active chains across the whole block of rubber. Distortion of the block causes the component strands between crosslinks to distort with respect to their equilibrium average shapes. As we have explored above, this costs a free energy due to the loss of configuration entropy. Departure from equilibrium corresponds to a reduction from the maximal entropy allowed by network constraints. The material is thereby a solid rather than a liquid, which would accommodate any distortion in its shape at constant energy. We shall return to this rather obvious remark later, since we shall find that it is

F IG . 3.9. A block of rubber with the underlying polymer network. (a) The chains of the network are shown linked. A test chain (heavy curve) has a span at formation R f between two successive crosslinks along its contour. (b) The block of rubber is extended by factors λii in the three principal directions. The test network span is now R = λ · R f .

64

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

not true for some distortions of nematic elastomers. Returning to simple rubber: without crosslinks it would be a polymer melt – a fluid that would eventually flow under stress. Relatively so few monomers are locally constrained by crosslinking that chains continue to have great mobility and explore the myriad of conformations characteristic of such a melt. Rubber is in effect a liquid in all regards except that it cannot flow! In nematic networks this observation is of central importance since the associated mobility of the director n is also great. The mobility of chains means, in particular, that they continue to explore many conformations and the drive to maximise entropy outweighs other influences. Change of average shape continues to be resisted, as in the single chain example given above. Consider the junction points in Fig. 3.9 to be fixed relative to the body. This implies that a selected strand’s end-to-end vector, connecting a pair of crosslinks, will deform in geometric proportion to the body’s deformation (the affine deformation approximation). Suppose a selected strand at network formation has been given an end-to-end distance R f . The deformation, λ , is defined such that any separation vector in the body, e.g. initially R f , will deform to a new value R given by: R = λ · Rf .

(3.20)

For instance in Fig. 3.9 if the sides are initially of unit length and deform to dimensions λxx , λyy , λzz in the directions x, y and z respectively (and no shear deformations are present), then each dimension of the chain is multiplied by the same geometric factors: Rx = λxx Rfx , Ry = λyy Rfy and Rz = λzz Rfz . The one test strand of Fig. 3.9 is shown in the deformed body in its affinely deformed state. The free energy of this particular strand is, as in the corresponding eqn (3.5): R) = kB T Fs (R

R2 3R 2R2o

 .

(3.21)

Recall that the mean square size, Ro , is the single parameter describing the Gaussian chain statistical properties. Using the affine relationship (3.20), we obtain R) = Fs (R

T 3kB T R f · λ · λ · R f . 2 R2o

(3.22)

Several constants, exposed and then neglected in eqn (3.5), such as the constant C from the normalisation of the Gaussian chain probability, have also been suppressed here. The current free energy of the selected network strand depends on the deformation λ and on the initial end-to-end separation R f (the subscript f denotes the state at the network formation). The overall elastic free energy of the block of rubber adds together contributions like eqn (3.22) for all other network strands. All different strands have their own initial end-to-end distance R f , but we know the proportion of chains with any

CLASSICAL RUBBER ELASTICITY

65

given R f among the whole ensemble – it is the probability distribution of chains having this end-to-end distance before crosslinking, at the moment of network formation: Rf ) = p(R

3 2π R2o

3/2

e−3(RRf )

2 /2R2 o

.

(3.23)

Naturally, it is the same Gaussian as in Sect. 3.1, eqn (3.3). Thus, the summing of individual chain free energies (3.22) in the deformed body is equivalent to the averaging of Fs over their distribution and then multiplying the resulting average free energy per strand by the total number of network strands in the system. Since R is derived from R f , the probability to find a strand currently with end-to-end separation R is simply the probability of finding the appropriate span R f at the moment of network formation, that Rf ). The average free energy per strand, F , is: is pN (R F=

3kB T Rf · λ T · λ · R f  p(RRf ) . R 2R2o

(3.24)

This amounts to averaging of a quadratic form (in R f ) with the corresponding Gaussian  2 Rf ) . The integration is of the form x2 e−α x dx, and yields the approdistribution, p(R priate averages: Rfi Rfj  = 31 R2o δi j .

(3.25)

We have assumed here that the mean square chain size at formation is the same as that which is current, R2o , when we are distorting the rubber, eqn (3.22). If for instance, temperature were to change between formation and current conditions, then the mean square size, R2f , at formation might be different from the current value, R2o . Substituting the average (3.25) back in to eqn (3.24) and multiplying by the average number of strands per unit volume ns , the free energy density (the free energy per unit volume) of a deformed rubber becomes   (3.26) F = 12 ns kB T Tr λ T · λ ≡ 12 ns kB T (λi j λ ji )   2 2 (3.27) + λyy + λzz2 . = 12 ns kB T λxx [We follow the Einstein convention of summation over the pairs of repeated indices in expressions involving matrices – such as in (3.26).] Equation (3.27) is the result of the particular extension shown in Fig. 3.9, that is the case where λ is diagonal, with no shear deformations. Note that the mean square size in each spatial direction, R2o , has cancelled out between expressions (3.24) and (3.25) and, as promised, the energy is just kB T times geometrical factors (λ 2 , the squares of the extensions). Nothing remains of the structure of the component chains, except that they must be long enough (and flexible enough) to satisfy laws of Gaussian statistics. Chapter 4 is concerned with classical elasticity. We show there that the free energy density of a material deforming at constant volume is of the form of eqn (3.26) where

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POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

the coefficient is 12 µ , with µ the solids’s shear modulus. Thus eqn (3.26) allows us to define, for the first time in this book, the characteristic rubber modulus:

µ = ns kB T . We shall constantly use this quantity. The magnitude of this static, equilibrium rubber modulus may vary greatly depending on the value of ns . It is, however, useful to give at least a crude estimate of µ . The number of chain strands per unit volume, ns , is equal to the inverse volume occupied by an average chain between two connected network crosslinks. Let us take a network of rather flexible polyethylene chains (see Fig. 3.1) with, ˚ on average, N = 100 units between crosslinks and a monomer size of, say a ∼ 3 A. 3 26 −3 Then ns  1/(Na ) ∼ 3 × 10 m . This is a high estimate – it leads to a modulus µ ∼ 106 Pa at room temperature. (The units of elastic moduli are commonly taken as Pa, which is the same as J/m3 or N/m2 ). In practice, the rubber modulus is often much lower. There are two common reasons, both serving to reduce the density of strands ns : ˚ often have much lower simple flexible chains with a small monomer size (a ∼ 2-3 A) crosslinking density and thus very long chain strands – often with monomer numbers N ≥ 104 between crosslinks. Polymers with a more complex molecular structure, in particular with long rigid rod elements necessary for liquid crystallinity, have the monomer volume 10-20 times greater then that of, say, polyethylene. Accordingly, one often finds rather weak rubbers with µ as low as 104 Pa, but hardly less than that. This identifies the characteristic range of possible magnitudes,

µ ∼ 104 − 106 Pa. At first sight the free energies (3.26) and (3.27) are unfortunate results. The lowest free energy density, F = 0 would apparently be at λxx = λyy = λzz = 0. Rubber should shrink to a point under the action of the entropic springs of the network! Of course, the repulsion arising when the molecules overlap eventually balances the attractive forces, as in any liquid or solid. The bulk modulus has the same dimensions (Pa) as the rubber modulus µ , and for a polymeric liquid, as for simple liquids, is roughly of the order 109 − 1010 J/m3 . The characteristic scale of rubber elastic energies is about 10−4 times that of the compressional modulus. Thus entropic effects of rubber elasticity are insignificant, compared with those causing or penalising volume change, and distortions of rubber must accordingly occur at constant volume (to within 1 part in 104 ). The difficulty that the rubber should shrink to a point under elastic forces is thus avoided. A more detailed discussion of small, subtle effects of volume change on formation and subsequent deformation of rubber is presented in Sect. 4.4. The example of Fig. 3.9 is instructive. Assuming the sides of the rubber block are along the coordinate axes x, y, z, the constancy of volume requires that the product of extensions is fixed:   (3.28) λxx λyy λzz ≡ Det λ = 1 . If we extend the sample by the factor λ in the z direction (λzz = λ ) and let the x and y dimensions simply be slaves √ to the condition of constant volume, then the constraint (3.28) demands λxx = λyy = 1/ λ and the free energy density (3.27) becomes:

MANIPULATING THE ELASTIC RESPONSE OF RUBBER

Elastic energy density F(l)

25

67

extension compression

20 15 10 5

3

l

0 0

1

2

3

4

5

6

F IG . 3.10. The plot of free energy density of incompressible rubber under uniaxial elongation and compression λ , in units of 12 µ ; the absolute minimum is F = 23 µ at λ = 1. Three curves correspond to the classical rubber-elastic expression (3.29), the middle solid line, and two its modifications (see Sect. 3.5), due to the ‘finite extensibility’ (curve labelled by solid circles), and the chain entanglements (solid squares).  2 2 1 . (3.29) F = 2µ λ + λ Having put in the conservation of volume by hand, this of course has λ = 1 as its relaxed, undeformed state. The higher energy scale associated with change of volume can be ignored if we always choose λ with the constraint (3.28) in mind, see more details on incompressibility in Sect. 4.4. Figure 3.10 shows the reduced energy density (3.29), in units of 12 µ , increasing from the ground-state level λ 2 + 2/λ = 3 in both elongation (λ > 1) and compression (λ < 1) modes of deformation. The two other curves in Fig. 3.10 represent the results of more advanced theories of rubber elasticity, briefly discussed in Sect. 3.5. The energy density as a function of deformation determines the force per unit area (see later for definitions of stress) as deformation is imposed. Take for example simple extension along the vertical axis, z, of a classical rubber, sketched in Fig. 3.9, that is initially a unit cube. Its current length in the z direction is λ . The force fz acting on the z ends of the body is the rate of change of energy, eqn (3.29), with length λ , times the area of the initial sample cross-section perpendicular to z-direction: fz = A

 ∂F 1 = Aµ λ − 2 . ∂λ λ

(3.30)

Taking a unit cube (A = 1) allowed us to consider forces and stresses interchangeably in this sketch. 3.4

Manipulating the elastic response of rubber

The effect of changing conditions (usually temperature, as the simplest but by no means the only control parameter) has been much studied for classical elastomers. We review

68

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

it here since quite dramatic effects from temperature or illumination will later emerge in nematic elastomers. The free energy, eqn (3.22), of a polymer strand with span R , deriving from R f at network formation, was averaged above with a probability (3.23) for R f assumed to be identical to that governing current conditions, eqn (3.3). Suppose at formation chains were Gaussian, but with a mean square extent R2f different from R2o . [We are abandoning here the simplification behind the rubber-elastic free energy density (3.26) that the two were equal. ] Thus the Gaussian would be characterised by the variance R2f and the R) would not have the neat cancellation of mean square sizes as result of averaging Fs (R in eqn (3.26), but would yield instead: F = 12 µ

R2f  T  Tr λ · λ . R2o

(3.31)

The coefficient R2f /R2o is sometimes denoted by αi2  in the classical literature (Dusek ˇ and Prins, 1969; Flory, 1969). The mean square sizes R2f and R2o reflect the stiffness of network chains, see Sect. 3.1, eqn (3.4) and the discussion of temperature dependence in R2o . One has to be resourceful to extract interesting phenomena from the additional behaviour in F introduced by αi2  since around λ = δ , the state of no distortion, changing αi2  has no mechanical effect, except perhaps small volume changes which we ignore. Exercise 3.2: Describe how a rubber-elastic ‘mechanical transistor’ might work. Solution: Imagine a load, say in the form of a suspended mass m, is applied to a unit sample of elastomer at temperature T . Then the force due to gravity is balanced by the rubber-elastic response, eqn (3.30), mg = µ αi2  (λ − 1/λ 2 ),

(3.32)

which gives the value for equilibrium extension, λo . Differentiating eqn (3.32) with respect to T , one obtains the thermo-mechanical response at this extension:

λo − 1/λo2 dλ =− dT 1 + 2/λo3



d lnαi2  d ln µ + dT dT

 ∼−

λo − 1/λo2 1 . 1 + 2/λo3 T

(3.33)

Only when there is a pre-imposed strain λo > 1 (from the load) is the factor λo − 1/λo2 non-zero. Then the response d λ /dT does not vanish. A temperature change, δ T , can be transduced into a mechanical effect, δ λ = (d λ /dT )δ T . As in the phenomenon of transferred resistance, a preimposed field can be modulated by a secondary field. The equivalences are: load ≡ source-drain voltage strain ≡ source-drain current temperature ≡ gate voltage. The analogy with the electronic device is a little forced since the ‘gate’ field is not of the same type as the ‘source-drain’ field.

MANIPULATING THE ELASTIC RESPONSE OF RUBBER

69

O

Exercise 3.2 shows how the application of a temperature change (the flow of heat into the sample) has done mechanical work (in lifting the load). The effects scale with the rate of change of µ αi2  with T . Experimentally there is little temperature variation in the αi2  term compared with the T factor in µ , which accounts for the final form of eqn (3.33). Also, a pre-strain is necessary to have a non-vanishing coefficient in the numerator of eqn (3.33). Thermo-mechanical effects are thus generally small in classical elastomers. We shall see later that none of these limitations hold for nematic elastomers. A larger and more unusual example of changing the rubber-elastic response by an external influence is that of photomechanical transduction in pre-strained photochromic elastomers (Eisenbach, 1980; Matejka ˇ et al., 1981). It arises when photons are absorbed into photochromic molecular moieties that form part of the elastomeric network. Such molecular groups undergo photoisomerisation, that is, they bend. Bending a monomer reduces its spatial span and thus also the effective step length o of the polymer strand it is part of. The mean square extent, R2o , of the network chains then also reduces; see eqn (3.4) for a reminder about the effective step length. The reduction for a given monomer can be very large (by more than 50%). Figure 3.11 shows an example of azobenzene photochromic moiety that is converted from the straight (trans) to the bent (cis) conformation on illumination by UV light of wavelength 365 nm. Such a photochromic moiety could be incorporated as a network crosslinker, Fig. 3.11(a), or as a side group of the polymer strand, Fig. 3.11(b). In both cases, it has been shown that the effective mechanical effect on illumination can be substantial. If sufficiently many of the photochromic groups isomerise from the straight trans form to the bent cis form on illumination, then the mean square extent of chains reduces and αi2  increases by a similar amount. For a given intensity of illumination, I, a new equilibrium state will be set up, with a corresponding associated αi2 (I). The incremental shrinkage of a loaded photosensitive rubber is analogous to eqn (3.33) with light intensity playing the role of temperature and the µ factor being inactive: trans N

O

O

N

N N

cis O

hw

trans

N

O

N

O

N

N

cis

hw O

(a)

(b)

O

F IG . 3.11. Photoisomerisation of azobenzene molecular moieties affecting the state of a classical, non-nematic elastomer network. Polymer strands have their natural length reduced on isomerisation of their monomer substituents: (a) the photochromic group incorporated as the network crosslink, and (b) as a pendant side group.

70

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

dλ λo − 1/λo2 d lnαi2  =− . dI 1 + 2/λo3 dI

(3.34)

Strain changes at constant load turn out to be rather small, of order just 1-2% (Eisenbach, 1980). Another experiment (Matejka ˇ et al., 1981) fixed the pre-strain at λ = 1.004 and observed a change in the response force on illumination, ∆ f / f of order up to 50%. Equation (3.32) at fixed λ then suggests that αi2  must have increased by a factor of 3/2. It is consistent with a very large fraction of photochromes being bent in the photostationary state, that is when the system has equilibrated to the illumination. (Subtle measures were taken to ensure that the response was optical and not thermal, that is, avoiding the radiation heating of rubber.) One can translate the results of this fixed strain experiment into the fixed load situation of Eisenbach by taking the ratio of eqn (3.32) at I = 0 and λo = 1.004 to that of eqn (3.32) at I and a new λI :

λI − 1/λI2 = 23 (λo − 1/λo2 ) . This would yield a hypothetical contraction to λI = 1.003 with respect to the original length, again a shrinkage of ∼ 8% with respect to the pre-strained state. Nematic elastomers will by contrast show very large thermal and photo effects, with strains up to 400%. This is because the analogous αi2  is not a single common prefactor, but different factors act separately on the different components of strain. For the same reason no pre-strain is required: induced network deformations can thus be spontaneous in non-distorted samples, see sections 5.4.1 and 5.4.2. 3.5

Finite extensibility and entanglements in elastomers

Rubbery polymer networks are highly complex disordered systems. The simplest theoretical model, discussed above, considers them as being made of ‘phantom chains’, where each polymer is modelled by a three-dimensional Gaussian random walk in space. One immediate question that arises in examining rubber elasticity is, what happens at larger extensions? Strictly, if the average length of network strands is very large, N → ∞, the chain will always remain in the Gaussian limit. However, in practice, rubbery networks are often formed with as few as 5-10 monomers between crosslinks. At large extensions the finite length of the chains makes itself felt. For a Gaussian chain, the typical network span R ∼ aN 1/2 is considerably smaller than the stretched-out arc length L = Na. The corrections thus become important when the deformation λ is suffi√ cient to take the chain to this extreme, that is when λ R ∼ Na, or equivalently, λ ∼ N. This limit can be very large for loosely linked systems. In other elastomers and gels, the ‘finite extensibility’ limit may be approached earlier. There are several approximations that describe the entropic free energy of a polymer (network strand) in the limit near its full extension (Treloar, 1975). The most commonly known is the so-called inverse Langevin function (Kuhn and Grun, 1942). The analysis has been extended to include the arbitrary strain tensors λ and presented the modified rubber-elastic energy in the form of 1/N expansion (Rusakov and Shliomis, 1992; Mao,

FINITE EXTENSIBILITY AND ENTANGLEMENTS IN ELASTOMERS

71

1999). In the particular case of uniaxial extension λ , the incompressible rubber with relatively short network strands responds with the elastic energy density    2 13 1 1 1 1− [3I12 − 4I2 ] F ≈ 2 µ 1 − + 2 I1 + N 5N 20N 5N  11 3 [5I − 12I1 I2 ] (3.35) + 350N 2 1 2 1 and I2 = 2λ + 2 where I1 = λ 2 + λ λ Obviously, in the N → ∞ limit one returns to the classical expressions (3.26)-(3.29). This approximation, for the case N = 10, is plotted in Fig. 3.10 to illustrate its deviations from the classical form at large extensions/compressions. Rubber elasticity has been intensively studied for over 50 years, asking questions beyond those of the simple classical picture. For instance, what are the effects of junction point fluctuations? Much depends on the functionality of a representative crosslinking point: the sketch in Fig. 3.9 has the crosslink functionality φ = 3, which is the minimum necessary for the network branching. One often finds chemical crosslinking mechanisms with higher functionality, φ = 4 and more. It is known that, for junctions of functionality φ , the end-to-end vector is not fixed, but fluctuates by the amount ∆ R ∆R 2  = 6R2o /φ , and that this is practically unaffected by deformations λ . Thus where ∆ the crosslink separation does not simply affinely follow that of the sample as a whole, except in the case of φ → ∞ (in that case the fluctuations ∆ R are killed by the requirement that a link should reconcile its motion with the large number of neighbours to which it is attached). However, it turns out that although these fluctuations for modest φ are large, their effect on the free energy of deformation is nevertheless simple: there is an overall reduction in F by a factor of (1 − 2/φ ) (Flory, 1976). The consequent renormalisation of rubber modulus µ is not vital to the exploration of nematic elastomers and we shall henceforth neglect it. Other effects of ordering intervene in networks as well, for instance, crystallisation can be induced by the chain alignment at large deformations and this too will lead to a hardening of the F(λ ) curve in Fig. 3.10. The coupling between imposed deformations and chain anisotropy is known as the stress-optical effect and also leads to non-classical corrections (Jarry and Monnerie, 1978; Deloche and Samulski, 1981). These are due to the effect of residual nematic interactions in an essentially isotropic phase. They have been calculated using the methods employed in describing nematic polymers and successfully model departures from classical rubber elasticity (Abramchuk and Khokhlov, 1987; Bladon and Warner, 1993). However, the most important and obvious effect missing in the classical theory is the effect of chain non-overlap, leading to the phenomenon of entanglement (Edwards and Vilgis, 1988; Kholodenko and Vilgis, 1998). To form a phantom network, polymer chains are crosslinked to each other at their end points, but do not interact otherwise. This has the unphysical consequence that the strands can pass through each other. If one tries to avoid this assumption, the theory is confronted with the intractable complexity of topological constraints. The mean field treatment of entangled polymer systems is the

72

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

F IG . 3.12. (a) A sketch of the ‘hoop model’ (Higgs and Ball, 1989) and the corresponding tube model picture, (b). A selected network strand is constrained to pass through a number of affinely fixed points as well as laterally constrained by the surrounding chains. In both sketches, r s indicates the position of a given monomer along the chain, while ∆ m is the step of the random walk of the primitive path between the entanglements. Both random walks, r s and ∆ m , have the same end-to-end distance the span R between crosslinked ends. now classical reptation theory (Edwards, 1967; de Gennes, 1971), later summarised in the monographs (de Gennes, 1979; Doi and Edwards, 1986), which spectacularly succeeds in describing a large variety of different physical effects in melts and semi-dilute solutions. However, one has to appreciate a significant difference in the entanglement topology of rubbers: in a polymer melt the confining chain has to be long enough to form a topological knot around a chosen polymer; even then the constraint is only dynamical and can be released by a reptation diffusion along the chain path. In a crosslinked network, any loop around a chosen strand becomes an entanglement, which could be mobile but cannot be released altogether. After a number of attempts by different authors, an approach has been adopted (Higgs and Ball, 1989) in which the entanglements localise certain short segments of a particular strand to a small volume. One can model this effect by describing a network strand as a free Gaussian random walk, which is, however, forced to pass through a certain number of hoops, which are fixed in space and deform affinely with the body, Fig. 3.12. The details of non-trivial tube-model calculations, also in the nematic context, can be found in the literature, (Kutter and Terentjev, 2001). The essence of the approach is to regard a strand as fluctuating around a certain trajectory, a mean path, which is called the primitive path in the reptation theory. The most intuitive way to visualise such a trajectory is to imagine that the chain is made shorter and thus made taut between its fixed crosslinking end points. The taut portions of the chain will form a broken line of straight segments between the points of entanglement, which restrict further tightening. This primitive path can be considered as a random walk with an associated typical step length, which is much bigger than the polymer step length, as sketched in Fig. 3.12(b). The number of corresponding tube segments M is determined by the average number of entanglements per chain (the situation with no entanglements corresponds to M = 1, a single straight end-to-end span). In each of these primitive path segments, the polymer

FINITE EXTENSIBILITY AND ENTANGLEMENTS IN ELASTOMERS

73

chain makes sm , m = 1, . . . , M steps, with the obvious condition, ∑M m=1 sm = N. We assume that all primitive path spans ∆ m deform affinely with the macroscopic strain: ∆ m = λ · ∆ m . This is the central point in the model: the rubber elastic response will arise due to the change in the number of polymer configurations in each segment of the distorted primitive path. The equilibrium number of monomers confined within a tube segment with the span vector ∆ m is obtained by the constrained optimisation of a statistical weight and gives the result: sm =

N∆m , M ∑m=1 ∆m

(3.36)

∆m | is the original length of the m-th step of the primitive path. On dewhere ∆m = |∆  formation, ∆m = |λ · ∆ m |, and the monomer steps have to re-distribute between different primitive path segments. The procedure of quenched averaging of the deformed state, with the probability distribution determined by the initial equilibrium state of the chain, produces a rubber-elastic energy density in the form F=

2 2M + 1 Tr(λ T · λ ) µ 3 3M + 1 3 2M + 1 (|λ |)2 + µ (M − 1)ln |λ |, + µ (M − 1) 2 3M + 1

(3.37)

with µ = ns kB T , as usual, and the following notations for angular averages of strains, see (Higgs and Ball, 1989; Kutter and Terentjev, 2001) for details: 

1 dΩ |λ · e| 4π |ee|=1  1 ln |λ | = dΩ ln |λ · e |. 4π |ee|=1 |λ | =

(3.38) (3.39)



The unit vector e takes all solid angles Ω in the integral dΩ. In the particular case of uniaxial extension/compression λ in a fully incompressible rubber, the three elastic terms in (3.37) take the form 1 Tr(λ T · λ ) = λ 2 +  λ   √ 1 1 λ 3/2 + λ 3 − 1 √ |λ | = λ+ √ √ ln 2 λ 3/2 − λ 3 − 1 2 λ λ3 −1 √ arctan λ 3 − 1 √ . ln(|λ |) = ln(λ ) − 1 + λ3 −1

(3.40) (3.41) (3.42)

In the limit of no entanglements, M = 1, we return back to the classical expressions (3.26) and (3.29). In the  opposite limit  of M  1 the rubber-elastic response can be 2 simplified to F ≈ µ M (|λ |) + ln |λ | and the effective rubber (shear) modulus is then

74

POLYMERS, ELASTOMERS AND RUBBER ELASTICITY

Reduced stress f*

1.4

1.2

1

0.8

compression extension

1/l

0.6 0

1

2

3

4

5

f∗

F IG . 3.13. Mooney-Rivlin plots of the reduced stress against the inverse extansion 1/λ . The classical rubber elasticity of Gaussian phantom networks gives the constant value of 1; the deviation due to finite extensibility is labelled by solid circles, while the entanglement corrections lead to the curve labelled by solid squares. Experimental data, open symbols, on weakly crosslinked natural rubbers are plotted on the same graph (Higgs and Gaylord, 1990). scaled up: G ≈ 11 15 µ M. Figure 3.10 shows the plot of this energy density, in units of rubber modulus. It is similar to the two other curves. To summarise, the question of deviations from the classical law of incompressible rubber elasticity (3.29) is very subtle and complicated. Experimentally these corrections are traditionally presented in a so-called Mooney-Rivlin plot (Treloar, 1975). In uniaxial extension geometry the classical rubber elasticity gives the response stress f = µ (λ − 1/λ 2 ). The deviations from this are best seen when the real data for the response stress f = ∂ F/∂ λ are represented by the non-dimensional reduced expression f∗ ≡

f µ (λ − 1/λ 2 )

and are plotted against the inverse strain 1/λ . Figure 3.13 illustrates the point for the two non-classical corrections discussed here. The interested reader can examine the various physical arguments in the context of a demanding multi-axial deformation scheme for confronting the models with experiment (Gottlieb and Gaylord, 1987). In general, however, we shall find that the distinctly nematic phenomena occur in rubbers at all deformations and they are best examined within a more simple and clear Gaussian limit. This is always the starting point of all studies in polymer physics and we shall follow the suit here. In Sect. 6.8 we shall generalise the nematic-Gaussian theory of rubber to accommodate finite extension and entanglement.

4 CLASSICAL ELASTICITY The deformation tensor λ has arisen naturally in the derivation of classical rubber elasticity. In the form given by eqn (3.20) it is appropriate for small and large strains alike, which is as well since rubber is capable for strains up to many hundreds of percent. Since the remainder of this book is concerned with elasticity of a new and unexpected form, with hitherto unsuspected phenomena to be summarised in the next chapter, we devote some space to reviewing the symmetry character of λ , from whence the effects will arise. We also examine the structure of non-linear elasticity and the connection with linear elasticity commonly used to describe solids at small strains. In contrast to classical elasticity, nematic rubber elasticity relies on the coupling of the rotations of internal degrees of freedom (the director n ) to not only elastic strains but also body rotations. We thus illustrate the geometry of deformations and local rotations in order to prepare for this new type of elasticity. We note that incompressible distortions are all essentially shears; even the simple extensions and compressions of the rectangular block in Fig. 3.9 are shears, just viewed from a rotated coordinate frame. Deformations, not in general symmetric or anti-symmetric, can be broken down into symmetric (pure shear) and rotational components. This will be useful in considering the mechanoorientational responses and instabilities of nematic elastomers. Intimately related to this symmetric/anti-symmetric resolution of strains are the square roots of tensors. We discuss them in this context, though principally to introduce them for the later treatment of soft elasticity. Finally, we discuss the (limited) role of compressibility in this elasticity. There is a large literature on the fundamentals of elasticity, for instance, the book by Atkins and Fox (1980) gives a good and compact overview, including the definitions of the various distortion and strain tensors and the more general requirements of invariance. Murnaghan (1967) discusses non-linear elasticity, symmetry requirements and the roots of tensors. Treloar (1975) also reviews elasticity in the context of rubber. 4.1 Deformation tensor and Cauchy–Green strain Consider a reference space SR of the relaxed body before deformation to a target space Ro ) in ST , see Fig. 4.1. The deformST . A material point R o in SR becomes R = R o + u (R ation records how differently neighbouring points are displaced (by u ) and hence how their relative separation is deformed from its relaxed value. The deformation gradient is defined as:

λi j =

∂ Ri , ∂ Ro j

(4.1)

see Fig. 4.1 [compare with eqn (3.20)]. It is clear that only the gradients of displacement contribute to physical effects: the uniform displacement field u corresponds to the 75

76

CLASSICAL ELASTICITY

RO

ST

SR

RO

u R

(V)

(U)

F IG . 4.1. The deformation of an elastic body. A point with the coordinate R o in a reference space SR is moved to a new position R in a target space ST . The deformation Ro ) at each point in the initial is fully described by a field of displacement vectors u (R Ro ) to R . The matrices body shape. The material point R o is thereby displaced by u (R V and U are the rotations relevant for SR and ST respectively. movement of the body as a whole. Some authors denote λ by F . We shall sometimes refer to it simply as the deformation tensor. If the target space transforms under rotations, represented by the matrix U , as R  = U · R , and the reference space deforms under rotations V as R o = V · R o , then the deformation tensor deforms as

λkl = Uki

∂ Ri T V ∂ Ro j jl

(4.2)

V T or conversely λ  = U · λ ·V

(4.3)



V. λ = U · λ ·V T

(4.4)

(See Sect. 4.3.1 for more on rotations, in particular the end of that section for explicit forms of matrix representations U and V of rotations.) Thus λ records the character of both the target and reference states, a property that will be essential in real nematic elastomers where an isotropic reference state cannot be reached. The connection with both spaces is quite different in character from the Cauchy tensors to be introduced below. Approaching large amplitude elasticity through λ makes dealing with non-linearities easier than via retaining non-linear terms in the Cauchy strain formalism of conventional elasticity which we outline in Sect. 4.2. Isotropic systems are invariant under rotations V of SR and the system’s final energy must be invariant under rotations of ST . (If SR is crystalline, the invariance under the relevant point group instead of under V is required). The rubber free energy (3.27) is a good vehicle to discuss this. F is a function of the combination 



U ·U U T · λ  ·V V = V T · λ T · λ  ·V V. λ T · λ = V T · λ T ·U Thus the product λ T · λ is invariant under body rotations U of the final (target) space ST ; it is called the right Cauchy–Green deformation tensor C = λT ·λ;

C  ·V V. C = V T ·C

Evidently C transforms as a second rank tensor in the reference space SR .

(4.5)

DEFORMATION TENSOR AND CAUCHY–GREEN STRAIN

77

The rubber elastic free energy in the initial (unprimed) frame can be expressed in terms of C  :     C  ·V V F = 12 µ Tr C = 12 µ Tr V T ·C     V ·V V T = 12 µ Tr C  = 12 µ Tr C  ·V

(4.6)

(by cyclical properties of the trace). F is invariant under rotations of the reference state because the trace of the product λ T · λ is. The form of the free energy in terms of C is then identical to that in terms of C  . For completeness, the left Cauchy–Green tensor is B = λ · λ T . In contrast to C , it is invariant under rotations of the reference state and transforms like a second rank tensor in the target state ST : U. B = U T · B  ·U (4.7) B = λ · λ T; Importantly, the elastic energy can be equivalently expressed in terms of the left Cauchy– Green deformation tensor, although this is much less common approach because the invariance of the current (target) state is often a more relevant condition (Lubensky et al., 2002). In general, the scalar free energy density F must be a function purely of the rotational invariants of C (or B , if this is the chosen representation). Such invariants of a second-rank 3 × 3 tensor Ci j are well known in linear algebra, and are usually called I1 , I2 and I3 . Explicitly:          2 C , I3 = Det C . (4.8) − Tr C T ·C I1 = Tr C , I2 = 12 Tr C Since these are rotational invariants, they are the same in all frames, including the diagonal frame. It is therefore sufficient to write their expressions in terms of the eigenvalues C1 , C2 and C3 of C . This yields I1 = C1 +C2 +C3 , I2 = C1C2 +C2C3 +C3C1 and I3 = C1C2C3 . We have seen that the separation of compressional and rubber-elastic energy scales   2 = 1. ensures that distortions are at essentially constant volume, that is I3 = Det λ Therefore in Sect. 4.4, I3 will be eliminated from further consideration. Classical molecular theory, eqn (3.26), produces F = 12 µ I1 which is a reasonable first approximation, valid up to surprisingly high extensions into the non-linear (large strain) regime. It is impossible to adopt a simpler result. The Mooney-Rivlin attempt to account phenomenologically for deviations due to entanglements and other causes, invokes the simplest possible correction, giving for F: F = c1 I1 + c2 I2 .

(4.9)

This modification is not entirely successful and gives no clue about the origins of the deviation from purely classical behaviour measured by the phenomenological coefficient c2 in eqn (4.9). Section 3.5 addresses how more complex forms of F arise. In this

78

CLASSICAL ELASTICITY

book we are concerned with much more dramatic departures from classical behavior than those like c2 I2 . We defer questions such as entanglements in nematic elastomers to Sect. 6.8. We shall re-examine the above rotational symmetry requirements for nematic elastomers in Sect. 7.2.3: there is, in fact, an additional hidden symmetry that leads to socalled ‘soft elasticity’ or a ‘Goldstone mode’ of their mechanical response. This is the subject of Chapter 7. For large deformations and for these new elastic modes, considerations at the level of I1 , or rather its generalisation for nematics, will turn out to be sufficient. Compatibility constraints Ro ) depends on the position in the Distortions are in general non-uniform and thus λ (R body where it is measured. There are then certain conditions of geometric compatibility that the components of λ must satisfy. The elements of this matrix cannot be completely independent because, in effect, there are only three independent components of Ro ) that determine all components of λi j . Mathematically, this geometric the vector u (R compatibility is most obvious when one calculates the second derivative, in which the order of derivatives is immaterial:

∂ λi j ∂ 2 Ri ∂ 2 Ri = ≡ ∂ Ro k ∂ Ro j ∂ Ro k ∂ Ro k ∂ Ro j ∂ λi j ∂ λik i.e. = ∂ Ro k ∂ Ro j

(4.10)

for all possible combinations of indices i, j, k from the set x, y, z. Of course, when the components of strain tensor are constant, all second derivatives are zero and compatibility is satisfied automatically. However, in many cases, for instance in modulated structures such as cholesteric elastomers or around topological defects, the deformations are naturally non-uniform and have to comply with this constraint. In cholesteric elastomers with the pitch axis along z, the director and the anisotropy associated with it rotate in the xy-plane as z advances. One might expect elastic deformations such as λxy (z) which would demand shears λxz (y) from compatibility, while λxz (z) can freely exist without the need for further attendant, compatibility-induced shears. 4.2

Non-linear and linear elasticity

To make contact with classical linear elasticity of solids, we break down the deformation tensor λ into a unit tensor δ (no deformation) plus a displacement gradient tensor ui j = ∂ ui /∂ Ro j :

λi j = δi j + ui j .

(4.11)

Since SR and ST are simple Euclidean spaces, then the distances between neighbouring R2o = dRoi dRoi in ST and SR respectR2 = dRi dRi ≡ λil dRol λik dRok and dR points are dR ively. The square of the change in separation is: R2 − dR R2o = (λik λil − δik δil )dRok dRol dR

NON-LINEAR AND LINEAR ELASTICITY

= (ukl + ulk + ukl ulk )dRok dRol ≡ 2εkl dRok dRol .

79

(4.12)

As ever, we use the convention of summation over repeated indices in matrix products. This difference, relative to the original element dRok dRol , is a measure of the distortion of the body at this point and is the right, finite symmetric strain tensor εi j :    ∂ ui ∂ u j ∂ ui ∂ u j T 1 1 . (4.13) εi j = 2 λ · λ − δ ≡2 + + ∂ x j ∂ xi ∂ x j ∂ xi ij The strain is symmetric by construction. For small displacement gradients, εi j reduces to the usual expression on dropping the u2i j term:

εi j = 12 (ui j + u ji ) .

(4.14)

The right strain tensor ε has the symmetry of λ T · λ – it is invariant under rotations U of ST and transforms like V T · ε · V under rotations of the reference state, SR . The left strain tensor ϕ could be derived from λ · λ T ; it would be invariant under rotations V of SR and transforms like U T · ϕ ·U U under rotations of ST . Thus when nematic order Q arises in SR and Q arises in ST , we can couple Q to ε to form true scalars, such as o o  Tr ε · Q , and Q couples to ϕ , since each transforms as tensors in the same space. o This is vital for developments of nematic rubber elasticity using Cauchy strains, rather than λ . We now identify the stress tensor σik , first in the linear case and then for large strains. Consider forces f acting on the surfaces of a small volume element in a body, see Fig. 4.2. The stress σik at this point is the component of force, fi acting in the ith direction, divided by the area of the element of the surface with normal in the kth direction on which the force is acting. Diagonal elements of the stress tensor are normal forces divided by the relevant surface area, the surface normal defined to be   vector 1 outwards from the body. The hydrostatic pressure is p = − 3 Tr σ , that is the average of the normal stresses with a minus sign since pressure acts inwards. By considering the variation of stress across an elementary volume, it is easy to show (Landau and Lifshitz, 1986) that the components of the resultant force per unit volume acting on the element due to its surface stresses are ∂ σik /∂ xk , that is div σ . We follow Landau and Lifshitz in a brief review of thermodynamic work. The density (per unit volume) of mechanical work produced when the force density div σ acts through a distance δ u is δ w = (∂ σik /∂ xk ) · δ ui . The total work done in the volume V of the body is, accordingly:      ∂ σik ∂ ui dV (4.15) δ W = δ w dV = δ ui dV = − σik δ ∂ xk ∂ xk (integrating by parts and ignoring surface terms in the last equation). The stress tensor for conventional solids is symmetric, otherwise infinite angular accelerations occur, see Appendix D. Using this symmetry we obtain for the work density δ w:

80

CLASSICAL ELASTICITY

F IG . 4.2. The definition of stress σ - a force per unit area of a body element. (a) A force f acting normally to the ith element of surface of a body element generates the diagonal (extensional) components of stress, σii , called normal stresses. (b) A force acting in the plane of the surface element yields a shear stress, an off-diagonal component of stress, σik .   1 ∂ ui ∂ uk ≡ −σik δ εik δ w = −δ σik + (4.16) 2 ∂ xk ∂ xi where the symmetric strain is defined by eqn (4.14). In nematic solids we shall find antisymmetric components of stress and antisymmetric components of strain (that is local torques and local rotations) entering the free energy. Appendix D gives the details of analogous arguments about mechanical work in a linear elastic material with local torques and rotations, the so-called Cosserat medium. From the thermodynamic derivative of free energy density dF = −sdT + σik d εik , with s the entropy per unit volume, we have:

σik = (∂ F/∂ εik )T .

(4.17)

We thus need F in terms of the strain tensor in order to obtain the stress σ . The free energy can only be a function of the rotational invariants, I1 , I2 and I3 of the strain ε . It can be developed in a power series, grouping terms of the same order in ε , that is F = F0 + F1 + F2 + . . . where Fn consists of terms like ε arranged in the form of invariants. Thus F0 is a constant and uninteresting for conventional elasticity (but will later carry nematic information and must be retained in nematic solids). F1 can only depend on the linear invariant I1 . At constant volume, its effect can be absorbed at quadratic order, as we shall show when discussing compressibility, and we ignore it for the moment. F2 is a combination of I12 and I2 since these generate ε 2 terms. It is usual (Murnaghan, 1967) to write F2 = 12 (λ + 2µ )I12 − 2µ I2 , where at this point only in this book, λ denotes the first Lam´e constant (standard notation in conventional Lam´e elasticity) and µ is the second Lam´e constant (and will turn out to be the shear modulus). The third term in this expansion, F3 , is conventionally written as F3 = 13 (l + 2m)I13 − 2mI1 I2 + nI3 , where l, m and n are the appropriate coefficients for the third-order terms. Inserting the expressions for I1 and I2 into F2 , one obtains the usual, Lam´e expression for the elastic free energy density at second order: F = 12 λ (εii )2 + µεi j ε ji

NON-LINEAR AND LINEAR ELASTICITY

  2   = 12 λ Tr ε + µ Tr ε T · ε .

81

(4.18)

  One can take the element of volume change, 13 Tr ε , out of the last term, so it is a pure shear, to yield the usual starting point for linear elasticity: ˜ εii )2 + µ (εi j − 1 δi j εll )2 F = 12 B( 3

(4.19)

where B˜ is the compression modulus B˜ = λ + 23 µ . Rubber, being essentially incompressible (B˜  µ ), has F ≈ µ εik εik . The stress linearly follows strain, that is as Hooke’s law σik = 2µ εik with the linear shear modulus µ = ns kB T , identifying with eqn (3.26) in the case of rubber. Strictly, as the free energy density (4.19) suggests, one must use the traceless form of symmetric strain, εi j − 13 Tr(ε )δi j . However, in incompressible materials, div u = 0 and there is no difference. The Poisson ratio of the transverse compression to the longitudinal extension is 1 3B˜ − 2µ ν = 12 ≈ , 2 3B˜ + µ the approximation being very close for rubber. Equations (4.18) and (4.19) are valid but incomplete when ε retains its non-linear character. Terms F3 and F4 must be added in for consistency (Murnaghan, 1967; Lubensky et al., 2002). As an example, we explore the rubber elastic free energy density (3.26) at small extensions which then becomes, explicitly,      . (4.20) F = 21 µ 3 + 2Tr ε + Tr ε · ε The reference value 3 in brackets corresponds to the value of F(λ ) for no distortion, the case of λ = δ (see Fig. 3.10). We shall suppress this irrelevant additive constant. The linear term would normally give a spontaneous compression since Tr(ε ) ≡ ∇i ui ≡ div u is clearly a dilatation. We have argued that dilatations are forbidden within rubber elasticity. The linear terms in (4.20) then disappear due to an identity that we derive from the Caley-Hamilton theorem algebra,see eqn (E.10) in Appendix E. The  linear   of incompressibility condition Det λ = Det δ + ε = 1, results in:   Tr ε =

     Tr ε · ε − (Tr ε )2 + . . .   → 12 Tr ε · ε + O(ε )3 . 1 2



(4.21)

    Thus the term Tr ε generates quadratic terms Tr ε · ε which are of the same order as those that interest us in F in eqn (4.20). The free energy is finally then:   F = µ Tr ε T · ε . (4.22) Comparison with eqn (4.18) shows that the constant µ of Lam´e elasticity theory and that of molecular rubber elasticity are identical. The details of compressibility are presented

82

CLASSICAL ELASTICITY

in Sect. 4.4 where additional bulk compression effects are added separately to expressions (3.26) and (4.20). For larger deformations, beyond the linear regime, conventional elasticity breaks down for large strains in two ways, one physical and one geometrical (Landau and Lifshitz, 1986). 1. The harmonic approximation to F is inadequate. Additional powers of invariants can be used in its expansion, but ultimately at hundreds of percent strain only a molecularly based F will be adequate. 2. One has to be more careful finding the stress than we have been above. The true stress is the ratio of the force to the current area, that is the area in the deformed state. The engineering stress is the ratio of the force to the initial area. Below we draw careful distinctions between the two. The process of using strains is problematic: the defining volumes for the integrals in eqn (4.15) should really be the current (i.e. deformed) rather than the initial ones (as in the distinction drawn between true and engineering stresses). Murnaghan, 1967, explicitly derives the expression for stress in terms of derivatives of the free energy density with respect to strain in the non-linear regime, taking into account changes in area. Take for example simple extension along the vertical axis (call it z) of a classical rubber, sketched in Fig. 3.9. We consider an initially unit cube so we can use the energy density in place of the total energy. Its current length in the z direction is λ . The force fz acting on the z ends of the body is the rate of change of energy, F, with length λ , eqn (3.29):  1 ∂F = Aµ λ − 2 fz = A ∂λ λ where A is the initial area of the sample cross-section facing the z-direction (A = 1 in the unit cube). However the real surface area of the cross-section in this xy-plane is of constancy volume (the two no longer unity, but has been reduced by 1/λ because √ perpendicular dimensions each contract by a factor 1/ λ ). The current area is thus Aλ = A/λ . Dividing the force by this current area we obtain for the true local stress in the elastic material:

σzz = fz /Aλ = λ ∂ F/∂ λ = µ (λ 2 − 1/λ ) .

(4.23)

Microscopically, as the rubber is extended there is the same number of strands (connecting crosslink points) crossing an xy-section of the rubber and conveying force (proportional to ∂ F/∂ λ ), but the sectional area is diminishing thereby increasing the force per unit area. Clearly λ ∂ F/∂ λ increases more rapidly than ∂ F/∂ λ . By contrast, the engineering stress is the current force divided by the initial area, A, which makes it much easier to measure. Whence

σzze = µ (λ − 1/λ 2 ) ;

fz = σzze A .

Accordingly, the production of mechanical work on such an extension is simply δ w = −σzze δ λ , using the engineering stress if we only measure the extension factor and not

GEOMETRY OF DEFORMATIONS AND ROTATIONS

83

F IG . 4.3. The geometry of rotations. the change in cross-section area. To account for this, and for the true stress, the work density should be written taking into account the changing area of the element at constant volume:  1 δ w = −σzz δ λ = −σzz δ (ln λ ). (4.24) λ Of course, both versions converge to the same linear expression at small deformations: when λ = 1 + ε with ε  1 one obtains δ λ ≈ δ (ln λ ) ≈ δ ε . (Both the stresses expand to σzz ≈ σzze = 3µε for a rubber in this limit.) 4.3

Geometry of deformations and rotations

In conventional elasticity of isotropic bodies, the energy is invariant under body rotations. We illustrate below how body rotations reflect the antisymmetric part, λ A , of the deformation tensor λ . To automatically render rotations irrelevant, one can take the symmetric part λ S of λ . Alternatively, one can work with C = λ T · λ which is symmetric by construction and where any rotations of the target state, which would appear as a multiplicative factor as U · λ , are also removed by construction. In liquid crystal elastomers there exists an internal rotational freedom (the director) with respect to which body rotations are important, see Chapter 1, page 4. Accordingly we recall the geometry of rotations and of pure shears in more detail than is usual in elasticity. We then give examples of the decomposition of λ into its two components (λ S and rotations). A more mathematical treatment, Sect. 4.3.3, on how this decomposition is achieved in general, can be skipped. It is however ultimately related to the question of the square roots of tensors, which we discuss. Roots of tensors are needed later in soft elasticity. 4.3.1

Rotations

Consider infinitesimal rotations about a particular axis, Ω , depicted in Fig. 4.3. The displacement u is u = Ω × r = r  − r where the magnitude of the displacement is u = r⊥ Ω, given by the rotation of an arm of length r⊥ by an infinitesimal angle Ω about the axis Ω . Returning to the definition λi j = ∂ ri /∂ r j we find the λ corresponding to body rotations to be:

84

CLASSICAL ELASTICITY

λi j = ∂ ri /∂ r j = δi j +

∂ Ω × r )i . (Ω ∂rj

(4.25)

One can write this more simply using the totally anti-symmetric Levi-Civita tensor
i jk Ω × r )i =
ilm Ωl rm . Using ∂ rm /∂ r j = δ jm we have since then (Ω

λi j = δi j +
il j Ωl ≡ δi j + Ωl
l ji .

(4.26)

We can invert this relation to give: Ωk = − 12
i jk λi j = −λiAj (i = j = k)

(4.27)

since the Levi-Civita tensor
i jk selects out the antisymmetric part, λ A , of λ . For instance, for infinitesimal rotations (about the axis y ), we have:  1 −Ωy = δi j + uAij (i, j = z, x) . λ= (4.28) Ωy 1 Here, and in the worked examples below in Sect. 4.3.2, we suppress for compactness y and just present the non-trivial, 2 × 2 part of matrices describing effects in the xz-plane.   The appearance in eqn (4.28) of the anti-symmetric component of strain, u A = Ω0 −Ω 0 is the signal that rotation is involved. The matrix representing rotation by a finite angle Ω about the axis y, in the xz-plane, is   cos Ω 0 −sin Ω WΩ =  0 1 0 . (4.29) sin Ω 0 cos Ω In the most general case, the rotation matrix W is determined by three Euler angles: two defining the orientation of the axis, Ω , and one specifying the amount of rotation, Ω, about this axis. It clearly agrees with eqn (4.26) at O(Ω). Recall that matrices W   W T = W T ·W W = δ and Det W = 1, that representing rotations have the properties W ·W is they are proper orthogonal matrices. The above example was of transforming a body, that is a position vector R transforms as R  = W · R . More discussion of finite rotations and their coordinate-independent representation is in Appendix E. Equally one could rotate the coordinate system by U with the concomitant change   Ω sin Ω in R → R  = U · R . Then U = −cos sin Ω cos Ω . Thus in this case U and W are rotations just differing in sense. Rotation of a vector R by Ω is equivalent in effect to rotating the coordinate system by −Ω. Second rank quantities, for instance the step length tensor  , U T. transform under rotations as   = U ·  ·U 4.3.2 Shears and their decomposition The symmetric part of λ , at constant volume, is pure shear. Despite appearances, simple extension and compression at right angles are in fact pure shear (at constant volume), but viewed in a rotated coordinate system. In the infinitesimal case the rotation is by 45o . For the illustration below, we are rotating SR and ST in the same way.

GEOMETRY OF DEFORMATIONS AND ROTATIONS

85

F IG . 4.4. Elongation and compression viewed at 45o is pure shear for an incompressible body. The arrows indicate the principal extensions and compressions. The original coordinate axes are shown unbroken. Those axes at 45o with respect to those in which one has simple extension and compression, are denoted by broken lines.   Exercise 4.1: Confirm that the pure, xz-planar shear λA λA , sketched in Fig. 4.4, is simply a uniform extension/compression viewed at 45o . When at constant volume,  and λ  in the associated diagonal frame. determine the values λxx zz W T , requires the roSolution: Transforming tensors between frames, λ  = W · λ ·W   cos θ sin θ tation matrix W given in 2 × 2 form by − sin θ cos θ . Here we have rotation about   1 1 y by θ = 45o and thus W (π /4) = √1 −1 . Applying this, in the new frame we 1 2 obtain the (diagonal) compression/extension form:    0 λ 0 A+λ ≡ . (4.30) λ = 0 A−λ 0 1/λ    √ A is fixed by Det λ = 1 = A2 − λ 2 . Thus A = 1 + λ 2 . At lowest order λ , when the deformation is small, λ  1, then A ≈ 1. In its original frame λ is recognisable   1 λ λ 1 . In the new frame, it corresponds to an

as a familiar form of pure shear

  extension λ  (= A + λ ) along x and a compression  λ (= A − λ ) along z , at  1/  constant volume, see Fig. 4.4. Confirm that Det λ = 1 in the new frame. It   must of course be, since Det λ is a rotational invariant of λ . Infinitesimally,   λ  ≈ 1+0 λ 1−0 λ .

When the original diagonal elements are not equal, then the principalaxesare not at 45o and 135o , but must still be orthogonal, since λ is symmetric. For λA λB , with A > B say, it is simple to show that the principal extensions are: 

λ± = 12 A + B ± (A + B)2 − 4 . (4.31) To derive λ± we have invoked incompressibility, that is λ 2 = AB − 1. The angles, χ± , of the principal extensions to the x-axis are:

86

CLASSICAL ELASTICITY



2(AB − 1)  cos χ± = ± 2 (A + B) − 4 ∓ (A − B) (A + B)2 − 4

1/2 .

(4.32)

We spare the reader the tedious algebra required to confirm that these two angles differ by π /2 (as they must if they are principal directions of a symmetric λ ). Exercise 4.2: As an example of shear strains in the context of rubber elasticity, evaluate the energy of the pure, planar x − z shear of magnitude λ (given as both a deformation gradient and the equivalent small strain):   λ A λ A−1 → ε= . λ= A−1 λ A λ Suppress the unchanging y components for compactness.    2 2 λ Solution: F = 12 µ Tr λ T · λ requires λ T · λ = A2+ λA shows directly F

2λ A A2 +λ 2

 . Taking the Trace   λ = A2 −

= µ (A2 + λ 2 ) = µ (1 + 2λ 2 ). Incompressibility, Det

λ 2 = 1, fixes A. The modulus associated with this distortion is 4µ which derives  from the curvature of the free energy density, ∂ 2 F/∂ λ 2 λ =0 at zero shear, that is at λ = 0. Viewed as a simple extension/compression λ  in a frame a 45o , that is λ  =    λ 0  0 1/λ , repeat the above exercise and find the free energy. Evaluate at zero

extension (λ  = 1) the rate of change of stress with extension, ∂ σxx /∂ λ  )λ  =1 , and correcting for area changes, show that again the modulus against this kind of extension and compression is 4µ .

Simple and pure shears, and rotations   Consider a deformation λ S = λa λa acting in the xz-plane. Then λyy = 1 and we suppress the y parts of λ in the discussion. Figure 4.4 shows the body initially and after a pure shear. At 45o these are shown to be simple extensions and compression, at least approximately for thisfinite  deformation. Figures 4.5(a) and (b) show simple   a b 1λ 1 0 shears λ = 0 1 and λ = λ 1 . For infinitesimal λ , we can regard the pure shear   λ S = λa λa , Fig. 4.5(c), as being made up of two such simple shears in the two relevant directions:     1 0 0 λ 0 0 a λ ≈ + + (4.33) 0 1 0 0 λ 0 λ a The entry a ∼ 1 + O(λ 2 ) in λ S ensures volume conservation:    Det λ S = 1 = a2 − λ 2 → a = 1 + λ 2

(4.34)

Material points R o at a θo not 45o , see Fig. 4.4, are transformed to points R which have a different final angle which increases, θ > θo , for initial angles θo < 45o . The

GEOMETRY OF DEFORMATIONS AND ROTATIONS

l

l

87 x’

z’

l/2

z

45o

(a)

(b)

x

l/2

(c)

F IG . 4.5. The two simple shears (a) and (b) that add to give the pure shear of Fig. 4.4. The simple shear can also be decomposed into the rotation and the pure shear components (c): One can view small simple shears λ as pure shears in a coordinate system (x -z ), followed by a rotation of - half the shear angle (λ /2) from the original coordinate system (x-z). angle diminishes, θ < θo , for initial angles θo > 45o . Thus vectors R o are drawn toward the extension diagonal and repelled from the compression diagonal. Overall, there is no rotation of the body. Recall Chapter 1, where a penalty D1 for director rotation relative to body rotation of the solid was anticipated. We can regard the D1 coupling as being that of the director to the anti-symmetric part of deformations. Figure 4.4 and the above discussion of the attraction of R o toward the extension axis suggests another coupling, D2 of n to λ , but this time to λ S rather than to λ A . This decomposition of the coupling will be revisited in Chapter 10. Simple shear has been seen to possess a component of pure shear. Additionally, simple shear has a component of rotation. For example in the limit of small distortions λ:    1 λ 1 0 λ /2 λ /2 = + (4.35) 0 1 1 λ /2 0 −λ /2 (4.36) ≈ W −λ /2 + ε λ /2 . Thus, λ a of Fig. 4.5(a) is a rotation W −λ /2 through an angle −λ /2 and a pure shear

ε λ /2 of amplitude λ /2, see Fig. 4.5(c). Recall ε λ /2 = 12 (∂ ux /∂ z + ∂ uz /∂ x) is the (symmetric) strain tensor. Simple shear is not so simple as pure shear! For finite deformations the decomposition is more complicated, but it is still true that any non-singular, non-symmetric λ can be broken down into products of a symmetric and an anti-symmetric deformation, in effect a pure shear preceded or followed by a body rotation: V λ = λ L ·V

or U · λ R

(4.37)

where the antisymmetric components λ A have been restricted to pure rotations, denoted as before by V or U depending upon which space they act on. The form of the accompanying symmetric deformations λ S will also depend on whether they precede or follow

88

CLASSICAL ELASTICITY

rotations and have been denoted by λ R and λ L because they yield the Cauchy-Green tensors C and B respectively. Since they are symmetric, a frame can be found in which they are diagonal, and in which the deformations are therefore simple:   λ1 0 0 λ S =  0 λ2 0  thus R1 = λ1 Ro1 , R2 = λ2 Ro2 , R3 = λ3 Ro3 . (4.38) 0 0 λ3 Hence all deformations are extensions (λi > 1) or compressions (λi < 1). Thus λ R and λ L are the right and left stretching tensors, respectively. An example of such a diagonal frame is in Fig. 4.4 at 45o for infinitesimal pure shears. Exercise 4.3: Show that simple shear can be broken down into a combination of symmetric distortion and body rotation, that is:    1 cos Ω sin Ω λ 2 1 λ S ·√ = 2 ≡U ·λ 0 1 − sin Ω cos Ω 4+λ2 λ 2+λ √ 2 where sin Ω = λ / 4 +  λ . Check that the symmetric shear tensor is volumepreserving, that is Det λ S = 1. Note that for small amplitudes the rotation is Ω ∼ λ /2, see Fig. 4.5. For large shears the rotation is Ω = π /2. Solution: Simple multiplication out of the rotation and simple shear matrices confirms the result. Exercise 4.4: Break down a general xz-distortion λ into a combination of symmetric distortion, λ s , followed by body rotation, U Ω , about y. Solution: Let λ be broken down as:

λxx δ

δ λzz



=

cos Ω − sin Ω

sin Ω cos Ω



a d

d b

 ≡UΩ ·λS .

Multiplying out the right-hand side and comparing with the corresponding elements on the left-hand side, one obtains four simultaneous equations. Eliminating between them in pairs one obtains equations for each of a and b and two equations for d. Equating the latter two, one obtains for the rotation: tan Ω = (δ − δ  )/(λxx + λzz )

(4.39)

 Thus sin Ω is equal to sin Ω = (δ − δ  )/∆ where ∆ = (λxx + λzz )2 + (δ − δ  )2 and similarly for cos Ω in the rotation matrix. Note, there is no body rotation for λ symmetric, that is δ = δ  . Restoring sin and cos to the expressions for a, b and d gives for the symmetric shear tensor:  1 λxx (λzz + λxx ) − δ  (δ − δ  ) λxx δ + δ  λzz . (4.40) λS = λxx δ + δ  λzz λxx (λzz + λxx ) + δ (δ − δ  ) ∆

GEOMETRY OF DEFORMATIONS AND ROTATIONS

89

We shall use this decomposition since in nematic elastomers the rotational component is of great physical significance.

V or U · λ R are examples of the polar decomposition theorem The results λ = λ L ·V (Horn and Johnson, 1991) which holds for non-singular matrices λ . One requires that   Det λ > 0, which means geometrically (i) that deformations do not shrink a body to     a point, Det λ = 0; or (ii) cause the body to pass through itself, Det λ < 0 (Atkins and Fox, 1980). Principal extensions Symmetric distortions λ have no rotational component and their principal extensions and compressions are perpendicular to each other and at ±45o to the laboratory axes (exercise 4.1). We have seen that any asymmetry in λ implies there is a component of pure (irrotational) shear plus a certain degree of local rotation. In nematic rubber elasticity it is useful to identify these components separately. It is also interesting to examine the principal directions of extension/compression of an arbitrary λ . These are the directions in which an element of the body is extended or compressed without being realigned; in the general case they are not perpendicular to each other. Such directions are also of significance for the rheology of polymers in flow fields with extensional components.3 Exercise 4.5: Find the eigenvectors e ± and eigenvalues λ± of a deformation λ =     √ A λ  λ  A in the xz plane, where A = 1 + λ λ ensures Det λ = 1. Solution: The eigenvalues are the roots of the characteristic equation for the nonsymmetric matrix λ :   Det λ − λ± δ = λ±2 − 2Aλ± + A2 − λ λ  ≡ λ±2 − 2Aλ± + 1 = 0 ,   also using Det λ = 1. This quadratic equation does not always have a solution (as not every non-symmetric matrix can be diagonalised), e.g. the case of simple shear, λ  = 0 is degenerate. When the solution exists, the principal extensions(+) or compressions(–) are:

λ± =

 √ 1+λλ ± λλ

and the corresponding (normalised) eigenvectors are:    1 1, ± λ  /λ . e± =  1 + λ  /λ The angle between these principal directions is: 3 We

thank S. Chu for helpful comments on this issue.

90

CLASSICAL ELASTICITY

χ = cos−1 (ee+ · e − ) = cos−1



λ −λ λ +λ

 .

The symmetric case λ = λ  clearly yields perpendicular directions. The limit of simple shear, λ  → 0, with pure shear and rotation in equal measure in the infinitesimal case, is degenerate.

We shall describe special, soft distortions of nematic elastomers where rotations are important. When determining directions associated with pure extension/compression the diagonal elements of λ will not be equal as in this simple example. 4.3.3 Square roots and polar decomposition of tensors The square root of a tensor is, most simply, a tensor that when multiplied by itself returns the original tensor. Representing tensors by matrices and adopting the principal frame, tensor multiplication is achieved by multiplying the corresponding diagonal elements. It is then easy to construct the square root tensor by taking the square root of each diagonal element. One can then subsequently rotate to a general coordinate frame. This procedure is only valid for the roots of symmetric tensors. More general results are required when such tensors are used to describe soft and semi-soft rubber elasticity. By the polar decomposition theorem we discuss below, we can find a symmetric tensor followed by a rotation to represent any deformation. In fact this theorem and the roots of tensors are deeply related. We conclude with the connection between decompositions of non-singular matrices in general and a practical algorithm for finding the pure shear plus rotation for general deformations. Exercise 4.6: Show in a general frame, where a symmetric matrix X is not necessarily diagonal, that the product of roots is as expected: X 1/2 · X 1/2 = X . (We shall only require the roots of symmetric tensors in this book.) Solution: Let U be the rotation that takes one from the current frame to the prin  x1 0 0 cipal frame where X is represented by the diagonal matrix X D =  0 x2 0 . 0 0 x3 The definition of the root in the diagonal frame is: √  x1 0 0 √ 1/2 1/2 1/2 x2 0 . X D = X D · X D , with X D =  0 √ x3 0 0 Post- and pre-multiply this equation by U and U T to give on the left hand side 1/2

X = U T · X D · U . On the right-hand side insert between the factors of X D the unity in the form δ = U · U T . Judiciously inserting some brackets to guide the eye, one obtains: 1/2

1/2

U = (U U T · X D ·U U ) · (U U T ·X X D ·U U ), U T · X D ·U  ! " δ

which can be rewritten as

COMPRESSIBILITY OF RUBBERY NETWORKS

91

X = X 1/2 · X 1/2 . The result is evidently true in all frames.

A sketch of the proof of the polar decomposition theorem brings us into contact with the more general properties of the square roots of tensors. Take C = λ T · λ , which       2 is symmetric by construction. Since Det C = Det λ T · λ = Det λ T > 0, the µ matrix C is also non-singular. A non-singular matrix has at least 2 non-similar roots, A and B are non-similar if they cannot where µ is the number of distinct eigenvalues (A be related by A = W · B · W  where W and W  are proper, orthogonal matrices). There are also not more than 2ν non-similar roots, with ν the number of Jordan blocks in C . At least one of these roots is a polynomial in C . Since in this case C is symmetric, then so will the polynomial roots be symmetric. Denote such a symmetric root by λ R . Now construct a matrix U = λ · (λ R )−1 . Then test whether this matrix is proper orthogonal U: by considering U T ·U  −1  −1  −1  −1 U = λR C· λR ·λT ·λ · λR = λR ·C U T ·U  −1  2  −1 ≡ λR · λR · λR =δ

(4.41) (4.42)

which confirms that U is indeed orthogonal. Inverting the definition of U , we recover the desired decomposition λ = U · λ R . Thus the utility of constructing Cauchy tensors from λ and that of roots of tensors are intimately related. 4.4

Compressibility of rubbery networks

Thus far we have assumed that therubbery  network is incompressible, that is  the  free energy density (3.26) F = 12 µ Tr λ T · λ is subject to the constraint Det λ = 1, and then examined the effects of remaining deformations at constant volume. Here we shall look at this question more closely. The nematic case is examined separately in Sect. 10.1.1 because of certain important differences. We conclude here that volume relaxation, both after formation and after imposed strains, is small – of order µ /B˜ ∼ 10−4 times the imposed strains. The elastic free energy density F, and equally its nematic extension (10.1), make no sense without the additional bulk-modulus term. Independently of the configurational entropy of polymer chains, volume changes are penalised by: 2  Det[λ ] − 1 .

1# 2B

(4.43)

Without it, the minimum of the elastic energy F ∝ λ 2 would be reached at a deformation λ equal to zero. Obviously, the state of no deformation corresponds to λαβ = δαβ , a unit matrix, but not a zero matrix! The large bulk modulus B# ∼ 109 − 1010 J/m3 in (4.43) is determined by interatomic forces in any condensed matter system. It resists the

92

CLASSICAL ELASTICITY

compression of any material,  including rubber. This large energy penalty constrains the  value of the determinant, Det λ ≈ 1. In Sect. 3.3 we analysed rubber elasticity with a rigidly volume-conserving constraint, arriving at the classical form (3.29). However, when the bulk modulus B˜ is not exactly infinite, one cannot exclude small volume changes (which will turn out to be of order µ /B˜  1). We must eliminate the possibility that the corresponding free energy penalty from the compressibility term (4.43) could be of the same order of magnitude as corrections to the rubber-elastic contribution, F, eqn (3.29). Let us separate the deformation tensor into two parts, with significantly different physical origin. Let us assume that, when the network is first formed, a small change of volume occurs. (Recall that the derivation of rubber-elastic energy density Rf ) quenched at formation, (3.26) involves the averaging over a chain distribution p(R see Chapter 3). After the network equilibrates, one may apply an additional, externally imposed deformation λ˜ , thus probing the actual rubber-elastic response. The full strain tensor, λ tot , to be inserted into F and into eqn (4.43), can be presented as a product of two sequential deformations

λ tot = λ˜ · λ (o) bulk

with λ (o) = (1 + α (o) )δ , bulk

(4.44)

where the, presumably small, correction α (o) reflects the initial volume change at formation represented by λ (o) . We now need to find the equilibrium value of this correction bulk and the effective elastic free energy of an equilibrated rubber. For this purpose, we can initially assume that there is no additional, externally imposed strain, λ˜ = δ . The comis in any bined rubber contribution F (best taken in the diagonal form (3.27) since λ (o) bulk case an isotropic deformation) and the compressibility contribution (4.43) are together: 2  (o) Fiso = 32 µ (1 + α (o) )2 + 12 B# (1 + α (o) )3 − 1 .

(4.45)

˜ we find by minimisation of eqn (4.45) the equilibrium value of For small ratio µ /B, spontaneous compression of the polymer network at formation: (o)

µ ˜ 2 + O(µ /B) 3B˜ ∼ 10−4 .

αeq = −

(4.46)

This effect is physically inherent in the whole construction of the classical rubber-elastic model: the network of phantom Gaussian chains, described by (3.26), would like to collapse to a single point (which maximises its configurational entropy), but the bulk energy (4.43) resists such a compression – the result of balancing these effects is the indeed small initial network compression (4.46). The effective rubber-elastic energy density that arises in an equilibrated network in response to the subsequently applied strain, λ˜ , is obtained by substituting the spontaneous compression (4.46) into the full free energy density, obtained for the complete strain λ tot , eqn (4.44). The result is:

COMPRESSIBILITY OF RUBBERY NETWORKS

F ≈ 12 µ

2  µ 2  ˜ T ˜  1 #  µ 3 1− Tr λ · λ + 2 B 1 − Det[ λ˜ ] − 1 . 3B˜ 3B˜

93

(4.47)

The energy expression shares the form of the initial rubber-elastic energy; the small correction of order µ /B˜ is insignificant but ensures there is no further volume change in the equilibrated network unless an external deformation is imposed. In order to complete the analysis, let us again separate the applied strain tensor into two parts, one describing a further possible small change of volume, (1 + α )δ , and the other, λ , the remaining volume-preserving part. In a typical experiment, one may externally impose some components of strain, for instance, an extension λzz = λ , and monitor the resulting stress. However, if other components of strain tensor λ are not mechanically constrained (and this includes the total volume of the equilibrated network), the elastomer will adopt the optimal, energy-minimising values for them – which we now need to find. Thus

λ˜ = λ · (1 + α )δ .

(4.48)   The remaining portion of deformation satisfies the condition Det λ = 1, i.e. it is purely volume conserving. We now find the equilibrium value of the bulk compression and the effective rubber-elastic free energy in response to the imposed strain. The effect of isotropic compressibility factorises in eqn (4.47) as before:    µ 2 (4.49) (1 + α )2 Tr λ T · λ Fiso = 12 µ 1 − 3B˜   2 µ 3 + 12 B# 1 − (1 + α )3 − 1 3B˜ and gives, at leading order in µ /B˜  1, the equilibrium value, αeq , of the additional compression:   µ  , or (4.50) αeq ≈ 1 − 13 Tr λ T · λ 3B˜    µ Det[λ˜ ] ≈ 1 + 1 − 13 Tr λ T · λ B˜ → 1 at λ = δ . This observation reflects what we already know about rubbery polymer networks. The thermodynamic equilibrium in an ideal network is determined by the entropy, i.e. the number of possible conformations the chain strands may adopt. On crosslinking, several constraints have been introduced and, as a result, a small decrease of volume has occurred, described by the relationship (4.46). When the network is further deformed, the number of possible conformations is reduced, which causes the usual rubber-elastic response, but also an additional small bulk compression. Note that, as long as µ /B˜  1, the dependence of αeq on λ remains harmonic even for large deformations λ . Substituting this back into the full free energy density (4.49), we obtain the effective rubber-elastic energy that arises in an equilibrated network in response to the isolated

94

CLASSICAL ELASTICITY

incompressible part, λ , of applied deformation. Collecting the leading terms, we arrive at the expression      µ 1− (4.51) F ∗ = 12 µ Tr λ T λ Tr λ T · λ + ... 9B˜          µ µ 1 (4.52) Tr λ T · λ 1 − Tr λ T · λ − 1 ≡ 12 µ 1 − 3B˜ 3B˜ 3 where, by construction, Det[λ ] ≡ 1 and in most cases one can safely neglect the small corrections of order µ /B˜  1. We have now turned a full circle, back to eqns (3.26), (3.27) and (3.29). The second form of  eqn (4.52)  is identical in form to (3.26),but with  µ a shear modulus renormalised to µ 1 − 3B˜ and with corrections to the Tr λ T · λ     ˜ vanishing as 1 Tr λ T · λ − 1 as λ → δ . It turns out form, again of order µ /B, 3 that it is indeed possible, and correct, to apply the condition of rigid incompressibil√ ity λxx = λyy = 1/ λ and work with the effective rubber-elastic energy density F ∗ = 1 2 ˜ 2 µ λ + 2/λ , as long as one neglects network compressions of the order µ /B  1. If, in some circumstances, one needs to specifically examine the effects of bulk compressibility, the full rubber-elastic energy would have to be used, 2  µ 2  ˜ T ˜  1 #  µ 3 1 ˜ Tr λ · λ + 2 B 1 − Det[ λ ] − 1 , (4.53) F = 2µ 1− 3B˜ 3B˜ with no condition on Det[ λ˜ ] except the minimum of total energy. When solvent is present, as in nematic gels, the compressibility becomes the osmotic compressibility which does not present such a rigid constraint on volume change. Interesting gel properties arise in the nematic case, including order-volume discontinuities as temperature changes. In the case of small symmetric strains, λ˜ i j = δi j + εi j with εi j  1, the linear elastic energy density is recovered: 2  2 (4.54) Fiso ≈ µ εi j − 13 Tr (εi j ) δi j + 12 B˜ Tr (εi j ) without the irrelevant additive constant 32 µ . This expression is obtained and discussed in the classical literature on the theory of elasticity (Landau and Lifshitz, 1986). We only need to make contact with standard definitions: the Young modulus E= and the Poisson ratio

ν=

9B˜ µ ≈ 3µ 3B˜ + µ

1 3B˜ − 2µ 1 µ ≈ − . ˜ 2 3B + µ 2 2B˜

As promised, the Poisson ratio deviates from the incompressible value ν = 1/2 by ˜ ∼ 10−4 . O(µ /B)

5 NEMATIC ELASTOMERS In this chapter we discuss for the first time examples of nematic elastomers which could be formed by crosslinking either or both main chain and side chain nematic polymers. Average molecular shape of long polymer chains, and its resistance to change, is at the heart of rubber elasticity. One then suspects that crosslinking liquid crystalline polymers, themselves capable of spontaneously changing shape already in the melt, will yield elastic solids displaying complex effects of coupling between the mechanical state of the sample and the state of its nematic order. Further, the high liquid-like molecular mobility in elastomers will allow the director to easily achieve gross changes in orientation. Since the molecular elongation of polymer chains relates to this direction, nematic rotation will be mirrored by equally gross, macroscopic shape changes at relatively low energy that we shall term ‘soft elasticity’. At the outset, when envisaging nematic polymers, de Gennes (1975) saw that networks would represent the most fertile outlet for the coupling of orientational order to average molecular shape in nematic polymers. As we shall see, this promise has been fulfilled both experimentally and theoretically. The principal phenomena exhibited by these materials will be introduced in this chapter and form a challenge to explain and model in the remainder of the book: 1. 2. 3. 4. 5. 6. 7. 8.

Spontaneous thermal distortions of hundreds of percent in strain. Reversible, optically-induced large strains. Coupling of strains to orientational order, principally to its direction. Stripe modulations and more complex spatial distributions of director. Rotational instabilities and their thresholds. The phenomenon of soft mechanical response. Critical dependence of soft elasticity on formation history. Polydomain-monodomain transition and effects of quenched disorder.

Without special precautions during fabrication, nematic elastomers always form very fine polydomain textures. In contrast to ordinary nematic liquid crystals (including polymers), where the Schlieren textures are only a kinetically delayed state, in elastomers director variation is crosslinked in and thus the polydomain state is a real thermodynamic equilibrium and very difficult to get rid of. The main progress in development and understanding of nematic elastomers has occurred only after a robust method of producing well-aligned monodomain elastomers was developed by K¨upfer and Finkelmann (1991). The following chapters will respectively introduce a simple generalisation of classical rubber elasticity to nematic elastomers, find soft elasticity in this new rubber elasticity, and then solve problems with spatial variation of nematic and elastic response. The 95

96

NEMATIC ELASTOMERS

F IG . 5.1. Typical crosslinking reactions of an acrylate side-chain polymer, illustrated in Fig. 3.4(a). Di-acrylate molecule participate in the formation of two polymer chains, thus linking them at both ends (a), and di-isocyanate molecules react with hydroxy-ester groups copolymerised into the polymer chains, (b). In both cases, the central part of the crosslinking molecule may be mesogenic itself, thus linking the two chains via a rigid rod, (c). methods developed will then be applied to cholesteric rubber elasticity and optics. Contact with usual linear continuum methods will be addressed in Chapter 10 where also the symmetry background to soft elasticity is developed. 5.1

Structure and examples of nematic elastomers

Polymer chains possessing liquid crystalline order were crosslinked to form networks a long time ago (Strzelecki and Liebert, 1973). The motivation was to create rigid matrices with nematic order permanently crosslinked in, analogously to its being frozen in by the formation of a nematic glass (with the crucial difference that the glass could be re-heated into the liquid state, while the densely crosslinked network makes a truly permanent thermoset). Samples were then cut and polished to expose topological defects and other nematic textures to microscopic examination (Bouligand et al., 1974). There might be many other uses for permanently anisotropic amorphous thermosets – e.g. the large body of work on ‘ferroelectric thermosets’, densely crosslinked ferroelectric liquid crystals (Hikmet and Broer, 1991), and on nematic epoxies (Barclay and Ober, 1993). However, not being molecularly mobile and therefore not rubbery, such rigid glasses do not display the rich mechanical and optical phenomena characteristic of nematic elastomers. We shall not discuss these dense networks any further. In thermotropic liquid crystalline elastomers, the ‘macro-Brownian’ motion of chains is prevented by a low density of crosslinks, but locally the polymers possess nearly the same level of configurational freedom as in the melt. Such networks, with their fluidity

STRUCTURE AND EXAMPLES OF NEMATIC ELASTOMERS

97

F IG . 5.2. Polysiloxane side-chain polymer chains of Fig. 3.4(b) crosslinked by bis-alkoxy divinyl molecules reacting with the free Si-H bonds of the two backbones, with (a) a rigid-rod or (b) a flexible central part. (c) the analogous tri-functional molecule linking together three chains. and rubbery response, renewed interest in these solids and has resulted in discoveries of a number of new mechanical effects. Pioneering work (Finkelmann et al., 1981) achieved the synthesis of weakly crosslinked networks from side-chain liquid crystalline polysiloxane polymers. In fact, Finkelmann et al. reported all three main elastomer phases – nematic, cholesteric and smectic. They also made the first experiments on the response of nematic order to applied strain. Excellent review articles (Finkelmann, 1984; Zentel, 1989; Barclay and Ober, 1993) tell the detailed story of this and the subsequent synthesis – we only briefly mention several key materials and techniques used over the years. Methods of crosslinking have varied from chemical, using copolymerisation with a small proportion of reactive groups on a chain and adding bi- or tri-functional crosslinking agents (K¨upfer and Finkelmann, 1991; Kundler and Finkelmann, 1998), to radiation processes using UV light with photoinitiators (Brehmer et al., 1994) or hard gamma-radiation (Yuranova et al., 1997). Acrylate or methacrylate polymer backbones with a number of mesogenic pendants have also been used by different groups to produce a variety of nematic (and smectic) elastomers; see representative examples in Fig. 5.1 (Zentel and Reckert, 1986; Legge et al., 1991; Zubarev et al., 1996). However, acrylate-based polymer chains have certain practical disadvantages, in particular, the high glass transition Tg ≥ 50o C and relatively low backbone anisotropy. Side-chain liquid crystalline polymers based on siloxane backbones, Fig. 5.2, have so far shown more dramatic mechanical properties due to a much higher chain anisotropy and are conveniently liquid crystalline at room temperature (with Tg ≤ 5o C). Lightly crosslinked rubbery networks of main-chain mesogenic polymers, for example, mesogenic epoxy resins prepared over the years (Carfagna et al., 1997; Barclay

98

NEMATIC ELASTOMERS

F IG . 5.3. Crosslinking of main-chain nematic polymers: (a) Vinyl-terminated chains of poly-[1-(4-oxydeca methyleneoxy)- biphenyl-2-phenyl] butyl, shown in Fig. 3.3(b), reacting with the free Si-H bonds of a siloxane chain or ring molecule; (b) Branching of a mesogenic epoxy resin acts as a low-functionality, point-like crosslinking. and Ober, 1993) were more difficult to synthesise. However, such materials are very attractive for physicists because of the expectation of dramatic mechanical effects due to the high anisotropy of their backbone chains. Figure 5.3 shows examples of main chain network formers. In a simple picture of rubbery networks, the crosslinks are often treated as points. Real networks often have large crosslinking groups, either themselves flexible or rigid and rod-like as shown in Figs. 5.1–5.3, and having a substantial restricting influence on their immediate neighbourhood. It appears that the most point-like crosslinks with the least radius of constraining influence are created by the random chemical bonds formed under gamma irradiation. Hard radiation produces free radicals on polymer chains, more or less at random. The chemical bonds are then formed between different chains, making a rubber network out of the initial polymer melt (Yuranova et al., 1997). It takes some fine tuning to have radiation flux strong enough to produce sufficient number of free radicals to create a network, but not too strong so that the chain scission will not be a dominant effect. The coupling of such different links with microstructure to the network elasticity and nematic order will be discussed in more detail in Chapter 8 when describing microscopic origins of random disorder. It is a question intimately related to the question of why the formation history can affect mechanical properties of nematic elastomers. Discotic phases arise when the pendant moieties are discs rather than rods. They instead order in stacks but the elastomers they form are similar in character to nematic elastomers (Disch et al., 1995). The element essential to nematic rubber elasticity is also present in discotics – the shape distribution of the backbone chains is distorted by the ordering of the associated mesogens. The elongation or flattening of the backbones

STRESS-OPTICAL COUPLING

99

F IG . 5.4. (a) The schematic chain structure and coupling to the pendant rods via evenand odd-length spacers. (b) The scheme of elastomer stretching and the taking of X-ray scattering images. (c) X-ray images in two projections for even and odd spacer lengths (stress axis is vertical in the picture plane). Even length spacers (e.g. n = 4) align the nematic director along the stretched backbone. Odd spacers (n = 3) force the pendant rods into the plane perpendicular to the stretching axis. We consider a monodomain case with the side chain director along I. in discotic phases yields the same new elastic phenomena that we describe below for nematic elastomers. 5.2 Stress-optical coupling Whether as side-chain or main-chain polymers, it is important to create anisotropy in the backbone chains since they are in effect the elastic elements of the rubbery network. The ordering of rod-like mesogenic groups creates this anisotropy, either directly as in main-chain polymers, or indirectly by coupling to the backbone via a spacer in sidechain polymers. Circular dichroism (Hammerschmidt and Finkelmann, 1989), as well as stress-optical and X-ray (Finkelmann et al., 1984) studies of the inverse situation, isotropic phases of side-chain polymers under strain, show that the external mechanical field affecting (and aligning) the backbones has a profound effect on the arrangement of mesogenic groups. This proves the presence of a coupling between rod-like side groups and backbone directions. The stress-optical experiments, where one measures the birefringence induced in the initially isotropic phase by applied uniaxial stress, have shown that on stretching the backbones the mesogenic groups may align along the axis of stress, or in the perpendicular plane, depending on the type of spacer connecting them to the chain. For instance, the elastomer shown in Fig. 5.2(a) with the spacer length of n = 3 (or, more generally, odd n) has a preference for the pendant rod to be perpendicular to the siloxane backbone. The same material with n = 4 (even) has a preference for the backbone and rods to have a parallel equilibrium alignment. Figure 5.4 shows this schematically as well as the two types of result expected from X-ray scattering from such polymers under stress. This represents the so-called ‘odd-even’ effect, well studied in the field of liquid crystals and their polymers. In nematic elastomers, where the average backbone anisotropy directly translates into the shape of the whole body and

100

NEMATIC ELASTOMERS Tni

60

(a)

Stress optical coefficient

40

20 Natural rubber

0

-20 (b) -40 T ni 60

80

100

120

Temperature ( C)

F IG . 5.5. Stress-optical properties of thermotropic nematic elastomers in the isotropic phase above their respective transition temperatures, Tni . (a) Even-length (CH2 )n spacers lead to a positive coupling between the pendant rods and the backbone orientation (and has an effect of enhancing the nematic field, thus increasing the Tni ). (b) Odd-length spacers lead to a negative coefficient, somewhat depressing the nematic order and making the rods align in the plane perpendicular to the stretching direction. For comparison, the stress-optical coefficient of linear natural rubber, with no particular nematic interaction, is always positive and 2-3 orders of magnitude smaller. its mechanical response, this effect determines the sign of the stress-optical coefficient. Figure 5.5 summarises the literature (Finkelmann et al., 1984; Mitchell et al., 1992) on this effect. A related odd-even effect of spacer length on the coupling of pendant mesogenic groups to the flexible backbone in side-chain nematic elastomers determines the effect of mechanical fields on the nematic phase transition (Mitchell et al., 1992). The local nematic order parameter Q(T ) in an external field was shown in Fig. 2.6. For even-length spacers, when the mechanically stretched polymer backbone exerts its positive aligning effect, increasing applied stress has the effect of bringing the first-order nematic transition towards the critical point and further into the supercritical regime. It is not straightforward to study this effect experimentally, because the elastomers usually form fine polydomain nematic textures and are not optically transparent below Tni . Nuclear magnetic resonance (NMR) is a reliable method to study the local nematic order of mesogenic groups Q(T ). Optical or X-ray methods by contrast provide a measure of the macroscopic alignment of the nematic director. NMR reveals the bias of orientational motions of selected chemical bonds on the molecule, showing as a characteristic splitting of the resonance peak (de Gennes and Prost, 1994). One can thus examine (Disch et al., 1994) the supercritical behaviour of nematic elastomers under increas-

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101

F IG . 5.6. Order parameter of nematic polymer melt, curve (a) and its corresponding crosslinked network, curve (b), as a function of temperature, from (Disch et al., 1994). Note the distinct features of the first-order transition in the melt, in particular, the jump Qni and the hysteresis on supercooling. In contrast, the crosslinked network has a continuous phase transformation and an overall lower magnitude of Q. ing mechanical load, obtaining the results fully analogous to Fig. 2.6 with stress taking the role of the external field. It is also interesting to compare the local order parameter Q(T ) in a nematic side-chain polymer melt and in the corresponding network made from the same melt. Figure 5.6 (Disch et al., 1994) shows that the presence of crosslink constraints makes the nematic-isotropic first order transition continuous and slightly reduces the order. A more recent study, also by NMR, fully confirms the supercritical nature of this transition (Lebar et al., 2005). In Chapter 8 we shall further discuss this effect, examining the role of randomly quenched impurities (the role played by network crosslinks) in modifying the nature of the nematic order. 5.3

Polydomain textures and alignment by stress

Unless special precautions are taken, liquid crystalline polymers always form a highly non-uniform, polydomain director alignment, often called Schlieren textures. There are different views on the subsequent evolution of such textures. On one hand, by analogy with ordinary liquid crystals always tending towards a macroscopically uniform global equilibrium state, one may regard polydomain textures as a kinetic effect – polymers simply take a much longer time to coarsen their textures. An alternative view might be that there are physical reasons, specific to long-chain polymers, that cause polydomain textures to be the equilibrium state in spite of their evidently high associated Frank elastic energy. Chain entanglements and a frequent possibility of chain branching are two of many possible causes. However, as with ordinary nematic liquid crystals, if one applies a sufficiently strong external field, uniform director alignment can be achieved in liquid crystalline polymers. Figure 5.7 illustrates the alignment of main-chain polymers by strong magnetic fields. Another traditional external aligning field is the effect of strong boundary conditions in rather thin samples.

NEMATIC ELASTOMERS

Average alignment S

102

Time (min)

F IG . 5.7. Alignment of a polydomain main-chain nematic polymer by magnetic field (Anwer and Windle, 1991). The curves show the increase with time of macroscopic alignment (the average director orientation parameter S) after a magnetic field was applied. The plots reveal the ‘field-on’ relaxation time τH ∼ γ1 /δ χ H 2 , eqn (2.40) and the saturation value, S(H). It is then not surprising to find that crosslinked nematic elastomers form polydomain textures too, unless measures are taken during the network formation. The next section describes the procedure which leads to monodomain nematic rubber (K¨upfer and Finkelmann, 1991). If, as a generic example, an elastomer was crosslinked in the isotropic phase with no additional fields imposed (mechanical, electric, or magnetic) – on cooling below Tni it forms an appropriately high local nematic order Q(T ) but the orientation of nematic director n is very non-uniform, polydomain with a characteristic length scale of the texture, ξD , often of order or even smaller than the wavelength of light. As a result, in such a generic system one has the sample completely transparent in the isotropic phase and completely opaque below Tni , due to the strong multiple scattering of light on strongly birefringent domains with randomly varying optical axis n (r). Note that this is not to be confused with the scattering from director fluctuations in ordinary nematic liquid crystals which render these also rather cloudy, even in monodomains. Many attempts at aligning or re-aligning polydomain nematic elastomers with electric and magnetic fields were made over the years, driven by the analogy with the ordinary liquid crystals. Generally, if an elastomer is held by mechanical constraints on boundaries, cf. Fig. 2.9 of Chapter 2, no response has been observed suggesting that the rubbery network presents too much a resistance for E or H-induced local rotation of rod-like side groups. Three notable exceptions all involved swelling an elastomer by nematic solvent, the molecules of which were similar to the side-chain polymer pendant rods. Small pieces of nematic elastomer were freely suspended in such a nematic solvent

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103

(Zentel, 1986), thus providing sample support without rigid mechanical constraints and allowing the solvent to respond to the external field. On application of an electric field, the shape of elastomer pieces noticeably changed. A similar experiment (Barnes et al., 1989) also observed shape changes on application of electric fields, and additionally registered solvent expulsion from the elastomer. For the latter observation side-chain nematic polymers, analogous to the elastomer strands but not chemically linked to the network, were a part of the solvent – one could optically detect the oozing of such free chains out of the elastomer sample on application of electric field above a certain threshold. Free solvent is possibly responding to the aligning action of the external field, thus changing boundary condition for the freely floating elastomer. An imbalance of chemical potential between outside and inside the network was generated when elastic resistance impeded solvent alignment. A comparison of the electrical and mechanical energy densities, 12 δ

0 E 2 and 12 µ respectively, suggests that electrical fields must be far too great to achieve an observable effect. A more subtle explanation is perhaps more likely: there is a drastic reduction of the elastic energy cost of shape change (the soft elasticity of Chapter 7) when director rotation accompanies strain. In another experiment (Chang et al., 1997a) a fixed glass cell constrained the overall shape of a rather weak nematic gel where only under 10% of the mesogenic rods were polymerised and participated in the percolating network. Surface effects aligned the director in the plane of the plates. Application of a perpendicular electric field caused a director rotation, much like in the classical Fredericks effect. However, in striking contrast with ordinary liquid crystals, a field rather than voltage threshold was observed – see eqn (2.28) for the voltage threshold in liquid nematics. The director response was registered only with a small amplitude, never a full 90o swing common in electrooptical nematic cells. Ordinary elastic energy costs prohibit such response – we discuss this effect in greater detail in Sect. 8.1. With the understanding that electric and magnetic fields are generally too weak to cause any significant mechanical distortion of nematic elastomers, the natural choice falls on mechanical stress as the appropriate external aligning field. The alignment of polydomain elastomers by stretching is one of the most characteristic effects in these materials. It is directly observed optically, even without resort to microscopes or polarising light: an initially opaque polydomain sample becomes clear and fully transparent after a certain degree of extension, see Fig. 5.8. This is easily understood: when the director alignment occurs, the strong scattering of light on misoriented domains is no longer present. Since thermal director fluctuations are suppressed in elastomers, the uniformly aligned birefringent medium is transparent to light. See Chapter 10, for details. The first synthesis of liquid crystal elastomers by Finkelmann et al. (1981) was accompanied by this very test: at high temperatures elastomers were isotropic and clear. They became opaque on entering the polydomain nematic state, but cleared again on the imposition of strains that evidently aligned the microdomains. Two other key aspects of this transition are important: the stress-strain anomaly and the rapid, singular increase of macroscopic director alignment above the threshold. For small deformations, below a certain stress threshold, the polydomain elastomer responds mechanically in a linear fashion, as an isotropic rubber (which, of course, the

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NEMATIC ELASTOMERS

F IG . 5.8. Polydomain-monodomain transition in nematic elastomers. A siloxane elastomer, similar to Fig. 3.4(b), crosslinked by a method of Fig. 5.2(b) at high temperature above Tni , is polydomain and opaque, (a). When stretched by approximately 30%, the elastomer becomes aligned and optically transparent, (b). The uniform nematic director n is here along the stretching direction (vertical), because the even-length spacer makes the pendant rods aligned parallel to the backbone. (Images courtesy of H. Finkelmann.) polydomain system is, on a large length scale  ξD ). Figure 5.9(a) shows the stress plateau that occurs when the transition threshold is reached. One can understand the existence of such a plateau (and the associated non-convex elastic energy) in a system with microstructure, when the local director in different domains starts to rotate towards the alignment axis causing local stresses to be relaxed. The other part of Fig. 5.9(b) shows the associated rapid increase in the parameter of average director alignment S =  32 cos2 θ − 12 , with θ the angle between the axis of uniaxial applied stress and the local nematic director. The details and a theoretical description of such a polydomain-monodomain transition induced by externally applied uniaxial stress will be examined in Chapter 8. The irregular structure of domains, with their intimate mechanical contact with each other, causing effects of elastic compatibility, present a great challenge. We wish initially to understand uniform elastomers. Large, monodomain nematic elastomers have been made by two methods.

5.4

Monodomain ‘single-crystal’ nematic elastomers

The thermal and mechanical history of network formation is crucial for the resulting director texture and mechanical response. If, for instance, the crosslinking reaction takes place in the nematic state with the director n of the polymer aligned by some external influence: boundary conditions, magnetic field, mechanical deformation – the companion uniaxial alignment of backbones will be remembered by the resulting network. Mitchell and co-workers (Lacey et al., 1998) aligned a nematic polymer melt with a strong magnetic field and then crosslinked it. The director appeared to be well anchored to the matrix. If heated to the isotropic phase, even for several days, the elastomer would cool to the same nematic monodomain configuration.

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105

F IG . 5.9. Polydomain-monodomain transition on stretching the prolate (even-spacer) siloxane nematic elastomer crosslinked by flexible tri-functional groups, cf. Fig. 5.2(c) (Clarke et al., 1998). The characteristic stress plateau σ = σc is very flat, plot (a); in many other materials the residual slope of stress-strain curve is more substantial, although always much lower than the pre- and post-transition moduli. Plot (b) shows the onset, at a threshold stress σc , and the rapid increase of the average macroscopic director alignment S – its variation is plotted against the measured stress (and not the imposed strain) and turned on its side to show the correspondence between the stress threshold and the plateau. Alternatively, one can adopt a two-step crosslinking strategy (K¨upfer and Finkelmann, 1991). At the first stage, the nematic polymer melt is lightly crosslinked to form a weak gel. This gel is then stretched in a uniaxial fashion, either in the isotropic phase at T > Tni or in the nematic phase (the latter case achieves the polydomain-monodomain transition described in the previous section). Keeping the sample stretched, one initiates a second crosslinking reaction which fixes the enforced uniaxial alignment. Provided the strain imposed between the two crosslinking stages is great enough, the final elastomer will be a nematic monodomain below a clearing temperature, Tni . This is the first example of what we term an elastomer with a complex thermomechanical history, here multi-stage crosslinking where the intermediate states can

F IG . 5.10. A monodomain nematic elastomer prepared by two-step crosslinking and the corresponding polydomain sample, which was not stretched before the second crosslinking stage (K¨upfer and Finkelmann, 1991). The free monodomain elastomer is perfectly transparent , while the polydomain sample is opaque unless stretched.

106

NEMATIC ELASTOMERS N1

Iso

cool

lm cool

l=1

N2

lm F IG . 5.11. Chain shape changes drive the shape changes of the network as a whole. For example, a chain extending from on average a sphere to a prolate spheroid induces a macroscopic elongation λm = L/L0 . For a chain that flattens to an oblate shape, the shape change would be a contraction, λm < 1, along the principal axis, that is along the nematic director n . either be isotropic or nematic. We shall discuss monodomain elastomers of both magnetic and mechanical genesis. They do not appear to be qualitatively different. Figure 5.10 (K¨upfer and Finkelmann, 1991) shows both poly- and monodomain versions of the same elastomer. The polydomain is opaque because of the strong light scattering by random director textures. The monodomain is optically clear, with its birefringence axis (nematic director n ) uniform over the whole sample. This is in contrast to monodomains of ordinary nematic liquid crystals, which are cloudy and turbid, because of director fluctuations. This is our first sign that the nematic elasticity of rubber is very different from liquid nematics. In Chapter 1, page 4, we already briefly discussed this difference between liquid and solid nematics. In elastomers, the director is anchored to the rubbery matrix and there is an energy penalty even for long wavelength (nearly uniform) director distortions. In ordinary liquid nematics these distortions are of vanishing energy and thus uncontrolled, whereupon they strongly scatter light. 5.4.1 Spontaneous shape changes In polydomain nematic elastomers the global average of chain extensions in all possible directions is zero over the whole sample. However, when a monodomain material is prepared, its macroscopic shape is directly determined by the state of nematic order in the system. Orientational order determines chain shape (see Figs. 3.5 and 3.7) and this in turn gives the macroscopic state, see the cartoon Fig. 5.11 (a picture to which we return). This connection is a principal theme of this book and the manipulation of the shape spheroids by temperature or by mechanical, electrical and optical fields is at the root of many the new effects that have been found in liquid crystal elastomers. If we take a side-chain polymer with even number n in their (CH2 )n spacers, the backbone will be extended in a prolate manner along the nematic director, see Figs. 3.5(d) and 3.7(a) [in a main-chain polymer this is the only natural possibility, Fig. 3.5(a)]. One then finds that the overall shape of a monodomain elastomer sample elongates by large

MONODOMAIN ‘SINGLE-CRYSTAL’ NEMATIC ELASTOMERS 0.5

1.5

(b)

Nematic order Q(T)

Relative length L/L0

(a) 1.4 1.3 1.2 1.1 1 20

107

40

60

80

Temperature ( C)

100

0.4 0.3 0.2 0.1 0 20

Tni 40

60

80

100

Temperature ( C)

F IG . 5.12. (a) Change of natural length of a monodomain nematic elastomer with temperature. As temperature changes, so does the nematic order Q(T ), shown in (b), and hence the chain anisotropy. Macroscopic deformation of rubber mirrors the change in underlying conformation of polymer backbone. Elongation on cooling indicates the prolate backbone anisotropy. amounts on cooling from the isotropic to nematic phase, or contracts on heating back to the isotropic state. In Sect. 6.2 we shall examine how the macroscopic shape change, the spontaneous deformation λm , follows the microscopic change in anisotropy of the nematic polymers in the network on, for instance, changing the temperature. The change of shape is not just at the transition, but continues to lower temperatures as the nematic order Q(T ) gets larger and the chains become more anisotropic. The measurements of natural length variation in a freely suspended sample of monodomain nematic elastomer are very easy. In a polydomain elastomer the analogous result may be obtained by extrapolating the high-modulus branch of the stress-strain curve past the aligning transition back to zero stress, Fig. 5.9(a): the point where this linear extrapolation crosses the axis gives the natural sample length if the polydomain material were, in fact, an aligned monodomain (Sch¨atzle and Finkelmann, 1989). Spontaneous deformation is more dramatic in naturally monodomain elastomers (K¨upfer and Finkelmann, 1991). Many experiments have seen this in a variety of different ways (Roberts et al., 1997; Thomsen et al., 2001). One can simultaneously measure (Finkelmann et al., 2001a; Clarke et al., 2001) thermal strains along the nematic director n and order parameter variation, see Fig. 5.12, and conclusively demonstrate their close correspondence. A mixed main chain-side chain monodomain nematic elastomer responding to temperature change is shown in Fig. 5.13(a) as a series of stills. The length changes are now very large, around 300% (Finkelmann and Wermter, 2000; Tajbakhsh and Terentjev, 2001). Note that a weight is lifted and work can be done. The associated quantitative measures of length change are shown in Fig. 5.13(b), increasing loads shifting the spontaneous distortion curves. More recent work on self-assembling thermoplastic nematic elastomers has allowed drawing thin fibers with a very high nematic alignment (Ahir et al., 2006), demonstrating spontaneous length change of 500% and more. In Sect. 5.2 we discussed how the coupling of mesogens can be parallel or perpendicular to the backbone, leading to prolate and oblate order respectively. The above spontaneous distortion examples show prolate (parallel) coupling effects translated into mechanical phenomena. The oblate case, see Figs. 3.5(c) and 3.7(c) is much less com-

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NEMATIC ELASTOMERS 4

Relative length L/L0

(a)

(b)

3.5

15g 3 10g 5g 2.5 0g 2 1.5 1

Isotropic

Heating - Cooling

20

40

Nematic

60

80

100

120

Temperature ( C)

F IG . 5.13. Shape changes in a mixed main-chain side-chain nematic monodomain elastomer, with the director aligned along the vertical axis. (a) Contraction with temperature depicted as a series of stills. (b) Quantitative measure of changing natural length; different curves correspond to the sample lifting an increasing weight (Tajbakhsh and Terentjev, 2001). mon. It has been observed in networks by X-ray scattering (Zentel et al., 1987) where factors beyond the usual odd-even effect in spacer couplings are discussed. Oblate ordering has also been studied in great detail by X-rays, stress-optical measurements, conoscopy and spontaneous distortions (Gleim and Finkelmann, 1987; Hammerschmidt and Finkelmann, 1989; Greve and Finkelmann, 2001). 5.4.2

Nematic photoelastomers

Photoisomerisable rod-like molecules undergo a molecular transition from the trans to cis state on absorbing an appropriate photon, Fig. 5.14. They strongly bend and no longer contribute to the nematic ordering; in fact, they act as an impurity and destabilIntermediate

Nematic

Isotropic

Spontaneous decay

UV

D

365nm

fc

hw¢

T O or UV (a)

465nm

trans

hw cis : I¢

Stimulated decay

:I

Excitation

f

trans

(b)

F IG . 5.14. (a) Scheme of the effect of trans-cis isomerisation on the nematic order. Bent cis rods dilute the straight trans rods and reduce the order. In the case of azobenzene molecular group, the resonant photon wavelength is 365 nm; the reverse cis-trans isomerisation occurs either spontaneously, as thermal relaxation, or can be stimulated by illumination at a wavelength of 465 nm. (b) Potential energy profile of a photo-sensitive moiety showing the cis and trans states and the intermediate state first reached on photon absorption.

MONODOMAIN ‘SINGLE-CRYSTAL’ NEMATIC ELASTOMERS T=25C T=30C T=35C T=40C

Relative length L/L

1.3

109

1.3

1.2

1.2

1.1

1.1

(b)

(a) 1

1

0

20

40

60

80

Irradiation time (min)

100

20

30

40

50

60

70

Temperature ( C)

F IG . 5.15. (a) The contraction of a nematic photoelastomer after it was exposed to UV radiation at various temperatures, from (Finkelmann et al., 2001). (b) The underlying thermal contraction L(T )/Lo of this photoelastomer: the temperatures, at which the UV-experiments in plot (a) were conducted, are labelled by arrows. ise the nematic order. An example of such a photosensitive azobenzene rod-like unit is shown in Fig. 3.11. In effect illumination plays a role equivalent to temperature elevation in reducing nematic order. One might then expect UV illumination to induce contractions in nematic elastomers containing such rods analogous to those observed thermally, which is indeed observed (Finkelmann et al., 2001; Hogan et al., 2002). Figure 5.15(a) shows a sizeable (> 20%) photo contraction and subsequent dark-state thermal recovery, while Fig. 5.15(b) shows the same photoelastomer exhibiting the usual uniaxial thermal contraction, for comparison. In this photoelastomer (Finkelmann et al., 2001), the rods that bend and thus disrupt the order are the crosslinkers. The complementary stress experiment is shown in Fig. 5.16. Here the photoelastomer is clamped. On illumination, the natural length decreases as in Fig. 5.15. Contrac-

F IG . 5.16. (a) The stress (force per unit area) produced by a clamped photoelastomer on irradiation, at different temperatures, from (Cviklinski et al., 2002). The stress is scaled by its maximum value at saturation, illustrating the difference in kinetics. (b) The repeatability of photo-mechanical effect, on several cycles of UV-irradiation and relaxation in the dark.

110

NEMATIC ELASTOMERS

tion of the natural length while the actual sample length is fixed represents an effective elongational strain which generates a stress. This increases with illumination until saturation is reached. The plot indicates how the rate of this reaction depends on temperature, while the saturation level of stress remains roughly the same for the same radiation intensity. Recovery follows when the UV radiation is removed. The forward reaction trans to cis rate for the population of molecules depends upon the rate of delivery of photons to the azo-bonds. The intensity of light and absorption coefficient at the resonant frequency determine this. The back reaction can be spontaneous, that is thermally activated in the dark, or stimulated by illumination by light in resonance with the cis bond. The rate for thermal recovery for the azobenzene molecular moiety of Fig. 3.11 will later be seen, Fig. 6.5(a), to be about 180 minutes. By variation of the units on either side of the azobenzene unit, for instance by straddling it with a donor acceptor pair, the thermal rate can be reduced to seconds (Camacho-Lopez et al., 2004). The photo-stimulated back reaction rate, deduced by the time taken to approach a photostationary state, is about 13 minutes for the same molecules (Finkelmann et al., 2001). Again, this rate can be tailored between fractions of seconds and hours. Returning our attention to unconstrained samples, not clamped in a device, the response of a photoelastomer to illumination depends on the penetration depth of light. If the material is heavily loaded with photochromic molecules, the light will be absorbed in a relatively thin layer on one side of the sample facing the light. In that case one expects, and indeed discovers a significant bending of photoelastomers due to the gradient of contractile strain across the sample thickness, associated with the gradient of light intensity in the sample, Fig. 5.17. How thin the absorption region must be relative to the elastomer thickness to optimise bend is an interesting question (Warner and Mahadevan, 2004). Too thin means that most of the volume is unaffected by the light and resists the contraction of the upper skin and thus also frustrates bend. Too thick, and the elastomer contracts uniformly with depth and there is overall contraction in favour of bend. Theory suggests that the absorption length optimal for bend is about 1/3 of the sample thickness, in the case of exponential decay of intensity. However, there are many possible sources of non-linearity and much more experiment and theory are required. So far we have considered monodomain elastomers – for instance in Fig. 5.17 the director is along the long dimension of the sample. The contraction in this direction determines the direction of bend. Polydomain nematic polymer glasses have been shown

F IG . 5.17. Bending of a nematic elastomer doped with a photoisomerising rod-like dye molecules. The full range of motion, on illumination, is achieved in fractions of a second (image: P. Palffy-Muhoray).

FIELD-INDUCED DIRECTOR ROTATION

111

(Yu et al., 2003) to bend too, the direction of bend being in the direction of the light’s polarisation. The direction of bend readily changes as the plane of polarisation of the light is rotated. Such photo-actuation in the equivalent elastomer polydomains would offer a greater degree of control than in the monodomain case. The specificity of bend direction to polarisation suggests strongly that such mechanical response is indeed due to photo-response and not simply a contraction in response to optically-generated heat which would be nugatory in a polydomain. The mechanism for contraction of polydomain photo-elastomers is thought to be subtle (Corbett and Warner, 2006). Contraction cannot be monotonic with increasing light intensity since the isotropic sample achieved at very high light intensity must have the same shape as its polydomain parent state that too is effectively isotropic. Contraction by director rotation of the passive domains with director at an angle to the light polarisation to accommodate those domains affected by the light is a possibility – the following section addresses the role of director rotation and it will be a recurrent theme in all of this book. More detailed experiments, possibly NMR, are suggested. Still more subtle control of mechanical bend in response to light has been achieved in nematic polymer glass beams where, for instance, the director is splayed in the bulk because it has planar alignment at one surface and homeotropic (normal) alignment at the other (Mol et al., 2005). Now, independently of from what side the beam is illuminated, the bend is towards the side with planar alignment since it is predominantly at that surface that the light can be absorbed. The dye follows the director’s local alignment and therefore finds itself at the lower surface in a plane the encompasses the light’s electric vector. One can also twist the director in the plane of the sample on traversing from one surface to another (Harris et al., 2005) and with light induce spectacular curling at an angle to the long axis of the sample. Mol et al compare the two mechanical responses. Another recent advance in this field has been in doping an ordinary nematic elastomer with a photochromic dye, molecules of which perform the same trans-cis isomerisation as illustrated in Fig. 5.14. There is a range of such dyes (many still based on an azobenzene core, but with various added donor-acceptor groups) that absorb in various parts of visible spectrum. New experiments on such dye-doped nematic elastomers (Camacho-Lopez et al., 2004) have demonstrated large-amplitude photo-induced bending, as well as dramatically increased speed of response. Section 6.4 explores the physics involved in photoisomerisation and the resulting mechanical deformations of such elastomers. The analogy between thermal and photo effects through their common influence on nematic order will be explored along with the dynamics of photoactuation. The thermal phenomena of Sect. 5.4.1, and the photo effects of Sect. 5.4.2, contrast greatly with the results of changing temperature and illumination in classical elastomers, Sect. 3.4. We explore the reasons for this in Sect. 6.2. 5.5

Field-induced director rotation

With all liquid crystalline systems it is always interesting, and often profitable, to induce a director rotation by external fields. The Freedericks effect of Sect.2.6 is an archetypical example, and also a foundation of liquid crystal displays. There have been many attempts, over the years, to induce an electrical response in liquid crystal elastomers.

112

NEMATIC ELASTOMERS

Interesting effects were seen in cholesteric and especially smectic C elastomers (see Fig. 13.6 for example), but the decisive experiments in nematic rubber have taken a long time to arrive. The reasons, discussed at greater length in Sect. 7.4.4, page 178, where electric fields are indeed seen to change an elastomer’s shape when the special combination of shears discussed below are able to develop. The response of nematic rubbers to mechanical deformations is very rich, in part because there are many possible components of strain tensor available, in contrast with just a single vector of electric field. Consider a sample having its shape changed by one component of strain being imposed upon it. It could react by changing the shapes of its constituent chains away from the distribution currently natural. As in a simple rubber, the shift away from the optimal distribution lowers the entropy and therefore raises the free energy. A nematic rubber could, however, adopt another strategy. Since the network chains are not naturally isotropic, rotating chains would cause a macroscopic shape change. Rotating a distribution of chain shapes without distorting it would cost no free energy. There is clearly an advantage in changing shape by, as much as possible, rotating the distribution of chains. If the strain accompanying rotation has a component matching the imposed component, the imposed distortion will be without energy cost. Macroscopically, rotation of anisotropic chains implies a rotation of the director. This was first seen when extensional strain was imposed perpendicular to the initial director of a polyacrylate nematic elastomer (Mitchell et al., 1993). We shall often revisit this stretching geometry in later sections of this book. As sketched in Fig. 5.18, a strip of elastomer is gripped at its ends and extended along its length. The director is initially perpendicular to the stretch direction. In a generic experiment producing the director rotation by the perpendicular extension, strain at first is resisted as in conventional rubbers, with no director rotation. When a critical deformation is reached, the director starts to rotate. At large deformations, well above the threshold, the material has its director aligned along the stretching axis. This is a reversible state: when the deformation is removed the elastomer returns to its natural equilibrium shape and the accompanying director orientation returns to n o . In the experiments of Mitchell et al., the director rotation was reported discontinuous: at a threshold strain the director n jumped toward the extension axis, see Fig. 5.19.

n

O

q

n l

z x

F IG . 5.18. A monodomain (single crystal) nematic elastomer extended perpendicular to its initial director, n o (coordinate axis z). The principal extension axis is along x, by λ = L/L0 , and the resulting director rotation in the plane of the paper. Shear deformation must accompany the director rotation, see Chapter 7.

FIELD-INDUCED DIRECTOR ROTATION

113

q (deg)

no

n

Director rotation

q

l

F IG . 5.19. The discontinuous rotation of the director through an angle θ on imposition of a perpendicular strain λ (Mitchell et al., 1993). The solid line gives the theoretical curve for a pre-tilt of 2o . The X-ray image and the sketch on the right suggest how the angle was measured.

Director rotation

q (deg)

100

q , deg

no

n

80

q

60 40 20

l

0 1

1.1 1.2

1.3 1.4

1.5 1.6 1.7 1.8

F IG . 5.20. The plot of director rotation θ after a strain λ was applied perpendicular to the original director n o , see Fig. 5.18. The several curves correspond to different crosslinking densities of the same polymer, and therefore also to different thermo-mechanical histories, or to elastomers of differing chemical composition (Kundler and Finkelmann, 1995). The X-ray image and the sketch on the right suggest how the angle was measured and illustrates the two opposite directions of rotation in stripes. central remnants around the beam stop are of no significance. Polysiloxane elastomers stretched in exactly the same way gave instead a continuous director rotation after a strain threshold λ1 (Kundler and Finkelmann, 1995). Their rotation followed a characteristic path θ (λ ) until at a certain higher elongation, λ2 , it was complete when the director had rotated by θ = 90o , see Fig. 5.20. It was found that the transverse stretch has generated director rotations in parallel bands, or stripes, taking opposite senses in neighbouring bands, in effect generating stripes of mechano-optical response, see Fig. 5.21 and the X-ray image in Fig. 5.20 for illustration. One can pic-

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NEMATIC ELASTOMERS

F IG . 5.21. Striped director texture in response to perpendicular extension in the experiments of Fig. 5.20. The extension is along the stripe direction; the initial director is perpendicular to the stripes. The stripes are regions of oppositely rotated nematic director, n , viewed through crossed polars. ture the continuous rotation generating mechanical distortion at low energy cost, see Fig. 5.22. The natural long axis of the body, as defined by the principal axis of the molecular shape distribution, rotates in an attempt to follow the apparent extension axis. To the extent that macroscopic shape change can thus be imitated, there is no real accompanying distortion of polymer shapes. We shall explore these key ideas of ‘soft elasticity’ in the next chapters, developing a theory of nematic rubber elasticity. The large body of data in Fig. 5.20 will then collapse onto one characteristic master curve θ (λ ), typified by the solid line on the figure. Spontaneous shape changes, molecular shape anisotropies and strain-induced director rotation have already hinted at an even more exotic phenomenon – shape change at very little energy cost must sometimes be possible. There is no rubber-elastic penalty for rotation of the distribution of chain shapes – it is distortion away from the optimal distribution of shapes that costs energy. However, shape change without energy cost is the hallmark of a liquid, not a solid and indeed stress-strain measurements on nematic elastomers show signs of a confusion between liquid and solid. Nematic elastomers can be thought of as solid liquid crystals, in as much as rubber is actually a solid. Here, however, we see the partial liquefaction of such solid liquid crystals, see Fig. 5.23 where some samples extend over many tens of

F IG . 5.22. If mechanical deformation of an ideal network can accommodate the rotations of the chain distribution without its distortion, such deformation takes place at no energy cost.

Stress s e (MPa)

APPLICATIONS OF LIQUID CRYSTALLINE ELASTOMERS

115

0.09

0.06

0.006

se

0.005 0.004

0.03

0.003 0.002 0.001 0.000

1.0

1.15

1.3

l 1

1.1

1.2

1.3

1.4

1.5

1.6

1.45

F IG . 5.23. Nominal elastic stress σe is plotted against uniaxial extension λ applied perpendicular to the nematic director.
and : elastomers crosslinked in the nematic phase.  and 2: crosslinked in the isotropic state. In all cases crosslinking was under uniaxial load, after (K¨upfer and Finkelmann, 1994). The low-stress region is expanded in the second graph, clearly indicating the regimes of pre-threshold elasticity, nearly zero modulus on the soft plateau and the subsequent modulus increase after director re-alignment. percent in strain with very little energy cost. We call this soft elasticity; it is the subject of Chapter 7. (During the alignment of polydomain nematic elastomers, the same stressstrain effect is indirectly observable.) The sketch of Fig. 5.22, gives the underlying reason why certain deformation can be soft. What is required is an additional, internal mode, the nematic director, which conventional solids do not possess. In Chapter 10 we shall return to symmetry arguments, which underpin the strange soft mechanical properties of nematic solids. K¨upfer and Finkelmann found this extreme softening depended critically on how the nematic monodomains had been formed. The pairs of open and closed symbols in Fig. 5.23 refer to chemically identical networks, but prepared with second stage linking in the isotropic and nematic states, respectively. Soft elasticity is evidenced much more by networks with a more isotropic genesis. The loss of total softness, arising from the formation process, gives rise to what we call semi-softness. We must explain how the conditions of genesis determine the degree of semi-softness. It will turn out that semisoft networks are qualitatively like soft elastomers and not like conventional ones. 5.6

Applications of liquid crystalline elastomers

The first applications one comes to with liquid crystalline elastomers are of course optical since they are analogues of the liquid crystals that have found wide application in display technology. Elastomers do not offer the same speeds of response, but the ma-

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NEMATIC ELASTOMERS

Ee Eo

Ee Eo

(a)

(b)

F IG . 5.24. (a) Schematic of a bifocal contact lens. The eyeball is shown as a shaded sphere behind the lens. The light from far away and from the reading distance has all polarisations, but the monodomain birefringent elastomer would select (by one of its principal refractive indices) one polarisation from each beam to focus on the eye retina. The human visual system contains filters that pass different spatial frequencies and thus the defocussed component of the image will have relatively little effect on the perception of the high spatial frequencies – the sharp edges – of the focussed component. (b) A single-crystal nematic hydrogel prepared as a bifocal contact lens (H. Finkelmann et al., patent No. WO 99/25788). nipulation of optical birefringence by mechanical means is quite unique and could lead to opto-mechanical sensors and other similar devices. The unusual highly non-linear elasticity of nematic rubbers (as well as of smectic/lamellar elastomers), with very high thermal sensitivity, very low plateau modulus and low acoustic impedance are other characteristic physical properties appropriate for applications. One area where these properties are relevant is that of shape-memory materials. For instance spontaneously shape-changing thermo-mechanical systems or rubbers with selective shape-memory could lead to applications in temperature sensors and temperature-controlled actuators. Light-driven actuation of this type is also possible with nematic photo-elastomers. A highly prospective area is that of rubbers with piezoelectric and non-linear optical properties which, again in contrast to traditional crystalline and ceramic materials, allow large deformations and manipulation of polarisation by mechanical means – we shall examine and discuss these effects later in the book. Another factor differentiating elastomers from their liquid nematic analogues is the optical clarity when in the monodomain state, due to suppression of thermal fluctuations of nematic director. In contrast to solid crystals and glasses, which can also be very optically transparent (for the same reason of damping of long-wavelength thermal fluctuations), elastomers and gels are rather soft mechanically – their typical modulus µ is at least 105 times smaller than in solids – and yet retain their shape as solids do. Finkelmann has exploited this property of mechanical stability and optical clarity, while being birefringent nematic. Bifocal contact and intra-ocular lenses can be made from nematic gels. The beauty of this idea is that only a single lens needs be made, but it has different focal lengths for light polarised in directions along (extraordinary) and perpendicular (ordinary) to the optical axis n of the material of the lens, in his case a soft, single-crystal nematic gel, see the schematic in Fig. 5.24(a). The limited power of

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117

a failing human eye to adjust between focusing near and distant images is aided by the lens which separates the focal planes of the extraordinary (e) and ordinary (o) beams. The eye no longer has to make large diopter changes to then change focus between objects. The fact that images are formed from light of differing polarisations does not seem to confound the retina. Figure 5.24(b) shows such a soft-gel bifocal lens and two images formed from it. The eye is able to distinguish between the two close images that are close in their focal planes but widely separated spatially (We are grateful to J.D. Mollon for his advice on this point). Such a lens can be stamped cheaply from a sheet of nematic elastomer and stuck to a conventional soft gel lens as a substrate. In a sense, this idea exploits the simplest possible property of a solid liquid crystal – it does not flow while displaying the same optical anisotropy utilised in display devices based on conventional liquid crystals and the same optical clarity as glass. Another attempt, again by Finkelmann and co-workers, to use the opto-mechanical response has been in the stamping of optical guiding patterns in sheets of nematic elastomer. If an optical fibre or similar fine body is placed on top of a sheet of elastomer and is then imprinted into it by the application of a weight, then there is a corresponding elastic distortion beneath it which mirrors the object. The body is then frozen below its glass temperature and the template removed. The elastic distortion is now permanent and so too is the distortion of the director. The pattern creates a light guiding pattern of refractive index variation below the path of the original object and can act as an optical link between devices assembled and integrated on the nematic glass substrate. These existing applications still do not exploit the molecular mobility and mechanical anisotropy of nematic rubbers that makes them fluid-like in their dynamics and ability to change shape softly. The fact that their optical properties couple strongly to their exotic mechanical properties suggest the application of nematic elastomers as sophisticated strain gauges. A range of possible applications could be envisaged using the effect of spontaneous shape change with temperature, the shape-memory property of nematic rubbers. One is a straightforward mechanism of artificial muscle (Herbert et al., 1997), Fig. 5.25(a), which contracts and extends on bringing the monodomain nematic rubber above and below its nematic-isotropic transition point – essentially following the plots of Fig. 5.12. The operating cycle of such a muscle involves fours steps: heating from low temperature T− to high temperature T+ (above Tni ), producing a nearly isothermal work, cooling back to T− and releasing work, again at nearly constant temperature. The net work produced by this cycle can be positive: W ∼ Ao (T+ − T− )Q¯ 2 h2 L, and the muscular efficiency (the ratio of work to losses) proportional to the cross-section size h (Herbert et al., 1997). Another interesting ‘muscular’ geometry is that in the ‘mono-morph’ conformation, a bi-rubber strip, called such by analogy with bi-metal strips commonly used in thermo-mechanical technology. Figure 5.25(b) explains the mechanism: when two strips of rubber are wielded together, one ordinary and the other monodomain nematic, the resulting beam will be bent very strongly on changing the temperature between T+ and T− and the associated spontaneous changing of natural length of the nematic strip. Many modern applications of shape-memory systems use this principle, sometimes extended to a ‘bi-morph’ conformation involving the actuating material on both sides of a

118

NEMATIC ELASTOMERS

h L

muscle fibre

n

iso

n L

T_

f iso iso

fg source T+

muscle W: work

source T_

(a)

T+

L0

(b)

F IG . 5.25. (a) The scheme of monodomain nematic elastomer muscle, performing work due to the natural length variation L(T ) during a thermal cycle between temperatures T+ and T− . (b) A bi-rubber strip of monodomain nematic and ordinary isotropic rubbers welded together: on nematic contraction the shape of such an elastic beam changes quite dramatically. support plate. The general area of actuation is one of growing importance and increasing technological interest, across a broad spectrum of applications. Clearly thermal response can be utilised for this. More novel are the photo-actuation possibilities – the supply of energy and control can be remote, via a light beam or via an optic fibre. A number of exciting novel devices, from micro-pumps (when a plate of mono-morph rubber is attached at a circular rim to a surface, covering an opening; on actuators, the plate bulges and creates a sucking action), to micro-manipulators and valves (for example for car carburettors or microfluidic devices) to an electrical generator engine based on a large wheel with spokes made of photoelastomer, half of which is exposed to a UV source, e.g. the Sun (contracting spokes move the wheel’s balance off axis and it rotates, bringing the ‘fresh’ spokes into the light and allowing the exposed ones to relax back). More involved and far less trivial properties of liquid crystalline elastomers have application potential as well. Later in the book we shall examine the dynamics of mechanical response of nematic elastomers: these properties show an anomalous dissipation (damping of mechanical energy) due to an underlying relaxation of director modes. The key characteristics of this dissipation are rivaling the record values of damping achieved in modern technology and offer many application possibilities. In addition, since this anomalous damping is highly anisotropic (in monodomain, aligned nematic rubbers), acoustic waves of different polarisation would be dissipated differently – suggesting a whole new area of polarised acoustic technology. Cholesteric elastomers have the same director distribution as their liquid cholesteric counterparts. The coupling between strains and the director has even more subtle consequences here. Since there is a periodic variation of their dielectric properties along the helical direction, light couples strongly to cholesteric. Indeed the elastomers are

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119

photonic in the sense that there is a band structure for light just as there is for electrons in metals and semi-conductors. In the gaps, no propagating light modes can exist and light of this frequency shone onto the elastomer will be reflected, giving it a characteristic colour. The underlying cholesteric structure can be distorted by applied strains and so also are the colours that are reflected, see the frontispiece for strips of rubber that change from red to green to blue and ultimately become colourless. One can anticipate applications that exploit a band structure that changes with strain. Lasing, which occurs near or at a band edge is seen to change in colour as the rubber laser is stretched. See Sect. 9.5 for a discussion of these effects. Since cholesteric elastomers are chiral and lack a centre of inversion symmetry, they are piezoelectric. They are unique among piezoelectric solids in that they are very soft and capable of very large distortions. Polarisation is physically dependent on the distortion of the underlying geometrical structure of a piezoelectric, that is upon the strain rather than the stress. For a given stress therefore the strain and thus the piezoelectric response of a soft solid is much higher than that of, say, a ceramic material. Additionally, softness means that the impedance matching between the device surroundings (e.g. air) is greater than for hard solids. This increases the effectiveness of energy transfer for instance in a piezoelectric microphone. Also these elastomers do not require poling, a consideration important in processing. The ability to take large strains offers not only larger polarisation responses, but also possible non-linear piezoelectric response. We shall see that cholesteric elastomers can be made to interact very differently with solvents of differing handedness. This is because chirality is topologically imprinted into networks. In some cases any intrinsic chirality can be removed, for instance by removing a chiral solvent that was present at crosslinking. The elastomer has the dilemma of whether to untwist to a more naturally unchiral state, or to remain twisted to satisfy the elastic constraints of chain connectivity. The result depends on the competition between the two effects, a balance that can be influenced by the solvent. Thus we see preferential absorption of one handedness over another from a racemic mixture. This separation, we stress is not a surface or a kinetic effect, but arises from the bulk mechanical stability of the elastomer. Preliminary experiments suggest new separation routes. Smectic elastomers (see Chapter 12) are materials with layer microstructure where the physics of elastomers, nematic order and that of layer deformations are all intimately coupled. There is evidence that their deformations are rubber-like in two dimensions and solid in the third. Such extreme mechanical anisotropy, in particular the anisotropy of the rubbery response, can be used to imitate bio-mechanical processes, for instance micro and peristaltic pumps and in vessels to contain fluids with pulsating pressure. The anisotropy also gives rise to the solid analogue of the Clark–Meyer–Helfrich–Hurault instabilities found in liquid smectics. There is then a coupling between strain and optical properties that one could perhaps apply to good effect in the optical measurement of strain. Chiral smectic elastomers can be ferroelectric in their Smectic C∗ phase, the degree of spontaneous polarisation being sensitive to temperature, light and strain. Strain and light sensitivity offer routes to piezoelectric and photo-ferroelectric devices that have been followed jointly by groups in Mainz, Leipzig and Stuttgart.

6 NEMATIC RUBBER ELASTICITY Nematic networks can be highly rubbery, that is capable of large extensions and composed of molecules with liquid-like mobility. The difference between nematic and classical elastomers is principally only that of molecular shape anisotropy induced by the liquid crystalline order. The simplest description of nematic rubber elasticity is essentially an extension of classical molecular rubber elasticity, reviewed in Chapter 3. We shall accordingly call it a ‘neo-classical’ theory of rubber elasticity, referring to the identity of approach. It will be introduced in this chapter and its richness explored in subsequent chapters of this book. This chapter will conclude with simple illustrations of the neo-classical free energy not involving rotations of the director and complex strains. Already remarkable phenomena arise, for instance we describe spontaneous thermal distortion, optical distortions, complex thermal history and relaxation, the Landau theory of transitions and the influence of mechanical stress on thermal effects. The nematic rubber elastic free energy is simple, even if the thermo-mechanical history has been complex. Readers unconcerned about the details of how this is so should omit Sect. 6.3, marked ‡, only reading its main conclusion which is stated in its opening paragraph. A mechanical instability emerges when the director is induced to jump by strains imposed perpendicular to the director, see Sect. 6.7. Strangely, the resultant distorted state can have an energy as low as the initial state – a precursor of the more general soft elasticity that is the subject of Chapter 7. For the first time we also meet convexity problems in the free energy and point in Sect. 6.7 to the microstructural routes between energy minima that will be discussed in Chapter 8. This chapter concludes by sketching the role of entanglement and other complex collective effects in nematic elastomers. 6.1

Neo-classical theory

The number of configurations of a test strand, connecting two crosslinks separated by a distance R in a nematic network, is proportional to the anisotropic Gaussian distribution we saw in Chapter 3, eqn (3.11):  R) ∝ p(R

1 Det()

1/2

 3 −1 exp − R ·  · R . 2L

(6.1)

The effective step length tensor  reflects the current nematic ordering in the network. At formation however, the span between the links was R f , say, and the shape of the chains at that time was described by a Gaussian distribution similar in form to eqn (6.1) but with a step-length tensor  o . One reason the distributions might differ is that the temperature, and hence the nematic order, of the two states differs. If the formation 120

NEO-CLASSICAL THEORY

121

condition was nematic, another reason might be the current director adopting a direction different from the director orientation at formation. Rf ), The probability of having crosslinked the current span R into the network is po (R that is the probability of finding at formation the span R f that R derives from. The distribution po is of the same form as eqn (6.1), but instead governs R f and has the step length tensor o . For the moment we take the subscript o to denote formation which we identify with the conditions before deformation is imposed. As in Chapter 3, we assume affine deformation: the total deformation tensor λ t from the formation to the current state is what took the initial span R f to become R , that is R = λ t · R f . The subscript t denotes total deformation, which is needed in case there were several deformation steps R). For instance there could be a spontaneous Rf ) and currently (R between formation (R distortion associated with changing conditions, followed then by an imposed distortion. The free energy of a strand, Fs , averaged over formation conditions, is then: R) po (RRf ) Fs = −kB T ln p(R

 Det() 3kB T kB T −1 Ri i j R j  + ln = 2L 2 a3 3kB T R · −1 · R + . . . . R ≡ 2L

(6.2)

The (ln Det)-term arises from the normalisation of the probability, the prefactor in (6.1). Inserting the a3 in the normalisation logarithm gives an arbitrary, additive constant. It is done simply to make the argument of the logarithm dimensionless. The ln Det contains the nematic order, via the step-length tensor , but not the deformation λ . We shall generally only display it when it plays a role, that is, when the magnitude rather than direction of nematic order changes. The free energy Fs can be simplified by using λ t and Rf for R: Fs =

3kB T Rf · λ Tt ·  −1 · λ t · R f  po . R 2L

(6.3)

The average over the initial span is easy – the second moment of the Gaussian distribuRf Rf  po (RRf ) = 13 o L whence the free energy of a strand is finally: tion is R Fs =

1 2 kB T

   Det() T −1 1 . Tr  o · λ t ·  · λ t + 2 kB T ln a3

(6.4)

This is what we call the neo-classical free energy of an average network strand since it is a simple generalisation of classical, Gaussian rubber elasticity. To convert it to a free energy density F we should simply count the number of such network strands, ns , per unit volume: F = ns Fs . We have seen that the combination ns kB T is µ , the linear shear modulus of an isotropic rubber with this density of network strands. The free energy density is then   F = 12 µ Tr  o · λ Tt ·  −1 · λ t

(6.5)

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NEMATIC RUBBER ELASTICITY

All the nematic rubber phenomena we describe will arise from this free energy and some non-ideal deviations from it. The deviations will arise from entanglements, compositional fluctuations, crosslinks of finite size and related random field effects. We shall frequently refer to it as the Trace formula. Ideally, it is valid for all, including very large, deformations and is only limited by extensions sufficient to so stretch network chains that they violate Gaussian statistics. It records, via o and , the initial and current directors no and n of the elastomer, unlike the free energy of a liquid nematic where F depends only on the current director. The initial and current magnitudes of the local nematic order parameter, Qo and Q, are also contained in F through the anisotropy of o and . Section 3.4 describes in classical elastomers analogous, but simpler effects of conditions changing between formation and the current state. In fact, since cholesterics are locally nematic, we will use this Trace result for cholesteric elastomers, taking into account the spatially varying director. Smectics have an underlying nematic order and in a molecular theory of such elastomers we again employ this free energy, but heavily modified by the dominating influence of the layers.  energy (6.5) has a very rich structure compared with the classical result The free T Tr λ · λ . The distortions appear now not as simple forms λ T · λ , but rather in the combination λ T ·  −1 · λ . Thus the current (after deformation) nematic state of the body is encoded into the elastic energy via the sandwiched factor of  −1 . The main principal direction of  −1 , the director n , is not necessarily along a principal direction of λ . Diagonal elements of λ (simple extensions and compression) can now couple with offdiagonal elements (shears), for instance giving terms like λxx λzx in the free energy. This does not occur in classical rubber elasticity. The structure λ T ·  −1 · λ also allows local torques and rotations to be applied to nematic elastomers, thereby coupling mechanical effects to the internal nematic freedom. Finally, the combination λ T ·  −1 · λ couples to  o , that is to the original director n o of the state before deformation, thereby coupling the current strains and director to the original anisotropy of the solid matrix, see Chapter 1, page 4. At first sight F seems rather tensorial, but it is easily dissected with a few examples where directions are unchanging (for instance that of the director and the step length tensors) and where the tensor structure is always diagonal and hence trivial. Most trivially, if the formation and current states are both isotropic, that is  o = aδ and    −1 = a−1 δ , then (6.5) collapses to the classical result F = 12 µ Tr λ T · λ . For the remainder of this chapter, we shall consider simple examples not involving continuous director rotation: 1. Spontaneous distortions on entering or leaving the nematic state, Sect. 6.2. 2. Effectively classical response if there is no director rotation and no order parameter change during the application of extensional strains along principal directions of the nematic order, Sect. 6.6. 3. Relaxation and the erasing of complex thermomechanical history by subsequent spontaneous deformations, Sect. 6.3. 4. Photo-elastic response of nematic elastomers, Sect. 6.4.

SPONTANEOUS DISTORTIONS

123

5. Thermal phase transitions in nematic elastomers, Sect. 6.5. 6. The influence of crosslinking and applied stress on thermal properties, Sect. 6.5. 7. The induction of nematic order by applied strains – the robustness of nematic order under mechanical fields – in analogy with fields acting on nematic liquids, Sect. 6.6. 8. Mechanical instabilities on imposing strain perpendicular to the director, Sect. 6.7. We defer until the next chapter questions of shear, continuous rotations of the director induced by electric and shear fields, and of torques in general. These lead to yet more new phenomena and in effect a new elasticity. 6.2 Spontaneous distortions Consider an elastomer formed in the isotropic state, that is with  o = aδ . It is cooled to its current, relaxed, monodomain nematic state. The chains now have a natural shape described by the tensor  r , the subscript {r} denoting ‘relaxed’. No stresses or constraints have been applied to it and thus any λ t is a spontaneous distortion, λ m . From the symmetry of the phase that has developed on cooling, the distortion must be uniaxial and directed along n . It must also be volume preserving. The extent to which volume is preserved is examined in detail in Sect. 4.4 and will be treated in nematic elastomers in Sect. 10.1.1. Taking the director to be along z , that is n = z , the deformation tensor λ must have its principal extension element along z . Call this component λ , whence the whole matrix λ can only be:   √ 0 0 1/ λ √ λ = 0 (6.6) 1/ λ 0  ≡ λ T . λ 0 0 The inverse step length tensor  −1 in the same system of coordinate axes is: r   0 0 1/r⊥ 0 . 1/r⊥ = 0  −1 r 0 0 1/r

(6.7)

Evaluating the free energy reduces to the trivial problem of multiplying four diagonal · λ and then tracing the result, that is summing along matrices that form (aδ ) · λ T · −1 r the diagonal. We take   0 0 a/(λ r⊥ ) 0 , 0 a/(λ r⊥ ) Tr  2 0 0 aλ /r which yields:

 F=

1 2µ

2 a a λ r +  λ r⊥ 2

 .

(6.8)

The free energy is close to that of a classical elastomer undergoing uniaxial extension, but instead of an overall prefactor accounting for chain size change between formation

124

NEMATIC RUBBER ELASTICITY

and current conditions (Sect. 3.4), there are separate factors a/r and a/r⊥ for the parallel and the two perpendicular directions – the imbalance between the directions will induce shape change. Unconstrained, the system will adopt a spontaneous extension λ minimising the elastic free energy density (6.8). Taking ∂ F/∂ λ = 0, we immediately conclude that on cooling from formation to current conditions, there must be a spontaneous uniaxial elongation λm , of (Abramchuk and Khokhlov, 1987; Warner et al., 1988): 1/3  λm = r /r⊥ , (6.9) offering possibilities for temperature controlled actuation. We have assumed in this example a chain elongated by its nematic order to a prolate shape. If by contrast the chain backbone were flattened by nematic order to an oblate shape, then the above elongation on leaving the nematic state would become instead a contraction. Exercise 6.1: If the elastomer were instead formed in the monodomain nematic state, with a uniaxial step length tensor  o , show that the spontaneous distortion on heating to the isotropic state with  r = aδ would be a uniaxial contraction 1/3  λm = o⊥ /o .  o  o Solution: Now eqn (6.8) becomes instead F = 12 µ λ 2 a + λ2 a⊥ since the roles of the two step length tensors have been reversed. The minimisation that led to eqn (6.9) now leads to the desired result .

Does the solid that the chains form allow them to become isotropic in shape on average when they leave the nematic state? One might doubt whether this is possible since the rubber is constrained to preserve volume as it reacts to changing order by deforming. Exercise 6.2: Does the topology established in an ideal network crosslinked in the nematic state create a memory of chain shape anisotropy on transition to the state of isotropic rod orientations (de Gennes, 1969a)? · R creSolution: A span R in the isotropic state derives from that span R f = λ −1 m ated at the moment of crosslinking in the nematic state. There has been a contraction λ m in making the transition to the isotropic state, see exercise 6.1. Thus the Rf ) for finding the strand R f at the genesis of the network Gaussian probability p(R R) for finding R : transforms into a probability p(R  R) ∝ p(R

· R) po (λ −1 m

∝ exp −  ≡ exp −

T −1 RT · λ −1 ·  o · λ −1 ·R 3R m m

2L  RT ·  −1 3R ·R e 2L



.

The crosslinked chains now have a Gaussian distribution in the isotropic state characterised by an effective step-length tensor,  e , defined by the above probability

SPONTANEOUS DISTORTIONS

125

R). From this distribution we can read off the current mean square distribution, p(R shape tensor: RR  = 13 Le = 13 λ m ·  o · λ Tm L . R Then  e = λ m ·  o · λ Tm is easily evaluated using frames as in eqn (6.6) and (6.7) for λ m and  o . The latter can also be expressed in their frame-independent forms

λ m

  1 1 n o n o and  o = o⊥ δ + (o − o⊥ )nno n o , = δ + λm  −  λm  λm 

and  e then evaluated in a frame-independent manner. This method is especially useful for the corresponding exercise 9.1, page 226, for cholesterics attempting to attain the isotropic state. The answer is RR  = 13 (o o⊥ 2 )1/3 L δ . R

(6.10)

Chains have an isotropic shape distribution after entering the orientationally isotropic state. There is therefore no memory of the chain shape anisotropy that pertained at crosslinking.

Despite the shape isotropy in the isotropic state, topology provides chains with a trivial R R = 13 aLδ . memory of their nematic genesis: isotropic uncrosslinked polymers have R Section 3.2.1 shows that the effective step length just derived, eqn (6.10), is (o o⊥2 )1/3 = a(1 − Q2o + . . .) in the freely jointed chain model of nematic polymers. The order parameter at crosslinking was Qo . Thus the constraints of crosslinking cause network chains in the isotropic state to be isotropically shrunken from their free condition. An important theorem, about the attainability of deformation without energy cost for nematic elastomers, will rest upon being able to obtain such perfectly isotropic states despite topological constraints, see Sect. 10.4. Exercise 6.3: If the formation state was nematic, with an  o characterised by principal values o and o⊥ , and the current state is also nematic (with  r ), show that the spontaneous distortion in going from formation to current states would be r 1/3  o λm = r ⊥o . Check that for prolate elastomers there is indeed spontaneous ⊥



extension on further cooling. Solution: Now there are four rather than three non-trivial diagonal matrices to   o o o  multiply in the free energy density (6.5). o is Diag ⊥ , ⊥ ,  . Multiplying out with (6.6) and (6.7)  to get the analogue of (6.8), we now instead obtain F = 1 2µ

o

λ 2 r + λ2 

formation λm .

o⊥ r⊥

. Minimising, ∂ F/∂ λ = 0, yields the optimal spontaneous de-

1/3  The spontaneous distortion λm =  /⊥ from the isotropic state turns out to be central to all of nematic rubber elasticity. Indeed, after relaxation has occurred, it

126

NEMATIC RUBBER ELASTICITY

is the only input to neo-classical rubber elasticity. For Gaussian chains it is a direct, and indeed the only, measure of chain anisotropy at current conditions. Since chains must be Gaussian to be rubbery, and since Gaussians are entirely determined by their second moments, the ratio of these second moments, r =  /⊥ , is in Gaussian rubber elasticity the only measure of chain anisotropy. Indiscussing the shapes of nematic  polymers in Sect. 3.2.1 we denoted the ratio  /⊥ by r. The step length ratio can be thus deduced from thermal expansion measurements, r = λm3 . It can be correlated with direct measurements from neutron scattering from labelled chains in nematic elastomers at the same conditions. This is a demanding situation for theory since there are finally no phenomenological free parameters in this theory of ideal nematic rubber elasticity. As an example, polyacrylate elastomers (Mitchell et al., 1993) suffered a distortion λm = 0.95 on going from nematic formation (crosslinking) conditions at T = 391◦ K to T = 393◦ K (just above the clearing temperature Tni ). Reversing the procedure, cooling through the clearing point, would see a λm = 1/0.95 ≈ 1.05, that is a spontaneous elongation of about 5%. The chain shape anisotropy was measured (Mitchell et al., 1991) on the same system. Scattering yields the average radii of gyration of chains. The radii   1/2  = 1.1. The sponof gyration ratio in the formation state was Ro /Ro⊥ ≡ o /o⊥ 1/3  2/3  = Ro /Ro⊥ = 1.06 which compares taneous distortion should be λm = o /o⊥ well with the actual mechanical observations. By contrast, Finkelmann’s side chain nematic elastomers are more anisotropic and yield higher spontaneous distortions, as we have seen in Fig. 5.12. Most anisotropic of all in shape are nematic elastomers composed purely or partly of main chain polymers. Figure 5.13 show spontaneous shape changes of 350% are easily achievable. From λs = 3/2 3.5 = r1/3 , we can conclude that R /R⊥ = r1/2 = λs must give a ratio of the radii of gyration of 6.5. This large value is consistent with the values that emerge for main chain polymers from neutron scattering, Fig. 3.6(a). Revisiting Sect. 3.4 that treats changing conditions in classical elastomers reveals a great contrast with nematic elastomers, thermally (Sect. 5.4.1) or photochromically (Sect. 5.4.2). The theory of this section shows that the difference between formation and current conditions is recorded in nematic elastomers by factors relating separately to the  and ⊥ components of strain, rather than in a common prefactor as in an isotropic system, eqn (3.31). Effects are huge and spontaneous, that is, no pre-strains are required. A simple model of spontaneous distortion In Sect. 3.2 we arrived at the connection between the step length tensor and the order parameter, eqn (3.16), in the simplest model of polymers, namely the main chain freely jointed rod model:     Q = a [1 − Q]δ + 3Q n n . (6.11)  = a δ + 2Q The step lengths, along the director  = a(1 + 2Q) and in the perpendicular plane ⊥ = a(1 − Q), determine the anisotropy r =  /⊥ = (1 + 2Q)/(1 − Q) ,

(6.12)

SPONTANEOUS DISTORTIONS 0.6

1.5

0.5

1.4

0.4 0.3

Q

1.2

0.2

1.1 1 20

(a)

0.1

l =L/L 0 Relative length l

Relative length

l

1.3

Nematic order Q(T)

l =L/L 0

1,7

60

80

1,5

0,10 1,4

Dn

l

1,3

0,05

1,2 1,1

0,00

1,0

0 40

0,15

1,6

Dn D n Birefringence

1.6

127

20

100

30

40

50

60

70

80

90

100

temperature [°C]

Temperature ( C)

(b)

Temperature ( C)

F IG . 6.1. Spontaneous distortion, λ , plotted (a) with the nematic order parameter, Q (Clarke et al., 2001), and with (b) the optical birefringence, ∆n, against temperature (Finkelmann et al., 2001a). The order parameter Q is measured from X-ray scattering, mainly from the aligned pendant side-chain rods. The birefringence is also a direct measure of Q, expressing the ordering of the most polarisable component of the elastomer, the mesogenic rods. see Sect. 3.2.1 and exercise 3.1. This characterises the chains and thus the elastomer, and determines the spontaneous distortion. In the freely jointed model of main chain polymers, the order parameter of nematogens, Q , and that of the chain backbone, Q , B are one and the same. But in side-chain polymers, the order of the backbone and of the side chains will in general differ. The pendant rods nematically order and indirectly induce order in the backbone, that is a Q induces a Q . It is Q that is revealed by B typical experiments, for example optical or X-ray. This question has been discussed in Sect. 3.2, where we have argued that, at least for small local nematic order Q, the chain anisotropy is a linear function  /⊥ − 1 = α Q. The constant of proportionality, α , accounts also for a difference between Q and QB , a distinction we must always keep in mind. Experiments on side chain nematic elastomers illustrate an intimate connection between macroscopic shape relaxation and rod order parameter. Figure 6.1 shows just how closely shape change and birefringence change (essentially change of side chain order parameter) are related since they fall on almost the same curve against temperature. We can now use eqn (6.9) for the spontaneous distortion to model this relationship more closely. Putting in the freely jointed rod model’s step length variation with order parameter, one finds: 

λm = → QB =

r

1/3

r⊥

λm3 − 1 λm3 + 2

=

 .

1 + 2QB 1 − QB

1/3 (6.13) (6.14)

Thus the model allows us to extract a notional backbone order parameter from the observed spontaneous distortion, and this is shown in Fig. 6.2. Superficially at least, the

128

NEMATIC RUBBER ELASTICITY

0,5

(c)

0.8

Backbone order Q B Q

0.6

(b)

0,4

0,3

b

Backbone order Q B

1

0.4

(a) 0.2 0 20

0,2

0,1

0,0

40

60

80

100

120

20

140

30

40

50

60

70

80

90

100

temperature [°C] ( C) Temperature

Temperature ( C)

F IG . 6.2. Backbone order, QB , of network strands as a function of temperature, calculated from the spontaneous distortion, eqn (6.14). Curves (a) and (b) show data from side-chain nematic elastomers with different spacers (Clarke et al., 2001; Finkelmann et al., 2001a), while the curve (c) is an example of highly anisotropic main– chain elastomer (Tajbakhsh and Terentjev, 2001).

Backbone order Q B

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nematic order Q

F IG . 6.3. Backbone order QB as a function of side chain nematic order Q, for the material Fig. 6.2(b) – a very good linear fit (Finkelmann et al., 2001a). variation of order parameter QB appears to follow the birefringence, ∆n, that is the anisotropy in the refractive index. We can now see whether this derived QB in fact has anything to do with the order parameter, Q, observed from the birefringence by plotting one against the other parametrically with temperature. The birefringence, by extrapolating it to zero temperature (where one presumes the nematic order parameter Q = 1), can be made to yield good estimates of the order parameter via Q(T ) = ∆n(T )/∆n(T = 0). Figure 6.3 shows the close relationship between the two order parameters as temperature is varied; backbone shape order is linearly induced by side-chain nematic order over a wide range of ordering. It is remarkable that in a side-chain system: (a) there is a simple relation between the two orders; and (b) the chain backbones can be considered as simple, freely jointed rod polymers (since the model relation eqn (6.14) is followed).

EQUILIBRIUM SHAPE OF NEMATIC ELASTOMERS‡

6.3

129

Equilibrium shape of nematic elastomers‡

We have seen that if distortions λ are measured with respect to the relaxed state, then the free energy simplifies considerably. It is also most natural to use the relaxed, current state as a reference from which strain is measured. Indeed the thermal and strain history of a given piece of rubber may not even be known to the experimenter! The reader unconcerned by details of the effect of thermal histories can skip this section by taking reassurance of its result: The relaxed state can be considered to be the formation state if all imposed strains are measured with respect to this state. In other words, both nematically and mechanically, the neo-classical Trace formula for the free energy density F, eqn (6.5), is of adequate generality if the strain λ is that measured from the relaxed state and o is, equally, taken to be the step length tensor of chains in the relaxed condition (and not to be the step length tensor of chains at the moment of formation of the network). The total strain entering F, as presented in eqn (6.5) is λ t = λ · λ m . The molecular shape  o , which is recorded at network formation, becomes the relaxed shape  r on changing temperature to the current condition and allowing the network to find a new nematic and mechanical equilibrium with a spontaneous strain λ m , see a schematic summary in Fig. 6.4. Both the mechanical relaxation λ m and the new nematic state  r

lt

lt = lm

= 1

l0

lr lm

formation

lt = l lm

l l

Relaxed at new conditions

Distorted by l with respect to relaxed state

F IG . 6.4. Formation, relaxation and subsequent imposed strain of a nematic elastomer. Total deformation λt is measured from the formation state whereas that in an experiment, λ , is measured from the relaxed state, a relaxation λm having occurred since formation. are coaxial with the original state  o , if that was itself nematic. The extent of ordering, i.e. the magnitude of the nematic order parameter rather than its direction, is changed on cooling or heating. If the formation state was only quasi-isotropic4 , then in practice 4 The qualification ‘quasi’ applies to non-ideal nematic elastomers when they have strains imposed during formation in the isotropic state, or are formed under fields in the nematic state. This is necessary to create a polymer network that will become an aligned monodomain nematic on cooling. Subtle effects arise from such crosslinking conditions, as we discuss in sections 7.4 and 7.6.

130

NEMATIC RUBBER ELASTICITY

cooling yields a monodomain; the elastomer will find a suitable director orientation and the step length (r ) and relaxation (λ m ) tensors will be coaxial with this direction. If the formation state was nematic, heating yields a quasi-isotropic state: the mechanical relaxation is coaxial with the original director and the relaxed step length tensor is very nearly isotropic. Let us take an example of an external uniaxial strain λ imposed coaxially with the director. Since the deformations are coaxial and uniaxial, we can characterise the total distortion by a single number, the principal extension λt = λ λm . Inserting λt into the r 1/3  o one obtains: free energy density F (6.5) and using λm = r ⊥o ⊥



   1/3  o o⊥2  Det( ) 2 r  F = 12 µ  r r 2 + ln λ2 + λ Det(o )  ⊥ 

≡

1 2µ

Det(o ) Det(r )

1/3

λ2 +

2 λ

 +... .

(6.15)

Now F(λ ) really does look like the free energy of a classical elastomer, with a prefactor, as in classical cases accounting for a change in chain size since formation. We emphasise that the strain λ is measured with respect to the relaxed state of the elastomer in its new conditions. This classical form of the free energy is only for strains applied along principal directions of  o , and where the director does not rotate. We shall see other examples of pseudo-classical response arising when there is no director rotation. In Sect. 6.6 the robustness of the magnitude of the nematic order is examined and it is found that when it relaxes under strains the moduli become anisotropic, see (Finkelmann et al., 2001a). The result for F also illustrates a general rule – if one deals with strains relative to the relaxed state, the memory of the prior history is largely relaxed away and is only found in prefactors, or in a step length tensor of an effective ‘formation state’ that a rubber finds itself just before the deformation is applied. Subsequent strains λ , more complex than in the above example, when applied to the relaxed state will in general change the step length tensor from  r to  . For instance λ can simply rotate the nematic director and thus the step length tensor while distorting the  √ 1/ λm

√0

0

1/ λm 0 0 with the body. We already know that spontaneous deformation λ m is 0 λm 0 r 1/3  o principal extension along the director λm = r ⊥o . Let us see that the subsequent ⊥



strains appear simply in a free energy of the neo-classical form. Exercise 6.4: By considering λ m in its principal frame, show that the relaxation can be written 

λm =

Det(o ) Det(r )

1/6  r1/2 ·  −1/2 . o

PHOTO-MECHANICAL EFFECTS

131

Show that by construction λ m conserves volume: Det(λ m ) = 1. Solution: r and o share the same principal frame as λ m ; write λ m out in full and  Det( ) 1/6 o o 2 1/6 = r ⊥r 2 from each term. Check the elements extract a factor of Det(o) r

 ⊥

against the result for λm of exercise 6.3.

We can now write down the free energy in terms of strains from the current state by ·  −1/2 . The Trace formula for the free putting in λ t = λ · λ m and by using λ m ∼  1/2 o r  T −1 1 energy density F = 2 µ Tr  o · λ t ·  · λ t then becomes:   1    3     Det( o )   Det( o ) F = 12 µ  Tr  r · λ T ·  −1 · λ − ln . Det(r ) Det()

(6.16)

·  −1/2 ·  o ·  −1/2 ·  1/2 resulting from We have simplified to just  r the product  1/2 r o o r T λ m ·  o · λ m , see Sect. 4.3.3. The Tr (. . .) part of the free energy is as if the elastomer had  1/3 been formed in the relaxed state. The prefactor Det(o )/Det(r ) to the part of F containing λ generalises the Flory prefactor accommodating changes in the equilibrium shape of chains since formation, see Sect. 3.4, especially eqn (3.31) for the isotropic rubber analogue of the effect of changing conditions. The additive logarithm term, as in eqn (6.2), only depends on the magnitude of the order and not its director. Nor does it depend on imposed strains λ . More complex crosslinking histories, for instance two or more stages of crosslinking with temperature and applied strain varying between stages, also have simple outcomes (Verwey and Warner, 1995). If strains are measured with respect to the relaxed state at a current condition, one can assume the network was effectively formed in this state. One can then safely use the Trace formula for F, eqn (6.5), with the identification  o =  r and an appropriate factor added to the modulus µ . In most of the book, we thus generally assume networks are formed at the current conditions of interest, unless we are specifically concerned about questions of history. 6.4

Photo-mechanical effects

Nematic elastomers containing photochromic rods contract when illuminated (Finkelmann et al., 2001; Hogan et al., 2002), see Fig. 5.15(a). If they are clamped, illumination generates a retractive stress (Cviklinski et al., 2002), see Fig. 5.16(a), because when the natural length diminishes, a fixed actual length is equivalent to a stress-generating elongation. The photochromic molecular moieties are azo-containing, mesogenic rods, Fig. 5.14, which bend on absorbing a photon. Contractions are large, in fact on the same scale as the spontaneous thermal changes of shape when nematic order is lost or gained due to heating or cooling. Figures 5.15(b) and 5.16(b) display the thermal strains of the corresponding materials. The similarity suggests that the common element between thermal and photo-induced shape change is the dependence of length on the nematic

132

NEMATIC RUBBER ELASTICITY

0

0.12

UV on

Stress (Ds) (kPa)

cis-fraction

0.11

-1

-1.5 0.10

-2 0

(a)

Irradiation time (min)

Birefringence (D n )

-0.5

(b)

20

40

60

80

100

120

Irradiation time (min)

F IG . 6.5. (a) The build up and decay of the fraction of cis monomers on illumination with UV light of appropriate wavelength, from (Eisenbach, 1978). (b) The simultaneous plot of the exerted mechanical stress, ∆σ – left axis, of a clamped irradiated sample (compare with Fig. 5.16) and the material birefringence ∆n – right axis, which is a direct measure of order parameter Q, from (Cviklinski et al., 2002). order parameter, whether Q be changed by temperature change or by illumination. The intimate relation between illumination, trans-cis isomerisation, change of nematic order and subsequent shape change is the subject of this section. Figure 6.5(a) shows directly how UV illumination increases the fraction of cis (bent) isomers with a characteristic time of about 10 min and how 465nm illumination stimulates the back reaction to the trans (rod-like) state. Of course, the characteristic times depend on the intensity of UV light. Simultaneous measurements of the mechanical response and order parameter change confirm that they accompany each other during photoelastic effects, see Fig. 6.5(b). We need few details of photisomerisation to model photo-mechanical effects. Figure 5.14(b) schematically shows the states, energies and paths between the states on an energy-configuration landscape. The energy to excite the trans state to the intermediate state is provided by a UV photon. The cis state that results then has either an activated decay over a thermal barrier of height ∆, or an optically stimulated decay. Thus ∆ determines the cis → trans return rate in the dark. The return rates balance the photondriven forward rate in the photostationary state. The underlying processes are seemingly clear for the photo-elasticity of nematic elastomers, but as yet both experiment and theory are at an early stage. We interpret the existing results with the simplest possible models that point to future developments. Photostationary elastic response of nematic elastomers The simple model of spontaneous thermal distortions discussed with the aid of Figs. 6.16.3 suggests that it is nematic order that determines the spontaneous deformation, that is, λm = f (Q). Then λm is a function of T only because Q is. If we change Q by illumination, I, instead of by temperature change, then λm will depend on I in an analogous way:

λm (T ) = f (Q(T ))

(6.17)

PHOTO-MECHANICAL EFFECTS

λm (I) = f (Q(I)) .

133

(6.18)

λm (T ) and λm (I) are measured with respect to the isotropic state where λm = 1. Thus contraction with respect to the starting state at T , ending up in a state at T  > T , is λeff = λm (T  )/λm (T ) < 1. In the cases of both T and I, deformation λm depends on the same function f (Q). Figure 6.1 shows simultaneous measurements of λm (T ) and Q(T ) which hence yield λm = f (Q). A particular model for f (Q) is, for instance, eqn (6.13) - which is evidently of considerable and unexpected precision, see Fig. 6.3. We do not need to know the form of f (Q) to proceed. Since it is the same function for thermal and optical response, then photo response can be predicted from an experimental knowledge of the thermal response. Every degree of illumination corresponds at equilibrium to an apparent elevation to an effective temperature, Teff (To , I), from the real temperature, To . One identifies equivalent drops in Q due to temperature rise with those due to illumination at constant temperature, To . The identification gives the effective temperature as a function of illumination. Q(To , I) ≡ Q(Teff , I = 0)

→ Teff (To , I) .

(6.19)

The change in order can be large. Figure 5.15(a) shows a 24% contraction of the elastomer on heating from 298K to the isotropic state. The final photo-contractions of Fig. 5.15(b) are comparable, around 20%. Thus in that experiment, illumination gives a photo-stationary state where Q is finally essentially eliminated. Modelling will require the full variation of f (Q). Other experiments (Cviklinski et al., 2002) show a much smaller shift in order and correspondingly a change in stress small compared with that developed on heating the clamped elastomer to the isotropic state, Figs. 5.16 and 6.5(b). Linear connections between all variables serve modelling well. We now simply model the illumination-temperature mapping, Teff (To , I), by first estimating the population of disruptive cis isomers. Let φo be the volume fraction of photochromic molecular moieties. Before illumination they are all in the trans state. Let φ be the current trans volume fraction. The rate of rod conversion into the kinked shape, the trans-cis transformation, depends on the illumination intensity, I, the rate of molecular absorption, η , and the volume fraction of trans rods itself. Thus φ is photo-depleted at a rate φ˙ = −η I φ . If the lifetime of the metastable cis state is τc , then the back reaction rate is φ˙ = (φo − φ )/τc , that is proportional to the volume fraction, φc = φo − φ , of rods in the cis conformation. The rate of back conversion is 1/τc which we assume for simplicity is not influenced by the current nematic order or mechanical tensions in the network. Note that these assumptions are not entirely realistic. Taking order and stress into account makes the dynamics highly non-linear – but we only examine the very basic ideas here. In the photo-stationary state, φ˙ = 0, the two conversion rates are equal and we have the equilibrium trans concentration:

φ = φo

1 1 + τc η I

with

(6.20)

134

NEMATIC RUBBER ELASTICITY

φc = φo

τc η I ∼ φo τc η I for small τc η I . 1 + τc η I

(6.21)

The bent, cis units dilute the other still nematic rods. For small enough concentrations of this cis impurity, the transition temperature will be diminished linearly: Tni → Tni (1 − φc ). This drop is equivalent to raising the effective temperature to Teff = Tni φc (To , I) + To thus τc η I ∼ To + Tni φo τc η I . Teff (To , I) = To + Tni φo 1 + τc η I

(6.22) (6.23)

The spontaneous thermal contraction curve, λm (T ), then provides the stationary photo-contraction by identification:

λphoto (To , I) = λm (Teff ) = λm (To + Tni φo τc η I)

(6.24)

The photo-response is revealed by traversing the thermal contraction curve to higher temperatures. The simplest model of the effective temperature when the dilution φc by cis units is not small would follow the Maier-Saupe/Landau-de Gennes pictures of nematic ordering (Finkelmann et al., 2001). The linear suggestion above of how the photo-stationary state, the plateau obtaining in Fig. 5.15(b) at long times, varies with the intensity of illumination and temperature has not yet been experimentally tested. But ideas close to these arise in the context of dynamics (see below). An important practical point emerges from the linearisation of eqn (6.24) for the photostrain at a temperature To : * d λm (T ) ** δ λphoto (I) = Tni φo τc η I (6.25) dT *To To obtain the maximum mechanical sensitivity for a given illumination, one should operate at a temperature where the gradient, d λm (T )/dT , of the thermal response is greatest. The ultimate photo-response possible at very large I, where conversion to cis is substantial, depends on the temperature. Starting with λm (T ), the largest reduction is to the isotropic state with λm = 1. The lower T , the greater λm (T ) and hence the greater the possible contraction 1/λm (T ) to the isotropic state. We have ignored two other subtle and interesting effects. The rate of absorption of light polarised, say, along the director of the rubber, depends on the degree of nematic order. For high order most rods have their transition moment along the optical electric vector of the light and absorption is easier than when the nematic order is low and many rods are misaligned with respect to the optical polarisation. Thus one might expect η to vary as η = ηo Q, ignoring internal field effects. This makes the dynamics non-linear, since Q depends on φ and thus φ would enter the dynamical equations more than once. The dependence would also shift the final point of equilibrium that we have used above. One can avoid this problem by using unpolarised light at normal incidence to the rubber

PHOTO-MECHANICAL EFFECTS

135

strip. It would be interesting to see if there was a difference in the photostationary states associated with polarised and unpolarised light. Photoisomerisation can instead lead to director rotation rather than simple depression of the order parameter. If rods rotate while they are in the cis state, they can reform to the trans state with their rod axis at various angles to the director. Those oriented so that their transition moments are aligned to the electric field of the light will suffer another photoisomerisation. Those oriented perpendicular to the light will not, and thus a population of rods perpendicular to the polarisation could increase. This is in effect a rotation of the director. It is not known whether this route is followed in nematic photoelastomers although it would be easy to test for. Director rotation would lead to anomalous mechanical responses perpendicular to the original director – the subject of Sect. 6.7 and of Chapter 7. Dynamics of photoelastomers The photo-induced trans to cis transition, with rate η I φ , and the thermal or stimulated back reaction, with rate (φo − φ )/τc , can be combined to form a simple, linear rate equation. One can assume that the barrier for transitions from the trans state is too large for thermal effects and thus neglect the rate of spontaneous, thermally driven trans to cis transitions. There is evidence that the photodynamics is largely linear, but perhaps with a richer structure than this simple model. Another worry is that the dynamics of photo-induced strain and stress could be determined not by the dynamics of photoisomerisation (which drives the process and which determines the stationary state), but by the dynamics of the network itself. Some aspects of nematic elastomer dynamics are discussed in Chapter 11. However, away from the glass temperature and for the slowish azobenzene photochromes thus far investigated, the polymer network response is clearly fast and the rate-limiting process is photochromic. Simultaneous measurements of the order parameter and stress changes on illumination, Fig. 5.16(b), show that stress response does not lag behind the photo-induced order parameter changes. Adding together the two rates, the rate of change in the population of trans isomers can simply be written as;

∂ 1 φ (t, To ) = −η I φ + (φo − φ ) ∂t τc

(6.26)

The initial condition is φ (0, To ) = φo (all photochromes in their rod-like trans conformation). The dynamic solution to eqn (6.26) is simply;   φo 1 + τc η I e−t/τeff (6.27) φ (t, To ) = 1 + τc η I τc . with τeff = 1 + τc η I The increasing concentration of kinked cis-isomers, which act as impurities for the nematic ordering, will eventually reach a plateau due to the opposing thermal (and possibly optically stimulated) back-reaction; the long-time limit of eqn (6.27) is identical to the photostationary state eqn (6.20). Both the rate and the final saturation value of the

136

NEMATIC RUBBER ELASTICITY

isomer density are crucially dependent on the irradiation rate η I; this differs between materials and with intensity of radiation. Temperature enters φ (t, To ) through activation in τc . The role of sample shape and gradients of radiation absorption is more subtle and would take us beyond this simple analysis. The relaxation process , after switching the UV radiation off (I = 0), is the optically stimulated or thermally driven flux of cis-isomers into the trans ground state. Either appropriate light is shone on the sample, or there is activation over the barrier ∆; the details are buried in the parameter τc which therefore may not necessarily be identical to the τc of the first irradiation phase. Assuming a fully saturated, photostationary state, with φ = φo − φ (∞, To ) = τc η I/(1 + τc η I) at t = 0 (when the UV illumination was switched off) results in the relaxation law   τc η I exp (−t/τc ) . φ (t, To ) = φo 1 − (6.28) 1 + τc η I Naturally, the relaxation time scale is now purely τc , a property of an individual photochromic group, which should remain at least approximately universal between all elastomers with the same active group and stimulated by the same colour and intensity of light. The basic features of growth and decay of the cis population are modelled above and correspond to the measured relative population dynamics φc /φo of Fig. 6.5(a). Translating the trans-cis population dynamics into strain or stress dynamics is analogous. While the cis population is varying, we have from eqn (6.22): d φc dTeff ≈ Tni → Tni φo η I dt dt

(6.29)

The latter limit is valid is for short times; recall that the rate of loss of trans volume fraction is −η I φ and thus the rate of gain of cis volume fraction must be +η I φ . The first part of eqn (6.24) then yields * * d λphoto (To , I) d λm (T ) ** d λm (T ) ** dTeff = → · Tni φo η I . (6.30) dt dT *Teff dt dT *To As an example, we have evaluated the short-time strain dynamics. The dynamical factor φo η I of this equation does not vary greatly with T . Then the initial strain rate should be determined by d λm /dT , the slope of the thermal expansion curve at the given experimental temperature, To . Figure 5.15(b) shows that the initial rates of strain are ordered in the sequence of the values of the gradient d λm /dT for the various operating temperatures. A fuller analysis using λm (T ) can describe the dynamics of photoinduced strain over the full range of contractions (Finkelmann et al., 2001) and thereby fits the photo-contraction and recovery curves Fig. 5.15(a). 6.5

Thermal phase transitions

An elastomer formed under conditions such that the chain step length tensor is  o , relaxes to a shape  r when conditions are subsequently changed. The zero of energy

THERMAL PHASE TRANSITIONS

137

then changes, but the form of the elastic free energy density, eqn (6.16), resembles a simple nematic elastomer. The elastic energy in the relaxed state modifies the underlying nematic energy and thereby influences the thermal properties. We use Landau theory to show: 1. How memory of the crosslinking conditions shifts Tni , the nematic-isotropic phase transition temperature. Crosslinking in the nematic state strengthens ordering and raises Tni , crosslinking in the isotropic phase does the reverse. 2. How the above effects depend upon crosslink density. For the sake of argument, we adopt freely jointed chains with the step length tensor Q). Section 6.2 explored how even side chain polymers can from eqn (3.16),  = a(δ + 2Q effectively follow such a simple model. Recall that the step lengths are  = a(1 + 2Q) and ⊥ = a(1 − Q) within the model. The determinant is Det() =  2⊥ = a3 (1 + 2Q)(1 − Q)2 = a3 (1 − 3Q2 + 2Q3 ) .

(6.31)

Corresponding expressions for the formation step length tensor  o arise from the order parameter Q that existed at network formation. The Q -dependence of the step length o tensor for general polymers is discussed in Sect. 3.2.1. In the relaxed free energy (6.16) we apply no external strain, that is λ = δ , and thus set  r =  , the step-length tensor of the relaxed state. The Trace is then equal to 3 and  1/3 the free energy depends only on the scalar combination W = Det(o )/Det() . We now expand this free energy about its minimum at W = 1:

(6.32) Fel = 32 µ [W − lnW ] = 32 µ 1 + (W − 1)2 /2 + . . . Elastomers formed in the isotropic state The step length tensor was  o = aδ and hence the determinant is Det(o ) = a3 . On substituting this for Det(o ) and eqn (6.31) for Det() into the expression for W , the expansion parameter then becomes (W − 1) = Q2 + . . .. Since (W − 1) appears squared in Fel , we do not have to worry about terms of order higher than Q2 in Det(). With all the provisos of Landau theory, Sect. 2.3, and discarding the additive constant 3µ /2 in Fel , we obtain Fel = 34 µα 4 Q4 + . . .

(6.33)

which is an elastic addition to the Landau free energy density Fnem (2.10) of the underlying nematic state. In eqn (6.33) the factor α accounts for different microscopic models and mechanisms of coupling between the order of the mesogenic rods and the polymer backbone shape, see Sect. 6.2. Combining the elastic and nematic contributions, the overall Landau energy density is: Ftotal = 21 Ao (T − T ∗ )Q2 − 13 BQ3 + 14 (C + 3µα 4 )Q4 + . . .

(6.34)

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NEMATIC RUBBER ELASTICITY 4

4

3

1

2

Shift DT ( C)

Shift DT ( C)

3

1 0 -1

2

-2 -3 0

1 0 -1 -2

2

-3

(a)

-4

1

2

(b)

-4 2

4

6

8

10

12

14

0

Ratio M W /MC

1

2

3

4

5

6

Ratio M W /MC

F IG . 6.6. Nematic-isotropic transition temperature, Tni against crosslinking density, in a polyacrylate side-chain nematic elastomer (Zubarev et al., 1997). In plot (a) crosslinks are chemical; data set 1 denotes linking in the nematic phase, 2 in the isotropic phase. In plot (b) crosslinks are established using γ -radiation, introducing fewer impurity sites; again data set 1 denotes linking in the nematic phase, 2 in the isotropic phase. Recall from eqn (2.12) that the weak first order transition temperature is Tni = T ∗ + 2B2 /(9CAo ). If C is increased as in (6.34), Tni must accordingly decrease. The change in the transition temperature is: 2B2 µα 4 . (6.35) 3AoC2 Not unexpectedly, the scale of the shift is set by the crosslinking density through the rubber modulus µ = ns kB T . There is considerable evidence to support this conclusion, that crosslinking in the isotropic state destabilises the nematic phase and an accordingly lower temperature is required before the nematic takes over from the isotropic elastomer, see for instance, Fig. 6.6. ∆Tni = −

Elastomers formed in the nematic state If there is nematic order Qo = 0 at formation, then the expansion in eqn (6.32) proceeds in powers of (small) deviation from Qo : W − 1 = (Q2 − Q2o ) + . . . .

(6.36) 3 4 2 2 2 µα Qo Q ,

in addition Now the free energy correction (6.33) has a quadratic term, to the quartic term. The effective Landau theory parameters, the coefficient C and the critical temperature T ∗ , thus change to new values: T ∗ → T ∗ + 3µα 4 Q2o /Ao C → C(1 + 3µα 4 /C) , whereupon there are two influences acting to shift Tni . Crosslinking itself represents constraints and thus increases C, thereby depressing Tni as considered above. Crosslinking in a memory of the order Qo shifts T ∗ and thus Tni since the latter is based on the former, see exercise 2.1:

EFFECT OF STRAIN ON NEMATIC ORDER

3µα 4 ∆Tni = Ao

 Q2o − 12



2B 3C

2  ≡

139

3µα 4  2 1 2  Qo − 2 Qni . Ao

The exercise shows that the final combination 2B/3C of constants is, for a nematic liquid, the magnitude of the order parameter at the transition point, Qni , see eqn (2.12). √ Thus if the order at formation is greater than Qni / 2, then crosslinking in nematic order raises the nematic-isotropic transition to higher temperatures – the nematic state is stabilised. A spontaneously formed nematic clearly has Qo ≥ Qni and the transition in the corresponding crosslinked network should be at a higher temperature, ∆Tni > 0. As we shall see, a small value of Qo could also be introduced by applying stress to an isotropic phase that is being√ crosslinked. An external field below the critical value would induce an order Qo ≤ Qni / 2 and the resultant elastomer would still see a depression in Tni . One would have to apply a supercritical field to an isotropic system in order to achieve an elevation of the transition temperature in the elastomer. Since the shift in T ∗ depends quadratically on Qo , it is increased by spontaneous order of both signs, prolate or oblate. Either way, it is the imprinting of order that increases the thermal stability of the nematic state. Real nematic elastomers are non-ideal in their thermal transitions. There are no jumps at a Tni (shifted or otherwise). Some show a blurred out, apparently supercritical behaviour, see Fig. 6.1 or curve (b) of Fig. 6.2. It is as if a linear Q term, an apparent external field, has been added to the free energy. See Fig. 2.6 for the parallel liquid nematic case. In fact, there are several possible reasons for an effective field, a linear Q contribution, to arise in the Landau free energy of non-ideal nematic elastomers, for example see Sect. 7.6. Other elastomers are quasi-critical in that their order vanishes as Q ∼ |T − Tc |β , see Fig. 6.1(a) and curves (a) and (c) of Fig. 6.2, with β ≈ 0.3 (Clarke et al., 2001). Randomly quenched fields that induce this type of non-ideality in nematic networks are described in Sect. 8.4. When shifts in Tni are discussed, for instance, in Fig. 6.6 or in (Popov and Semenov, 1998), then the transition temperature is defined by the order parameter crossing an arbitrary, set value of Q. 6.6

Effect of strain on nematic order

Network formation modifies the free energy and shifts transitions, even when no strain is imposed. How robust is the nematic order when strains are applied? Electric and magnetic fields induce small changes in nematic order and only shift transitions very little. Mechanical fields affect the magnitude of the order much more if the elastomer cannot rotate the director in order to accommodate the perturbation (internal rotations will be the main theme of later chapters). The regime when no director rotation occurs is of actual relevance in nematic elastomers and gels. There is no incentive to rotate the director in response to elongations along the director, which includes the spontaneous deformation λm discussed earlier. Even when the extension axis is perpendicular to the nematic director, a number of experiments (e.g. of Mitchell, Fig. 5.19, and of Finkelmann, Fig. 5.20) show that there is range of strains (up to 10%) before the director rotates in response to a perpendicular elongation. When the director stays fixed on deformation, the local nematic order might be altered. Here we examine:

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NEMATIC RUBBER ELASTICITY

1. strain-nematic coupling 2. consequent stress-optical effects. Classical response and strain-induced relaxation If the nematic order parameter Q does not change and the director does not rotate, the response of a nematic elastomer to longitudinal and transverse elongations is that of a classical rubber. Consider the experimental geometry of perpendicular extension, Fig. 5.18, where one of the principal extensional strains λxx (perpendicular to n ) or λzz (along n ) will be imposed. The other component of these deformations and the third principal strain λyy will relax in response. The strain is simply λ = Diag (λxx , 1/(λxx λzz ), λzz ): there is no reason for shears if we extend along principal directions of the nematic order and prohibit the director rotation. In this case of unchanging Q, the step length tensor  is simply the initial form  o . The Trace formula eqn (6.5) involves the trivial   = multiplication of four diagonal matrices: λ = λ T ,  o = Diag o⊥ , o⊥ , o and  −1 o   Diag 1/o⊥ , 1/o⊥ , 1/o . The result is simply Fel =

1 2µ

2 λxx + λzz2 +

1 (λxx λzz )2

 .

(6.37)

Fel is entirely classical irrespective of whether extension is along or perpendicular to the director. From the symmetry of eqn (6.37), either imposed λ = λzz or imposed λ = λxx , gives5 a relaxed Fel = 12 µ (λ 2 + 2/λ ). Thus the moduli for extension in these directions must be the same, despite the solid being of uniaxial symmetry. Expanding in either of the two directions λ = 1 + ε for small (linear) deformations gives Fel = 32 µε 2 with an extension (Young) modulus of E = 3µ . The free energy and modulus take their classical forms, as in the isotropic rubber, since in each case we are imposing a distortion on a simple, albeit anisotropic, random walk and paying the same entropic penalty in each case. The relative change in the number of configurations is the same in each case, a property unique to rubber elasticity. Since the elastic addition (6.37) to the nematic free energy has no dependence on Q, there can be no nematic-mechanical coupling. Experimentally, nematic order Q does indeed vary with strain. In the isotropic state this is the classical stress-optical effect, but evidently order should also vary when strains are imposed on the nematic state. P-G. de Gennes first envisaged strain-nematic coupling (de Gennes, 1975), concentrating on the high temperature state of the nematic elastomer (thus order was small and only the A term in the Landau-de Gennes energy was needed). Others (Deloche and Samulski, 1981) used nematic order changing with imposed strain to explain departures from classical rubber elasticity. Strains along and perpendicular to the director perturb nematic order in subtly different ways necessitating a tensor treatment of how nematic √ instance if λ = λzz is imposed, then minimisation of (6.37) over λxx yields λxx = 1/ λ . On sub2 stitution of this λxx we get the classical form for Fel of ∼ λ + 2/λ . The result is the same if λ = λxx is imposed. 5 For

EFFECT OF STRAIN ON NEMATIC ORDER

141

order responds to strain. We first present a simplified analysis and then sketch the full problem. Scalar mechanical-nematic coupling We here look phenomenologically at the scalar problem of the responses of the isotropic phase to strain and of the nematic phase to strain along its director (Rusakov and Shliomis, 1985; Halperin, 1988; Kaufhold et al., 1991). To the nematic free energy we add an isotropic elastic energy density, generically 12 µε 2 , a phenomenological coupling between the strain ε and the nematic order, −UQε , and if we wish to consider the imposed stress, σ , instead of strain, then we also add the external work −εσ . The elastic energy is taken to be isotropic in this scalar approximation. The addition to the nematic free energy is then 12 µε 2 −UQε − εσ . The optimal strain comes from minimising over the strain ε to give

ε∗ =

σ +UQ . µ

(6.38)

The first part (σ /µ ) is the strain of a passive elastomer with σ applied, the additional part arises from the induced order Q. Returning this ε ∗ to the free energy density we ˜ σ , Q), now dependent on σ rather than on ε , replacing have overall a free energy F( F(ε , Q): F˜ = 12 Ao (T − T ∗ )Q2 − 13 BQ3 + 14 CQ4 − σ UQ/µ − 12 U 2 Q2 /µ − 12 σ 2 /µ . (6.39) We have in effect performed a Legendre transform from the independent variable ε to σ . The − 12 U 2 Q2 /µ contribution can be combined with the first term and thus increases the critical temperature T ∗ . It elevates the actual nematic-isotropic transition, as we saw in Sect. 6.5. Elongation always stabilises the nematic phase, as if the elastomer had been crosslinked in the nematic state. The −σ UQ/µ term, linear in the field σ and in the order parameter Q, is the classical coupling of a nematic to an external field. The induction of paranematic order and shifting of the transition increasingly to a critical point with increasing stress is as for liquid nematics, compare with Fig. 2.6. At high temperatures an elastomer has minimal energy for zero nematic order. A curvature (or stiffness), A, of the energy at this minimum penalises changes away from Q = 0. Since the induced order Q is small, we just take F = 12 AQ2 − σ UQ/µ . The induced order (minimising F with respect to Q) is: Q∗ =

σU σU ≡ . µA µ Ao (T − T ∗ )

(6.40)

The susceptibility of the nematic order to the applied stress, ∂ Q∗ /∂ σ , gets very large as A vanishes at the pseudo second order transition temperature T ∗  Tni . The divergence is of the form ∼ (T − T ∗ )−γ with a classical mean field exponent γ = 1. We have neglected any fluctuations which usually dominate the response close enough to a true second order phase transition. Kaufhold et al. have analysed elastomers under stress close to, but above, their nematic-isotropic temperature. Birefringence is a good measure of the

142

NEMATIC RUBBER ELASTICITY

F IG . 6.7. Birefringence induced by stress or temperature (Kaufhold et al., 1991). (a) Birefringence against true stress, σW , at a series of fixed (reduced) temperatures just above an elastomer’s nematic phase . (b) Average birefringence and nematic order parameter (right axis) against reduced temperature from the nematic to paranematic states at various fixed nominal stresses. Behaviour is clearly supercritical, analogous to that of Fig. 2.6. order parameter; its induction by stress at fixed (reduced) temperatures, and the increase of birefringence with (reduced) temperature at a given stresses, are shown in Figs. 6.7(a) and (b). The would-be divergence, eqn (6.40), as the temperature approaches T ∗ is best seen by plotting reciprocal birefringence against reduced temperature, see Fig. 6.8. This data derives (Kaufhold et al., 1991) from vertical cuts through graphs like Fig. 6.7(b), and experiments to even higher temperature, at two different true stresses. The linear dependence of the inverse shows that one can indeed neglect critical fluctuations. Since Kaufhold et al. (1991) analysed polydomain elastomers, the optical birefringence in the nematic phase was impossible to measure in the stress-free state. Instead they measured the average birefringence against temperature at many values of fixed imposed stress (when the polydomain state begins to align under extension), see Fig. 6.7(b), and extrapolated to a stress-free birefringence versus temperature relation. One sees, for a given stress, typically states of residual nematic order above Tni , varying with temperature. The extrapolation back to zero stress sees an apparent total loss of order above a transition temperature. In neither of the experimental graphs, Fig. 6.7(a) and (b), is there a jump that one might have expected from the classical first-order transition response to an applied field in Fig. 2.6: in elastomers one finds a significant blurring out of the transition. Later chapters address this and other departures from ideality. Nevertheless Kaufhold et al., by careful analysis, could find a linear shift of

Inverse birefringence Dn -1 (X10-3 )

EFFECT OF STRAIN ON NEMATIC ORDER

143

sw= 6.10-3 N.mm-2

6 5 4 3 2

sw=2.5 .10 -3 N .mm-2

1 0

1.01

1.07 1.05 1.03 Reduced temperature T/Tni

F IG . 6.8. Reciprocal birefringence against reduced temperature T /Tni , showing a good linear fit extrapolating to the critical temperature, see eqn (6.40). The two data sets are for applied stresses of σ ≈ 2.5 and 6 kPa. a transition temperature (defined as the temperature where, at a given stress, the average order parameter reaches an arbitrarily set value of Q = 0.2) with applied stress. They also determined a linear connection between elongation (at fixed stress) and order parameter (found by separately determining elongation and order parameter against temperature). This ingenious study of polydomain elastomers confirms the picture of chain order determining macroscopic shape and the action of applied fields in shifting nematic transitions (albeit in a blurred fashion). The authors were able to then find the Landau coefficients and the coupling constant U, which took the value U ∼ 105 J/m3 , not surprisingly of the same magnitude as the rubber modulus µ . The connection between order and mechanical distortions presaged the investigations on monodomains that were to follow. Tensor mechanical-nematic coupling Nematic order is induced by strain because order causes chains to naturally elongate via orientation effects. This natural elongation thereby reduces the elastic cost of strain. The network benefit must be weighed against the cost of nematic ordering deviating from the value natural at this temperature, a cost expressed by the stiffness A in the foregoing. Directional effects enter in two ways – in the nematic cost and in the induction of polymer shape change by the (tensor) order. Extension along the director n will be cheaper if the order increases, δ Q > 0, and the chains are naturally longer in this direction. Extension perpendicular to the director causes contraction along the director. The uniaxial order would be induced to decrease so the chains become less elongated, hence a δ Q < 0. The magnitudes of the changes, |δ Q| and |δ Q |, are not the same for the same strain: a change δ Q < 0 by increasing polar angles θ of rods away from n causes elongations in both directions perpendicular to n , one of which matches the applied extension, but one of which is in fact a direction of contraction. The strain energy is further reduced by developing biaxial order perpendicular to n . Rods at a given θ to n find their azimuthal angles φ no longer uniformly distributed, but biased towards the extension direction and away from the contraction direction. See Fig. 2.4 for the angles involved in defining the uniaxial and biaxial order

144

NEMATIC RUBBER ELASTICITY

Q and b. Chain shapes follow the order changes of their constituent rods. Thus distortions along and perpendicular to n induce quite different changes in nematic order. Equally the nematic costs are very different. Sect. 2.3 discusses general nematic ordering. The stiffness of uniaxial order (FQ ) is given in exercise 2.2, that is, more than just the A term is required when nematic order is large. The biaxial stiffness (Fb ) against changes in b is given by the coefficient of 12 b2 in eqn (2.11). Uniaxial order stiffness is much reduced near T = Tni where order weakens. Hence chain elongational relaxation due to change in Q can take place more readily and the modulus against mechanical elongation along n is lower than at low T where the order stiffens. By contrast the induction of biaxial order is relatively less temperature dependent and thus one expects the perpendicular elastic modulus to vary less. Order is encoded in the nematic rubber elastic free energy via the  −1 term in the trace formula. Changes in the natural shape  in response to δ Q and δ b differ and thus the rubber elastic benefit from order-induced polymer shape changes also differ for the two distortions. Again, the effects on the elastic moduli are different. All these influences come together in a tensor analysis (Finkelmann et al., 2001a) of the variation of moduli that uses the estimates of Sect. 2.3 and exercise 2.1 of the Landau coefficients. For instance Fb can be simply shown to be BQ(T ) and then B is estimated from the temperature, order parameter and latent entropy of the liquid crystal transition. This stiffness varies with T only through Q(T ), rather than through A(T ) as the more sharply varying FQ does. The theory is given in Appendix A. Experimentally, the moduli do behave differently, see Fig 6.9(a), in contrast to the Young modulus E (kPa)

2.5

E

E

2.0

1.5

1.0

30

(a)

Temperature ( C)

(b)

40

50

60

70

80

90

Temperature ( C)

F IG . 6.9. The temperature variation (Finkelmann et al., 2001a) (with Tni ≈ 72o C) of (a) the Young moduli for extensions parallel and perpendicular to the director, E and E⊥ respectively and (b) the ratio E /E⊥ . Gaussian rubber elasticity gives 1 for this ratio. Deviations occur via the induction of strain-induced order changes. Gaussian prediction that they should be the same when order and director are unchanging. Taking the ratio of the moduli, Fig. 6.9(b), betrays fundamental differences in how uniaxial and biaxial order respond to mechanical fields and thereby change the relevant moduli. The extent of mechanically-induced biaxiality consistent with Figs. 6.9 is large, an interesting response that could be tested by NMR.

MECHANICAL AND NEMATIC INSTABILITIES

6.7

145

Mechanical and nematic instabilities

Consider once again the ‘classical’ experimental geometry used by Mitchell and many others, Fig. 5.18, where extension λxx is applied along a rubber strip perpendicularly to the initial director n o which is along z . Taking the imposed λxx to be λ and the zrelaxation to be λzz , then the deformation is simply λ = Diag (λ , 1/(λ λzz ), λzz ). We suppress here the possibility for the clamped strip of rubber to shear. There are then only two possible scenarios, that we label A and C to be consistent with later figures: (A) The director does not rotate and the nematic order Q does not change. The step length tensor  clearly remains equal to the initial tensor  o and the response is again classical as in eqn (6.37) since the principal directions of strain and nematic order remain aligned:   Fel = 12 µ λ 2 + λzz2 + 1/(λ λzz )2 . We ignored order parameter relaxation, Sect. 6.6, assuming the nematic √ order is sufficiently robust to render this effect small. The z-relaxation√λzz = 1/ λ minimises Fel . The y-relaxation is the same as in the z direction, λyy = 1/ λ . The free energy, FA , is:   FA = 12 µ λ 2 + 2/λ .

(6.41)

(C) The director jumps to the axis x of extension. The new director is n = x . The inverse   current step length tensor is instead Diag 1/ , 1/⊥ , 1/⊥ : note the 1/ term has moved to the xx position in  −1 . We meet for the first time elastic effects with a change of director. But the Trace formula is still trivial since its 4 tensors are still diagonal. Multiplying them out and summing the diagonal elements of the result yields:  1 2 ⊥ 2  1 . (6.42) + λzz + Fel = 2 µ λ  ⊥ (λ λzz )2 √ 1/4  (1/ λ ). Let us reThe transverse relaxation minimising Fel is now λzz = ⊥ /  write this as 1/λ λ2 where the reference deformation λ2 is λ2 ≡ ( /⊥ )1/2 = r1/2 .  The y-deformation λyy = 1/(λ λzz ) = λ2 /λ is different, so in this sense the response is quite non-classical. Fel takes the form FC :    λ 2 λ2 1 , (6.43) +2 FC = 2 µ λ2 λ which seems classical, but where the natural state is λ = λ2 rather than λ = 1. Figure 6.10 shows the two free energy densities FA and FC corresponding to the unrotated and rotated chain distributions. They have the same minimum energy, F = 23 µ , even though at the origin of FC there has been a distortion, λxx = λ2 . The equality of energy of the states at λ = 1 and λ = λ2 with n rotated by 90o was addressed with Fig. 5.22. If the distortion of the sample is sufficient to accommodate

146

NEMATIC RUBBER ELASTICITY

F

q=

p/ 2

FC

FA

0

q=

1

l

FB r 1/6

r 1/3

r 1/2

F IG . 6.10. The free energy densities FA of the unrotated nematic director (denoted by θ = 0) and FC of the rotated state (θ = π /2) against strain applied in the x direction, perpendicular to the original director n o . The minimum of the rotated branch is at λ2 = r1/2 and takes the same value as the minimum at λ = 1. See Sect. 6.7.2 for hints as to how to traverse λ = 1 → λ = r1/2 without energy cost, the branch FB . The limit to stability of FA is at r1/3 and that of FC is at r1/6 . These limits are reached when the state becomes unstable with respect to small director rotations.

F IG . 6.11. Accommodation of chains by the shape changes of the embedding solid. A solid with dimensions in proportion to chain dimensions changes by the given factors when the director jumps by 90o . the rotated chains, there is no distortion of the distribution of molecular shapes  and thus no energy cost. The object being rotated reflects the sizes of chains: R ∝  and √ R⊥ ∝ ⊥ . In effect, it is  1/2 that is being accommodated, see Fig. 6.11. The square root √ of the nematic anisotropy of chain step lengths, r, is vital. We now see how peculiar, in terms of classical elastomers, the response is. In addition to the energy not differing between λ = 1 and λ = λ2 , the transverse relaxation λzz accommodating the rotation of  1/2 is λzz = 1/λ2 , while the other transverse strain λyy does not change between these states. For distortions larger than λ2 , case C is entirely classical if λ is measured from λ2 , that is, λ /λ2 is the effective distortion. After rotation √ is complete the rubber behaves classically and the transverse relaxations are again ∼ 1/ λ . 6.7.1

Mechanical Freedericks transition

Returning to the Mitchell experiment and to Fig. 6.10, the elastic energies of the two branches with different orientation of n, FA and FC , cross each other when λ 2 + 2/λ =

MECHANICAL AND NEMATIC INSTABILITIES

147

√ λ 2 /r + 2 r/λ , that is, at a crossover extension:

λc = r

1/3

2 √ r+1



1/3 ≡ λm

2 √ r+1

1/3 .

It is advantageous to flip the director by θ = 90o at this value of extension λxx . However the θ = 0 (FA ) branch of the free energy only becomes unstable against rotations at λ = r1/3 ≡ λm , slightly larger than λc . The state with θ = 0 no longer corresponds to a minimum, albeit metastable, of the free energy, but to a maximum. The same remarks apply to the state with θ = π /2 at r1/6 as λ is reduced. We explore this later when we consider general rotations of the director. Again the special value of strain λm arises, the spontaneous elongation on going from the isotropic to nematic state. At this deformao tion, then, the director must finally jump √ to 90 and the system descends, on further extension, to the minimum at λ = λ2 = r. We can now discuss the experimental result of Mitchell et al. since the director orientation θ (λ ) should show a jump at λc , or at the latest λm = r1/3 , see Fig. 6.12.

F IG . 6.12. Extension by λ perpendicular to the initial director n o inducing a director jump by an angle θ = π /2. The jump is probably at the limit to stability λ = r1/3 of the unrotated state, or possibly at λc where the energies of the initial and final states are equal. The rotated state is unstable when strains are reduced to λ = r1/6 . A practical difficulty is evident in Fig. 6.10 for the rotated state FC , since √ between the instability point of phase A at λ = r1/3 and the minimum of FC at λ = r the slope of the free energy dFC /d λ is negative. The stress in this interval of imposed strains must thus be negative. The sample would try to spontaneously expand from λ = r1/3 to λ = r1/2 , but, since it is a clamped rubber strip, it would suffer a what is known as Euler strut instability. A rod or strip subjected to longitudinal compressive force at its ends will eventually buckle sideways. The same problem of Euler instability renders academic the achieving of the instability at λ = r1/6 on reducing strain from the θ = 90o state to get back to the θ = 0 state. It is thus unlikely that one could ever exactly observe Fig. 6.12. We shall see later in Chapter 7 departures from ideality and soft elasticity that modify Fig. 6.10 and reduce or eliminate regions of possible negative stress. Another possibility is that n indeed starts jumping from the phase A at λ = r1/3 . A part of the sample adopts the larger strain λ = λ2 = r1/2 such that overall the length change imposed on the sample is accommodated. The fraction of sample with λ = λ2

148

NEMATIC RUBBER ELASTICITY

would increase slowly until an overall deformation λ2 is achieved, see the dotted line in Fig. 6.12. This would transform a strictly discontinuous transition into one spread over this interval of strains. The ratio of the starting and completion strains is r1/6 and is a measure of the width of the interval. In the Mitchell experiment, see Fig. 5.19, the onset is at a strain of r1/3 = 1.06 (assuming this is the instability point λm ). It should be complete at λ ≈ 1.09 – a much wider transition than the experimental observation and we discount this proposed scheme of transition. This putative path between the A and C branches involves disproportionation of the sample into two regions of differing extension. Such necking (thinning) of the sample is in fact not seen. We discuss this possibility again in Sect. 7.5, and again discount it. However the experimental data does show rounding before and after the transition. The most likely explanation is of some deviation of the initial director from being closely perpendicular to the extension. The analysis of oblique initial directors shows rounding of the discontinuous jump as seen in experiment and summarised in Fig. 6.12. A third possibility of transition between the two free energy branches in Fig. 6.10 is that, until the director jumps, energy is lowered by reductions in the order parameter Q and the induction of biaxial order b. Measurements (Roberts et al., 1997) of the order parameter during the transition show a rather small change, not enough to offer a low energy route between the two states. Associated with the jump in director over an interval of changing λ must be a change in λzz in order that the √ right accommodation of chain shape be made. It must be faster than the classical 1/ λ response since there is no change in the y direction needed when the ellipsoid representing the average chain shape is rotating in the xz-plane and so, for reasons of constancy of volume, everything must be taken up in the z direction. The scenario of no yy-changes (constant thickness of the rubber strip) during the transition from λ = r1/3 to r1/2 would make the zz variation λzz ∼ 1/λ , distinctly faster than classical. We see such an effect in the record of the zz strain taken (Roberts et al., 1997) during the transition, see Fig. 6.13(a). It is possible that there is an even more complicated scenario where λyy recovers from 1/r1/6 (reached before the critical strain for the transition of A to C is achieved) back to 1, as λ extends from λ = r1/3 to r1/2 , necessitating a drop in λzz even more rapid than 1/λ . See Fig. 6.13(b) for a scheme of distortions. One should perhaps call the director jump instability seen by Mitchell an ‘antiFreedericks transition’. Unlike the classical Freedericks transition in a nematic liquid in a cell subjected to an external field, the field (stress) is applied here at the surfaces of the sample rather than the bulk. The resistance to rotation is bulk anchoring of the nematic director to the rubber-elastic network (rather than on cell surfaces). The director response is (ideally) discontinuous (rather than continuous). 6.7.2

The elastic low road

Chapters 7 and 8 discuss alternative routes of evolution between the two key branches of elastic energy, FA and FC , with their equal minima at λ = 1 and λ = r1/2 . Taking the ‘low road’ between the two strains would correspond to shape changes without free energy cost – normally an option for liquids but forbidden to solids. This is the heavy

MECHANICAL AND NEMATIC INSTABILITIES

Transverse contraction

1.02

149

l zz , l yy

1.00 0.98

1

0.96 0.92

l yy= r

1/ l

0.94

1/4

l

r -1/6

0.90 0.88

l zz = 1/ r

0.86 0.84 1.00

1.05

1.10

1.15

1.20

Imposed extension, l

1.25

1.30

1

r 1/3

r 1/2

1/4

l

l xx = l

F IG . 6.13. (a) Transverse contraction λzz against extension perpendicular to the initial et al., 1997). Before and after the director jump the response is director no (Roberts √ classical (1/ λ ). One sees, in an interval during the jump between the initial and final regimes of λ , a much more rapid variation, at least as fast as 1/λ . (b) Speculations on the transverse contractions λzz and λyy against extension perpendicular to the initial director n o . During a transition in the range of λ between r1/3 and r1/2 , the transverse relaxation strain λyy must increase to 1. Therefore λxx must diminish faster than 1/λ in order to conserve volume. line of Fig. 6.10. The special case of a distortion λ = 1 → λ2 being accommodated at no energy cost by a 90o director jump, Fig. 6.11, can be generalised to intermediate director rotations, θ < 90o . The shape changes accommodating the chain shapes are no longer simple extensions and contractions, but have non-symmetric shears added in to more simple distortions. The rotation of the director relative to their associated component of body rotation are significant. Thus an entirely new type of elasticity emerges where shape can change at zero energy cost and where body rotations are significant. Chapter 7 is dedicated to this soft elasticity and gives concrete examples of such deformations that comprise the soft branch, with FB = 3µ /2 connecting FA and FC . The loss of convexity in the F(λ ) versus λ plot evident in Fig. 6.10 for λ between 1 and λ2 led to problems such as negative stress. The loss of convexity and its associated problems are circumvented by the soft elasticity sketched above. In practical problems, sample clamps or compatibility constraints with neighbouring bodies may constrain one to, say, the simple extensional form of λ in Fig. 6.11. Such distortions without shears are only soft at one extension, namely λ = λ2 . Thus clamps suppress soft modes with an extensional component λ < λ2 and an intermediate n . However, even with clamps a more complex soft route can generally still be found: Soft deformations with a given extensional component λ and with equal and opposite accompanying shears occurring in neighbouring regions, can be added in pairs to give the right (constraint satisfying) global deformation but still at low energy cost. For instance one might globally have λ = Diag (λ , 1, 1/λ ) with λ = r1/2 and hence not at the minimum of FC . Globally no shears are present and hence constraints are satisfied. But since λ is composed of soft shears added together, it too is soft. This process is

150

NEMATIC RUBBER ELASTICITY

called ‘quasi-convexification’ of the free energy. It will turn out that there is an infinity of soft modes to choose from and thus such recovery of soft response in the face of constraints is easy. The cost is the development of microstructures (which are the neighbouring regions of equal and opposite soft response). Chapter 8 first explores the simplest example of soft microstructures – the stripe instabilities of nematic elastomers. It then sketches the formal quasi-convexification process (DeSimone, 1999), taken over from crystal systems such as shape memory alloys (Ball and James, 1992). 6.8

Finite extensibility and entanglements

In Sect. 3.5 we examined two main effects leading to deviations from classical rubber elasticity. The situation in nematic elastomers is exactly the same. The neo-classical approach assumes the network is made of phantom, Gaussian, spontaneously-anisotropic chains and produces the Trace formula, eqn (6.5). In the main body of this book we look at many examples of successful application of this simple model; similarly, the classical isotropic rubber-elastic free energy density F = 12 µ (λ 2 + 2/λ ) is good first approximation for many physical situations. However, one still must appreciate a number of obvious model limitations. Phenomenologically, the Mooney-Rivlin approach (Treloar, 1975) attempts to account for corrections to classical Gaussian rubber elasticity by writing the free energy density in terms of strain invariants. With a sufficient number of phenomenological parameters, an adequate description of many practical deformations can sometimes be achieved. From the physical point of view, it is important to understand the microscopic background which leads to such corrections. If the network strands between crosslinks are not long enough, the effects of finite extensibility (deviations from the Gaussian limit) will become noticeable. In nematic elastomers, the resulting free energy density then takes a rather complicated form (Mao et al., 1998)   27µ    2 Tr λ ·  o · λ T ·  −2 : B :  −1 Fel ≈ 12 µ Tr λ ·  o · λ T ·  −1 + N   −Tr  −1 · λ ·  o · λ T ·  −1 : B :  −1 · λ ·  o · λ T ·  −1

(6.44)

where N is the number of steps the average chain makes between the crosslinks. The new 4-rank tensor B in eqn (6.44) is a small difference between the second and fourth moments of the chain link distribution which is a measure of deviations from nonGaussian behaviour [compare with the definition of  in eqn (3.13)]: Bi jkl =

1 2 24 a (ui u j uk ul  − ui u j ui u j ) ,

where a = (2 + ⊥ )/3 is the mean step length of the polymer backbone.  √ √  An important case is simple uniaxial extension λ = Diag λ , 1/ λ , 1/ λ . The corresponding linear elastic modulus (the Young modulus E) now takes different values when the principal extension direction is applied along the nematic director n , or perpendicular to it:

FINITE EXTENSIBILITY AND ENTANGLEMENTS

151

F IG . 6.14. A nematic polymer strand confined to a tube. The tube segment m contains sm monomer steps, where the index m runs from 1 to M. Since the direction of chain steps is, on average, uniaxial (as illustrated by the ellipsoidal shape of a free chain gyration on the right), the number of steps in each tube segment depends on the orientation of ∆ m with respect to the nematic director n .  1 E,⊥ ≈ 3µ 1 ± h(Q, ) (6.45) N where h(Q, ) is a complicated function of order parameter and chain step lengths (Mao et al., 1998). Importantly the difference due to non-Gaussian statistics is proportional to the small factor 1/N; in the neo-classical limit of N  1 the linear modulus is, of course, the same for extensions in all directions. The obvious fact that polymer chains cannot cross each other (the excluded-volume effect) leads to restrictions for the lateral motion of a chain, especially pronounced in a dense melt. It also leads to chain entanglements. As already mentioned in Sect. 3.5 (Edwards et al., 2000; Kutter and Terentjev, 2001) the role of entanglements in a crosslinked network must be much greater than in the corresponding melt because the reptation motion, leading to disengagement, is not possible here and so even a relatively short strand could be strongly entangled. The number of conformations of a chain confined to an effective tube around the primitive path connecting the entanglement nodes, Fig. 6.14, is determined by the number of steps the chain makes within each tube segment. The calculation shows (Kutter and Terentjev, 2001) that this number is determined by the segment length ∆m (as in the isotropic case of Sect. 3.5) but also on the degree of nematic anisotropy and the orientation of n with respect to the tube axis. The average number of steps the chain of the length N makes in a given tube segment is given by sm =

N|−1/2 ∆m| o −1/2

o ∑M i=m |

∆m|

.

(6.46)

On deformation of the network several things happen. First of all, not only the crosslinks but the entanglement points are assumed to move affinely with deformation, transform ing ∆ m into ∆ m = λ · ∆ m . Deformation could also affect the nematic order: the director

152

NEMATIC RUBBER ELASTICITY

n could rotate and the degree of average chain anisotropy r may change as well. In other words, the step-length tensor  o transforms into a new tensor  with possibly different eigenvalues  and ⊥ in a reference frame rotated by the angle θ . Hence |−1/2 · ∆m| o

∆m | on deformation. The quenched averaging procedure gives transforms into |−1/2 · λ ·∆ the elastic free energy density 2M + 1 Tr(o · λ T ·  −1 · λ ) 3M + 1 2 2M + 1  −1/2 | · λ · 1/2 | + 32 µ (M − 1) o 3M + 1

Fel = 23 µ

(6.47)

+µ (M − 1)ln |−1/2 · λ ·  1/2 |, o where µ = ns kB T , M is the number of tube segments (M = 1 corresponds to no entanglements) and the notation as in eqn (3.38) for angular integration is used: |−1/2 · λ ·  1/2 |= o

1 4π

1 ln |−1/2 · λ ·  1/2 |= o 4π



dΩ |−1/2 · λ ·  1/2 · e| o

(6.48)

dΩ ln |−1/2 · λ ·  1/2 · e| o

(6.49)

|ee|=1



|ee|=1

Network strands that are too short to be entangled (M = 1 case) would respond to deformation in the classical way, returning the Trace formula as the only remaining term in eqn (6.47). However, very short chains may end up in the regime of finite extensibility, with relevant corrections (6.44). In the opposite situation, when relatively long network strands (N  1) are Gaussian, the correspondingly large number of entanglements (M  1) makes the non-classical terms in eqn (6.47) dominate the response:   2 −1/2 1/2 −1/2 1/2 · λ ·  o | + ln | · λ · o | . (6.50) Fel ≈ µ M |

Exercise 6.5: Evaluate the entanglement corrections to the elastic free energy density for a case of uniaxial extension along the nematic director. Solution: In this geometry of deformation no director rotation will occur; no shears will accompany the imposed extension. On the other hand, an induced change in chain anisotropy could be expected. eqn (6.47)   √ Accordingly,  involves √ four diagonal matrices: λ = λ T = Diag 1/ λ , 1/ λ , λ ,  o = Diag o⊥ , o⊥ , o   and  −1 = Diag 1/⊥ , 1/⊥ , 1/ . The two new entanglement terms require the evaluation of  1  o  √   ⊥ /⊥ 0 0 λ sin θ cos φ    1 o √ 0 ⊥ /⊥ 0   sin θ sin φ  Λ ≡ −1/2 · λ · o1/2 · e =  λ  

o cos θ λ  / 0 0 



FINITE EXTENSIBILITY AND ENTANGLEMENTS

153

where (θ , φ ) are the angles e makes with the axes of our matrices. The length of this composite vector is . . o o λ 2 Λ| = / ⊥ sin2 θ + cos2 θ . |Λ ⊥ λ 

(6.51)

The angular averaging in (6.48) is now 1 2

 π 0

λ Λ(θ )| sin θ d θ = |Λ 2

0

o

(6.52)



0 1

+ √  4 λ λ 3 − r/ro

o⊥ r ln ⊥ ro



λ 3/2 + λ 3/2 −

 

λ 3 − r/ro



λ 3 − r/ro

where the ratios are, as usual, r =  /⊥ and ro = o /o⊥ . The second term is evaluated in the similar way: 1 2

 π 0

Λ(θ )| sin θ d θ = ln(λ ) − 1 + 12 ln(o / ) ln |Λ (6.53) 1



r 1 + arctan (λ 3 − r/ro )/r 3 λ − r/ro ro

When the chain anisotropy is not changed by the deformation (r = ro ), these expressions are exactly the same as those for the isotropic entangled rubber, eqn (3.41)(3.42). Adding the square of (6.52), and (6.53), to the free energy density (6.47) gives the answer, a rather complicated expression for the elastic free energy of a uniaxially stretched entangled nematic rubber.

An interesting and important conclusion of this analysis is the prediction for the spontaneous elongation of uniaxial rubber on increasing its nematic order, see eqns (6.9) and (6.14) derived from the Trace formula. Explicit minimisation of the modified free energy density (6.47), with the two entanglement corrections given in the exercise above, returns a remarkable result: both corrections (6.52) and (6.53) are minimised by exactly the same strain as the neo-classical Trace formula and, therefore, the spontaneous thermal expansion along the nematic director is still given by

λm =

r ro



1/3 ≡

 o⊥ o ⊥

1/3 .

Experimental observations of such a relationship have been quite unambiguous, over the years. Curiously, the important effects of excluded volume and chain entanglements modify the theoretical description in such a way that this relationship remains in force.

7 SOFT ELASTICITY Nematic elastomers possess a mobile internal degree of freedom, the rotations of a director. Unusual properties result, for instance spontaneous distortions, instabilities, and deformations at low energy  generalisation of the rubber elastic energy,  cost. A nematic T −1 1 the Trace formula 2 µ Tr  o · λ ·  · λ , is harmonic in distortions as in conventional rubber elasticity. But there are differences. The current nematic director n and order Q, in the form of  −1 , combine with the distortion tensor as λ T ·  −1 · λ . Further, the current director plus distortion connect with the original director via the initial step-length tensor  o . So both the history and the current state of nematic order are encoded in this more complex elasticity. We now investigate the elasticity of nematic elastomers, now allowing director rotations and shears. These influences render them unlike any other solid. The new element is that changing the director orientation allows a significant reduction in the elastic energy. The first example, Sect. 6.7, was the nematic-mechanical instability seen by Mitchell where the director jumps discontinuously to allow the rubbery network to accommodate the imposed strain. More generally, the director rotates in a continuous fashion. In this chapter we initially illustrate the role of rotation by discussing the relative rotation coupling constant D1 (briefly mentioned in Chapter 1). The analogous coupling D2 of symmetric network strains to relative rotations is also discussed here. We then address the singular nematic rotational response to strains and the apparent liquid-like mechanical response (low storage moduli) observed for some strain geometries. We call this effect ‘soft elasticity’. Section 6.7 sketched some deformations that apparently cost little or no energy, provided the director changed. Experiments confirm it is indeed nematic rotations that make the new elasticity possible. In general, the deformation λ does not have to be symmetric. It has eight independent elements, one being lost to the incompressibility constraint. Therefore the manifold of elastic trajectories is very large. Soft elasticity was first discovered (Warner et al., 1994) by straightforward minimisation of the Trace formula eqn (6.5), searching in a restricted set of possibilities. However we start with a more elegant and general method (Olmsted, 1994) that draws upon the roots of tensors. The explicit form of the soft response is then examined in a practical geometry where an extension is imposed perpendicular to the director, with director rotation and only simple shear permitted in response. Finally, in this chapter we shall look at departures from ideality and as to why thermomechanical history is so vital in determining the extent of non-ideality. We analyse an explicit form of the resulting ‘semi-soft’ response by returning to the experimentally important geometry explored in the soft case. The beautiful and general case of soft elasticity in biaxial elastomers is treated in Appendix B. 154

DIRECTOR ANCHORING TO THE BULK

155

In the following chapter we further illustrate semi-soft elasticity, rotational instabilities and characteristic quartic elasticity by selecting experiments where the geometry determines the peculiar responses observed. Why do we adopt a molecularly based picture for nematic solids, depicted by the Trace formula, and not a generalisation of continuum elasticity theory, the subject of Chapter 10? For small, elastically hard deformations this could be an adequate path (though with rubber, deformations need never be particularly small). But when softness occurs, its effect can extend up to 50-60% strain and much more than that (> 300%) in highly anisotropic main-chain nematic elastomers. Then, ideally, even an infinitesimal applied stress will induce such large strains before any elastic resistance is felt that the response to a stress is always non-linear – far beyond the validity of continuum theory. Chapter 10 develops such a continuum theory for the cases of imposed strains and for fluctuations, but it will be necessary to go to cubic order to explain many phenomena in nematic rubbers. Indeed, in this and in the next chapter on practical examples of distortion, we shall see that the response, if not exactly soft because of geometrical constraints, can be quartic in small deformations. Within linear elasticity, this is equivalent to being soft. In any event, the full non-linear possibilities of the Trace formula offer the simplest way forward. 7.1

Director anchoring to the bulk

Non-classical elasticity arises when we allow the director to rotate continuously during distortion. The tensorial character of the Trace formula records these rotations which we illustrate by considering: 1. The coupling of a rotating director to a clamped matrix, that is where there is no elastic deformation, λ = δ . The scale of the energy penalty for such rotation is set by the coefficient D1 of relative rotation coupling, see the discussion of Chapter 1. 2. The coupling between director rotation (relative to an unrotating background matrix) and pure, symmetric shears. For small shears and rotations, the scale of the coupling is given by D2 . 7.1.1

Director rotation without strain

Let the director rotate by an angle θ about the y axis. This could be achieved in principle by applying an electric or magnetic field obliquely to the initial director n o . More practically it can be achieved by instead rotating the matrix while anchoring the director, say to a surface – all that is important is the relative rotation of the director and matrix. The free energy density with λ = δ is then   Fel = 12 µ Tr  o ·  −1

(7.1)

where, ignoring changes in the magnitude Q, we have the current step-length tensor U θ , rotated by θ from its original configuration before deformation. Recall  = U Tθ ·  o ·U that  = ⊥ δ + ( − ⊥ )nnn . If we take out a factor of ⊥ , we have  = ⊥ (δ + (r − 1)nnn )

156

SOFT ELASTICITY

where r =  /⊥ measures the anisotropy of the average chain shape spheroid, that is the deviation from a sphere. Equally, the inverse is  −1 = 1 δ + ( 1 − 1 )nnn and one can take out a factor of 1/⊥ to give

1 ⊥ (δ

⊥



⊥

− (1 − 1r )nnn ) where again there is a negative

deviation from spherical (1− 1r ) for the inverse  −1 of a prolate (r > 1) spheroid. Back in the Trace formula the ⊥ and 1/⊥ factors cancel and the whole result is characterised by the single parameter, the ratio r. One sees from the  −1 expression that when we rotate from n o → n , it is the anisotropic part −(1 − 1/r) that is modified. Clearly rotations of the spherical part, δ , are not visible! Returning to eqn (7.1), we obtain, using reduced step-length tensors,  1 (7.2) Fel = 12 µ Tr [δ + (r − 1)nno n o ] · [δ − (1 − )nnn ] r  1 = 12 µ 3 + (r + − 2)(1 − (nno · n )2 ) (7.3) r (r − 1)2 2 sin θ r ≡ 32 µ + 12 D1 sin2 θ . = 32 µ + 12 µ

(7.4) (7.5)

For small θ we get, apart from the ground state constant value 32 µ , Fel ∼ 12 µ

(r − 1)2 2 1 θ ≡ 2 D1 θ 2 . r

(7.6)

The coefficient is identified as D1 = µ (r − 1)2 /r, giving the harmonic penalty for small rotations θ of the director with respect to the matrix. We return to such continuum expressions at small deformations in Chapter 10. The distribution of chain conformations (and hence its principal axis, the director n ) is rotated with respect to the background as in Fig. 7.1 and without the shape of the solid changing to accommodate the spheroid as it rotates, there is clearly an elastic penalty to be paid. When the rotation is by θ = π , in the nematic sense the system has recovered its initial state and the energy returns to zero, as we indeed see in 21 D1 sin2 θ (since

z q

(r-1)

x l=1 F IG . 7.1. Rotation of the anisotropic part (r − 1) of the step-length tensor with respect to the fixed rubber matrix; see exercise 7.1 for a discussion.

DIRECTOR ANCHORING TO THE BULK

157

sin π = 0). We explore such large deformations, making necessary the full non-linear form of the nematic rubber-elastic energy involving sin2 θ rather than θ 2 . As the rubber becomes isotropic, r → 1, the rotation of anisotropy loses its meaning and D1 ∼ (r − 1)2 vanishes, as it must. Note also that both prolate (r > 1) and oblate (r < 1) elastomers have a positive cost, D1 > 0, of rotating their respective anisotropy directions n with respect to the rubber matrix, Fig. 7.1. Exercise 7.1: Find an explicit form for the matrix of effective step lengths  after rotation from  o by angle θ about the y axis, see Fig. 7.2. Show that the shape distribution transforms like 2θ .

F IG . 7.2. Director rotation about the y axis from z to x .

Solution: Consider just the (x, z) parts of the  tensor and take the long axis (r) of   (y) θ sin θ  o to be along z . The y-rotation matrix is U θ = −cos sin θ cos θ . Then  − sin θ cos θ  1 + (r − 1) sin2 θ (r − 1) sin θ cos θ = 2 (r − 1) sin θ cos θ r − (r − 1) sin θ   sin θ cos θ 1 0 sin2 θ ≡ + (r − 1) sin θ cos θ cos2 θ 0 1

U Tθ =  = U θ ·  o ·U

cos θ − sin θ

sin θ cos θ



1 0 0 r



cos θ sin θ

Of course the spherical part remains spherical under rotations. Changes come from 0 rotating the aspherical part 00 r−1 , see Fig. 7.1. In analogy (with only r changed to 1/r), the inverse step-length tensor, −1 transforms to:    1 sin2 θ sin θ cos θ 1 0 −1 +  −1 = sin θ cos θ cos2 θ 0 1 r     1 0 − cos 2θ sin 2θ 1 1 1 1 +1 +2 −1 ≡ 2 sin 2θ cos 2θ 0 1 r r where in the last expression we have used trigonometric double angle formulae. One sees explicitly that a second rank tensor transforms as 2θ . Its transformation is complete after half a revolution, θ = π , reflecting the fact that the director recovers

158

SOFT ELASTICITY its original physical state after this rotation. The average part is of size   and the deviations from spherical are 12 1r − 1 .

7.1.2

1 2

1 r



+1

Coupling of rotations to pure shear

Let us apply a symmetric shear to a nematic elastomer in a plane that includes the director n . Section 4.3 shows there is no body rotation component to the corresponding deformation tensor λ S . However, we did see in Sect. 4.3.2 and Fig. 4.4 that such shears are represented by a local combination of extensions and compressions and thus induce director rotation toward the extension diagonal in spite of there being no rotational component in the strain tensor. To preserve volume, the tensor of pure shear must be written as   1 + λxz2  λxz , λS = λxz 1 + λxz2   the diagonal parts ensuring that Det λ S = 1; nothing is assumed to happen in the third direction, out of the shear plane – see the exercises in Sect. 4.3. We shall collect and examine terms at O(λxz2 ) in the Trace formula expression for Fel , so it is important  to retain the 1 + λxz2 contributions. We must now multiply the tensors  o · λ S ·  −1 · λ S together and take the trace of the resulting matrix. Explicit matrix forms of  o and  −1 are given in exercise 7.1. The rotation by θ of the director in response to λ S can thus be accounted for. Alternatively diadic forms can be taken, see Sect.  10.1.3, where this problem is discussed at greater length. The diadic form of λ S is 1 + λxz2 (xxx + z z ) + λxz (xxz +zzx ), and diadic forms for  o and  −1 were given in Sect. 7.1.1, eqn (7.2). Taking either route to evaluate the trace, one obtains:  r −1 (r − (r − 1) sin2 θ )(1 + λxz2 ) + Fel = 12 µ 1 + r + 2(r + 1)λxz2 − r

 (1 + (r − 1) sin2 θ )λxz2 + 2(1 + r)λxz 1 + λxz2 sin θ cos θ . Expanding to take terms at harmonic order, that is λxz2 , θ 2 and λxz θ only, one obtains: 2 1 (r + 1) µ λxz2 2 r

≡ 4C5 λxz2

(7.7)

2 1 (r − 1) θ2 2µ r

≡ 12 D1 θ 2

(7.8)

1 µ (r − )λxz θ ≡ −D2 λxz θ . r

(7.9)

We have already met the linear relative rotation coupling constant D1 above. The imposed shear induces the director to rotate with respect to the unrotating background matrix with an energy cost D1 . There is also a purely elastic penalty, 4C5 λxz2 to symmetric shears in a plane encompassing n o which we discuss in Sect. 10.2. The shear

SOFT ELASTICITY

159

modulus C5 does not vanish as r → 1; in isotropic solids pure shear still costs energy 4µλxz2 , as we saw in the exercises of Sect. 4.3. The new coupling −D2 λxz θ between the elastic shear and the director rotation demands a new elastic constant (de Gennes, 1982), D2 = µ (1 − r2 )/r. As expected, D2 → 0 as isotropy is approached, r → 1, and there is nothing to rotate. On going from prolate (r > 1) to oblate (r < 1) chains, the sign of D2 reverses – in contrast to the always positive D1 . This means that the sign of a rotation θ induced by a given λxz will be the opposite for prolate and oblate elastomers. Prolate elastomers have their director rotated by λ S to the extension diagonal, as is suggested by Figs. 4.4 and 10.3. Oblate elastomers have their director attracted to the contraction diagonal – it allows them to put a long dimension of their shape ellipsoid along the extension direction and thereby lower the elastic energy, see also the discussion of Sect. 10.1.3. We see another astonishing elastic effect. The term −D2 λxz θ of eqn (7.9) is bilinear which means its overall sign can always be made negative by a suitable choice of the sign of the response θ to a given imposed λxz . For instance for D2 > 0, taking both θ and λxz to be positive, or both negative, lowers the energy by −D2 λxz θ < 0. Thus, although the C5 λxz2 and D1 θ 2 terms are positive, the D2 term offers a mechanism for reducing these penalties. The next section derives this effect for general geometries and large amplitudes. It is found ideally that the three terms can cancel overall to give no net energy cost to such symmetric shears encompassing the director! We call this soft elasticity and it rests behind most of the nematic and cholesteric elastic phenomena we discuss in this book. The above continuum approach to soft and semi soft elasticity is completed in Sect. 10.4. We continue here with a non-linear approach. 7.2

Soft elasticity

A gas adopts the volume and shape of its container. A liquid adopts its container’s shape but has its own volume which then costs energy to change. A solid has both its own volume and shape, both of which cost energy to change. Let us see how nematic elastomers fit into this classical categorisation of the states of matter. Consider the deformation gradient represented by the expression, obscure as it may appear at first glance (Olmsted, 1994) W α ·  −1/2 λ =  1/2 ·W , o

(7.10)

where W is an arbitrary rotation by an angle α . The current and initial chain step-length tensors  and  o specify the current state, characterised by its director n and order parameter Q, and the initial state with n o and Qo . Considering the rotations connecting n and n o and those associated with W α , this λ represents a large number of potential distortions. If we insert such a deformation into the Trace formula, as well as its transpose W T ·  1/2 since the  are symmetric, we obtain: λ T , equivalent to  −1/2 ·W o   1/2 −1 1/2 −1/2 T W W ·W ·  ·  ·  ·W ·  Fel = 12 µ Tr  o ·  −1/2 o α  ! " α o δ

160

SOFT ELASTICITY

  ≡ 12 µ Tr δ = 32 µ .

(7.11)

The middle section  1/2 ·  −1 ·  1/2 gives the unit matrix δ , by definition. The rotation matrix W then meets its transpose to also give unity: W T · W = δ . Likewise disposing of the  o terms, one obtains the final value Fel = 32 µ . This is identical to the free energy of an undistorted network. The non-trivial set of distortions λ of the form eqn (7.10) have not raised the energy of nematic elastomer! We saw pictorially in Fig. 5.22 that, on applying a extension perpendicular to the initial director, rotation of the chain distribution is accommodated by the very elongation we have applied, together with a shear. If this picture of nematic elastomer response via rotation is correct, two remarkable consequences immediately follow: • All the distortions accompanying the imposed extension λxx must be in the plane of rotation, that is a transverse contraction λzz and a shear (only λxz is evident in the figure, but λzx would also accommodate the rotation of the distribution). No distortions perpendicular to this plane, that is involving the y-direction (λyy √, λyx , . . . etc.), are needed. For a classical isotropic elastomer λyy = λzz = 1/ λxx is demanded by incompressibility, whereas in soft elasticity there is no shrinkage in the y direction (λyy = 1 and the appropriate Poisson ratio is zero). • Softness must come to an end when the

rotation is complete and the z dimension has diminished in the proportion λzz = ⊥ / and the x dimension extended in

√  the proportion λxx =  /⊥ . The original sizes  and ⊥ have transformed √  to ⊥ and  respectively. Thus softness would cease and director rotation 3/2

be complete at λxx = r1/2 ≡ λm . The familiar characteristic deformation λm = ( /⊥ )1/3 is the spontaneous extension suffered on cooling to the nematic phase, see Sect. 6.2. This final deformation we denoted by λ2 = r1/2 in the mechanical Freedericks transition of Sect. 6.7. It minimises the free energy FC to which the director jumped in the Mitchell experiment. Here it appears at the end of the ‘low road’ of soft deformation, Sect. 6.7.2. Likewise one can imagine from an oblique form of Fig. 5.22 that if the initial director n o (the long axis of the shape ellipsoid) is not at 90o to the imposed strain, then rotation and softness is complete at a smaller λxx < λ2 . Shape change without energy cost suggests that nematic elastomers fit the classical category of liquid! However their non-soft deformations are rubbery, that is at least notionally solid-like. 7.2.1 Soft modes of deformation We now explore the general character of soft modes. Exercise 7.2: What distortion does the soft mode λ soft =  θ ·  −1/2 represent? o (The arbitrary rotation matrix W α is absent in this example.) 1/2

SOFT ELASTICITY

161

Solution: The soft mode is characterised parametrically by the angle θ by which  o is rotated to  θ , that is by which n o is rotated to n : √ 1 λ soft = (δ + ( r − 1)nnn ) · (δ + ( √ − 1)nno n o ) r √ √ = δ + (1/ r − 1)nno n o + ( r − 1)nnn √ √ +(nn · no )(2 − r − 1/ r)nn no .

(7.12)

If n o is along z and is rotated by θ toward x , it becomes n = z cos θ + x sin θ , Fig. 7.2. We can write down a particular representation of λ soft : √ 1 λ soft = (1 − (1 − √ ) sin2 θ )zzz + (1 + ( r − 1) sin2 θ )xxx + y y r √ 1 +(1 − √ ) sin θ cos θ x z + ( r − 1) sin θ cos θ z x r   √ √ 1 + ( r − 1) sin2 θ 0 (1 − 1/ r) sin θ cos θ . 0 1 0√ ≡ √ ( r − 1) sin θ cos θ 0 1 − (1 − 1/ r) sin2 θ

(7.13)

Figure 7.3 illustrates soft deformations of a nematic elastomer. The distortions are parameterised by the director rotation, θ , which ranges between 0 and π /2. The x√ extension is√λxx = 1 + ( r − 1) sin2 θ ≥ 1 and the perpendicular contraction is λzz = 1 − (1 − 1/ r) sin2 θ ≤ 1. Both are proportional to sin2 θ . Thus the infinitesimal diagonal strain components
zz = λzz − 1 and
xx = λzz − 1, at small rotations θ , are proportional to θ 2 . By contrast the shears λxz and λzx are proportional to sin θ cos θ and hence the infinitesimal strains
xz and
zx are linear in small rotation angle θ – a lower order than
xx and
zz . The soft mode, eqn (7.12), starts at no strain, λ = δ , and as the director rotates from √ √ θ = 0 all the way to π /2, the √ soft regime eventually ends at λ = Diag ( r, 1, 1/ r), that is a x-extension λxx = r, a z-transverse contraction √ λzz = 1/ r and no remaining shear. Director rotation is taken up by shape change so that there is no entropically expensive deformation of the chain distribution as when a conventional elastomer deforms. The anisotropy of step-length tensor r =  /⊥ characterises the ratio of the mean square size √ along the director to that perpendicular to the director. The square root of this ratio, r, gives the characteristic ratio of average (r.m.s.) dimensions of chains in the network. During a soft deformation, the solid must change shape such that the rotating ellipsoid 1/2 , characterising the physical dimensions of the distribution of chains6 , is accommodated without distortion, Fig. 7.3. In the isotropic limit (r = 1) both chain step-length tensors,  o and  θ , reduce to a unit matrix and the general soft deformations matrix (7.10) reduces to W α , an arbitrary body rotation. Certainly, we would expect no elastic energy rise when we turn and rotate the sample as a whole! The soft modes become real, non-trivial deformations when the material becomes a nematic elastomer. 6 The ellipsoid is described by the condition R · −1 ·R R = 1, or in the initial principal frame, x2 +y2 +z2 /r = √ 1, that is, the xz-sectional ellipse has semi-major axes of 1 and r.

162

SOFT ELASTICITY

F IG . 7.3. Soft deformations of a nematic elastomer with anisotropy r = 2.78. The deformations correspond to director rotations of θ = 0, π /6, π /4, π /3, 5π /12 and π /2 which parametrically generate the distortions as discussed above. Note that the distribution of chains, embedded in the distorting solid, when rotated by θ can be accommodated without distortion. The nematic order, so crucial for the availability of internal orientational microstructure leading to soft deformations, does not change through such a distortion. At the outset the value Q minimised the nematic component of the free energy. Since the elastic component of the energy does not rise, then the initial optimal magnitude of Q remains optimal during the soft deformation. Unchanging Q implies an unchanging distribution of chain shapes, that is  is strictly a rotated form of  o . Thus  θ = M T ·  o · M   where M (θ ) is a rotation, represented by an orthogonal matrix with Det M = 1. Note   that one cannot simply take an orthogonal matrix, M T = M −1 , without Det M = 1. Even though such matrices M would also generate a form of strain λ yielding Fel = 32 µ , they would at the same time change the shape of  from that of  o which already has the optimal aspect ratio. The consequent change in Q during such a deformation means the thermodynamic nematic part of the free energy, Fnem , would rise and deformation could not be soft. Chapter 6.6 shows that, if director rotation does not occur and no low energy trajectory is available, then strain modifies the magnitude of nematic order – the opposite of the present soft scenario.

SOFT ELASTICITY

lo

-1/2

163

lq

1/2

q

F IG . 7.4. Soft deformations  1/2 ·  −1/2 of a nematic elastomer can be broken down θ o into two component deformations that reduce the initial anisotropic state to isotropy and then recreates it again at angle θ with its associated deformation. Thus elastomers can be soft through a changing macroscopic shape of the elastic body by rotating the distribution of chains at constant average shape of network chains. This implies that chain entropy, normally at the root of rubber elastic response, does not drop. Equally there is no change of the nematic order; it has merely been rotated in alignment direction. The energy change is zero. The overall change in free energy density, ∆F = ∆U − T ∆S, is thus also zero (U and S being the internal energy and entropy). There is another way to view the soft deformations of the cartoon, Fig. 7.3. The ·  −1/2 can be broken down into two deformations (that are deformation λ soft =  1/2 θ o multiplicative rather than additive since they may correspond to large strains). Figure 7.4 takes the original solid illustrates their successive action. The first deformation  −1/2 o into a cube and the anisotropic chain distribution to the isotropic spherical one. The intermediate distribution, being isotropic, can be rotated through an angle θ without , then acts to energy cost or physical effect. The second part of the deformation,  1/2 θ restore the current anisotropic distribution, but at the new angle θ . The cube suffers a non-trivial distortion to the new shape which exscribes the rotated ellipsoid. An identification of great significance to the symmetry arguments of Sect. 10.4.1 is that  1/2 is related to the spontaneous elongation on cooling, λ m , by a simple constant. Likewise  −1/2 is related to the inverse spontaneous deformation on heating: o

 1/3   r 0 0 0 0 r 1/6 =  0 1 0 ≡ r  0 0  ≡ r1/6 λ m r−1/6 0 0 r−1/6 0 0 1  −1/3   √  0 0 r 1/ r 0 0 = 0 0  ≡ r−1/6 λ −1 . (7.14) 1 0  ≡ r−1/6  0 r1/6 m 0 0 r1/6 0 0 1 √



1/2

 −1/2 o

The two tensors are expressed in their principal frames, the first being rotated by θ with respect to the second. The prefactors cancel when we take their product and we obtain: U (θ ) · λ −1 ·  −1/2 = λ m (θ ) · λ −1 = U T (θ ) · λ m ·U .  1/2 θ o m m

(7.15)

164

SOFT ELASTICITY

The notation λ m (θ ) means we take the spontaneous distortion along a director that has been rotated by θ . The final form of eqn (7.15) explicitly displays the rotation matrices. The existence of a (virtual) intermediate isotropic state, as in Fig. 7.4, is the basis of symmetry arguments that prove the soft response associated with Fig. 7.3 is universal. This state is often taken as the reference state, although it is far from the physical reference state associated with experiment. Indeed such arguments yield explicit forms of non-linear soft strain (Lubensky et al., 2002), analogous to eqn (7.13), and are also useful in the case of smectic C elastomers (Stenull and Lubensky, 2005), as we later discuss Sect. 12.6.1 and in an associated appendix. 7.2.2

Principal symmetric strains and body rotations

The soft deformations of Fig. 7.3 and eqns (7.12) and (7.13) of the body are in general non-symmetric. The shear is neither simple nor pure, but a mixture. We decompose λ soft into pure shears plus rotations. The element of body rotation, irrelevant for conventional solids, is vital for nematic elastomers. Results (4.39) and (4.40) of exercise 4.4 break the soft modes down into a symmetric shear λ S followed by a body rotation U Ω through an angle Ω about the axis

perpendicular to the shear. Thus, λ soft = U Ω · λ S . We continue to parameterise the soft modes by the director rotation θ . The body rotation angle Ω is √ √ sin θ cos θ ( r − 1)2 ( r − 1)2 tan θ √ tan Ω = √ ≡ (7.16) √ 2 r + (r + 1) tan2 θ 2 r + sin2 θ ( r − 1)2 (to get the correct sign of rotation, one must be careful about how to reduce the 3 × 3 matrix of eqn (7.13) into a 2 × 2 form: the order of elements should be z − x to retain the cyclic permutation rule). For small director rotation the component of body rotation is also small: √ ( r − 1)2 √ . Ω∼θ 2 r

For large rotations, θ → π /2, the rotation Ω vanishes as we have seen in Fig. 7.3. Body rotation evidently plays some part in accommodating the rotating chains. The corresponding symmetric shear strain, the off-diagonal component of λ S , is sin θ cos θ (r − 1) λxz ≡ λ S = √ . r(4 + sin2 θ (r − 1)2 /r)1/2 For small distortions it too is proportional to θ :

λS ∼ θ

(r − 1) √ . 2 r

Symmetric shear also vanishes at the end of the soft regime, Fig. 7.3 (θ = π /2), where no further accommodation of the shape tensor by body rotation and shear is possible.

SOFT ELASTICITY

165

0.4 0

S

Shear l and rotation W

0.8

-0.4 -0.8

0

p/2 p/4 Director rotation

q

3p/4

p

F IG . 7.5. Symmetric shear, λ S (solid lines), and body rotation, Ω (in radians – dashed lines), plotted against the director rotation angle θ for the soft deformations of Fig. 7.3 for a nematic elastomer with anisotropy r = 2.78 (filled diamond) and r = 20 (open circle). For r = 20 the maximum body rotation occurring during this soft deformation is 23o , see eqn (7.17), for a director rotation of 33o , see eqn (7.18). The symmetric shear and body rotation reverse when the director rotates beyond 90o and recover initial conditions at 180o . Figure 7.5 shows the rotation and symmetric shear occurring during soft deformations between θ = 0 → π . The maximum body rotation during the soft deformations, Ωm is √ ( r − 1)2 tan Ωm = √ . (7.17) 2 2r1/4 (r + 1)1/2 It occurs at a director rotation θm given by: tan2 θm =

√ 2 r . (r + 1)

(7.18)

More general soft deformations have the arbitrary rotation, W α , embedded in the form of soft deformation eqn (7.10) We had removed W α in the above illustration. Its effect is trivial if its rotation axis is along y or n o . If the axis vector has components not along either of these directions, then it gives shears involving the y direction. Pictures of this λ soft are qualitatively similar to those of the biaxial soft modes we illustrate in Appendix B. The directions of principal extension and compression can be identified  as in ex ercise 4.5. The 2 × 2 active part of λ soft , eqn (7.13), is of the form λA λB where the

distorrelation AB − λ λ  = 1 confirms the incompressibility. In a simple  case of small √ √  θ ( r−1)/ r 1 √ , tions, θ  1, at leading order this deformation becomes λ soft ≈ θ ( r−1) √ √ √ 1  that is exactly of the form in exercise 4.5 with λ = θ ( r − 1)/ r, λ = θ ( r − 1) and A = 1. The principal extension vectors and the angle between them are, in this case: √    r−1 1 1/4 −1 √ ; χ = − cos e± =  . √ 1, ±r r+1 1+ r

166

SOFT ELASTICITY

n

n

n n

F IG . 7.6. The nematic director and the unit vectors of principle extension(+) and compression(–) associated with soft deformations parametrised, as in Fig.7.3, by the director rotation, here θ = 0, 150 , 450 and 870 . Note that the principal directions are not perpendicular until θ = π /2. √ Figure 7.3 illustrated soft modes for chains with r = 5/3. As we discussed earlier, there is a component of rotation in λ soft . Accordingly, the eigenvectors for principal extension and compression are not perpendicular √ to each other as for any symmetric deformation; e.g. in the limit θ  1, for r = 5/3, they are at an angle of χ = cos−1 (1/4) = 75o . Figure 7.6 shows soft deformations, eqn (7.13), with the current director and the vectors of principal extension(+) and compression(–). 7.2.3

Forms of the free energy allowing softness

Symmetry arguments, Sect. 10.4.1, dictate softness exists in solids with an internal degree of freedom and where an isotropic reference state is accessible. The underlying reason is the invariance of the free energy under both rotations V of the reference state R o and U of the target state R . Recall, eqn (4.3), that the deformation tensor transforms like: VT . λ  = U · λ ·V The first index, see eqn (4.2), of λ refers to R which is transformed by U and the second index refers to Ro which is transformed by V . There are thus two sets of symmetry operations, referring to the reference and final states independently. One route to elasticity theory we saw in Sect. 4.1 was to construct the Cauchy Green tensors C = λ T · λ and B = λ · λ T which transform as second rank tensors under reference state rotations as C  = V · C · V T or under target state rotations as B  = U · B · U T . They can be used to form suitable, scalar expressions for the energy. In nematic elastomers there are now Ro R o } and {R RR }. other tensors to draw upon, namely  o and  with the character {R     T T Invariant expressions now include Tr λ · λ ·  o and Tr λ · λ ·  . The first nontrivial expression that records  the structureof both initial and current states is our fundamental Trace formula, Tr  o · λ T ·  −1 · λ . More complex possibilities exist that have the correct invariance properties under U and V , for instance the trace of a product m     n Tr λ ·  o · λ T ·  −1 . . . and other scalar functions constructed from combinations of powers of the tensor expressions λ ·  o · λ T and λ T ·  · λ . These all, as we see at the end of Sect. 7.6.2, have the capacity to possess soft elasticity. Such more

OPTIMAL DEFORMATIONS

167

complicated forms arise when one considers deviations from the concept of an ideal Gaussian network – for example the effect of finite chain extensibility, that is where chains are sufficiently short and extensions sufficiently large that they no longer behave as Gaussians. Other even more more complex forms arise when considering the effect of entanglements in nematic elastomers, Sect. 6.8. The above symmetry considerations show these elastomers must remain soft despite these new constraints. Indeed the precise form of the soft response, eqn (7.30)-(7.33), emerges independently of the form of the free energy adopted, see Sect. 10.4.1. 7.3

Optimal deformations

Having seen soft deformations generally and a little of their shear and rotational character, we now look at two methods for calculating their form in more practical situations. The first is more general and offers insight into how the director angle follows the best direction set by imposed and relaxational strains. The second is a concrete example of how the best deformations are found for the practically important geometry of extension imposed perpendicular to the director. 7.3.1

A practical method of calculating deformations

In some situations the nematic elastic free energy is straightforward to calculate and to optimise. For instance, the Freedericks transition of Sect. 8.1 has only a single (simple shear) distortion and a director rotation. Others we have seen to be more complex. Section 7.3.2 calculates soft modes by a direct attack in a practical geometry, but seems quite involved since one minimises over a large number of components of λ and the director rotation. A more elegant way is to describe the distortion λ parametrically by the associated director rotation, the method of Olmsted in Sect. 7.2, but then one does not have control over aspects of the distortion that may be constrained. A more straightforward way exists to evaluate the optimal free energy and this offers insight as to how the final director rotation is achieved. Moreover, for planar distortions (with, in general, relaxation in the third direction) finding the free energy only involves solving a quadratic equation. Consider the Trace free energy density, with its tensor components conveniently cyclically permutated:   (7.19) F = 12 µ Tr λ ·  o · λ T ·  −1 . The first three tensors can be combined to form a tensor, S , that is symmetric by construction and thus also has a frame in which it is diagonal:   s1 0 0 (7.20) S = λ ·  o · λ T →  0 s2 0  , 0 0 s3 the latter form displaying the three eigenvalues si of S which have been ordered so that s1 ≥ s2 ≥ s3 . The principal frame of S is in general rotated from the starting frame of

168

SOFT ELASTICITY

nematic order in which  o was diagonal and in which we conveniently define λ . We now have to multiply S by −1 and take the trace in order to find the free energy. The  1/r 0 0 trace will be minimised by aligning  −1 = 00 10 01 (as it appears in its own principal frame) with the principal frame of S such that the smallest element of  −1 (that is 1/r) meets the largest element of S (i.e. s1 ). This is equivalent to demanding that the final director n is aligned with the eigenvector of S corresponding to s1 . The free energy is F = 12 µ (s1 /r + s2 + s3 ) .

(7.21)

The difference between this approach and others is that n simply follows the direction established by S , that is by the distortions combined with the original director. It is a slave to these and is not minimised over. Now F can be minimised over the elements of λ that were not imposed or clamped, but which are allowed to relax. Only after this should the eigenvectors of S be explicitly determined in order to determine n. This is best illustrated by an example: Exercise 7.3: Find the relaxations and director rotation associated with an extension λ imposed perpendicular to the initial director. Only shears consistent with displacements along the extension direction are allowed. Solution: As before, take the general strain and initial (reduced) step-length tensors: 

λ λ =0 0

 0 λxz 1/(λ λzz ) 0  λzz 0



;

1 o =  0 0

 0 0 1 0 . 0 r

From these we construct S : 

  λ 0 λxz λ 1 0 0 S =  0 1/(λ λzz ) 0   0 1 0   0 λzz λxz 0 0 0 0 r  2  2 λ + rλxz 0 rλxz λzz 0 1/(λ 2 λzz2 ) 0  = 0 rλzz2 rλxz λzz

 0 0 1/(λ λzz ) 0  0 λzz

and the determinant condition for its eigenvalues:      2 + rλzz2 )s + rλ 2 λzz2 Det S − sδ = 0 = 1/(λ 2 λzz2 ) − s s2 − (λ 2 + rλxz where we have gathered terms in and simplified the final factor somewhat. Because shears were limited to the xz-plane, the matrix is blocked and one eigenvalue emerges trivially and the other two can only come from the roots of a quadratic. They are s1,2 =

1 2

 

2 2 + r λ 2 )2 − 4r λ 2 λ 2 λ 2 + rλxz + rλzz2 ± (λ 2 + rλxz zz zz

(7.22)

OPTIMAL DEFORMATIONS

169

s3 = 1/(λ λzz )2 whence the free energy density is:    r−1 r+1  2 1 √ 2 F = 12 µ λ + rλxz + rλzz2 − ... + 2 2 . r r λ λzz 2 , yielding: This is most easily first minimised with respect to λxz

(r + 1) − (r − 1)

2 + rλ 2 λ 2 + rλxz zz =0 √ ...

which on squaring and simplifying gives: 2 λ 2 + rλxz + rλzz2 = (r + 1)λzz λ

(7.23)

the left hand side of which pervades F and can clearly Re be used to advantage.

turning this combination to F then gives F = 12 µ 2λzz λ + 1/(λ 2 λzz2 ) . The optimal λzz is then trivially λzz = 1/λ . Returning this λzz to eqn (7.23) for λxz gives 2 = 1 (λ 2 − 1)(r − λ 2 ), compare with eqn (7.31) below. The matrix S is now: λxz rλ 2  2  2 λ + rλxz 0 rλxz /λ S= 0 1 0  0 r/λ 2 rλxz /λ and its eigenvalues are: s1,2 =

1 2

 

(r + 1) ± (r − 1)2 = r or 1

s3 = 1 on returning the left hand side of eqn (7.23) and λzz = 1/λ to eqn (7.22). The eigenvectors e i can now be trivially found. That corresponding to s1 = r, and thus to n , is (x, 0, z) where x and z are related by z(r/λ 2 − s1 ) + xrλxz /λ = 0 using the bottom line of the matrix equation S · e1 = s1 e1 . Each of the three equations in S · e = see is equivalent when s is set equal to a particular si . The simplest can be chosen to find the connection between the x and z components of the associated e i . Since x/z = tan θ = (λ − λ1 )/λxz , it is straightforward to find eqn (7.31) for the director rotation angle θ , that is sin2 θ =

r λ2 −1 . r −1 λ2

This method offers the advantage that θ , the director rotation angle, is not a variable to be optimised over. It emerges naturally as the director rotates towards the direction of biggest extension. 7.3.2 Stretching perpendicular to the director Although it has been easy to construct soft deformations parametrically through θ , it is not easy this way to visualise strains imposed in practical experiments. Usually one

170

SOFT ELASTICITY

component is directly imposed by clamping and stretching the sample. The other strains, plus n , relax to their optimal values. To the long rectangular strip of nematic elastomer of Fig. 7.7 we impose an xextension λ perpendicular to the initial director, n o , which is along z (Verwey et al., 1996). The cartoon of Fig. 7.3 suggests that soft deformation requires rotation of n o toward x and that shears λzx and λxz accompany the imposed λ . We limit ourselves to λxz which derives from x-displacements, ux . Shear λzx is suppressed by the turning moment, fx uz , generated by a z-displacement, uz , in the presence of an x force, fx (should it arise on extension in the x-direction). Equally, the large length to width ratio of the strip in Fig. 7.7 allows us to defer to Chapter 8 the effect on shear of the clamps gripping the sample when imposing λ . Assume there are no impediments to any simple shear λxz necessary for optimising deformations. The vanishing of the  shear  λzx (and also the shears involving yx, yz etc.) means the incompressibility, Det λ = 1, is ensured by taking λyy = 1/(λ λzz ). The reduced inverse step-length tensor, on director rotation of θ from the z axis, is    1   1 + 1r − 1 sin2 θ 0 r − 1 sin θ cos θ .  −1 =   (7.24) 1 0 1 0  2 1 − 1 sin θ cos θ 0 1 + − 1 cos θ r r See exercise 3.1 for how to evaluate  . One can think of  −1 and  o as in effect 2 × 2 matrices spanning (x, z) since it is only their x − z components that get mixed up by rotations about the y-axis. The free energy density, putting −1 and λ in the Trace, becomes (in units of 12 µ ): 2 1 Fel 2 2 = λ + λ + + rλxz2 + zz 1 λ λ µ zz 2    1 2 2 2 2 (7.25) λ − λzz + λxz sin θ . − (r − 1) 2λzz λxz sin θ cos θ + r The first group is the energy of deforming an isotropic rubber (r = 1). The second group represents additional effects due rotating the anisotropy (r − 1).

l xz

n0

n l (=lx x)

z x

lzz

F IG . 7.7. The extension of a long strip of nematic elastomer perpendicular to its initial director. One shear component, λxz , develops. The other, λzx , is suppressed by the counter torque that would develop from such a distortion in the field of a force applied along the x axis.

OPTIMAL DEFORMATIONS

171

The elastomer relaxes its transverse dimension, λzz , and its shear, λxz , to minimise this elastic energy – which leads to the conditions

∂ Fel = 0 → (r − (r − 1) sin2 θ )λxz = (r − 1) sin θ cos θ λzz (7.26) ∂ λxz ∂F 1 = 0 → λzz (1 + (r − 1) sin2 θ ) − 3 2 = (r − 1) sin θ cos θ λxz . (7.27) ∂ λzz λzz λ Putting eqn (7.26) for λxz into eqn (7.27) yields λzz4 λ 2 = (r − (r − 1) sin2 θ )/r. Putting λxz from eqn (7.26) into eqn (7.25) gives Fel (λ , λzz2 , s) which can then be simplified by injecting the above expression for λzz . The free energy, at a fixed imposed extension λ , is now a function only of the director rotation angle (through its sin2 θ ):   r−1 2 2 1 . sin θ ) +

(7.28) Fel (λ , θ ) = 12 µ λ 2 (1 − r λ 1 − r−1 sin2 θ r

On symmetry grounds rotations, ±θ are not distinguished - the extension λ is imposed at 90o to the initial director and it does not matter which way it rotates. Being a nematic, θ = 0 and θ = π states are also identical. The optimal angle condition, best examined as the derivative with respect to sin2 θ rather than θ itself, yields 1 r−1 2 ∂F sin θ = 1 − 2 . =0 → 2 r λ ∂ (sin θ )

(7.29)

The resulting director rotation angle, and the accompanying shear and transverse contractions take the form 0 r λ2 −1 −1 1 θ = sin ; (7.30) λzz = ; (7.32) r −1 λ2 λ 2 2 λyy = 1 . (7.33) (λ − 1)(r − λ ) λxz2 = ; (7.31) 2 rλ One can verify these results give Fel = 32 µ for the free energy density, which is also the value in the relaxed state (λ = δ ), even though the mechanical shape of the elastomer has manifestly changed. The shear λxz and the angle θ√both start at zero in a singular fashion when λ = 1. When the extension reaches λ = r, the rotation is complete: θ → π /2. The shear λxz returns to zero since off-axis shape accommodation associated with oblique director angles is no longer required. Director rotation, θ (λ ), is central to the new effects and is of a distinctive √form. Its singular form at λ = 1 arises from the square root. The singular form at λ = r and the saturation level θ = π /2 arises from the sin−1 function. This free reflects sin θ , rather than the angle itself, being the natural variable of the elastic √ energy (7.28). The accompanying shear is also singular at λ = 1 and λ = r. The θ

172

SOFT ELASTICITY

p/2

r=2.78

Relaxing strains

Director rotation q

1 r=10

p/4

r 0

1

(a)

1.5

2

2.5

3

(yy)

0.8 0.6

(zz)

0.4 0.2 (xz)

0

3.5

Imposed extension l

1

(b)

1.5

r 2

2.5

Imposed extension l

F IG . 7.8. (a) Director rotation θ against the imposed extension, for the chain anisotropy r = 2.78 and r = 10, see eqn (7.31). (b) The shear strain λxz (star) and two transverse relaxations λzz√(diamond) and λyy (circle), for r = 2.78. The soft region spans from λ = 1 to λ = r = 1.67, beyond which the response is conventional.

Elastic free energy

response, shown graphically in Fig. 7.8(a), is already reminiscent of the experiments of Fig. 5.20 to which we return in discussing semi-softness, Sect. 7.4. In the soft interval, the z-transverse relaxation is λzz = 1/λ and the y-dimension is unchanged, λyy = 1, see Fig. 7.8(b). As in the cartoon, Fig. 7.3, the shape spheroid rotates in the xz-plane, No y-dimensional change is needed since no y molecular shape change has to be accommodated. The free energy is constant and thus the stress is zero – ‘soft deformation’, see Fig. 7.9. This is the ‘elastic low road’ of Sect. 6.7.2. We have thereby returned the free energy of Fig. 6.10 to convexity by taking as the central section the flat part of Fig.7.9. √ When λ > r the rotation is complete (θ = π /2) and the imposed shape change cannot be further accommodated by directing the long dimension of the molecules toward x. In eqn (7.28) set sin θ = 1, whence

3 2

1

m

r 1.5

2

2.5

Imposed extension l

F IG . 7.9. Deformation λ applied in the x direction does not cause the free energy to rise √ r whereupon the energy rises as for a classical elastomer, of apparent until λ = √ natural size r, see eqn (7.34). Compare with the ‘low road’ in Fig. 6.10.

SEMI-SOFT ELASTICITY

Fel =

1 2µ

√  λ2 r +2 . r λ

173

(7.34)

The free energy now rises with λ and the stress is non-zero. √ The rubber responds as a normal elastomer, but with an apparent natural length of r. If the strain along the√x axis is measured as λ  with respect to this √ state, that is if we apply and extension r first followed by λ  , then overall λ = λ  r. We can rewrite the free energy density as Fel (λ  ) = 12 µ (λ 2 + 2/λ  ). The elastomer now appears entirely conventional. Nematic elastomers distorted without order change respond classically, Sect. 6.7, region FC given by eqn (6.43). 7.4

Semi-soft elasticity

Softness is a delicate phenomenon. It depends on being able to rotate a chain distribution at constant entropy by accommodating anisotropic chains with suitable extensions and shears of the body the chains comprise. We would partially lose softness if for instance we had a mixture of chains in the network with differing anisotropies. An optimal soft λ for one population might not be optimal for another and such chains would then cost energy to deform along the trajectory selected for the first population. We shall introduce and illustrate so-called ‘semi softness’ via this particular effect (Verwey and Warner, 1997a). There are many other causes of semi-softness, for instance crosslinks being themselves rod-like and therefore able to record orientational order (Verwey and Warner, 1997b; Popov and Semenov, 1998). See Sect. 7.6 for a discussion of more general aspects of semisoftness. W α ·−1/2 are intimately related to the structure The general soft modes λ soft =  1/2 ·W o of the Trace formula for the elastic free energy density. Some additions or modifications to the Trace formula preserve softness, others lead to deviations from ideality, while still preserving the lower-energy path of deformations – which we call semi-softness. We sketch below a way in which semi-softness can arise, and examine the mathematical requirements for semi-softness in Sect. 7.6.2. The requirement for softness is that an isotropic reference state be in principle achievable. If there is always a residual anisotropy, even at high temperatures, then the nematic phase cannot be truly soft. Indeed we have seen in Fig. 5.23 that chemically identical networks with differing thermomechanical histories can have drastically different stress-strain characters, in particular in their semi-soft regions of deformation. This is evidence for the dependence of semi-softness on residual order. The softest networks were formed in the isotropic state, the least soft were prepared in the nematic state and had more anisotropy permanently imprinted into them. Another signal of this imprinting is the only gradual loss of nematic order at Tni for some nematic elastomers compared with the discontinuous jump to zero order for the melt at this point, see Fig. 5.6. Section 7.6 explores with examples the subtleties of thermo-mechanical history and how softness is and is not lost according to whether the genesis is anisotropic or not. The phenomenon of threshold strain is related to this problem as well. Nematic rotation induced by an imposed extension, Fig. 5.20, and the nearly completely flat soft stress plateau, Fig. 5.23, do not onset directly at λ = 1, as the stretch starts, but instead

174

SOFT ELASTICITY

at a noticeable threshold λ1 > 1. It is as if the matrix first holds back the rotation of chains: the memory of an intrinsic or imprinted anisotropy must first be overcome. We shall discover that elastomers, despite their non-ideality and partial loss of softness, nevertheless retain the qualitative aspects of soft elasticity, namely the same universal form of the director rotation θ (λ ) and the non-classical transverse contraction characteristic of the soft state: λzz ∝ 1/λ and λyy = const. 7.4.1 Example: random copolymer networks Many nematic polymers used for networks are random copolymers. The backbone, side chain (nematic and non-nematic) and crosslinking elements are put together at one time. Inevitable attendant compositional fluctuations between strands cause chains to vary the extent of anisotropy r in their step-length tensor,  ν for the ν th chain. We deal with monodomains, that is where all chains see the same director. A putative soft mode W α ·  o(ν )−1/2 , λ soft =  (ν )1/2 ·W

(7.35)

good for the chain ν , will clearly not be optimal for other chains. In general it will be impossible to find a mode soft for the elastomer as a whole. The free energy density is an average over all strands ν    Fel = 12 ns kB T Tr  o(ν ) · λ T ·  (ν )−1 · λ . (7.36) ν

The reduced step-length tensors are, as usual:  o(ν ) = δ + (r(ν ) − 1)nno n o  (ν )−1 = δ − (1 −

1 r (ν )

)nnn

(7.37)

and the average is in effect over r(ν ) . Putting  o(ν ) and  (ν )−1 into the Trace of eqn (7.36) and averaging, one obtains    1   1 T −1 Tr (δ − n o n o ) · λ T · n n · λ (7.38) Tr  o · λ ·  · λ +   − r r and thus a semi-soft free energy density:     Fss = 12 µ Tr  o · λ T ·  −1 · λ + 12 µ α Tr δ (tr) · λ T · n n · λ

(7.39)

The first term in eqn (7.39) is that of an apparently homogeneous system of average chains. Now  o and  −1 are step-length tensors with the mean anisotropy r appearing 1 n where we had a particular r(ν ) appearing before, for instance  −1 = δ − (1 − r )n n . The addition to this familiar Trace formula is a term proportional to the factor 2 3 1 1 − , (7.40) α= r r which is not zero if the microscopic structure, determining the values  and ⊥ for each chain, fluctuates throughout the network. By convexity, 1/r is always greater than 1/r, so the measure α of non-ideality is positive.

SEMI-SOFT ELASTICITY

175

The symmetry of deformations relevant to this additional term is determined by the tensor δ (tr) = δ − n o n o , which is a perpendicular projector and inhabits the plane perpendicular to n o n o . For instance the xy-plane is perpendicular to n o and the projector takes the form δ (tr) = x x + y y . It selects out the x and y components of objects it encounters, see exercise 7.4 for shears important to semi-softness. Exercise 7.4: What shears are vital to the semi-soft fluctuation term? Assume for concreteness that n rotates in the xz-plane. Solution: As n rotates in the xz-plane starting from n o = z , it becomes cos θ z + sin θ x . Since n n is sandwiched between λ T and λ in the new semi-soft term of eqn (7.39), the ·nn and n · operations bind it to the z or x legs of λ T and λ . There are no λxy , λzy , λyx and λyz elements of λ . But δ (tr) lives in the xy-plane and thus the only part of δ (tr) that can be active is x x . Recall that the diadic form of λ is λ = x x λxx + x z λxz + . . .. The vector n · λ = (λxx sin θ + λzx cos θ )xx + (λxz sin θ + λzz cos θ )zz contracts with x x , selecting out the shear λzx and the imposed extension λxx . The non-ideal term in Fss , eqn (7.39), is then: 2 2 1 2 µα (λxx sin θ

2 + λzx cos2 θ + λxx λzx 2 sin θ cos θ ) .

(7.41)

The extension λxx perpendicular to n o and the shearing displacements along n o generate semi-softness.

7.4.2

A practical geometry of semi-soft deformation

Reconsider the long strip of Fig. 7.7 with an x-extension λ = λxx imposed (Verwey et al., 1996). Shears λzx are suppressed by the turning moments they generate in the presence of an external force in the x direction. In this case the non-ideal correction of eqn (7.41) is simply α sin2 θ λ 2 . It must be added to the soft free energy eqn (7.25) of the stretched strip, Fig. 7.7, in the simple geometry of Sect. 7.3.2. Since the transverse and shear relaxations λzz and λxz are not involved in the semisoft α -term arising due to compositional fluctuations, the minimisation over these distortions in Fel is identical to the soft case, Sect. 7.3.2. One obtains instead of eqn (7.28):  Fss = 12 µ λ 2 (1 −

 1 r−1 2 2 sin θ ) +

+ αλ 2 sin2 θ  r λ 1 − r−1 sin2 θ r

(7.42)

(recall that r now denotes r, the anisotropy parameter averaged between the network chains with fluctuating composition). Semi-softness changes the soft analysis very little; the condition (7.29) for the optimal director rotation angle is only amended to 1 r−1 2 sin θ = 1 − 2 r λ



r−1 r − 1 − αr

2/3

≡ 1−

λ1 λ

2 .

(7.43)

176

SOFT ELASTICITY

The solution for sin2 θ is exactly as before, but as a function of the reduced extension, (λ /λ1 ), instead of λ . Accordingly, the onset of director rotation and all other features of soft regime will now take place not at λ = 1 but at a threshold λ = λ1 , with 1/3 r−1 λ1 = . r − 1 − αr Clearly λ1 ≥ 1 (since the semi-soft factor α > 0). Initially, below the stretching threshold, the response of a semi-soft nematic elastomer is exactly the same as of a conventional √ rubber. The transverse contraction is the usual 1/ λ for both y- and z-directions and neither shear λxz nor√director rotation arise. The semi-soft regime starts at λ = λ1 and is complete at λ = rλ1 . The strains and the director rotation take forms very similar to those in eqns (7.33): 0 1/2 λ1 r λ 2 − λ12 −1 θ = sin ; λ = ; zz r −1 λ2 λ 1 (λ 2 − λ12 )(rλ12 − λ 2 ) λyy = 1/2 . (7.44) λxz2 = ; λ1 rλ 2 λ13 Both θ and λxz have exactly the same singular response as in the ideally soft case. All the strains and rotations are thus as in Fig. 7.8, if λ is scaled by λ1 . Soft and semi-soft response are qualitatively the same, except that the elastic energy rises slightly in the latter case (see below). The threshold and the whole semi-soft elasticity depend generally on the form of correction to the ideal free energy of the type α in eqn (7.39). For the particular model of compositional fluctuations that we have chosen as an illustration, α = 1/r − 1/ r, restoring for the moment the symbol . . . to the mean anisotropy r. We can thus rewrite λ1 as:

λ13 =

1 − 1/r 1 − 1/r

(7.45)

where the difference between 1/r and 1/ r is clearly seen to be at the root of the effect. Alternatively one can regard the threshold as the essential measure of non-ideality. Rearranging the connection between α and λ13 , we can derive the extent, α , of semisoftness from the observed threshold and the anisotropy: r − 1 λ13 − 1 . (7.46) r λ13   The measure of anisotropy remains the average r =  /⊥ , but would be now extracted experimentally from the reduced width of the semi-soft interval, the √ ratio between the final and the initial threshold strain of the semi-soft regime ≡ ( rλ1 )/λ1 . More anisotropic chains compel greater shape change of the rubber before their rotations are complete.

α=

SEMI-SOFT ELASTICITY

7.4.3

177

Experiments on long, semi-soft strips

Kundler and Finkelmann took several nematic elastomers of differing thermomechanical histories and differing chemical compositions. They stretched them, in the form of long strips, perpendicular to their initial director, see Fig. 5.18. They followed director response by both X-ray scattering and by observing their elastomers in the microscope through crossed polars. Their results in Fig. 5.20 for the director rotation θ (λ ) of each elastomer were qualitatively similar in shape but differed in the values of the threshold where rotation started, and in the values of extension at which rotation was complete. Figure 7.10 instead plots a particular function of the rotation angle, eqn (7.43), against deformation reduced by the corresponding threshold, that is against λ /λ1 , and takes out √ the observed r in each case from the observed semi-soft range. All the data collapses onto the universal curve, essentially confirming the results given by eqns (7.43) and (7.44). Elastomers with differing compositions, effective anisotropies and semi-softness thresholds all conform to the characteristic soft behaviour. There is, in fact, an essential difference between the textures observed in (Kundler and Finkelmann, 1995) and the analysis above, which assumes a uniform director rotation throughout the whole stretched sample of nematic rubber. Kundler and Finkelmann observed the characteristic parallel stripe domains, in which the director rotates by the angle θ (λ ), but in alternating directions. Later this observation has been confirmed by other groups and other types of materials and geometries (Zubarev et al., 1998; Finkelmann et al., 1997). It turns out that such stripe microstructures are an integral part of soft response in the presence of constraints, in this case the mechanical clamps with which the strip is gripped for stretching: see Chapter 8 for examples of how nematic elastomers respond in practical geometries.

Reduced director rotation f(q)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

1.2

1.4

1.6

1.8

Reduced extension l l 1

F IG . 7.10. A function of the director rotation f (θ ) = [(r − 1)/r] sin2 θ plotted against the reduced deformation λ /λ1 applied perpendicularly to the initial director. Data from a wide range of samples, Fig. 5.20, collapse onto a master curve when plotted according to eqn (7.33).

178

7.4.4

SOFT ELASTICITY

Unconstrained elastomers in external fields

As soon as new liquid crystal systems come onto the scene, an expectation arises of electrooptical effects, or other transitions driven by external fields. After all, much of liquid crystal research was driven by, e.g., display applications that now surround us in every walk of life. However, with nematic elastomers the response has been disappointing. Probably every group involved in elastomer research have tried electric or magnetic fields, usually with little success. Early reports (Zentel, 1986; Barnes et al., 1989) of shape change in response to modest applied electric fields were somewhat inconclusive. Today we understand better what was seen, for instance, when small pieces of elastomer floating in a suspending liquid were shown to change shape in a field. Two key properties have to be achieved: monodomain alignment of the initial elastomer sample, and the absence of mechanical constraints. The first is required to provide a sufficient torque on the director from the dielectric (or diamagnetic) anisotropy in a field; the second is needed to allow for soft deformations. The possibilities have been delineated within a description of the electric field response of constrained and freely suspended nematic elastomers (Terentjev et al., 1994). The local torque provided by an electric or magnetic field is almost always too small to overcome the rubber-elastic resistance for the director to rotate (a few exceptions from this rule are discussed later in this book in different contexts). However, if a sample is mechanically unconstrained, a soft deformation can occur in association with director rotation – and the only resistance to dielectric torque would come from the (presumably weak) semisoftness. Figure 7.11 shows the results of (Urayama et al., 2005; Urayama et al., 2006), where a monodomain elastomer was floated in silicon oil. On switching on the electric field perpendicular to the director, a shape change has been registered: the sample contracts in the x-direction (the axis of original director n o ) and does not change its length in the transverse y-direction. The latter is always a signature of soft deformation response! In experiments (Urayama et al., 2005) a small threshold was recorded, certainly due to the (a)

(b)

E n n0 z

l xx

(c) x

F IG . 7.11. Unconstrained film of monodomain nematic elastomer, viewed between crossed polars, appears light if the director is aligned in-plane at 45o to the polars: image (a). When an electric field is applied along the z-axis, the director rotates to homeotropic position and the sample becomes dark between crossed polars: image (b). The associated change in length λxx is marked on the plot; no transverse contraction (λyy ) has occurred (image: K. Urayama). (c) Schematic.

SEMI-SOFT FREE ENERGY AND STRESS

179

weak semisoftness of the elastomer sample. It is easy to see that a 90o rotation would √ cause a contraction λxx = 1/ r, if r is the underlying chain anisotropy, see Fig. 6.11. So in the experiments of Urayama, Fig. 7.11, λxx ≈ 0.85 and r ≈ 1.38. 7.5

Semi-soft free energy and stress

The optical rotation and mechanical relaxations of semisoft elastomers are essentially the same as for soft elastomers. However non-ideality modifies the free energy so that it rises, at least slightly, when the nematic rubber undergoes a deformation from the soft family. We quantify this rise, thereby determining the stress in the semi-soft interval. Elastic free energy Before there is nematic rotation, λ < λ1 (region A of Fig. 7.10 withe corresponding region of the unscaled Fig. 5.20), the response must be classical, that is θ = 0, λxz = √ 0, λzz = λyy = 1/ λ , with FA√= 12 µ (λ 2 + 2/λ ). Between the semi-soft threshold λ1 and the end of semi-softness, rλ1 , we express the elastic free energy density Fel (λ , θ ) in terms of λ13 rather than α , as these two parameters measuring the degree of semisoftness are directly related by eqn (7.46). Thus re-expressing eqn (7.42) one has instead    r−1 2 2 µ 2 1 . λ 1− sin θ +

Fss = 2 λ 1 − r−1 sin2 θ rλ13 r

Using the optimal director rotation from eqn (7.44), the semi-soft free energy density in the region B of Fig. 5.20 takes the form:  1 3 2 1 . (7.47) FB = 2 µ λ (1 − 3 ) + λ1 λ1 For λ1 = 1, the case of ideally soft rubber, one recovers the unchanging F = 32 µ as in Sect. 7.2. The larger the threshold λ1 , the harder FB becomes; the modulus, the coefficient of 12 λ 2 , is µ (1 − 1/λ13 ). √ Exercise 7.5: What is the free energy for λ > λ1 r, that is when the director rotation is complete?

Solution: Recall that the free energy is (Ideal Trace) + αλ 2 sin2 θ . The rotation of n by 90o to align along the stretching direction x interchanges the non-trivial diagonal element entry 1/r in  −1 from the zz to the xx position. All other matrices in the ideal trace formula, cf. eqn (6.42), are diagonal as well; after a rotation of π /2 in  there are no remaining shears. Adding on the semi-soft α -term with θ = π /2 yields overall:

 1 2 1 λ + rλzz2 + 2 2 + 12 µαλ 2 r λ λzz  √  2 λ r−1 2 r 1 (r − 3 ) + . ≡ 2µ r λ λ1

FC =

1 2µ

(7.48)

SOFT ELASTICITY

Elastic free energy

180

3 2

C B A

1

(a)

1.2

1.4

1.6

l1

C

r

B

m

1.8

Imposed extension l

2

1

r l1

1.2

(b)

1.4

1.6

1.8

2

Imposed extension l

F IG . 7.12. Free energy density, in units of µ , against deformation for (a) soft elastomers with r = 2.78 and for (b) semi-soft elastomers with r = 2.78 and λ1 = 1.1. The branches of free energy for the director fixed at angles θ = 0 (regime A) and θ = π /2 (regime C) are so labelled. One can see the zero-stress regime B in the soft case and the corresponding non-zero modulus in the semi-soft situation. √ To obtain the last formula one needs to minimise over λzz to give λzz2 = 1/( rλ ) and then return this strain to the FC expression. FC looks almost classical, with λ 2 and 1/λ terms but with modified factors, as does the ideal Fel in eqn (7.34).

Figure 7.12 shows the overall free energy F(λ ) given by FA , FB or FC according to which regime one is in. The continuations of FA and FC are labelled θ = 0 and π /2, respectively, since director √ rotation has either not begun or is complete. FB only exists in the interval (λ1 → λ1 r). The soft free energy density is plotted for comparison in (a); the semi-soft version – in (b). The two elastomers have the same anisotropies. Elastic stress We imposed an extension λxx = λ in the x direction, with other strains and rotations being a natural optimal response of the nematic elastomer under a uniaxial extension. What is the stress needed to make this imposition? Taking a sample of initially unit dimensions (and hence also unit volume), the work done by a force normal to the x surface in extending the sample by d λ is −σxx λzz λyy d λ . The (λzz λyy )-factor is the crosssection area reduction which, when multiplying the force per unit area σxx (the stress), yields an actual force which does the work. The (−) sign indicates the reduction in energy when the system extends (d λ > 0) in the direction of the force. If this work is added to the change in free energy per unit volume dF, then dF − σxx d λ /λ must vanish in equilibrium for the body (volume conservation gives λzz λyy = 1/λ ). Thus the true stress is:  ∂F . (7.49) σxx = λ ∂λ The continuity of stress is assured since the free energies FA and FB have the same tangent when they meet at the deformation λ1 ; likewise FB and FC have matching slopes

SEMI-SOFT FREE ENERGY AND STRESS

181

Nominal stress s e

0.8 0.6

C B

0.4

A

0.2 0

r l1

l1 1

1.2

1.4

1.6

1.8

2

Imposed extension l e , in units of µ , in a uniaxially stretched semi-soft elastF IG . 7.13. Nominal stresses σxx omer, plotted for the values r = 2.78 and λ1 = 1.1. The corresponding director rotation angle and the accompanying shear and transverse contraction are the same as in Fig. 7.8(a,b) – but shifted along the strain axis to begin at λ1 . √ and hence stresses at λ1 r. The true extensional stress in each of the three regimes of semi-soft deformation is

 1 A σxx = µ λ2 − λ  1 B σxx = µλ 2 1 − 3 λ1 √  r r−1 C 2 . σxx = µ λ (1 − 3 ) − λ λ1 r

(7.50)

A (λ ) = σ B (λ ) and σ B (√r λ ) = σ C (√r λ ). One can easily confirm that σxx 1 1 1 xx 1 xx xx The nominal or engineering stress is the force per unit initial area, in which case one does not correct for the transverse shrinkage as the strain proceeds. Without this e = ∂ F/∂ λ ≡ σ /λ for our decorrection, the nominal stress from eqn (7.49) is σxx xx formation. Figure 7.13 plots nominal stress against strain in the three regimes of semisoft deformation and reveals (A) the initial classical rubber response until λ = λ1 , (B) √ the neo-classical response in the semi-soft regime, λ1 < λ < rλ1 and (C) the finally hard rubber response characteristic of conventional rubbers, after the director rotation is completed.

Stability under deformation The stress is continuous on deforming between regions A, B, C. Equally important is that the free energy dependence on the imposed strain is everywhere convex, especially in the semi-soft region B. Here the curvature is ∂ 2 FB /∂ λ 2 = µ (1 − 1/λ13 ) ≥ 0. Convexity rules out strain-necking and related classical instabilities known in the polymer physics that might otherwise be invoked to explain the semi-soft constitutive relation in region B.

182

SOFT ELASTICITY

Elastic free energy

C B

A 1

(a)

1.2

1.4

1.6

1.8

Imposed extension l

2

1

1.2

(b)

l’’

l

l’ 1.4

1.6

1.8

2

Imposed extension l

F IG . 7.14. Free energy against deformation (a) for semi-soft nematic rubbers with their characteristically convex form and (b) for a solid possessing a concave region in its free energy. The common tangent construction in (b) indicates the path of the Considere instability. By contrast, a material with a concave free energy could lower its total free energy when extended by λ by separating into regions of one extension, λ  say, and into regions of another, λ  , with the total sample extension λ being bounded by these two extensions, that is λ  < λ < λ  . If a fraction w of the original length extends by λ  , and 1−w by λ  , then the extension as a whole is wλ  +(1−w)λ  . The fraction 1−w is more extended and the sample appears necked. Figure 7.14(b) shows the energy reduction at λ where a putatively concave FB (λ ) drops to λ [F(λ  ) − F(λ  )] /(λ  − λ  )+constant. The usual lever rule gives w = (λ  − λ )/(λ  − λ  ). The effective nominal stress σ e = (F(λ  ) − F(λ  ))/(λ  − λ  ) is constant, and could be quite low, as λ increases from λ  to λ  as the necking or generation of L¨uders bands proceeds. This is known as a Considere instability. These are not seen in nematic elastomers and are not expected since their energy is convex7 . Mechanical experiments The stress-strain data of Finkelmann et al., Fig. 5.23 and reproduced as part of Fig. 7.15, is qualitatively as in Fig. 7.13. On extension perpendicular to the initial nematic director, the measured nominal stress initially rises with λ . Then at a certain threshold the stress reaches essentially a plateau or a region of lower slope. Finally, after the plateau, the stress increases again as the imposed deformation increases further. Such experimental data is highly significant since it unambiguously illustrates the effect of soft elasticity and also allows to extract several key material parameters. We shall return to this characteristic stress-strain behaviour and its variations for samples of different shape in Chapter 8 and also correlate it with the mechanico-optical response θ (λ ) we first investigated. Figure 7.15 shows nominal stress against strain for a long and narrow strip of nematic rubber stretched perpendicular to the initial director (K¨upfer and Finkelmann, 1994; Clarke et al., 2001). This shape is important to minimise the effect imposed by 7 We

thank A.H. Windle for pointing out this issue and its implications.

THERMOMECHANICAL HISTORY AND GENERAL SEMI-SOFTNESS 16

Nominal stress (kPa)

6

183

1600

(a)

(b)

(c)

5 12

1200

8

800

4

400

4 3 2 1 0

0 1

1.1

1.2

1.3

1.4

Imposed extension l

1.5

1.6

1

1.1

1.2

1.3

1.4

1.5

Imposed extension l

1.6

0

1

2

3

4

5

6

Imposed extension l

7

F IG . 7.15. Nominal stress (in units of kPa for all three graphs) is plotted against deformation for three different nematic elastomers (experimental data from Freiburg and Cambridge groups). The composition of side-chain polysiloxane rubbers in (a) and (b) is very similar, but the materials differ in thermal history of crosslinking resulting in a different threshold and stress plateau, while the chain anisotropy r is similar. The sample in (c) is based on the main-chain nematic polymer, Fig. 3.3(b); √ a much higher chain anisotropy r ∼ 25-30 is apparent ( r ∼ 5). The straight lines are fits to the nematic elastomer constitutive relations, eqn (7.50). the rigid clamps, where the non-uniform deformation would cause many complications, and certainly deviations from the simple constitutive equations (7.50). Straight line fits exhibit the three regions we have explored, see the discussion of eqns (7.50) (Warner, 1999). The value of semi-soft threshold strain λ1 is directly related to the residual nonzero slope on the stress plateau, while the average chain anisotropy r =  /⊥ can be estimated from the plateau end position. A few practical difficulties are immediately obvious from the data. First, in all three plots, one can note a characteristic hump at the semi-soft threshold – this is clearly an effect of very slow stress relaxation at the onset of the plateau (in these experiments the extension is imposed very slowly, but still never in a fully equilibrium conditions; we shall review some effects of dynamics and relaxation in Chapter 11). Secondly, the end of the semi-soft plateau in all cases is much less abrupt that its beginning. Several reasons contribute to this, including perhaps variable uniformity of some samples, but the main reason for this rounding is believed to be due to the increasingly inhomogeneous deformations in the region of rigid clamps, see Chapter 8, in particular Sect. 8.3.3. 7.6

Thermomechanical history and general semi-softness

Non-ideality arises when there is not an isotropic reference state. Random copolymer networks formed in the nematic state were explored as an example in Sect. 7.4.1. This requirement of isotropy is the basis of the Golubovic and Lubensky (GL) theorem governing soft elasticity. Section 10.4 discusses symmetry and mechanisms required for this unusual response. Conversely, we have seen that ideal systems can attain isotropy – example 6.2 proves the de Gennes statement of nematic chains reaching isotropy – and with this comes softness, as we demonstrate by example below. In this section we will also sketch other routes to non-ideality and then explore the minimal modifications to the Trace formula that produce semi-softness. (Many addi-

184

SOFT ELASTICITY

tions simply produce mechanical relaxation to a new, but ideal, state.) 7.6.1

Thermomechanical history dependence

An example of an isotropic reference state could be the formation state itself of an elastomer. To demonstrate the role of thermomechanical history we revisit the problem of compositional fluctuations in a random copolymer network which can vitiate ideal softness because given strands do not always want to respond as the mean strand’s ideal response would dictate (Verwey and Warner, 1997a). We show that softness is not vitiated as a result of fluctuations, if formation was under isotropic conditions. Formed this way, any order associated with non-ideality can be relaxed away. A second example shows that an elastomer with the same fluctuations but with a nematic genesis is non-soft. Thus the loss of softness depends not only on, say, fluctuations, but also on the conditions under which the network was made. Random copolymers formed in the isotropic state Recall that the elastic free energy depends on: 1. The total strain, λ t , imposed and developed since formation. 2. The step-length tensor at formation – here it is  f = aδ reflecting the isotropic genesis of the network. 3. The current nematic step-length tensor  (ν ) which varies between chains, ν . Thus the free energy density (in units of µ /2) is:      (7.51) = Tr aδ · λ Tm · λ T · (ν )−1  · λ · λ m F = Tr aδ · λ Tt ·  (ν )−1 · λ t ν

where we average over chains ν and decompose the total strain as λ t = λ · λ m . The spontaneous distortion associated with from formation state to the   cooling √ √ the isotropic relaxed nematic state is λ m = Diag 1/ λm , 1/ λm , λm with n o along z and λm3 =     (ν ) (ν ) 1/⊥ / 1/ . The free energy density for strains λ away from the relaxed,   nematic state is then F = Tr  o · λ T · (ν )−1  · λ where from (7.51) the effective steplength tensor is identified as  o = aλ m · λ Tm . This F is clearly ideal, with soft deform-

, since for λ soft the free energy density takes its ations λ soft = (ν )−1 −1/2 · W ·  −1/2 o ideal value, despite fluctuations:   F = Tr (ν )−1 −1/2 · (ν )−1  · (ν )−1 −1/2 = 3 .

Despite the apparent complexity of F, the GL theorem is obeyed and no stress is required for distortion, in contrast to the non-ideality in Sect. 7.4 and which we find below. One can consider more general disorder. We assumed in taking a single formation  f that all chains had the same effective step length a in the isotropic state. In effect shape variation due to compositional fluctuations only expressed itself when chains became nematic. If now the (isotropic) shape varies in magnitude, a(ν ) , from chain to chain, we

THERMOMECHANICAL HISTORY AND GENERAL SEMI-SOFTNESS

185

can absorb this scalar in the accompanying  (ν )−1 and then average as we have above. Such elastomers are therefore also soft. Random copolymers formed in the nematic state If the network’s genesis is nematic, we expect loss of softness, but must check that a mechanical relaxation does not occur which could conceivably take the elastomer to a soft state. The nematic formation step-length tensor  f(ν ) expresses the variability of chain shape distributions from species to species ν . Now, after any strain relaxation, the effective step-length tensor  o(ν ) = λ m ·  f(ν ) · λ Tm still varies from chain to chain. The relaxation, λ m , should there have been any change in conditions between formation and (ν )

(ν )

(ν )

(ν )

imposing strain, is of the same form as before but with λm3 = ⊥f /⊥ /f / .    on returning λ t = λ · λ m to the Trace The free energy is Tr  o(ν ) · λ T ·  (ν )−1 · λ ν

formula and using the definition of  o(ν ) . We demonstrated in Sect. 7.4.1 that such a free energy is incapable of softness. Indeed one can easily show that this free energy produces terms linear in Q when expanded in the order parameter to give the Landau-de Gennes free energy (Verwey and Warner, 1997b). With this apparent ‘external field’ acting, there can be no absolutely isotropic state at any temperature. The de Gennes observation, exercise 6.2, of recovery of isotropy is not fulfilled and, via the GL theorem, there can be no soft elasticity. 7.6.2

Forms of the free energy violating softness

We explore what additions to the trace formula must be made in order to destroy soft response. To the ideal free energy density we add a harmonic form to give:     (7.52) F = 12 µ Tr  o · λ T ·  −1 · λ + 12 µ Tr A · λ T · B · λ . We assume for simplicity that relaxation has occurred and is accounted for in a prefactor, or that the formation and current conditions are identical. The medium is uniaxial, initially characterised by a director n o and, when λ is imposed, by the current director n . If deformation is soft, there is no driving force to deviate from uniaxiality, simply to redirect it. Assume it is A that depends on n o and B on n . Softness requires a deformation that restores both −1 and B to their values they took when λ = δ , n = no and the energy was minimal, corresponding to the relaxed state. They must be returned to  −1 and o T −1 −1 B o respectively. For instance λ soft ·  · λ soft must give  o where λ soft is given in the usual way by expressions of the form in eqns (7.53) below. When  −1 and B are both restored, the free energy will not have risen. We therefore require a soft deformation λ soft simultaneously satisfying: V ·  −1/2 U · B 1/2 λ soft =  1/2 ·V and λ soft = B −1/2 ·U . o o

(7.53)

It happens that in the ideal term the first factor  o is the inverse of the tensor  −1 when n = n o ; the argument given clearly holds when this is not so, in analogy to how we

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treat the B factor. The free energy is unchanging with director rotation since if these are   inserted into eqn (7.52) one obtains 3 + Tr A · B o which is independent of n and is indeed the starting value before distortion and any possible director rotation. Equating the two expressions for λ soft we obtain: V = B 1/2 U T · B 1/2 ·  1/2 ·V ·  1/2 . o o

(7.54)

  √ 1 0 1 √0 ≡ δ + ( b − 1)nnn and in the n n frame, then B 1/2 = b 0 b 0 1/2 1/2 equally for B 1/2 ,  and  . Multiplying out (7.54) one obtains: o o

Thus, if B =

√ √ V = (δ + n o n o ( rb − 1)) U T · (δ + n n ( rb − 1)) ·V

(7.55)

which is immediately evident directly from (7.54) when one notes that all terms depending on n o are on the right-hand side and all those depending on n on the left-hand side. The two sides can only be equal if U = V = R θ , where R θ is exactly the rotation that takes n o to n . The rotation is through the angle between n and n o , that is cos θ = (nn · n o ). Then λ soft is a trivial body rotation:

λ soft =  1/2 · R θ ·  −1/2 = R θ · R Tθ ·  1/2 · R θ ·  −1/2 = R θ ·  1/2 ·  −1/2 = R θ . (7.56) o o o o The only way to avoid triviality is for B = −1 , whence B1/2 = −1/2 and both parts o o of eqn (7.53) are all satisfied without any restriction on V . Once B =  −1 is assumed,   then one can write the free energy as Tr (o + A ) · λ T ·  −1 · λ and, as we have seen in Chapter 6.3, subsequent relaxation restores the Trace to an ideal, soft form. Thus the conclusion is: A necessary and sufficient condition for softness is that not both A and B can be different from  o and  −1 respectively. The argument is of greater generality than the harmonic example suggests. For instance, terms like A p · (λ T · B · λ )q can have the same arguments applied to them – the action of a λ o must be to restore B to the value it had when λ = δ , that is to B o , say. Another specific example of molecular non-ideality that provides non-soft additions to the free energy is that of rigid rod crosslinks. Because of their shape, such crosslinks can sense orientation and can remember it after crosslinking because the network chains tethered to the ends of the rods localise them angularly. One can show (Verwey and Warner, 1997b) that a linear Q term in the Landau-de Gennes free energy arises and that such elastomers are non-ideal if they have been crosslinked more than once. This is subtly different from non-ideality due to compositional fluctuations where oncecrosslinking is sufficient for semi-softness, provided it was done in the nematic state.

8 DISTORTIONS OF NEMATIC ELASTOMERS Soft or semi-soft deformations are energetically the best response to shape changes imposed on a nematic elastomer. When the director can be rotated, then an elastomer will always deform softly, if the accompanying relaxations can be reconciled with the boundary conditions. In this chapter we examine distortions of nematic elastomers where the imperative to deform softly is in conflict with the external constraints imposed on them. There are generically two ways to resolve this conflict. 1. A sample deforms almost but not quite softly, for instance with an energy cost quartic in the deformation or the director rotation (and not quadratic, as usual). For small distortions, the rubber is thus essentially soft or semi-soft, the anchoring effect of the matrix being first felt at large amplitudes. We treat an example of this by first dealing with the Freedericks effect for nematic elastomers. The effect is different from classical liquid crystals since the transition occurs at a critical field rather than critical voltage. 2. A sample deforms softly, but with a local strain λ soft that differs from region to region. For example a given soft extension, λ of Sect. 7.3.2, comes with simple shears of either δ (λ ) or −δ (λ ), and director rotations ±θ (λ ). By judiciously putting together neighbouring regions with shear deformations of opposite sense, one can obtain an extension that overall has no net shear and hence may satisfy zero-shear boundary conditions in some gross sense. Much finer microstructures are required in mechanical experiments to achieve global softness. We examine such microstructures in a refined treatment of a clamped version of the simple extension of Sect. 7.3.2 which produces the stripes first seen by Finkelmann and coworkers. Coexisting neighbouring regions of differing λ soft create inhomogeneous interfaces in the nematic elastomer. Their energetic cost turns out to be extremely small and does not hinder elastomers resorting to microstructures to eliminate the otherwise considerable elastic costs of deformation. Ignoring interfacial energies and reducing elastic energy by judicious choices of sets of coexisting strains is called ‘quasi-convexification of the free energy’. It was invented in a more difficult problem of discrete sets of low energy crystallographic distortions in martensite, a shape-memory alloy (Ball and James, 1992). The same geometric and physical ideas were independently applied (Verwey et al., 1996) to the very much simpler problem of a nematic elastomer with an ideal form of clamping. We shall examine this problem in this chapter, including the details of the interfaces. We then review the formal quasi-convexification of the nematic elastomer free energy and describe the general microstructures that arise (DeSimone, 1999). The 187

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method has been applied to give a full numerical solution of the extension of a nematic elastomer strip and the evolution of its complex microstructures (Conti et al., 2002), in effect a multi-scale analysis. Towards the end of this chapter the problem of randomly quenched disorder in nematic elastomers is addressed. Quenched disorder seems to be an inherent feature of liquid crystalline polymer networks, where the crosslinks are not specifically enforced to be aligned. With such randomly oriented local sources, which are unable to participate in thermal motion characteristic of polymer chains and mesogenic moieties, the polydomain textures result as an equilibrium state of liquid crystal elastomers. We examine properties of such states and transitions they undergo. 8.1

Freedericks transitions in nematic elastomers

Nematic fluids have anisotropic dielectric and diamagnetic susceptibilities. Energies are lower when their director is aligned with the field E or H . Thus liquid nematics follow the field unless the anchoring at their boundaries is too strong. A large enough field will rotate the director in the bulk to lower the field energy, despite the Frank penalty for distorting n from the bulk to its original orientation fixed on the boundaries – see the conventional Freedericks transition, Sect. 2.6. Anchoring, fields and the penetration depth Nematic elastomers also have anisotropic susceptibilities and hence a desire to align with a field. However, their director anchoring is not only at the boundaries, but also in the bulk, as sketched in Chapter 1 and derived in Sect. 7.1.1 from a molecular picture of director-rubber coupling. For small rotations of the director by θ relative to the matrix, the penalty is 12 D1 θ 2 , see also continuum theory in Chapter 10. The relative rotation modulus D1 = µ (r − 1)2 /r is of the order of µ except for very weakly anisotropic elastomers where r → 1. This rubber-elastic penalty, giving bulk anchoring of the director, leads to prohibitively high electric fields to induce dielectric response: equating the elastic and electric field energy densities,
o ∆
E 2 ∼ D1 gives E ∼ (r − 1)(µ /
o ∆
)1/2 ∼ 107 V/m, for typical values of rubber moduli, of chain anisotropy and of dielectric anisotropy ∆
. It is as if there were effectively a very strong aligning field µ acting along the axis n o , unless the sample is mechanically unconstrained and an appropriate soft deformation can be found (Terentjev et al., 1994). The competition between surfacing anchoring and fields led in Sect.2.6 to the liquid  1/2 , with K an appropriate Frank constant. nematic penetration depth ζ = K/(
o ∆
E 2 ) The analogous solid nematic characteristic length, where the “field” is D1 ∼ µ , was also introduced: ξ = (K/D1 )1/2 ∼ (K/µ )1/2 ∼ 10−8 m if the chain anisotropy is nonvanishing, [r − 1] ∼ 1. This penetration length ξ is a characteristic size for the director distortions to be confined in the nematic solid. This length scale is much smaller than the value ζ for nematic liquids with typical electric fields applied. Freedericks experiment and analysis with rubber elasticity Given the powerful anchoring D1 of the bulk elastomer, one might conclude that electric and magnetic Freedericks effects are impossible in nematic elastomers. However,

FREEDERICKS TRANSITIONS IN NEMATIC ELASTOMERS

189

director rotation relative to the matrix is what makes the soft response possible, see Fig. 7.3. In reverse, if we find suitable relaxing combination of strains, then the corresponding director rotation should also be soft, as briefly illustrated in Sect.7.4.4. Indeed, the penalty 12 D1 θ 2 is for rigidly clamped elastomers where any elastic relaxation is forbidden. In Chapter 10 we show within continuum theory how rotations lower energy and lead to softness, Sect. 10.4.2. Thus the secret of apparently impossible director response to electric fields lies in the accompanying strain relaxations and explains the early observations (Zentel, 1986; Barnes et al., 1989) of shape change in response to modest applied electric fields. Detailed experiments on the response of unconstrained elastomers, capable of exploring soft deformation routes, are now available (Urayama et al., 2005; Urayama et al., 2006) and are discussed in the context of soft elasticity, see Sect. 7.4.4. The classical Freedericks experiment with confined cell boundaries has been first performed (Chang et al., 1997a) on a slab of nematic gel, with a positive ∆
and with initially planar director alignment between two transparent electrodes. An electric field applied perpendicular to the initial director, see Fig. 8.1, induced director rotation which was observed by conoscopy. Shear in the sample plane was also apparent from the translation of dust and other imperfections imbedded in the gel. Rotation started at a critical field, Ec , which Chang et al. determined by taking cells of varying thickness. Classical Freedericks transitions occur at a critical voltage, Sect. 2.6 and eqn (2.28), since the gradient cost of distortion between the anchoring plates diminishes with their separation. The field required to generate the distortion also accordingly diminishes, but the product of the field and thickness, the voltage, remains constant. With intrinsic anchoring to the bulk of the elastomer matrix, independent of sample thickness, it is clear that rotation in nematic elastomers starts at a critical field E rather than a critical voltage V = Ed. Another difference from the classical transition is anticipated by Fig. 8.1; instead

F IG . 8.1. Field-induced director rotation in a conventional, liquid nematic (a) and in a nematic elastomer (b). The liquid has its director anchored at the surfaces x = 0 and x = d to be along z. The solid has its initial director everywhere aligned along z. The electric field E is applied across the cell. The shear strain λzx accompanying the director rotation in nematic elastomers is shown on the right. The conventional Freedericks effect has one half wavelength of director rotation between the plates, while the solid nematic Freedericks effect has the full wavelength.

190

DISTORTIONS OF NEMATIC ELASTOMERS

of a 1/2 wavelength of director variation, we require a whole wavelength. Shears of opposite sign in different halves of the cycle must add together to give zero translation of the upper plate with respect to the lower. Indeed, this too was deduced from the conoscopy, and observed in computer simulation (Skacej ˇ and Zannoni, 2006) which also presents the expected NMR spectra and birefringence. The semi-soft elastic free energy density, eqn (7.39), is:      F = 12 µ Tr o · λ T · −1 · λ + α Tr δ (tr) · λ T · n n · λ The first term is the ideal elastic energy for ‘average’ chains, for instance  −1 = δ + (1/r − 1)nnn . (For shorter notation, in the following we will drop the   symbols on the average of r.) The additional semi-soft α term departs from ideality; in it, the matrix δ (tr) = δ −nno n o projects out components of vectors in the plane perpendicular to n o , see Sect. 7.4.1. The simplest deformation consistent with unchanging positions of boundary electrodes is the simple shear of Fig. 8.1, namely:   1 0 0 λ =  0 1 0  ≡ δ + λzx z x . λzx 0 1 It will be required to vary across the cell with the coordinate x, the variation λzx (x) presenting no particular compatibility problems. In fact other shears such as λxz (x) could also induce a low energy rotation of the director and, provided they varied through a full cycle in the interval x = 0 → d, could satisfy boundary conditions too. However the compatibility condition ∂ λxz /∂ x = ∂ λzx /∂ z implies a variation of λxx (z), in the plane of the sample, which presents obvious complications that  we shall not analyse. The suggested deformation is incompressible, Det λ = 1. The z-displacement is uz (x) =

 x 0

dx λzx (x )

which clearly vanishes at the top plate, x = d, if λzx oscillates through a full cycle between x = 0 and x = d. This is the first example we see of spatial variation being determined by boundary conditions. Exercise 8.1: What is the elastic cost of the above distortion λ when n is rotated by an angle θ ? Solution: The initial director was n o = z and is currently n = x sin θ + z cos θ . By putting in the corresponding expressions for o and −1 into the first (ideal) part of the free energy, one obtains:   (r − 1)2 2 (r − 1) 1 + (r − 1) sin2 θ 2 1 sin θ − 2 λzx sin θ cos θ + λzx . Fel = 2 µ 3 + r r r The semi-soft part of the elastic free energy density is, by a similar contraction of matrices:   Fss = 12 µα Tr (xxx + y y ) · (δ + λzx x z ) · n n · (δ + λzx z x )

FREEDERICKS TRANSITIONS IN NEMATIC ELASTOMERS =

1 xx + y y + λzx x z ) · (nnn + λzx n x cos θ )) 2 µα Tr ((x

=

1 2 µα

=

1 2 µα [sin θ

191

  2 sin2 θ + 2λzx sin θ cos θ + λzx cos2 θ + λzx cos θ ]2 .

The shear λzx is free to relax, given the fixed director rotation, to a value that optimises the elastic free energy. Solving ∂ (Fel + Fss )/∂ λzx = 0 gives the shear associated with such a rotation is thus:

λzx =

(r − 1 − α r) sin θ cos θ . 1 + α r + (r − 1 − α r) sin2 θ

(8.1)

The dielectric energy of the uniaxial nematic elastomer is: E .nn)2 = − 12
o ∆
E 2 sin2 θ . − 12
o ∆
(E This expression ignores the differences between D and E , that is induced polarisation modifying the field distribution. Treatment of the electrostatics in this particular Freedericks context (Terentjev et al., 1999) justifies the neglect of this subtlety. Otherwise, one should simply imagine that this analysis is for magnetic fields. Optimal rotations in the uniform case Initially we ignore director variation since the barrier to director rotation will turn out to have little to do with Frank elasticity. The optimal shear, eqn (8.1), associated with rotation is returned to the elastic energy above which then only depends on angle θ . It can then be combined with the electrical energy which too only depends on angle, to give the overall energy FE (θ ): 4  (r − 1)2 sin4 θ
o ∆
E 2 2 sin θ + (8.2) FE = 12 µ − µ 1 + (r − 1) sin2 θ 5 α r2 sin2 θ + (1 + (r − 1) sin2 θ )(1 + α r + (r − 1 − α r) sin2 θ ) 6 7 1 (8.3) ≡ 2 µ −E 2 sin2 θ + α A sin2 θ + B sin4 θ The reduced electric field in the first term denotes the non-dimensional ratio 0
o ∆
. E =E µ The elastic energy in eqn (8.2) has been expanded in sin θ to give eqn (8.3). The ideal part of eqn (8.2) yields the quartic penalty B sin4 θ (where B is a constant). Evidently the Freedericks geometry cannot offer complete softness, even for ideal elastomers (no elongations along x have been permitted). However the quartic form starts softly and plays no role in thresholds. The semi-soft part of (8.2) gives the quadratic penalty, α A sin2 θ , with the constant α A = α r2 /(1 + α r).

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DISTORTIONS OF NEMATIC ELASTOMERS

The symbolic form eqn (8.3) shows directly how the director rotation starts when the electric advantage −E 2 sin2 θ outweighs the semi-soft penalty α A sin2 θ at a reduced field Ec2 =

α r2 . 1 + αr

(8.4)

Minimising the full FE (θ ), eqn (8.2), over sin2 θ gives the optimal angle of rotation for reduced fields above the threshold, E > Ec : √ αr + 1 αr + 1 +√ . (8.5) sin2 θ = − √ r − αr − 1 r − 1 − E 2 r − αr − 1 At the threshold field the rotation starts continuously from zero and, while it is small, one can expand about θ = 0, E = Ec :

θ2 ≈

Ec (E − Ec ) , (r − 1 − Ec2 )2

θ∝



E − Ec .

(8.6)

The evolution of director orientation, eqn (8.6), is singular, as we met in the mechanical equivalents of (anti) Freedericks transitions. Eventually the rotation is complete, sin θ = 1, that is θ = π /2, at a field: Eu2 =

(r − 1)2 (r + 1) r − 2 − αr . −α r2 r

(8.7)

Influence of Frank elasticity The uniform analysis above ignored the role of spatial variation in the director, as in the semi-soft elongation of a nematic rubber strip in the previous chapter. If θ = 0 is rigidly preserved at x = 0, x = d, and if there can be no relative displacement of the plates at x = 0, x = d, then the director has to vary along x. The associated Frank cost may then delay the onset on director rotation. Let θ (x) vary as θ sin(2π x/d) just as rotation starts. Later, at higher fields, the profile will coarsen first to elliptic functions and then finally to a single sharp wall in the middle of the sample (Terentjev et al., 1999). In the initial phase, just above the threshold, the Frank free energy density is FFr = 12 K(d θ /dx)2 = 12 K(2π /d)2 θ 2 cos2 (2π x/d) . The average Frank energy density, FFr  = K(π /d)2 θ 2 , must be added to the elastic penalty. Take the simplified form eqn (8.3), which is valid near the transition: 4    5 K π 2 (8.8) − E 2 + Ec2 sin2 θ + B sin4 θ . F = 12 µ µ d We have re-expressed α A by Ec2 , the value of threshold field (squared) in the absence of gradients.

FREEDERICKS TRANSITIONS IN NEMATIC ELASTOMERS

193

 The nematic penetration depth K/µ = ξ measures the competition between Frank and rubber elasticity and is a length of order ξ ∼ 10−8 m. This is how far director orientation anchored at one point will spatially persist if it is in conflict with the natural direction of the elastomer. The smallness of ξ arises from the strength of the effective field µ presented by the matrix. It is much stronger than that exerted by electric fields,
o ∆
E 2 . Although here we discuss nematic elastomers, and not liquid crystals, we nevertheless have adopted characteristically liquid nematic values for the Frank constant K in estimating ξ . Light scattering studies (Schmidtke et al., 2000) show that side chain nematic polymer melts indeed have Frank constants of the same order as those of simple liquid nematics. One can also argue that similar values are also expected in nematic networks on physical grounds: the molecular fluidity of an elastomer means nematic order is little perturbed by network formation. Moreover, nematic energies are of the order of kB T per monomer, rather than kB T per group of N monomers on an average network strand as it is for the elastic component of energy. Since N can be large, elastic energies are generally subservient to nematic energies which are then expected to be close to their liquid state values, including energies associated with director gradients. Because of the D1 coupling of the director to the elastic matrix in elastomers, modes involving spatial gradients of the director are masked in light scattering studies – all one can say is that the nematic penetration depth is much less than the inverse scattering vector involved. The failure to find direct methods to measure the Frank constants in elastomers has led to different approaches to describing nematic elastomers. However a conclusive constraint on modelling is provided by experiment, in particular that on stripe domains (see below, Sect. 8.2) – interfaces should be at all times coarsened, that is the interfacial width is very small. These constraints tend to support the idea that Frank constants are not abnormally large in elastomers. The sin2 θ term in F, eqn (8.8), changes sign at a new critical field: E 2 = Ec2 +

πξ d

2 .

(8.9)

At this critical field the director rotation starts. How great is the shift in the transition due to the additional (πξ /d)2 arising from Frank elasticity? In experiments on nematic gels the thickness was in the range d ∼ 10−4 -10−3 m, whence (πξ /d)2 ∼ 10−7 -10−9 . The square of the reduced, uniform critical field eqn (8.4) is roughly Ec2 ∼ r2 α . The anisotropy r is typically 1.1-3, and the degree of semi-softness is in the interval α ∼ 0.010.1, depending on how ideal the elastomer is. Thus Ec2 ∼ 0.01-0.3. It seems that Ec2 and (πξ /d)2 differ by a factor of ∼ 106 . Frank effects are very small when fields are large enough to compete with elastic anchoring to the bulk. Boundary conditions will determine the distributions of θ (x) and λzx (x), but not the threshold. We will reach the same conclusion about the microstructure in elongating strips arising from boundary conditions – microstructure does not affect the form of director rotation nor the macroscopic stress-strain curves. In particular here, the threshold should remain a critical field rather than voltage. In nematic elastomers, we can neglect the second, classical liquid crystal term in eqn (8.9).

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DISTORTIONS OF NEMATIC ELASTOMERS

F IG . 8.2. The conoscopic length, δ (E), against electric field (Chang et al., 1997a) for a 62 µ m thick sample (crosses) and a 125 µ m thick sample (squares). The δ values for the 62 µ m sample have been doubled, collapsing the data from two samples onto the same theoretical plot, solid line. Figure 8.2 shows optical path length δ against electric field for two slabs of differing thickness. The parameter δ depends on the light wave phase, which in turn depends on the director orientation sin2 θ . The results, when normalised by thickness are superposable, indicating that a critical field rather than voltage dominates. The threshold relation can be rearranged to express the electric voltage applied to the cell: µ 2 d +VF2 V 2 = E 2 d 2 = α r2
o ∆
where VF is the classical Freedericks voltage determined by the usual nematic parameters, VF2 = π 2 K/
o ∆
, see eqn (2.28). When α r2 d 2 (µ /
o ∆
) ≤ VF2 we return to the old situation of critical voltages determined by Frank elasticity. It could happen for very soft elastomers, α → 0, or for very weak gels where µ is very small. The condition of course is as before; it does not involve the dielectric anisotropy
o ∆
, but rather the comparison of (ξ /d)2 with α – one would require large penetration depths (from small µ ) or small semi-softness, α . 8.2 Strain-induced microstructure: stripe domains Elastomers elongated perpendicularly to their director deform (semi) softly if they shear. The cartoon in Fig. 7.3 shows graphically how elongation must be accompanied by shear if energy cost is to be eliminated. However clamps, through which stretch is imposed, prohibit shear in their vicinity. Then microstructure, in the form of stripe domains, offers the best compromise between the drive for soft deformation and the constraining boundary conditions. Figure 8.1 in the Freedericks case illustrates the solution to the problem when the shears are less complicated: the upper plate is fixed with respect to the lower plate. If shear is required to soften the response (to make it quartic in that case), there must be two compensating shears in order that they create no net displacement. For imposed mechanical fields, such as the uniaxial extension, semi-soft simple shear is a good example with which to illustrate the emerging microstructure. See

STRAIN-INDUCED MICROSTRUCTURE: STRIPE DOMAINS

195

Fig. 8.3 where a strip is extended beyond the semi-soft threshold for director rotation λ = λ1 . In the bulk of the strip, the local shear and director rotation follow the optimal, semi-soft values consistent with the extension λ . A compensating pair of stripes is shown magnified; on traversing the pair (in the z-direction, along the initial n o ) the total x-displacement averages out. Real systems have a collection of many stripes stretching along the elongated elastomer strip, see Figs. 5.21 and 8.4. The elastic softness is unattainable only at the ends and in the sharp interfaces between stripes. The bulk of the elastomer deforms softly, at least until the director rotation is complete at the extension λ2 . This is seen macroscopically in the stress-strain and opto-mechanical relations. The precise details of the clamp constraints will determine how the stripe domains evolve. In general the problem with simple, realistic constraints is extremely complex, see Sect. 8.3.3. A simplification is to consider rigid but sliding/rolling constraints or clamps that themselves deform at a compensating rate. Shear is suppressed at the end while allowing any necessary transverse λzz relaxation required to conserve volume while extension λxx proceeds, see Fig. 8.5. Some residual curvature at the ends of the rubber strip may still occur even if z-motion in the clamp is free: in the bulk the relaxa√ tion is λzz ∼ 1/λxx , as with all soft modes, whereas at the ends it is only λzz ∼ 1/ λxx , since the response is hard in the absence of shear. Additionally, the extension λxx is

prohibited shear z

no

l

h clamp

n

n

n

n

x l

local shear

F IG . 8.3. Microstructure in a nematic elastomer strip being extended perpendicular to its initial director n o , assumed along the z-axis. A section with only two neighbouring stripes of width h and opposing shear, λx z, and rotation is shown. At the ends the displacement associated with soft shear is shown suppressed by the clamps.

F IG . 8.4. Stripe domains in a nematic elastomer extended as in the schematic Fig. 8.3. All three images are the same stripe system at a fixed extension but viewed at different angles (0o , 15o and 77o ) with respect to a crossed polariser-analyser pair. Different details of the stripe substructures then emerge (images: I. Kundler).

196

DISTORTIONS OF NEMATIC ELASTOMERS

z x

F IG . 8.5. Ends of a strip are rigidly constrained, forbidding shear but allowing for transverse relaxation λzz . itself smaller at the ends than in the bulk because the nominal stress has to be conserved. Appendix C explains the conservation of nominal stress requirement and what the energy cost of distortion at the ends is. We ignore curvature in the clamp region and use the idealisation of Fig. 8.5 as a model for how the uniform texture of Fig. 8.3 is finally terminated far enough from the ends that the complications of static clamps can be ignored. The problem is examined in more detail in Sect. 8.3.3. If the imposed elongational deformation in the x-direction is λ (greater than the rotation threshold λ1 ), the optimal semi-soft shear and director rotation within the individual alternating stripes are ±λxz and θ± = ±θo (λ ), with (λ 2 − λ12 )(rλ12 − λ 2 ) λxz2 = rλ 2 λ13

0

θo = sin−1

r λ 2 − λ12 , r −1 λ2

(8.10)

see Sect. 7.4.2, eqn (7.44). We now see how alternating stripes can be fitted together. Stripe structure and energy An interface region must separate two domains of optimal deformations obtaining in one sense (+) and in the opposite sense (−). An area A of interface has an energy ∆F ∼ γ A where γ is an effective interfacial tension, which we shall derive shortly. The director generally responds to an imposed strain. Non-uniform mechanical distortions such as the above stripe structure generate non-uniform directors which in turn cost a Frank nematic elastic penalty. We therefore, indirectly, have an energetic cost to non-uniform elastic strain. As usual in elasticity, we ignore the direct elastic cost of gradients of strain (∂ λi j /∂ xk ). We proceed as in the description of Freedericks transition: to combine Frank elasticity with nematic rubber elasticity, one must reduce the problem so that the director angle is left as the single variable upon which the energy depends. The strains that can relax are set equal to their minimal value, subject to a given θ and to the strain components that are imposed. The total elastic energy then depends on θ for the elastic part and on ∇θ for the Frank part. Take strains and director rotations to vary in the z-direction, but to be basically of the simple shear type we have already considered in Sect. 7.4.2; also see Fig. 8.6(a). At a fixed imposed x-strain λ , the elastic free energy density for a given θ , with the shear λxz and transverse strain λzz relaxed, was given in eqn (7.28). We give the free energy a subscript λ to emphasise that it is obtained for a fixed imposed value of λ . It is

STRAIN-INDUCED MICROSTRUCTURE: STRIPE DOMAINS

z

l xz

q =qo

Effective potential

fl (q)

l

w

qo

x

(a)

qm

n

q =-q o

197

q

(b)

F IG . 8.6. (a) An infinite block of nematic elastomer deformed in opposite senses in the two half spaces z > 0 and z < 0. In the transition region of width w the deformation is not soft. (b) The effective potential of dynamical motion equivalent to the stripe interface. The amplitude θm may be less than the optimal rotation θo required by the elastic medium at a given extension λ .   1 r − 1 2  ≡ µ fλ (θ ) ) sin2 θ +

Fλ (θ ) = 12 µ λ 2 + λ 2 (α − r λ 1 − r−1 sin2 θ r   2 r−1 2 λ 1 , sin2 θ +

fλ (θ ) = 12 λ 2 − 3 (8.11) λ 1 − r−1 sin2 θ λ1 r r

where fλ (θ ) is the dimensionless part of the elastic free energy density, that is with the rubber elastic energy scale µ taken out. In eqn (8.11) we have re-expressed α in terms of the threshold strain, r − 1 λ13 − 1 α= , r λ13 in order to give a simpler expression for fλ (θ ). Far away from the interface where optimal rotations and shears can be achieved, the energy (8.11) is minimised by the strains and rotations (8.10) – our problem is finding the shears, rotations and energies through the stripe where the bulk optimal values cannot be achieved. The director varies with z. It rotates in the xz-plane, making a local angle θ with the z-direction, n = (sin θ , 0, cos θ ). The Frank energy density involves only splay (K1 ) and bend (K3 ) in this geometry, see Sect. 2.6. It is: FF =

1 2

 2  dθ 2 dθ 1 K1 sin θ + K3 cos θ → 2K . dz dz



2

2

The latter simplification arises in the single constant approximation K1 = K3 = K. Scale the length zby the penetration depth ξ , identifying a dimensionless length  u = z/ξ , whereupon dz becomes ξ du and d θ /dz becomes ξ1 d θ /du ≡ ξ1 θ  . The

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total free energy Fsurf associated with the transition region between domains with ±θo is     Lz /2ξ K du µ fλ (θ ) + 2 θ 2 Fsurf = dxdydz (Fel + FF ) = Aξ 2ξ −Lz /2ξ  

(8.12) = A K µ du fλ (θ ) + 12 θ 2 . 8 Here A is the area of the strip’s xy-section, √ over which there is no x or y variation. The geometric mean of the two energies, γ = K µ , can be interpreted as an effective surface tension and provides the energy scale of all interfacial problems in nematic elastomers.  The integral du[. . .] is dimensionless. It depends on λ and on the profile of director variation θ (u). Length scales are setby the nematic penetration depth ξ . Any structure that θ (u) develops from minimising du . . . will be multiplied by ξ to give it a physical dimension. For instance ξ will set the scale of the thickness of the interface between stripes, w in Fig. 8.6(a) (Finkelmann et al., 1997). If we take some typical values, µ ∼ 105 Pa and K ∼ 10−11 N, the ‘surface tension’ can be estimated as  γ = K µ ∼ 10−3 N/m . (8.13)

The interfacial energy scale γ is relatively small. The surface tensions of liquids, by comparison, are in the range 40 × 10−3 N/m (benzene) and 72 × 10−3 N/m (water), at least an order of magnitude higher. In the interface between the two stripe domains the director varies (θ  = 0) as it makes the transition from θ+ to θ− through the value θ = 0 at z = 0. There is a Frank energy which can be reduced by making the variation slow, that is, making the interface w thicker. The elastic energy is higher in this case, since this is a region in which θ and λxz do not adopt their optimal values ±θo and ±λxz . For instance at the centre of the interface there is no director rotation or shear, θ = λxz = 0, and the energy density is of order µ , that is deformations are elastically hard. The reduced elastic energy in this regime is fλ = fA = 12 (λ 2 + 2/λ ), eqn (6.41). This is an imperative to make the interface narrower and acts against that of the Frank energy. The optimal θ (u) is that which mimimises the interfacial integral in eqn (8.12) to, say, fint . The actual interfacial energy ∆F is the F of eqn (8.12) minus the body’s energy when the surface is absent, that is when θ = +θo or −θo everywhere. We subtract fλ (θo ) to give ∆F = Aγ fint .

(8.14)

When, at the very beginning of director rotation, the optimal values ±θo are very small, one obtains a modulation of the director θ (z), initially simply sinusoidal. As the amplitude of θ increases, the stripes enter the coarsened regime. This is a classical problem  L /2  L /2

is the original area, A = −Lx x /2 −Ly y /2 dxdy = Lx Ly , and is appropriate for the elastic part of (8.12). Strictly speaking, for the Frank part we should use the current area Axy = Aλ λyy which will introduce some additional z-variation in the interfacial regions. 8A

STRAIN-INDUCED MICROSTRUCTURE: STRIPE DOMAINS

199

of coarsening that we discuss in greater detail in Appendix C (Finkelmann et al., 1997). It is solved by using the analogy with a dynamical system where the θ 2 represents the kinetic energy of the particle moving in the potential fλ (θ ). Expanding in powers of small θ , the dimensionless part of the free energy eqn (8.12) becomes:      r − 1 2 4 1 2 3 1 2 1 3(r − 1)[λ − λ1 ] .(8.15) Fsurf ≈ du − 2 θ + θ + 2θ Aγ 8λ1 r λ12 r In the absence of gradients (i.e. far from the interface) the value of the director angle θ minimising the eqn (8.15), is indeed the solution θo of eqn (7.44) for λ close to λ1 . The actual non-uniform profile of the director orientation minimising the free energy (C.4) is given by the Euler-Lagrange equation θ  = ∂ fλ /∂ θ . This is a classical problem of motion in an anharmonic potential well with coordinate θ , Fig. 8.6(b). ‘Motion’ ceases at the turning point θm , the amplitude of oscillation (θ  = 0 when θ = θm ). Half the period is just the width of the stripe interface. If the director rotation is small, θm  θo (i.e. the director does not reach its optimal orientation) the ‘motion’ becomes harmonic (sinusoidal) and the period (stripe width) becomes independent of the amplitude. However, the amplitude of ‘oscillations’ θm is large, the ‘period’ becomes larger since the particle starts off from rest at θ = θm where the potential is very flat and only accelerates slowly. When θm = θo the elastomer is soft on both sides of the interface. The full solution of the problem of stripe coarsening (Finkelmann et al., 1997) requires the elliptical functions analysis; its conclusion is that the jump from the sinusoidal director modulation (when most of the sample is not in the optimal soft conformation) to the state with θ = θm occurs almost immediately above the stripe the threshold λ1 . The interface becomes more narrow with increasing strain: 0  2λ12 r 1 √ . (8.16) wξ 3 r−1 λ − λ1 Experimentally it was indeed found that in stretched nematic elastomer the stripes are immediately coarse when they form (Kundler and Finkelmann, 1998; Zubarev et al., 1999), that is the majority of the sample is taken by the regions of relatively uniform director rotation, alternating in neighbouring stripes, with interfaces narrow compared with stripe width. Theoretically, the optimal director profile θ (u) in this coarse limit gives a value for fint ∼ 1 and an interface width, in units of u, also of ∼ 1. This means that the width, in units of real length, is w ∼ ξ . Experiments could not optically resolve the interfacial width – it was evidently very small. The director angle, determined from X-ray studies, was clearly distributed in two uniform states, θ = ±θo , almost everywhere in the sample. It is this variation, θo (λ ), that has been plotted in Fig. 5.20 and analysed in Fig 7.10. The mechanical response senses the stored free energy, for instance through the nominal stress σ = ∂ F/∂ λ . Could stripes, with their attendant interfacial energy, mask the underlying soft response? The stripe width h is easily resolvable, see Fig. 5.21, and was in the range between 1 and 100 µ m, depending on the material and the initial shape of rubber strip. The volume

200

DISTORTIONS OF NEMATIC ELASTOMERS

of elastomer responding semi-softly is of the order A(h − ξ ) with an energy density ∼ µ (1 − 1/λ13 ), only different from zero by the measure of semi-softness expressed through λ1 . The volume of interfaces is ∼ Aξ ; the material here responds to deformation with an energy density ∼ µ . The latter energy is insignificant compared with the elastic energy stored in stripes, if the following inequality holds:   1 1 or ξ /h  1 − 3 Aξ µ  Ahµ 1 − 3 , λ1 λ1 (neglecting the width w ∼ ξ compared with h in estimating the semi-soft volume). This condition is often satisfied since the ratio ξ /h is of the order of 10−4 , while the semi-soft threshold λ1 ∼ 1.05 − 1.1 whence 1 − 1/λ13 ∼ 0.7. Thus we could expect no mechanical effect of stripe interfaces unless the stripes are considerably finer than those found experimentally (and suggested by the analysis below), or the nematic elastomer is nearly ideally soft. The stripe width We now give a similar scaling argument to estimate the stripe width h. It is only appropriate for the ideal clamps considered in this section, but the magnitude that results is close to the observed values and the argument illustrates further the idea of competition, this time between the interfacial and the clamp energy, see Fig. 8.3. In Appendix C we show that the energy cost for a region of size ∼ h at each end of the sample, extending with hard response FA , is roughly ∆Fend = Ayz h µ (λ − λ1 )2 (1 − 1/λ13 ) where λ is the bulk strain and very closely equal to the strain imposed onto the softly deforming central section of the strip. To get this, we ensured that the force passing through the end sections matches the force passing through any cross-section of the central region. In further considering estimates of end energies, Appendix C, we argue that the extent of the end regions of the stripes is of the same order h as their width. Ayz is the y-z cross-section area of the rubber strip. The total energy for creating Lz /h stripes across the sample, each with an added interfacial energy γ A, is ∆Fstripes = ∆Fend + Axy γ (Lz /h) :   γ h 2 3 1 + µ (λ − λ1 ) (1 − 1/λ1 ) , ∆Fstripes = 2 V h Lx where the sample volume is V = Lx Ly Lz = Axy Lz = Ayz Lx . There is now a compromise between having narrow stripes to minimise the elastic energy penalty for hard extension in the end region, which decreases for smaller h, and having fewer stripes (larger h) so that there are fewer interfaces between them. The optimal stripe width is:  ξ Lx . (8.17) h= (λ − λ1 )(1 − 1/λ13 )1/2  The nematic penetration depth ξ = γ /µ ≡ K/µ enters again, as the single materialdependent length, in the geometric mean with the strip length Lx . Analogous problems

GENERAL DISTORTIONS OF NEMATIC ELASTOMERS

201

of strain-induced microstructure occur for instance in martensitic alloys and in minerals. Where low-energy shape change is in conflict with boundary conditions, different coexisting ‘variants’ are induced (corresponding to our two ±θo regions). Nematic elastomers, even for small rotations θo , just above the semi-soft threshold λ1 , show coarsened stripes of a width h that hardly changes as λ further increases. Apparently the critical region immediately around λ ∼ λ1 is bypassed and no divergent behaviour survives in expressions such as h(λ ), eqn (8.17). The critical region and the first order transition to coarse stripes (Finkelmann et al., 1997) is analysed in Appendix C. We can, accordingly, simply think of an unchanging stripe width of h∼

8

 ξ Lx 1 − 1/λ13

(8.18)

where the λ1 factor reminds us that stripes are wider when the threshold is small and the elastomer is soft, λ1 → 1. A strip of length Lx ∼ 1 cm and with a threshold at λ1 = 1.05 would then ideally have stripes of width h ∼ 5×10−5 m, that is 50 µ m. However, the real mechanically rigid clamps disrupt the stripe patterns and the microstructure does not extend unbroken for the entire length of the sample, as we have assumed; see Sect. 8.3.3 for more discussion. These are questions also addressed in the equivalent problems in 1/2 martensite (Bhattacharya, 2003) where it is often seen that stripe widths scale as Lx . However, more refined arguments (Kohn and M¨uller, 1992) can give a stripe width 2/3 that scales like Lx : there is an energetic advantage in having fine stripes meet the constraints. However, in the bulk, fineness is not an advantage, indeed it means that more interfacial energy cost is paid. Twins that coarsen away from an interface with austenite are indeed observed in martensite and it is possible that such behaviour, and a consequent 2/3 power scaling, could occur in nematic elastomers too. 8.3

General distortions of nematic elastomers

Two examples of preceding sections have shown nematic elastomers deforming softly (or with energy quartic in strain) even when soft modes are in conflict with boundary conditions. The answer is to satisfy boundary conditions on average, by the establishment of inhomogeneous microstructure. The ideal clamps of the above example allowed the whole sample, except in a small volume near the clamps, to deform softly – for instance the elongations λ and the transverse relaxations λzz were uniform. Shears differed between stripes, but averaged to zero, as demanded by the clamps.

F IG . 8.7. Successive steps in the elongation of a rubber strip.

202

DISTORTIONS OF NEMATIC ELASTOMERS

A more realistic clamping and extension scenario is sketched in Fig. 8.7. The clamps do not permit transverse relaxation in their vicinity. The sample develops curved edges and a complicated shear pattern as a result. Combinations of soft shears to give a soft, non-uniform response are now more complex than in our example above. Moreover we have spatial non-uniformity both at the scale of the stripes and on the scale of the whole sample strip – it now becomes a problem of multiscale compatibility. We discuss the general problem of constructing the appropriate λ from combinations of various λ soft , to achieve a macroscopic situation that is also nearly soft – so-called ‘quasiconvexification’. We then sketch the full numerical solution to the problem of general distortions and compare it with experiment. 8.3.1

One-dimensional quasi-convexification

In the previous section we have, in effect, presented an example of quasi-convexification of the nematic free energy, considering stripe modulation of the sample only along the z-axis. This serves as an illustration of  a more general problem. Suppose one wants to  achieve a deformation λ =

λ 0 0 0 1 0 0 0 1/λ

without paying any energy cost. Without shear, √ this deformation is only without cost for λ = 1 and for λ = r, that is at the two ends of the ‘low road’, that is where the soft √ plateau starts and finishes in Fig. 7.3. For all intermediate deformations with 1 < λ < r, we have achieved, Fig. 8.3, an overall soft deformation by splitting into equal volumes with opposite shears, ±λxz , and rotation:       λ 0 0 λ 0 λxz λ 0 −λxz   1  λ = 0 1 0 1 0 + 0 1 0  0 =  2 0 0 1/λ 0 0 1/λ 0 0 1/λ ? @ = 12 λ soft (+λxz ) + λ soft (−λxz )

λ = λ soft  ±λxz

(8.19)

The effectively soft outcome λ is achieved by a suitably weighted mean of ‘real’ locally soft strains. If we make the microstructure sufficiently fine, then this overall strain λ has the status of a uniform deformation that is soft, despite having no visible shears associated with it as one might have expected. With this assumption of indefinite fineness (certainly finer than any scale of the problem one will later attempt to address) one has: FQC (λ soft  ±λxz ) = F(λ soft (λxz )) = 0

(8.20)

(the additive constant of 23 µ being ignored). One ignores the cost of interfacial energy of the narrow regions that separate the individual soft domains. We showed in practical cases that there is only a negligible volume of hard deformation associated with the interfaces. The free energy F is said to have been quasi-convexified to FQC . Finally, in a sense the FQC points to more general ‘low roads’, elastic trajectories between the initial and the final states of deformation, FA (λ ) and FC (λ ), of Fig. 6.10. In the soft-deformation expression  1/2 · W ·  −1/2 , eqn (7.10), we were able to identify an infinite number of o

GENERAL DISTORTIONS OF NEMATIC ELASTOMERS

203

such deformations, the cartoon of Fig. 7.3 being a special case where the matrix matrix W is simply W = δ . However with FQC one creates more freedom, for instance to eliminate shears from soft modes. One further preliminary consideration is required; the pairs of deformations λ soft must be chosen to be kinematically compatible. Deformation gradients λ displace the material points R o of the initial body to new positions R in the target state, R = λ · R o . Figure 8.3 shows two stripes within the sample, separated by an interface. The accumulated x-displacement is only that corresponding to the externally imposed λ , while the additional modulation due to local shears averages to zero, in going from the bottom of the lower stripe to the top of the upper stripe. One sees this also in Fig. 8.1 for the Freedericks effect, where there is no overall shear generated between the lower and upper plates which are imposing the overall boundary conditions. Moreover, at the interface between the stripes, the position R is given equally by the displacement in going from the top of the upper stripe or from the bottom of the lower stripe - the deformations are compatible. The decomposition of the deformation gradient presented in eqn (8.19) trivially satisfies this condition. In general the λ s on the two sides of the interface must be “rank-1 connected”, that is, the deformations applied using either λ on a material point in the interface must agree so that the interface’s deformations are uniquely defined (Bhattacharya, 2003). The compatibility requirement on the λ s of the quasi-convexification arises less trivially already in the simple example (Verwey et al., 1996) where the initial director n o is not perpendicular to the imposed extension λ along x, but has a pre-tilt angle φ , see Fig. 8.8. The concept of soft deformations as a low energy route for director reorientation remains valid. For a uniform system generating only simple shear and the transverse relaxation 1/λ in response to the imposed extension, the optimal director rotation to angle θ with respect to z , and the associated shear λxz , take the form in alternating (±) stripes: z f

no

x

l (b)

(a) n n

(c)

F IG . 8.8. A strip of nematic elastomer with initial director, n o , at angle φ to the z-axis (a). On extension by λ along x it breaks up into a microstructure which avoids macroscopic shear (b). Pairs of stripes suffer soft deformations with director rotations to angles ±θ , and shears, ±λxz , such that there is no net x-displacement on passing through two stripes (c).

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sin2 θ

λxz

2

1 r(λ − 1) + (r − 1) sin2 φ λ (r − 1) 1 − λ 2 (r − 1) sin φ cos φ ± = λ [r − (r − 1) sin2 φ ] 



± r(λ 2 − 1) + (r − 1) sin2 φ r − λ 2 − (r − 1) sin2 φ . =

(8.21) (8.22)

Note there are two modes of shear in consecutive stripe domains for each value of sin2 θ , (+) (−) that is λxz and λxz corresponding to director orientation angles +θ and −θ respectively. Before the deformation is applied (λ = 1) one has θ = φ , the initial orientation (+) of n o . When director rotation begins, the ‘positive’ domain of shear λxz > 0, in which the existing director pre-tilt φ is in the same direction as the rotation can start its shear deformation continuously from λxz = 0. However, in order to form the ‘negative’ stripe (−) with shear λxz < 0 of opposite sense, it is necessary to overcome a barrier. Stripes with −θ must jump to that state from the initial orientation +φ . If the transition takes place (−) immediately as λ exceeds 1, the jump in λxz takes the value (Verwey et al., 1996): (−)

∆λxz (λ = 1) = −

2(r − 1) sin 2φ . r + 1 + (r − 1) cos 2φ

(8.23)

To satisfy the requirement of no net transverse displacement after traversing a pair of stripes, see Fig. 8.8, one needs the connection between the width, h, of stripes and their shear: (+)

(−)

h+ λxz + h− λxz = 0 . (−)

(8.24) (+)

Thus, as extension begins and λxz is effectively finite while λxz is still zero, then the (+) (−) ratio of stripe widths h− /h+ = −λxz /λxz must be zero and increase as λ is greater than 1. However, the relative width of the opposite stripe domains remains different – which would also be reflected in the different intensity of the two pairs of X-ray scattering lobes, cf. Fig. 5.20 The mean deformation is still without shear, on average, but is composed as a mean of the two soft deformations with modified weights, according to their relative volume in the system:       (+) (−) λ 0 0 λ 0 λxz λ 0 λxz h h + − 0 1 0 1 λ = 0 1 0 = 0 + 0  h+ + h− h+ + h− 0 0 1/λ 0 0 1/λ 0 0 1/λ ? @ 1 (+) (−) (−) (+) λxz λ soft (λxz ) − λxz λ soft (λxz ) = λ soft  (±) . (8.25) ≡ (+) (−) λxz λxz − λxz The quasi-convexified energy is volume-averaged over the energies of the two component distortions of λ :

GENERAL DISTORTIONS OF NEMATIC ELASTOMERS

FQC (λ ) =

1 (+) (−) λxz − λxz

205

? @ (+) (−) (−) (+) λxz F(λ soft (λxz )) − λxz F(λ soft (λxz )) = 0 . (8.26)

 λ 0 0   (±) The deformation λ = 00 10 1/0λ is still effectively soft since the F λ soft (λxz ) both vanish, but the details of the microstructure are not as before – the volumes taken up by the two new types of λ soft are now different and the interfacial structure problem is modified. The full problem is much more difficult than simply taking an initial director at an angle to the principal stretch. The third dimension becomes involved and sample shape as a whole plays an important role. 8.3.2

Full quasi-convexification

So far we have only dealt with planar soft problems. The director has rotated in the zx-plane and hence there has been no y-relaxation: λyy = 1. We may need to make soft imposed deformations that are not restricted to this special value of λyy . For geometrical reasons (the ideal clamping), we so far only considered simple shears, whereas the full geometric representation of soft deformations, Fig. (7.3), tells us that more complex local strains may be needed, even for the planar problem. The quasi-convexification problem has been solved in complete generality for ideal nematic elastomers (DeSimone, 1999; DeSimone and Dolzmann, 2002) and applied in a numerical multiscale analysis of the response of a strip with realistic clamps and suffering extension (Conti et al., 2002). Nematic elastomers admit of a fuller analysis of their soft deformations than Martensite and other crystalline transformation problems (Bhattacharya, 2003) since their soft modes are described by the continuous rotations of a director, rather than by discrete crystal symmetries. We sketch the philosophy of the quasi-convexification of nematic elastomers and then examine the response of samples with practical geometries.9 It is difficult to represent the free energy density, even schematically, since it is a function of eight variables (when deforming at constant volume). We attempt this in Fig. 8.9 where we display a free energy density as a function of externally applied extension λ in any of the two directions perpendicular to the initial n (but with no shear). The energy has a central minimum at (λxx , λyy ) = (1, 1) representing no distortion, F = 0 on ignoring the 3µ /2 constant. Without shears, we saw in Fig. 6.10 that the free energy √ rises on distortion as it would in a classical elastomer. At simple extensions of λ = r the free energy density is again naturally minimal, making contact with the large space of soft deformed states which are generally of greater complexity than these simple extensions and contractions. Between the origin and the simple extension of √ r, there is a finite energy cost F > 0 since we do not allow shear in this scheme. It would be obvious in a depiction of F in higher dimensions that there are soft routes around this barrier, the cartoon of Fig. 7.3 offering one of an infinity of such routes that require shear. One can therefore replace this interval of concavity in F by FQC = 0 as in eqn (8.26). This introduces inhomogeneous microstructure of no elastic cost, as we have seen in the previous sections. Our earlier, simple examples of convexification 9 We

are grateful to A. DeSimone and S. Conti for their help in the material of these sections.

206

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F

0.5 0.25

l xx

0 1

l

l yy (a)

1.2

-0.25

1.4 1.6

-0.5

l xz (b)

F IG . 8.9. Schematic of the free energy density F against (a) simple extension in two directions perpendicular to n , with no shear. At the√centre is the minimum associated with no distortion. The minimal values at λ = r corresponds to soft distortion o associated with director √ rotation by 90 . The concave interval of F connecting the centre and the ring at r can be flattened to zero by quasi-convexification via the creation of microstructure. An example of a continuous soft path around the barrier was given in Fig. 7.3. (b) Soft paths using the component deformations of eqn (8.19) are shown in a plot (S. Conti) using λ and λxz as variables (with the other in-plane strain 1/λ not shown). Two dots and the connecting path correspond to the λxz (λ ) curve in Fig. 7.8(b). corresponded to traversing along one axis only in Fig. 8.9. More complex geometry, including shears induced in more than one direction, is required to quasi-convexify F √ in all directions. Outside an ultimate distortion (at most an extension of r applied perpendicular to the initial n , and less for oblique directions) nothing more can be done – the director is fully aligned with the direction of principal stretch and no mechanism of soft deformation now exists: this elastically hard region is convex and cannot be quasi-convexified away. This is represented by the large strain regions of Fig. 8.9, but is of course much more complex in eight dimensions. The need for the most general quasi-convexification can be seen from Fig. 8.7. With extension and transverse contraction in the bulk of the sample, but not near the clamps, the sample develops curved edges and the local principal stretch is not uniformly along x; examples of where this is so are indicated by arrows on the figure. In some regions, especially near the clamps, it may be impossible to find soft combinations of distortions at all – the free energy density may then be quartic or even harder, for instance if λzz is constrained to be λzz ∼ 1 near a completely rigid clamp. On the other hand, the obliquity of the local principal stretching direction may require shears for softness that do not demand inhomogeneous microstructural variation at all: there could be regions of soft response without stripes. One can see that the precise pattern of deformation depends on the macroscopic shape of the sample, in particular its aspect ratio (length to width ratio). Depending on this shape, oblique stripes will occur in different places at different macroscopic extensions (externally applied λ , distinguishing these from the local extensions in the

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207

material) and that the stripe pattern will shift spatially as extension continues. For instance, Fig. 8.10 shows an enhanced view of the clamp region of an elastomer that was stretched√well beyond the end of the soft plateau denoted by λ2 , which was determined as λ2 = rλ1 in Sect. 7.5; however, there are several regions where the stripes remain (seen as white areas, strongly scattering light, in contrast to the transparent areas of uniformly aligned nematic director). If there is a continuous path of regions across the sample that can deform softly, then they will do so first until sufficient are exhausted that the path of soft regions across the sample is broken. The macroscopically soft response will harden due to the non-soft regions through which stress can now pass. Thereafter, other regions of the sample that did not initially have softness available to them will also start deforming. In doing so there is a change of geometry and it is possible that some other regions can then start deforming softly while others are deforming non-softly. Hard and soft deformations can coexist whenever there is a path of non-soft deforming material along the sample so that there is an ultimate continuity of force transmitted along the sample. 8.3.3

Numerical and experimental studies

A numerical solution of strip elongation (Conti et al., 2002) reveals the non-uniformities that arise because of the clamp constraints and which are intensified when soft deformations in the bulk come to their end. Figure 8.11 shows the force against deformation for an ideal elastomer. The ‘affine curve’ corresponds to a model case where no clamp effect is exerted on the stretched elastomer sample. Naturally there is no force until dir√ ector rotation is complete at λ2 = r with θ = ±π /2, depending on which stripe one is in. The clamped sample with aspect ratio of 3 can get closer to the ideal soft cut off than that with the aspect ratio = 1 (the square shape), since the former has a relatively smaller volume fraction influenced by the constraints exerted by the clamps. A salutary lesson emerges – the apparent length of the soft region is a function of the macroscopic aspect ratio of the sample. Fortunately experiments discussed before, Sect. 7.4.2, were carried

F IG . 8.10. (a) The sequence of images illustrating the stretching of an elastomer film. The middle image shows the sample with stripe domains strongly scattering light, the bottom images shows the sample beyond the soft plateau, still retaining scattering regions near the clamps. (b) The expanded image of the clamp region (at λ > λ2 ), showing the complicated pattern of areas with local stripe microstructure.

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out on samples with large aspect ratios, long rubber strips with aspect ratio ∼ 10-12. This sensitivity to aspect ratio was explored in experiments (Zubarev et al., 1999) and revealed a spatial distribution of microstructure that depended on strain differently in samples of different aspect ratios. As we have seen in Fig. 8.10, when samples are stretched beyond the onset of hard response, λ > λ2 , the stress patterns are non-trivial, especially near the clamps. Figure 8.12 shows larger stresses diagonally towards the corners and much less in the central clamp regions, as anticipated in Fig. 8.7. The macroscopic shape is also rather different from that of a classical elastomer undergoing the same macroscopic extension. The edges near the clamps tend much more directly and with less curvature to the central, straight region. This is a consequence of the microscopic constitutive relation and its macroscopic quasi-convexified form (Conti et al., 2002). Inhomogeneous microstructure is evident not only by direct microscopic observation, but also through X-ray scattering. Since the beam area is generally large compared with the width of individual stripes, of several microns, both nematic directions are revealed in the somewhat averaged picture. Figure 8.13 shows X-ray patterns taken from different regions of a sample at a fixed extension. Central regions of the rubber strip (points C2 and C3 ) already have their director rotation complete. In the fringes of the clamp region, where microstructure still exists, point C1 , the rotation within stripes is not quite complete and so the maxima associated with the directors in neighbouring stripes are not quite coincident, leading to an apparently broadened nematic azimuthal distribution of scattered X-ray intensity. The extreme regions A and E along the rigid clamp have the directors pointing along the maximum extension directions, that is towards the corners and there is no microstructure. At the middle-point C, there are still fully developed stripes with directors at ±π /4 to the extension direction, x, and thus four azimuthal maxima (see for comparison Fig. 5.20). The X-ray pattern is that of two nematics with orthogonal directors. B and D represent more oblique regions. The

F IG . 8.11. Force against extension calculated numerically (Conti et al., 2002) for an ‘affine’ sample, with no clamp effect, and for clamped samples of aspect ratio (AR) of 3 and 1, respectively. The anisotropy parameter was taken r = 2. The two arrows point at the specific extensions examined in Fig. 8.14.

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209

F IG . 8.12. Spatial distribution of stress is indicated by levels of shading in a sample with r = 2, of initial aspect ratio = 3, stretched to λ = 1.38, see Fig. 8.11. distribution of extension and microstructure emerges from the numerical solution of the problem. Figure 8.14(a) shows the microstructure of the top right hand quarter of a stripe with the type of patterns developing at different places shown as insets. The deformation is still soft overall (determined by the stress needed to deform the central region). Closer to the clamp there is reaction but still not hardness at this value λ = 1.31, see Fig. 8.11. Beyond the hard threshold at λ = 1.38 , see Fig. 8.14(b), the director rotation at points along the centre of the sample (bottom of the figure) is as found in experiment, Fig. 8.13. Where the director rotation is complete, that is at θ = ±π /2, one might expect a homogeneous director distribution since in a nematic θ = π /2 is equivalent to that at θ = −π /2. However, neighbouring stripes are separated by a narrow inhomogeneous wall that is drawn in Fig. 8.6 and such walls separate the otherwise identical regions of θ = ±π /2. A uniform nematic texture can be made to appear completely dark under suitably oriented cross polarisers. With a dark background, any possible remaining regions of deviating director will appear very bright and will be detected with great sensitivity. One could indeed see bright, unresolvably thin lines where the separation A B C D C3

C2 C 1

E

F IG . 8.13. Different labelled regions of an extending nematic elastomer with their correspondingly labelled X-ray patterns, indicating the local director orientation, after (Zubarev et al., 1999).

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F IG . 8.14. Numerical calculations of the distribution of microstructure in a sample of aspect ratio = 3, stretched to (a) λ = 1.31, and (b) λ = 1.38, these strains being labelled in Fig. 8.11. Levels of shading indicate the extent of microstructural development. Director rotation in the stripes at the central clamp region is less than in the bulk which in (a) is undergoing essentially unconstrained soft deformation with the expected director rotation. In (b) the rotation in the bulk is clearly complete and the director apparently uniformly points along x. walls between stripes used to be. These were probably the interfaces of Fig. 8.6 with the director traversing between the equivalent states θ = ±π /2 over a short distance, w ∼ ξ , this volume of sample then not being dark under the polarisers. The effect of initial sample aspect ratio has been seen in the force extension curves, Fig. 8.11, and has also been visualised in experiment. In a square sample (AR=1) one first finds pronounced stripe regions in the middle (the region of highest local extension), which then grow and migrate across the sample towards the clamps, as strain develops (Zubarev et al., 1999). This is qualitatively the sequence of results seen in the numerical solution (Conti et al., 2002), of which Fig. 8.14 (a) and (b) are examples at two strains. 8.4

Random disorder in nematic networks

It has been recognised long ago that, if no special precautions are taken to preserve the monodomain director alignment in the network, the liquid crystal elastomers always form with a highly disordered director texture, cf. Sect.5.3. In this field such a texture has been historically called ‘polydomain’ although this might be considered a misleading term: the system does not have pronounced uniform domains separated by accentuated walls, as for instance in ferromagnets or polycrystalline solids. This disordered texture may appear similar to liquid crystalline polymers and low-molar mass liquid crystals, where the quenching from isotropic to the nematic phase results in a dense defect texture (called Schlieren in nematics), which then coarsens with time. This evolution of topological defect structure has led to the famous analogy with the cosmological

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211

theory of the early Universe undergoing symmetry-breaking transitions (Chuang et al., 1991). It has been exhaustively investigated in liquid crystals as well as in other ordered media (Volovik, 2003) and also reviewed in the context of mixtures (Bray, 2002). The major difference with crosslinked liquid crystal elastomers is that their disordered director textures represent a true thermodynamic equilibrium. If such a material is made uniformly aligned by application of an external field or mechanical stretching (what we call polydomain-monodomain transition, cf. Sect.5.3), it always returns back to its disordered state after the external aligning influence is removed. If a disordered, polydomain nematic rubber is heated to the isotropic phase, on return to the nematic state the texture is re-established with the same characteristic length scale (the ‘domain size’, ξD ). This feature is preserved for the networks that are crosslinked in the high-temperature isotropic state and then cooled into the nematic (or other) liquid crystal phase. Therefore, it is not the case that certain topological defects are permanently crosslinked into the network – the texture forms spontaneously: the system ‘knows’ the precise length scale, i.e. the size of the regions in which it can afford having the aligned director, and even how this size reversibly changes with temperature (Elias et al., 1999). Many experimental observations report that this domain size ξD is of the order of microns, so each small aligned region contains many crosslink sites. (The average distance between crosslinks is only a few nanometers in a typical rubber. The simplest argument, not even involving the particular chemistry, is about the rubber modulus: to achieve the typical µ = ns kB T ∼ 105 Pa at room temperature (4 × 10−21 J), one needs to have cross−1/3 ∼ 4 nm.) links separated by the distance ns A concept of quenched sources of random disorder and their influence on the overall structure has been put forward long time ago (Larkin, 1970; Imry and Ma, 1975) and fruitfully explored in many areas of modern physics, in particular, spin glasses, superconductor vortex lattices and neural networks, see e.g. (Mezard et al., 1987; Dotsenko, 1994). The basic result is the destruction of long-range correlations in the ordering field, which nevertheless remains locally strong and well defined: the system breaks into uncorrelated domains and has no overall order or alignment. Applying the knowledge accumulated in this field to nematic systems, which differ from magnets by their quadrupolar ordering symmetry, one must identify quenched sources of random orientational disorder. Another related productive area is that of liquid crystals confined in randomly porous media, for example, in silica gels (Bellini et al., 1992). Here the source of disorder stems from the director anchoring on random interfaces, which is strong and has its own (rather large) characteristic size. In nematic elastomers, the microscopic crosslinks could be the entities carrying the quenched orientational disorder. Figure 8.15(a) illustrates this idea: even for idealised point crosslinks one can always identify the direction of anisotropy, which is quenched since the crosslink is not totally free to rotate under thermal noise (Terentjev, 1997). In practice, crosslinking agents are always anisotropic and frequently deliberately made of mesogenic rods themselves. Freezing their orientation by linking to several tethered chains would restrict their orientational freedom, but their coupling to the surrounding mesogenic units would of course remain in force. In crude terms, one can say that each such crosslink adds an energy ∼ − 21 g k · Q · k to the system Hamiltonian, where g is

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xD (a)

(b)

(c)

F IG . 8.15. (a) Schematic of the orientational effect of network crosslinks. (b) A computer-simulated image of the polydomain nematic elastomer; the contrast represents the difference in director orientation and indicates the length scale ξD , from (Uchida, 2000). (c) A polydomain texture image between crossed polars (the bar = 10 µ m), from (Elias et al., 1999). the coupling strength, k the local orientation of the source and Qi j is the nematic order parameter at this point in space (Fridrikh and Terentjev, 1999). Adding all the crosslinks together, introducing their continuum density ρ (rr ) = ∑x δ (rr − r x ) and separating the magnitude of local nematic order from its principal direction n , one obtains the random field contribution to the nematic free energy density FRF = −



2 3 1 r k 2 gQρ (r ) (k · n ) d r .

(8.27)

Here both the density of sources and their orientation are the randomly quenched variables. When points are uniformly distributed in space their density adopts a Gaussian distribution: 5 4  [ρ − ρ0 ]2 , (8.28) P[ρ (rr )]  exp − d 3 r 2ρ0 where ρ0 is the mean density of impurities (Edwards and Mutukumar, 1988), in our case proportional to the crosslink density ns . As the defects are at the same time randomly oriented in three dimensions, we have the probability of their orientation as simply P[kk ] = 1/4π . Obviously, when a monodomain elastomer is prepared by crosslinking in the aligned state, the distribution P[kk ] would become increasingly biased, so the crosslinks would then support the equilibrium macroscopic alignment, as opposed to disrupting it in the disordered case. 8.4.1

Nematic ordering with quenched disorder

How can we describe nematic ordering in a system with quenched sources of orientational disorder? The problem has many parallels with magnetic ordering in a lattice doped with impurity atoms that maintain rigid, and random, orientation of their magnetic moments (‘random-anisotropy magnets’). Such systems are thoroughly studied in the modern theory of spin glasses. However, in the nematic case the situation is aggravated by the first order (discontinuous) nature of the phase transition dictated by the quadrupolar symmetry of the order parameter Qi j , see Sect.2.3. We have seen in the review of experimental results, in Fig. 5.6, how the same nematic polymer, exhibiting the expected jump in the order parameter on the transition

RANDOM DISORDER IN NEMATIC NETWORKS T3 T2 T1

F

T4

F

213

T3 T2

0.05

0.1

0.15

0.2

0.25

0.3

Q

(a)

(b)

T1

Q 0.1

0.2

0.3

0.4

F IG . 8.16. Plots of free energy density against Q for several values of temperature: (a) at very weak disorder, still preserving the features of isotropic-nematic transition; (b) at higher values of ns g2 , reflecting the continuous nature of transformation between the low-Q and the high-Q states. Dashed lines are the classical and the rendom-disorder portions of F(Q). into nematic phase, suddenly appears to have a continuous variation of order through the transition when it is crosslinked. Many studies, e.g. Fig. 5.12, suggest a continuous, apparently critical transition, and also a number of studies report a completely diffuse, non-critical change of Q(T ). Experimentally it is very hard to measure the nematic order in a naturally polydomain state, NMR being one of the very few techniques available. However, to date, most experiments have been performed on permanently aligned monodomains – in which case the transition is certainly affected by the internal field and should be in the supercritical regime (Lebar et al., 2005). Simulations (Pasini et al., 2005) and theory of quenched disorder (Petridis and Terentjev, 2006a) predict different regimes of the nematic transition affected by quenched disorder. In theory, the homogeneous in space, nematic order parameter Q is set in the system, while the principal axis of Q (the director n ) follows an equilibrium randomly quenched texture with the characteristic size of correlated regions ξD . The continuum Landau-de Gennes free energy density, eqn (2.11), gets an additional Q-dependent term. In the limit of small Q and weak strength of disorder, this renormalisation leads to F ≈ 12 AQ2 − 13 BQ3 + 14 CQ4 +

Vans Γ , 8π 2 K 3

(8.29)

where ns is the density of crosslinks, a is the small-distance length scale cutoff (crudely, a monomer size), and Γ ≈ (g2 /kB T )Q2 ) is a measure of disorder strength. At Q  1 this addition scales as Q−4 and evidently prevents the system ever achieving a state with Q = 0, Fig. 8.16. Nevertheless, depending on the strength of disorder, one may still see a sharp transition between the free energy minima, or a continuous transition that may resemble supercritical behaviour. 8.4.2

Characteristic domain size

In the continuum theory, any deviation of nematic director from a uniform orientation is penalised by Frank elasticity. The random field energy density, eqn (8.27), has to be added to it and together these two terms capture the main features of systems with quenched weak orientational disorder

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FFr + FRF = 12 K (∇nn)2 − 12 g(Q)ρ (kk · n )2 ,

(8.30)

where K is Frank elasticity constant in one-constant approximation and g(Q) is a coupling constant depending on the local nematic order parameter. For the free energy containing these two terms one can now apply the famous ImryMa argument (Imry and Ma, 1975). For a texture with the correlation distance ξD we can estimate the gradient ∇nn as ∼ 1/ξD , immediately getting an estimate for the Frank energy (per domain of the size ξD ): FFR ∼ KV ξD−2 ∼ K ξD , when the volume is V ∼ ξD3 in the normal three dimensions. The number of sources of random anisotropy Nx in such a domain is proportional to the domain volume, Nx  ρ0 ξD3 . To minimise the free energy the system will tend to have nematic director parallel to the direction most of the crosslinks (vectors k ) within this volume are parallel to. To visualize this picture, one can take all k i and make a chain from them by connecting them head to tail. The end-to-end vector of such a ‘chain’ will give the preferable alignment direction for the nematic director. This is the same construction as that of an ordinary random walk, Fig. 3.2, and we are looking for the net value of the orientational field. The mean square 1/2 3/2 length of this end-to-end vector will be proportional to N 1/2  ρ0 ξD . This value indicates the magnitude of the mean field of quenched sources, averaged over the chosen domain volume. Thus the excess of the crosslinks looking in the preferable direction will be of the order of N 1/2 and the system will gain a random-field energy of the order of FRF  −g (ρ0 ξD3 )1/2 by aligning in this direction. Hence the domain free energy FFr + FRF can be estimated as: 1/2

3/2

F  K ξD − g ρ 0 ξ D .

(8.31)

Free energy density (a.u.)

Minimising this with respect to the correlated domain size (∂ F/∂ ξD = 0) we obtain the characteristic length

Frank Total

RF xD

Possible domain size x D

F IG . 8.17. Plots of free energy density contributions (eqn (8.31) divided by the domain volume ξD3 ) against the size ξD . Frank elasticity wants the regions of uniform alignment be as big as possible; the random-field effect would prefer the domains to be as small as the source separation. The balance between these two conflicting demands results in the optimal, thermodynamically equilibrium size ξD .

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215

F IG . 8.18. Measurements of the characteristic texture size ξD . (a) The data extracted from light-scattering studies (Clarke et al., 1998) on a siloxane side-chain elastomer with relatively low Tni ; (b) Polarised microscopy on a main-chain material, cf. (Elias et al., 1999). In both plots, open symbols show the evolution of ξD on heating towards Tni ; filled symbols cooling, clearly indicating the equilibrium nature of ξD . The lines show a power-law fit with ξD ∝ |T − Tni |−1 .

ξD ∼ K 2 /ρ0 g2 .

(8.32)

Figure 8.17 illustrates the energy balance argument pictorially. This simple scaling argument is known to be very robust and is confirmed by explicit calculations within several different models by many authors over the years. It shows that no matter how weak is the random disorder, it will eventually win at sufficiently long distances and destroy the long range correlation of director orientation, breaking the system into the misaligned nematic regions of size ξD . One needs to emphasise that, similarly to the analogous effect in spin glasses and other physical systems with quenched disorder, the structure is not one of uniformly aligned domains with sharp boundaries (as the word ‘polydomain’ might imply). ξD represents the size of correlated regions within which the local nematic order Q is high and the director n is more or less aligned. The resulting equilibrium texture is characterised by rapidly decaying correlation of orientation, crudely nn(rr ) · n(rr  ) ∼ exp [−|rr − r  |/ξD ]. It is most important to realise that this length scale is macroscopic, although the average distance between quenched sources of disorder (e.g. crosslinking points) could be measured in few nanometres. At this time, there are only few direct measurements of characteristic domain size of randomly disordered (polydomain) nematic elastomers. It appears that in various different systems and materials this length scale is, roughly, between 0.5 and a few microns. Accordingly, such materials scatter light very strongly, between the misaligned regions of very high birefringence. Figure 8.18 shows how this size ξD evolves with temperature, as the local nematic order Q decreases. The temperature dependence of local order parameter Q(T ) has not been measured in these experiments (it is, generally, very difficult to access – perhaps only by NMR), but we could, as a guess, consider the order parameter measured for similar monodomain elastomers, see Fig. 6.1, giving

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Q ∝ |T − Tni |0.3 . Mapping ξD (T ) onto Q(T ) and assuming the power-law fit of the data in Fig. 8.18, this leads to a scaling ξD ∝ 1/Q3 . If one safely assumes the Frank constant varies as K ∝ Q2 , this very crude discussion and the Imry-Ma estimate eqn (8.32) suggest that the random-field coupling constant should be g ∝ Q3.5 , or a similarly high power. This, of course, is in contrast with the first-order approximation of eqn (8.27), indicating that much remains unknown and unclear about the nature of nematic coupling to sources of quenched disorder. A more recent study of the spin-glass ordering of nematic elastomers, using more advanced replica symmetry breaking methods (Petridis and Terentjev, 2006b), predicted a more accurate dependence of the characteristic length scale ξD on the nematic order. It turns out that director correlations decay differently depending how the separation |rr − r  | compares with the length scale ξD : 4 exp(−r/ξD ) , for r  ξD nn(0) · n (rr ) ∼ . (8.33) (r/ξD )−1 , for r  ξD Far from the isotropic phase, and when the disorder is sufficiently strong, the expression (8.32) is a valid limiting case for ξD . However, near the transition (at Q  1) as well as at ρ0 g2  1 this size takes on a different scaling limit:

ξD ∼ (K 2 /ρ0 g2 )e2kB T a/π

2K

2

∝ Q2 e1/Q .

(8.34)

This estimate is taken under the more logical assumption that g ∝ Q, see eqn (8.27). Here a is the continuum-limit cutoff scale, of the order of the mesogen size that enters the dimensional estimate of the Frank constant, K ∼ (kB T /a), see section 2.5. Evidently this ‘domain size’ ξD diverges as Q → 0, as the experimental data in Fig. 8.18 indeed demonstrates. 8.4.3

Polydomain-monodomain transition

Another interesting fundamental feature of disordered, polydomain nematic elastomers, also important practically, is their alignment under applied uniaxial stress. Again, there is a direct analogy with alignment of random-anisotropy spin glasses in a strong magnetic field and frustrated vortex lattices under the influence of underlying crystalline order (Chudnovsky et al., 1986; Emig and Nattermann, 1997). The basic result says that, on increasing the external field, the globally misaligned texture orders in the direction of the field – via rotating the individual domains towards a common axis, Fig. 8.19(a) (rather than, for instance, changing of relative size of correctly and wrongly aligned domains). There are fundamental and as yet unresolved differences between theoretical models as to whether the transition itself is critical or continuously diffused (Fridrikh and Terentjev, 1997; Emig and Nattermann, 1997), but the essence of the phenomenon is more or less clear. The case of elastomers is special because when each nematic domain attempts to rotate, there are associated shape changes – which we extensively discussed in Chapter 7, for example see Fig. 7.3. Of course, the reason for each region of uniformly aligned director (of size ξD ) to rotate towards the extension axis is the attempt to follow the low-energy soft deformation path. (This is the reason for an extended soft plateau in

RANDOM DISORDER IN NEMATIC NETWORKS

217

F IG . 8.19. (a) A sketch of the polydomain-monodomain transition under stress, compare with Fig. 5.8 and the discussion in Sect. 5.3. (b) A sketch of mechanical incompatibility between neighbouring domains rotating in a different way: the mismatch between the parallel and perpendicular dimensions makes the previously broad intermediate region localise into a narrow domain wall w. stress-strain curves during the polydomain-monodomain transition, cf. Fig. 5.9.) However, the neighbouring domains have their own, different rotation to perform and their resulting shape changes could be incompatible, see Fig. 8.19(b). A simple argument gives an estimate of the additional mechanical energy penalising this incompatibility and localising the remaining director distortions in narrow walls. Let us say that under the extension ε , the domain B in Fig. 8.19(b) cannot rotate softly and thus will generate the elastic energy of the order of 12 µε 2 ξD3 , in proportion to its volume. The wall region of the width w will be deformed as well, with the corresponding elastic penalty 12 µε 2 ξD2 w. However, the domain A may be able to deform softly, with director rotating by up to 90o , with ideally no elastic energy raised in its volume. The mismatch between the two domains (the difference between their natural dimensions along and across n ) will cause additional shear

deformation ε˜ inside the domain wall which can be estimated as (ξ − ξ⊥ )/ξD  (  /⊥ − 1), with ξ and ξ⊥ the natural dimensions along and across the local nematic director: the aspect ratio r =  /⊥ is reflected in the local shape of the network and provides the geometric mismatch. Hence √ the additional strain in domain walls is√(ξD /w)( r − 1) and the corresponding energy cost in the wall of volume ξD2 w is 12 µ ( r − 1)2 ξD4 /w. Thus the elastic free energy per unit volume can be expressed as ∆F  12 µε 2

√ w ξD 1 2 + µε . + 12 µ ( r − 1)2 ξD w 2

(8.35)

The first part in eqn (8.35) favours a smaller width w, thus allowing more volume to be taken by the soft domain A. The second part has the opposite effect, tending to increase the width w over which the newly generated size mismatch has to be accommodated (the third part is simply the energy of stretching the domain B). Optimizing with respect to w gives the equilibrium domain wall width:

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DISTORTIONS OF NEMATIC ELASTOMERS

w∗ =

ξD √ ( r − 1), ε

(8.36)

which decreases with growing applied strain ε (wall localisation). This feature is very similar to what we saw in eqn (8.16) for the interface width of stripe domains. Note that the chain anisotropy (r − 1) is proportional to the nematic order parameter √ Q. Substituting (8.36) into eqn (8.35), we obtain an additional energy term ≈ µ ε ( r − 1) responsible for the localisation of domain walls. This energy density, linear in macroscopic strain, should be added √ to the external stress term −σ ε , providing the value # ≈ σ − µ ( r − 1). This shift can be substantial and it usually deof effective stress σ termines the value of the threshold stress σc for the polydomain-monodomain transition (Fridrikh and Terentjev, 1999). Theoretical calculations also show how the macroscopic order increases in the system with increasing external stress. As the domains become more and more aligned, the overall can be characterised by an alignment parameter Smacro =   nematic alignment  Q V1 d 3 r 32 cos2 θ − 12 , where Q is the (assumed constant) local nematic order and θ the angle between an individual domain orientation and the direction of applied stress. The parameter S may appear as the genuine nematic order parameter to an observer who is unaware of the spin-glass like structure of locally well-ordered polydomain nematic elastomer. As we have already mentioned, the transition point at a critical stress σc is poorly described by the theories, but the subsequent evolution of S = S(σ ) is predicted as a sharp approach to saturation (Fridrikh and Terentjev, 1997): 4 Smacro  so + (Q − so ) exp −

5 const . (σ − σc )1/2

(8.37)

Macroscopic alignment S

The small constant so accounts for the jump at the vicinity of the transition point, which is only described by a more complicated analysis of replica symmetry breaking

Nominal stress (kPa)

(a)

Nominal stress (kPa)

(b)

Nominal stress (kPa)

(c)

F IG . 8.20. The onset of macroscopic alignment Smacro during the polydomain– monodomain transition, for: (a) main-chain epoxy elastomer, after (Ortiz et al., 1998a), (b) siloxane side-chain network, after Clarke et al (1998) and (c) acrylate side-chain elastomer, after Zubarev et al (1998). Solid lines are the fit to eqn (8.37). Note a very low critical stress for the acrylate material and the very similar values of underlying local nematic order parameter Q (the saturation level of Smacro ).

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219

F IG . 8.21. Stress-strain response of nematic rubber undergoing polydomain-monodomain transition under imposed uniaxial extension: (a) polyacrylate side-chain elastomer (Hotta and Terentjev, 2001); (b) epoxy main-chain elastomer with predominantly nematic order (Ortiz et al., 1998b). The soft stress plateau is evident; its approximate extent is marked by arrows. Note also how the plateau stress decreases as the elastomers are heated and the order parameter Q is reduced. or functional renormalisation group. At σ˜ → ∞ , the macroscopic order Smacro tends to Q, the thermodynamic value of the nematic order parameter in a fully aligned elastomer. In the above discussion, we have assumed that the material is given sufficient time to relax and reach its mechanical equilibrium. In some cases, this may be a condition difficult to implement in practice. The main-chain nematic polymers are known to exhibit particularly slow relaxation of stress and director orientation due to their hairpin nature. During the transition, the barriers between domains are high and at a high strain rate the system does not have time to find its equilibrium deformation. Instead the stress response rises high, as if no transition were taking place, and only at much higher deformations the domain are finally forced to rotate and soften the system. When the strain is imposed very slowly the transition proceeds much closer to the presumed equilibrium critical stress and the flat plateau, however, the study (Clarke et al., 2001) has still reported significant relaxation at strain rates as low as 6 × 10−7 s−1 . In practice, the soft plateau of the polydomain-monodomain transition (at σ = σc ) can span a great distance in strain, if the network chain anisotropy r is high – as in mainchain elastomers, see Fig. 8.21. The structural change occurring during this transition under an imposed uniaxial stress (or extension) is illustrated by the rise in the macroscopic alignment parameter Smacro , Fig. 8.20; the data is usually obtained from the analysis of azimuthal scans of wide-angle X-ray scattering. These plots show the results of different groups on very different nematic elastomer networks, yet the trend remains evident: initially there is no alignment (the domains are completely un-correlated), but after a clear stress threshold σc (which is temperature dependent and may take very different values in different materials) the rise in alignment is very rapid. Characteristically, all plots saturate at a very similar value Smacro = Q ∼ 0.5-0.6, which one would have expected of the ‘real’ (local) nematic order parameter.

9 CHOLESTERIC ELASTOMERS

Liquids of rod-like units, which possess a certain handedness in their shape or polarisability, spontaneously form cholesterics, or twisted nematic phases. The breaking of symmetry such that an object differs from its mirror-image (such as right and left hands) is called chirality. Liquid cholesterics were reviewed in Sect. 2.8. Chiral, nematogenic polymers also form cholesteric phases. Crosslinked networks made from such polymers accordingly have a spontaneously twisted director distribution; they are noncentrosymmetric in their symmetry and possess corresponding physical properties, such as the piezoelectricity. Their director distribution being helical with characteristic length scale in the optical range, they display their chirality in their brilliant colours. Liquid nematics respond to electric and magnetic fields, while nematic elastomers are most sensitive to mechanical deformations. We shall see that cholesteric rubbers respond to imposed mechanical fields too since the matrix that we distort, and also let relax, is coupled to the director distribution. The periodic non-uniformity of the initial director texture makes the mechanical response complex; elastic softness is only partially achieved. After deformations, the modified director distribution is usually no longer simply helical and the characteristic length scales can be modified too. We shall describe here some peculiar chiral-mechanical effects leading to changes in optical and photonic properties. Pieces of cholesteric rubber change colour on extension since their helical pitch changes. They can act as tunable lasers, which change their emission wavelength in response to strain. New photonic bandgaps open up in certain geometries of deformation. On large stretches, the characteristic cholesteric colour is essentially lost due to mechanical alignment of the director. This loss requires a topological barrier be surmounted, a problem not encountered for a liquid cholesteric in an external field where twist can be lost continuously by period lengthening. Another possibility not open to a liquid is ‘topological imprinting’ of the director distribution. Being anchored to the rubber matrix, it can remain helical even when all chiral species are removed from the elastomer. This can happen if the twist was merely provided by a chiral dopant present at crosslinking and then later removed. Internally stored mechanical twist can cause cholesteric elastomers to interact selectively with solvents according to its imprinted handedness. Separation of racemic mixtures, or stereo-selectivity, can then arise as a mechanical elastic effect. We review the properties of cholesteric elastomers. The remainder of this chapter then describes these many effects in terms of simple models of director rotation and strain accommodation in nematics, since locally cholesterics are essentially nematic. 220

CHOLESTERIC NETWORKS

9.1

221

Cholesteric networks

Crosslinking cholesteric melts or solutions of chains gives an elastomer with a chiral director pattern. If the chains contain chiral (handed) centres themselves, then the driving force for twist remains and is reinforced by the permanent linking of chains which tend to anchor the director pattern to that distribution pertaining at the moment of formation. We call these intrinsic cholesteric elastomers. The alternative is to link ordinary nematic chains in a presence of a chiral dopant (a solvent) which induces the director twist. After linking and thus creating a memory of the director distribution, one can remove the chiral solvent to leave behind only centrosymmetric molecules, albeit in a body with a possibly non-centrosymmetric (helical) director distribution. We call these imprinted elastomers. Each type will be separately discussed since they present rather different phenomena. 9.1.1

Intrinsically chiral networks

The nematic networks we described in Chapter 5 are all susceptible to the induction of cholesteric twist. Side-chain or main-chain rods order nematically and if a proportion of them are chiral, then twist follows. Indeed, the polymers first crosslinked to form nematic elastomers (Finkelmann et al., 1981) were in some cases copolymerised with chiral groups so that cholesteric elastomers were also accessible. They had the motherof-pearl and shimmer characteristic of their linear and low molecular weight counterparts. These were siloxane side chain polymers; polyacrylate cholesteric materials were made shortly after (Freidzon et al., 1986; Shibaev et al., 1989). One can also make combined main chain-side chain cholesteric polymers and elastomers, for instance with chiral polyesters (Zentel et al., 1987; Bualek et al., 1989). In many cases smectic C* phases also emerge – see Chapter 12. Cholesteric elastomers were seen to be fundamentally different from nematics. They were polydomain, but required huge imposed strains to reach a state with uniform director. Typically external strains of 30-100% are required for aligning nematic polydomains whereas strains upward of 150% were required for analogous cholesterics (Zentel, 1988). Not only do domains require alignment, but also the director distribution internal to each domain must be overcome if one is to achieve uniform alignment. As we shall see in Sect. 9.2, higher strains are required for global alignment because the director is much more tightly constrained in a cholesteric than it is in the case where there is simply a nematic domain structure. Compressional strains on a cylindrical sample of cholesteric elastomer can be most effective in realigning cholesteric polydomain texture so that all the helical axes point along the compression axis (Meier and Finkelmann, 1990). For locally prolate nematic chain order, directing the helix axis pˆ along the compression axis presents, at every point along the helix, the smallest chain dimension to the compression axis, thereby lowering any elastic strain cost, Fig. 9.1. While domains reorient, stress is very low in both elongation and compression. This is reminiscent of the low stress seen when aligning polydomain nematic elastomers and goes back to soft elasticity. When the sample has its helix axis aligned, stress rises with further increasing compression. From Fig. 9.1 one sees that once aligned by compression, further strain does not unwind the helical

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CHOLESTERIC ELASTOMERS

l zz

z

p y x

F IG . 9.1. A compression λzz = λ applied to a cholesteric elastomer will ensure that the helix axis is along z if the chains are naturally prolate and extended along the local director. The shortest chain dimension is then presented to the sample compression axis. This is the probable route to helix alignment of polydomain cholesteric elastomers. structure, nothing soft can happen since no director rotation takes place, and stress must rise. We shall see, by contrast, that extension deformation is altogether more subtle. We shall exploit the ideas of soft response when analysing the mechanical and optical properties of monodomain cholesteric elastomers when director rotation arises. Monodomain cholesteric elastomers are now available (Kim and Finkelmann, 2001). If a strip of cholesteric polymer solution starts crosslinking and losing solvent at the same time, then distortions are imposed that get recorded by the crosslinking. Strips of crosslinking gel are typically held in band-shaped troughs in a centrifuge so that the rubber strip is formed with its outer flat surface stuck to a solid substrate by the centripetal acceleration. The other flat surface is free and thus able to expel solvent. Being stuck to the outer surface, the strip cannot change its two lateral dimensions in the plane to accommodate the loss of volume; only the thickness can change. These constraints in effect apply a large uniaxial compression (equivalent to a biaxial extension in the flat plane) and establish a unique helical axis along the strip normal. The strains are equivalent to those set up in compressing polydomain cholesterics to a helical monodomain, see Fig. 9.1, but the director distribution is permanently recorded by the concurrent crosslinking. On removal from the substrate, perfect cholesteric monodomains of free standing elastomers emerge. As for nematic elastomers, this advance in preparation techniques has led to huge progress in the understanding of cholesteric rubber elasticity, phase behaviour and photonics. Elastomers do not need any mechanical support (e.g. glass plates) to retain a stable liquid crystalline structure. One can modify their helicoidal structure with mechanical fields. Their optical properties respond accordingly – mechanically induced colour change, shifts in lasing colour from these elastomers and modifications of the photonic band structure are three examples we shall mention. Thus the advent of monodomain cholesteric rubbers is leading to subtle and beautiful applications. 9.1.2

Chirally imprinted networks

In a 1969 paper entitled “Possibilit´es offertes par la reticulation de polym`eres en presence d’un cristal liquide”, de Gennes proposed setting up liquid crystalline order in

CHOLESTERIC NETWORKS

223

networks by dissolving conventional polymers in a liquid crystal phase of the appropriate type.10 Crosslinking then imposes a permanent topology on the chains which may then imprint a memory of the liquid crystal order into the network that survives the subsequent removal of the solvent. The memory effect was dismissed (de Gennes, 1969a) in the nematic case; an ideal Gaussian network relaxes away any memory of chain shape anisotropy, as has been illustrated in exercise 6.2, and the network has an isotropic distribution of chains at high temperatures. For this reason ideal nematic networks are soft, even when crosslinking was in the nematic state. Cholesteric elastomers are different. Mechanical relaxations associated with chains losing shape anisotropy in neighbouring planes along the helix must be compatible with each other, a restriction which impedes chains finding an isotropic shape, even in the absence of nematic order, see exercise 9.1 below. An intermediate chiral imprinting strategy is to take nematic polymers with the addition of chiral dopant that induces a cholesteric texture and then crosslink them to topologically imprint the texture. Now the removal of chiral dopant leaves behind local nematic order and, possibly, also strong cholesteric order (rather than just a chiral pattern of chain backbone shapes). Monodomain cholesteric networks have been made from a composite of both rodlike synthetic polypeptides and acrylate polymers (Tsutsui and Tanaka, 1981). The former are chiral and introduced a twisted nematic structure in the network that persisted even after this component had been chemically removed. The surviving twisting power of the elastomer network was lost in the dry state (at least on the length scale of light), but could be recovered when it was re-swollen by simple solvents, suggesting that the fixed topology of the chains permanently recorded the chiral order. Nematic polymers induced into a cholesteric texture by a chiral solvent and aligned into monodomains by a magnetic field have been crosslinked to freeze in the helical director distribution (Hasson et al., 1998). The texture persisted on removing the chiral solvent. Evidence, Fig. 9.2(a), comes from apparent light absorption which has a peak (actually due to Bragg-like reflection) at a wavelength corresponding to the cholesteric pitch. The mixture before crosslinking and the crosslinked cholesteric gel (with its chiral solvent still in) are practically identical. The dry elastomer retains its locally nematic order and also its global cholesteric order, with a pitch simply shortened in line with the volume change associated with solvent loss (seen in the absorption peak shifting to lower wavelengths). This is not quite the imprinting as discussed by de Gennes since it is a director twist with a surviving underlying nematic order, at least at low enough temperatures. At the clearing temperature of the corresponding nematic elastomer, where the nematic order is lost, the absorption in the imprinted cholesteric elastomer is also lost. It shows that nematic order can be lost from the imprinted elastomer even within the constraints of twisting mechanical properties and, with this loss, one also loses cholesteric reflections. Figure 9.2(b) shows an alternative way of demonstrating the chiral imprinting via 10 A concept that was later productively exploited in a completely different context: in developing microphase separated, polymer-stabilised liquid crystal displays (Yang et al., 1992).

224

CHOLESTERIC ELASTOMERS

the measurement of rotation of linearly polarised light passing through the sample. In the initial state, the crosslinked network with the chiral dopant still inside has a high rotating power (as expected). To remove the dopant, the material is placed in a large volume of non-chiral solvent (the moment of time t = 0 in the figure). Highly swollen gel is not liquid crystalline and thus registers no optical rotation on the coarse scale of this plot. However, the inset shows how the optical rotation increases in the bulk of the solvent, as the released dopant concentration increases and then saturates – a much more sensitive method is needed to detect this! Finally, as the solvent is removed, taking with it all of the chiral dopant, the network dries and returns to the highly optically active cholesteric state, in which no chemically chiral entity is present. Retention of cholesteric twist below the nematic clearing temperature on removing the chiral agent could be surprising. The lack of chiral molecules means the system, if free, would no longer be spontaneously twisted; indeed the twist now costs an energy density 12 K2 (nn · curl n )2 since qo = 0 in eqn (2.44). Can the director untwist to relieve this Frank elastic cost? Elongations, contractions and shears accompany rotation of the director in the uniform nematic state. However in a cholesteric the ideal strains associated with rotation of n at one point differ in their principal axes from those at the next point, since the initial directors differ. The spatial incompatibility of the strains means that many components of λ cannot occur and thus reaching the uniform state can be costly. Thus the loss or retention twist after imprinting depends on the balance between the cost (K2 ) of retaining twist and the cost of undoing it, measured by the coupling constant D1 discussed in Sect. 7.1. We calculate the extent of topological imprinting of the director in Sect. 9.4. Another similar measure of imprinting occurs in intrinsically cholesteric networks. 10

(ii)

Optical rotation (deg)

Absorption

0

3

2 1

(a)

0 -0.1

-10

-0.2

-20

-0.3 0

5

10

15

-30

20

25

30

35

40

Time (min)

-40

(iii)

-50

(i)

-60 -70 -10

l (nm)

0.1

-5

0

5

10

15

20

25

Time (h)

(b)

F IG . 9.2. (a) Absorption spectra of (1) a mixture of nematic polymer with chiral dopant, (2) the crosslinked mixture with dopant still present, (3) the imprinted network with chiral dopant removed (Hasson et al., 1998). (b) Optical rotation of an imprinted cholesteric network in different states of chiral doping: the initial rotation by over 60 deg; at point (i) in time the large amount of non-chiral solvent is added to wash the dopant out (the inset shows the increasing rotation in the bulk of solvent induced by the released dopant); at time (ii) the solvent is removed and the remaining dry elastomer (iii) contains no chiral dopant, but still retains a substantial rotating power due to imprinted cholesteric order (Courty et al., 2003).

CHOLESTERIC NETWORKS

225

On temperature changes that cause a substantial pitch variation in a non-crosslinked cholesteric polymer melt, the corresponding network suffers essentially no variation, see Fig. 9.3 (Maxein et al., 1999). The twist is firmly retained. The inverse case arises when a nematic network, actually crosslinked in isotropic solution in the presence of a twisting agent, is cooled and dried (Meier and Finkelmann, 1990). A cholesteric elastomer results. Evidently, chiral guest molecules that remain lower the Frank energy by twisting. The network resists twist (the de Gennes D1 penalty) but some small twist must always be induced.

F IG . 9.3. Temperature dependence of selective reflection before and after crosslinking of intrinsic cholesteric polymers (Maxein et al., 1999). Selective reflections are lost at the point where the underlying nematic ordering vanishes and there is little local optical anisotropy. Finally, one can add the effect of imposed strain into the imprinting problem. Consider an elastomer that is a cholesteric monodomain at temperature T1 and heat it to T2 where it is isotropic. Then stretch it uniaxially and cool it back to T1 while in the extended state. On releasing the strain one should find a nematic monodomain elastomer, that is the twist has not been restored. There are two possible barriers to be overcome in order to restore the original helical state. 1. With a uniform nematic director field established by cooling while extended, the elastomer has to pay a D1 energy penalty to rotate the director with respect to the matrix to regain cholesteric order. 2. Restoring twist presents a topological problem. If for instance the director were anchored at the surfaces normal to the forming helix axis, then twist has to be established by the insertion of π -twist walls which are topological defects. One such wall is needed for each half period of the helix. The activation cost of this may inhibit helix reformation, even if the D1 energetic cost can be met. An analogous topological barrier is encountered in the sequence of stretched states of a cholesteric elastomer of Kim and Finkelmann (2001). The helix period diminishes as stretch λxx is increased – see Fig. 9.4 below where it is the transverse relaxation λzz that

226

CHOLESTERIC ELASTOMERS

shrinks the helix period. With this decrease comes a change in colour from red towards blue. Each increment in stretch is followed quickly by the change in colour. As we shall see in Sect. 9.2.1, eventually the twisted state is rendered unstable by the strain and the network eventually becomes an untwisted (transparent) nematic. The recovery of twist is also possible by simply heating the material to the isotropic state and then letting it cool without a strain. We now examine the second de Gennes’ topological imprinting case as an exercise. (Recall that the first case, exercise 6.2, was about the loss of nematic memory on heating.) The underlying drive to cholesteric ordering can be lost from a network with no imposed strain either because the cholesteric solvent is removed from a conventional network crosslinked in its presence, or because an intrinsic cholesteric network is heated above its clearing temperature. Associated with chains wishing to lose their anisotropic shape, there must be mechanical elongation λch along pˆ and contrac tions 1/ λch in the two directions perpendicular to pˆ . We discuss in Sect. 9.2 why there is no compatible variation of the perpendicular elongations with distance z along the helix, and why shears areprohibited. Alternatively, one can view the simple form λ ch = 1/ λch δ + (λch − 1/ λch ) pˆ pˆ as a consequence of coarse-graining to a state where pˆ is the only distinguished direction.

Exercise 9.1: Find the mechanical strains accompanying the loss of cholesteric order. What chain shape anisotropy survives into the orientationally isotropic state from topological imprinting in the cholesteric state (de Gennes, 1969a)? Solution: Since λ ch has cylindrical symmetry, without loss of generality one can take for the axes of λ ch the local principal axes of  o (z) = ⊥o δ +( o −⊥o )nno n o , that is the directions pˆ , no and k = pˆ × no , the latter being the vector orthogonal to n o in the plane  perpendicular to the pitch axis pˆ . The free energy density is 1 F = 2 (µ /a)Tr o (z) · λ Tch · λ ch where we have used that in the isotropic state  −1 = (1/a)δ . The matrices representing  o and λ ch are, locally: 

 o o =  0 0

0 ⊥o 0

 0 0  ⊥o

√ 1/ λ λ ch =  0 0 

and

0 √

1/ λ 0

 0 0 λ

(9.1)

whence F = 12 µ (⊥o /a) λ 2 + (r + 1)/λ . The free energy density is independent of z, given our assumptions on λ ch . It is minimised by λch 3 = 12 (r + 1). The Gaussian shape distribution of crosslinked chains on entering the isotropic RR  = 13 A L where the step-length state is characterised by the second moment R tensor was generally given in the solution to exercise 6.2 (the equivalent problem posed by de Gennes for nematic networks): A = λ ch ·  o (z) · λ Tch  1 r−1 1 2 = ⊥ δ + pˆ pˆ (λch − ) + no no λch λch λch

MECHANICAL DEFORMATIONS = ⊥

2 r+1

1/3



r 0 0 1 0 0

227

 0 . 0 1 (r + 1) 2

In the matrix representation of A the elements are in the order no , pˆ × no , pˆ . In r 0 0 the cholesteric state one had  o = ⊥ 00 10 01 . It is then clear that the first two (unequal) dimensions of  o shrink by the same factor, [2/(r + 1)]1/3 , and the third expands by [ 21 (r + 1)]2/3 on reaching the new step-length tensor A . One expects this since the prolate spheroid, initially pointing along n o in the plane perpendicular to pˆ fattens in its attempt to reach isotropy, but it is constrained and does not become a sphere, instead becoming biaxial. Of course, if chains are distorted in this way, then a totally isotropic state of underlying orientational order cannot obtain. In fact the high temperature phase of the chirally imprinted network must have weak residual, twisting biaxial order, but this is too weak to cause selective (coloured) reflections or absorption beyond the underlying cholesteric to isotropic transition of the network polymers (Maxein et al., 1999).

9.2

Mechanical deformations

Consider a monodomain cholesteric elastomer with an ideal helically twisted director n o (z) in the xy-plane, initially making angle φo = qo z with the x axis, Fig. 2.11. We shall examine two specific cases of imposed extension λ : (i) T RANSVERSE UNIAXIAL EXTENSION in the x direction, λxx = λ , that is in one direction in the plane transverse to pˆ . (ii) L ONGITUDINAL DEFORMATION along the helix axis λzz = λ . The symmetry obvious from Fig. 2.11 requires that in case (i) the director remains in the xy-plane and its azimuthal angle variation is modified to a φ (z) = qo z, while in case (ii) one may expect a conical texture with n (z) inclined towards the stretching axis z and, therefore, described by two angles θ and φ , see Fig. 2.12. In ordinary liquid cholesterics subjected to, for example a magnetic field Hz , such conical states are not generally seen. The direction of n can be freely changed; only its gradient is penalised. Depending on the ratio of the twist to the bend Frank constant, the conical state can be preempted by a 90o rotation of the helix axis to being perpendicular to H , and then untwisting in the newly created ‘transverse’ geometry (Meyer, 1968; de Gennes, 1968). We shall see that in elastomers, due to the director anchoring, this switching is not possible and the conical director configurations should occur. We adopt the simplest form for the locally nematic rubber elasticity   of cholesteric elastomers, the Trace formula for the free energy density, F = 21 µ Tr  o · λ T ·  −1 · λ , where the local step-length tensors (with factors of ⊥ and 1/⊥ , respectively, extracted) are: o = δ + (r − 1)nno n o ,

−1 = δ + (1/r − 1)nn n .

(9.2)

228

CHOLESTERIC ELASTOMERS

For this qualitative discussion we ignore entanglements, finite extensibility and other sources of complexity, such as semi-softness. We have assumed, in adopting the steplength tensors (9.2), that the cholesteric is locally uniaxial about n in its order and thus in its rubber elasticity. The assumption about locally uniaxial order is usual in cholesterics. Extension of it to the rubber-elastic degrees of freedom is plausible since rubber elasticity is determined on the scale of network crosslink separations (a few nanometers), whereas cholesteric pitches are 103 times longer. We ignore Frank elasticity The  initially.  nematic penetration depth extensively dis1 K/µ is the length scale below which director cussed in Chapter 8: ξ = K/D1 = r−1 variation costs more energy than the nematic rubber elastic penalty of rotating the director with respect to the matrix. In most cases the pitch is long, po  ξ , and the energy scales are separated enough, that we can ignore the Frank energy. Later we consider when and how this breaks down. When an extension λ is imposed, a locally nematic elastomer normally adopts relaxations of the other components of the strain tensor λ , and the appropriate rotation of its director, so that its response is soft, that is there is no (or as little as possible) energy cost associated with shape change. If the rather special combinations of shears and director rotations associated with a soft deformation are not consistent with boundary conditions, then the elastomer still deforms softly, but with possible stripes or other microstructures caused by such external mechanical constraints. Cholesteric elastomers are however more complicated. Potential soft strains (and director rotations) associated with a single imposed λ would have different forms at different planes z=const. along the helical axis since the initial director n o has different orientations φo . Thus the elements of λ (z) would vary along pˆ . When strains are soft, the cartoon Fig. 7.3 shows there is a subtle relation between the components of λ and the initial director. Compatibility, see Sect. 4.1, however requires relations between the components of λ when they are spatially varying. For instance the shear λxy would accompany an extension λ in the x direction, if it were to be soft. However, since n o varies with z, then so does λxy and it would have to satisfy the condition ∂ λxy /∂ z = ∂ λxz /∂ y. That is, a shear in the xy-plane, which varies with z (that is along the helix of initial directors n o ), must be accompanied by a shear in the xz-plane which must vary in the y direction, λxz (y). Shears more pertinent to potential soft elasticity in the conical case of λ imposed along z are λmz and λzm (see Fig. 2.12 for the direction m in the xy-plane). We shall see that not all these shears can be matched from place to place along the pitch pˆ . The deformations cannot be soft. However the partial satisfaction of soft relaxation conditions, to an extent dependent on position along the helix, renders the response highly non-classical. 9.2.1

Uniaxial transverse elongation

Consider an imposed stretch in a direction perpendicular to the pitch axis, λxx = λ say, see Fig. 9.4. One expects the director rotation to be in the azimuthal plane x-y, but the associated shear strains λxy (z) and λyx (z), necessary to generate softness, are suppressed. The shears λxz (z) and λyz (z) are not subject to compatibility requirements. However, they should not appear on symmetry grounds, which is easily confirmed by direct minimisation. The original director at a given value of z, n o = {cos φo , sin φo , 0},

MECHANICAL DEFORMATIONS

229

is rotated after deformation to n = {cos φ , sin φ , 0}, still remaining in the xy-plane. Note that the helix is φo = qo z in the initial undistorted material. After deformation, because of the uniform affine contraction λzz , the material frame shrinks and the effective helical ˜ It is φo that gives the z dependence wave-vector becomes q˜ = qo /λzz and thus φo → qz. below of φ (z) and F⊥ (z), respectively the new director orientation and the free energy density associated with perpendicular (⊥) extensions. Inserting o and  from eqn (9.2) and λ above into the Trace formula for the free energy density, eqn (6.5), yields  r−1 2 2 2 2 1 (r − 1)(λ 2 + λyy )(1 − cos 2φo cos 2φ ) (9.3) F⊥ = 2 µ λ + λyy + λzz + 4r  2 . )(cos 2φo − cos 2φ ) − 2(r − 1)λ λyy sin 2φo sin 2φ + (r + 1)(λ 2 − λyy The appearance of terms linear in 2φ (or equivalently sin 2φ ) indicates that local director rotations can always lower the energy for λ = 1 (Mao et al., 2001); we give a sketch here. Minimization of F⊥ with respect to the current director angle φ (z), at a given imposed extension λ , depends on the initial phase of cholesteric helix, φo . It gives for the new azimuthal direction: tan 2φ =

2λ λyy (r − 1) sin 2φo . 2 2 ) cos 2φ + (r + 1)(λ 2 − λ 2 ) (r − 1)(λ + λyy o yy

(9.4)

Since Frank elasticity is being temporarily ignored, the free energy is optimised at each point z, independently of that at neighbouring z. Optimal contractions along and perpendicular to the pitch, λzz and λyy minimise  π /2 the total free energy of half a helix repeat, F⊥ 1/2 = 0 d φo F⊥ , (in effect coarsegraining) with respect to λyy . Then λzz = 1/λ λyy is fixed by incompressibility. Taking 2 = 0 gives rise to a condition for λ : ∂ F⊥ 1/2 /∂ λyy yy

π (r + 1)2 π − 4 2= 4 r λyy λ    π /2 r−1 λ = d φo [(r + 1) − (r − 1) cos 2φo ] cos 2φ − (r − 1) sin 2φo sin 2φ 2r 0 λyy

F IG . 9.4. A uniaxial transverse stretch λxx = λ applied to a cholesteric elastomer.

230

CHOLESTERIC ELASTOMERS

F IG . 9.5. (a) The transverse contraction, λyy , as a function of imposed λxx = λ . The line associated with the experimental data points√is the analytical form λ −5/7 . The two other lines show the classical (hard) λyy = 1/ λ and the soft 1/λ response. (b) The predicted director angle φ , for chain anisotropy r = 1.9, against the cholesteric helix phase qo z for increasing strain; the curves are labelled: λ = 1 (◦), 1.15 (), 1.23 (2), 1.25 () and 1.5 (•). At λ ≥ λc the director pinning at φ = π /2 breaks down and a discontinuous transition occurs, after which the director continuously rotates with further increasing λ towards the final uniform φ = 0. =

r−1 2r

 π /2 0

d φo

a1 [(r + 1) − (r − 1) cos 2φo ] − 4rλ 2

2 a21 − 4rλ 2 λyy

(9.5)

We have trivially integrated the terms not involving cos 2φ and sin 2φ . The integrand was simplified by incorporating tan 2φ using | cos 2φ | = 

1 1 + tan2 2φ

,

| sin 2φ | = 

tan 2φ 1 + tan2 2φ

with appropriate signs. The factor a1 (φ ) under the integral in (9.5) denotes:

2 2 a1 = 12 (r + 1)(λ 2 + λyy ) + (r − 1)(λ 2 − λyy ) cos 2φo .

(9.6)

Equation (9.5) can be solved numerically (Mao et al., 2001). Alternatively, one can expand for small strains, λ − 1  1. It is then easy to show analytically that the relaxation goes as λyy  λ −5/7 and λzz  λ −2/7 . Figure 9.5(a) shows this analytic approximation to the transverse √ contraction and for contrast also shows (i) that of an isotropic rubber λyy = λzz ∼ 1/ λ and (ii) that of (semi) soft response λyy ∼ 1/λ , λzz = const. The response is a compromise between classically hard (i) and soft (ii) behaviour, reflecting that at some points along the helix the director rotates toward the extension direction x to lower the energetic cost, at others it cannot and responds as if hard (e.g. when n o is already along x ). The contraction λzz convects the helix with it, thus the pitch contracts by a factor of λ −2/7 whence the new wavevector is q = λ 2/7 qo . The wavelength of the light giving the characteristic reflections in the material, Λ, is essentially the cholesteric pitch. It will thus shrink as Λ → λ 2/7 Λo , which is seen a the colour

MECHANICAL DEFORMATIONS

231

change from red through green to blue on extension. See Sect. 9.5 for a more quantitative experimental analysis of the pitch change with λ . The experimental data points in Fig. 9.5(a), from two different cholesteric materials (Cicuta et al., 2002), record the shift of the selective reflection with elongation, translated into a transverse relaxation of λyy = 1/λzz λ . This is described in more detail in Sect. 9.5 on photonics, in particular in Fig. 9.18 which records the shift of the photonic band gap that gives the change in colour of these elastomers. ˜ < π /2 (first helical half-turn) are induced Initially for λ < λc , all directors at 0 < qz ˜ < π (second helical to rotate ‘backward’ towards φ = 0, and all directors at π /2 < qz half turn) equivalently rotate ‘forward’ towards φ = π , as the imposed deformation λ increases, see Fig. 9.5(b). In a nematic φ = 0 and π describe equivalent directors. However, there is a twist wall, centred at qz ˜ ≡ φo = π /2, separating these two states. The response of cholesteric elastomers to a mechanical field is quite different from that of a cholesteric liquid to a magnetic field: Here, there is no change of the helical pitch apart from the affine contraction p = po λzz , whereas in liquid cholesterics one finds an increase in pitch (Meyer, 1968; de Gennes, 1968). However, in both cases the helix coarsens. The twist wall can be defined as the region along the helical axis where the director rotates between the values φ = π /4 and 3π /4. As the applied strain increases towards the critical value λc , the width of such a wall decreases to zero and a discontinuous transition occurs. The director in the mid-point of the wall jumps from φ = π /2 to φ = 0, that is to be along the strain axis. For this to happen a disclination loop is presumably generated and expanded in the plane of the wall. The passage of the disclination is needed to remove the topological character of the twist wall. From this point there is no barrier for director rotation with further increasing strain towards the final uniform orientation with φ = 0, as the last two curves in Fig. 9.5(b) indicate. At λ > λc the director rotation is first forward and then backward on going through one cycle, keeping close to φ = 0. The lack of net twist induces qualitatively new features into the photonic band structure for a strained cholesteric elastomer, see Sect. 9.5. The scaling of the critical value of strain with chain anisotropy is λc  r7/24 . It can be estimated by examining eqn (9.4) for φ near the twist wall, that is as φo → π /2± . The limiting values of φ are π /4± and hence one requires the right hand side of eqn (9.4) to tend to ±∞ respectively since tan 2φ → ±∞. This is simply achieved by setting the 2 at the critical point. denominator in eqn (9.4) to zero, yielding the connection λc2 = rλyy Taking the analytical form λyy  λ −5/7 , which is strictly only valid for λ ∼ 1, one can 2 = λ 2 /r to eqn (9.5) gives a more precise, but more directly solve for λc . Returning λyy c complicated, estimate for the critical strain λc as a function of r. Section 6.7 discussed a discontinuous director jump at a critical strain when nematic elastomers are stretched at exactly 90o to their initial director n o (Bladon et al., 1993; Mitchell et al., 1993; Bladon et al., 1994). In a cholesteric elastomer, one can always find planes along the helix where the initial n had an exact phase angle φo = π /2. On stretching with λ < λc this is where the center of narrowing twist walls become pinned from both sides. These points in the cholesteric experience a discontinuous jump analogously to in a uniform nematic. Thus λc is an upper bound on the stability and the

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CHOLESTERIC ELASTOMERS

F IG . 9.6. Theoretical predictions for (a) the nominal stress σxx (λ ) for chain anisotropy r = 1.3 (dashed line) and r = 1.9 (solid line). The equal-area dividing line indicates the corresponding critical stress σc . (b) The associated calculated contraction of the sample width, λyy (λ ) for the same values of r; the thin line is the small-strain estimate λ −5/7 , for comparison. transition. However, as λ approaches λc the twist wall becomes very narrow, the Frank twist energy density gets ever greater and eventually becomes as significant as the elastic energies. It will raise the energy of the twisted state still further and make the equality of energy condition occur at a still slightly lower values of λ < λc . At the discontinuous jump in the distribution of director angles there is also a kink in the λyy (λ ) curve, which one sees in the full numerical solution (Mao et al., 2001), rather than in the λ −5/7 approximation in Fig. 9.5. This is reasonable – recall the optimal λyy arose from minimising the energy of a whole cycle of the cholesteric. This energy changed at the jump and its minimising relaxation does too. The stress is the derivative of the energy with respect to the imposed strain; the nominal stress σxx (λ ) is obtained by differentiating the energy eqn (9.3) with respect to λ . Since the energy changes at the jump, so too will the form of the stress-strain relation. The theoretical results for σxx are plotted in Fig. 9.6. We can see that the stress exhibits a dip corresponding to the kink in the transverse relaxation. If a controlled-stress device is used, the material will undergo a sudden jump in elongation, ∆λ around λc , the mechanical instability being accompanied by the discontinuous jump in director orientations. The Maxwell equal area construction will determine the critical stress, σc and the λ interval, see Fig. 9.6. The equal area construction has the usual motivation: A Legendre transformation takes one from the Helmholtz free energy as a function of deformation (the extensive variable), F(λ ), to the Gibbs function G(σ ) which has instead the stress as its independent intensive variable (Landau and Lifshitz, 1970). Now instead of σ = ∂ F/∂ λ , we have λ = −∂ G/∂ σ . The values of Gibbs free energy G(σ ) at different values of its variable are connected by G(σ ) =

 σ σ1



d σ (∂ G/∂ σ ) + G(σ1 ) = −

 σ σ1

d σ  λ (σ  ) + G(σ1 ) .

For two phases to coexist at the same stress, σ , their free energies G must be equal. Obtaining the free energy Gu of the untwisted phase in terms of Gt of the twisted phase:

MECHANICAL DEFORMATIONS 

233



Gu = Gt − σσ1 d σ  λ (σ  ), implies that the d σ  λ (σ  ) along the λ (σ ) curve between the points of coexistence through the transition is zero, thereby giving the equal area rule for the coexistence stress. 9.2.2

Stretching along the pitch axis

We now consider an imposed stretch along the helical pitch axis, λzz = λ , see Fig. 9.7(a). With the geometry of currently available cholesteric monodomain elastomers, this is not an easy strain to impose – the helix axis is typically along the short axis of the solid. The deformation tensor in its simplest form would be:   √ 0 λxz 1/ λ √ (9.7) λ =  0 1/ λ λyz  , 0 0 λ which is volume preserving, Detλ = 1, by construction. No compatibility problem for the shears λxz (z) and λyz (z) arises from their z-variation, along the helical axis. We shall optimise them differently at each z according to where the local director points. By contrast, their conjugate strains λzx and λzy , which would also have to vary with z, lead to a serious compatibility mismatch, e.g. ∂ λzx /∂ z = ∂ λzz /∂ x. We therefore deduce that λzx and λzy are suppressed even though in other settings, Chapter 7, these are the generators of soft elastic response. Conceivably, λxy and λyx could exist, but numerical tests (Mao et al., 2001) suggest that this is unlikely. In this geometry one expects the director to rotate out of the xy-plane to make an angle θ with pˆ , see Fig. 9.7(b). The initial director orientation is, as before, in the plane perpendicular to the helix axis n o = {cos qo z, sin qo z, 0}. After deformation the rotated director is aligned along the surface of a cone or hour-glass: n = ˜ sin θ sin qz, ˜ cos θ }. We expect the two shear strains λxz (z) and λyz (z) to {sin θ cos qz, be in proportion so that they describe a shear in the plane of n o , n and pˆ so that their effect expansions and contractions in this plane help accommodate the rotation of the chain distributions (see the cartoon of softness, Fig. 7.3) and thereby keep the elastic energy low. As the original director and thus the plane of rotation of n changes down the

x

x

fo y

n (a)

l zz

f=qo z y

w n

no

l zz

(b)

F IG . 9.7. (a) A stretch λzz = λ applied along the pitch of a cholesteric elastomer. The director now sits on a cone of semi-angle θ , having been rotated by an angle ω = π /2 − θ from the xy-plane, (b).

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CHOLESTERIC ELASTOMERS

helix, then so will the proportions of the shear. However, we optimise the rotations and shears independently at each position z down the helix (the continued ignoring of Frank ˜ =0 effects) and so without loss of generality solve the problem at z = 0, that is qo z = qz (all other values of qo z being related by rotational symmetry). As in the case of a transverse stretch, all physical dimensions in the deformed sample are scaled by the affine strain. In particular, here z → λ z and the cholesteric pitch expands; the wave-vector contracts: q˜ = qo /λ . As usual, o and  are defined by no and n, see eqn (9.2). The Trace free energy density (6.5) then yields F (where  denotes the longitudinal stretching case): 2 1 F = 2 µ λ 2 + + λyz2 − (9.8) λ  r 1 − [r + 1 + (r − 1) cos 2θ ](λ 2 − − λxz2 ) + 2(r − 1)λ λxz sin 2θ . 2r λ Optimising for the shears λxz and λyz gives

λxz = λ

(r − 1) sin 2θ ; (r + 1) + (r − 1) cos 2θ

λyz = 0.

This represents a strain made up of displacements in the direction of n projected into the xy-plane, in fact along x for the case of z = 0 examined above. Generalising to the qz ˜ = 0 case, we have   (r − 1) sin 2θ cos qz ˜ λxz = λ , (9.9) ˜ λyz r + 1 + (r − 1) cos 2θ sin qz in phase with the azimuthal angle along the helical pitch. Equation (9.9) describes distortions in the xy-plane, perpendicular to the helix axis and following the initial orientation n o . On substitution of these optimal shears back into the free energy density one obtains  2λ 2 2 + r + 1 + (r − 1) cos 2θ . (9.10) + F = 12 µ r + 1 + (r − 1) cos 2θ 2λ F expands at small tilt angle ω (recall that we defined ω = π /2 − θ as the angle of director rotation in the polar plane) F ≈ 12 µ (λ 2 + 2/λ ) − 12 µ ω 2 (r − 1)(λ 2 − 1/λ ), that is the director starts to rotate down to define a cone of semi-angle θ immediately as the strain λ > 1 is imposed. (The coefficient of ω 2 is negative as soon as λ > 1.) The equilibrium director tilt is obtained by minimisation of the full free energy density F (ω ): 0 λ 3/2 − 1 r + 1 − 2λ 3/2 ; . (9.11) ω = arcsin cos 2ω = r−1 r−1 The rotation is complete, with the director aligned along the extension axis (θ = π /2) at λ2 = r2/3 which, for some elastomers, can be a very large extension. Director rotation

MECHANICAL DEFORMATIONS

235

starts and ends in a characteristically singular fashion, Fig. 9.8(a). One of course is reminded about the similar response during the soft (and semi-soft, with a threshold) director rotation in nematic elastomers, Fig. 7.8.

F IG . 9.8. (a) The angle ω = π /2 − θ of director tilt plotted against the imposed deformation λ along the helix axis, Eq. (9.11), for r = 1.3 (dashed) and r = 1.9 (solid line). At deformation λ2 = r2/3 the alignment is complete, ω = π /2, and thereafter the director points uniformly along the former pitch axis. (b) The nominal stress σzz vs imposed strain, again for r = 1.3 and r = 1.9. After cos 2ω ≡ − cos 2θ in eqn (9.11) is returned to eqn (9.10), the free energy density is only a function of the imposed strain: F =

1 2µ

2λ

= 12 µ

1/2

1 + λ



λ2 r +1 + r λ

1 < λ < λ2

(9.12)

λ > λ2

(9.13)



The free energy is not soft as is perhaps suggested by the director rotation, Fig. 9.8(a). In fact F ∼ 32 µ (1 +
2 /4 + . . .) which is quadratic in small strains
= λ − 1. Thus the response is qualitatively hard as in a conventional rubber, but the coefficient of
2 (proportional to the modulus) is only 1/4 that of the same rubber in the isotropic state, see the discussion on page 83. Some softening of the response is possible since there is director rotation, allowing the shape of the polymers to take up some of the imposed shape change. The corresponding stress-strain plots are accordingly flatter while director rotation is still possible. Two are shown in Fig. 9.8(b). The material hardens significantly after reaching λ2 , which reflects the fact that there is no longer a freedom to rotate its director in its attempt to lower distortion free energy. The stress-strain relations are independent of anisotropy r for λ < λ2 , which can be confirmed from the independence from r of the free energy eqn (9.12). After rotation is complete, the stress does depend on r, elastomers with larger r being softer (flatter stress-strain relations) which can be confirmed from eqn (9.13).

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CHOLESTERIC ELASTOMERS

It is possible that conical states might appear spontaneously in imprinted elastomers, that is where there is no chiral material present, but a cholesteric helix established at crosslinking. On cooling from the isotropic state, a conical configuration would reduce the amount of twist penalty associated with the helix, while not incurring the full D1 penalty associated with not twisting at all back to the helical genesis (Pelcovits and Meyer, 2002; Warner, 2003). This remains a possibility to search for experimentally.

9.3

Piezoelectricity of cholesteric elastomers

Non-centrosymmetric molecules and phase structures admit the possibility of piezoelectric response, that is the generation of electric polarisation from the imposition of mechanical strains. Handed molecules lack a mirror plane of symmetry. Left and right handed enantiomers of the same molecule are related by a mirror reflection but not by rotations or continuous distortions. The helical texture of cholesterics also lacks a mirror plane. Cholesteric liquids cannot sustain a static shear and are thus not piezoelectrics. Cholesteric elastomers, as we have seen, do sustain shear and we now explore their piezoelectric properties. In principle, by suitable imprinting, a chiral, untwisted nematic elastomer could be envisaged – with only the molecular moieties and not the phase structure chiral. These too should be piezoelectric, but in a subtle way we do not pursue here. Equilibrium piezoelectric response of elastic networks with a genuine cholesteric twist has been first predicted in (Brand, 1989). This has been an important work because, although the polarisation has been predicted in the direction of pitch axis pˆ (which, of course, violates the cholesteric symmetry), it stimulated the experiments of Meier and Finkelmann (1990,1993) that unambiguously demonstrated the electric voltage generated across the deformed sample in phase with the strain. Terentjev (1993) developed the continuum theory with correct symmetry, but did not properly account for the helical twisting of the structure and local polarisation. Pelcovits and Meyer (1995) completed this development, deriving the piezoelectric free energy and polarisation of a coarsegrained cholesteric elastomer, but attributed the effect to the local flexoelectricity in a non-uniformly distorted director texture. True piezoelectricity in chiral polymer networks must exist on the grounds of symmetry and much remains to be learned in this field, especially because of many attractive practical applications. Cylindrical symmetry and coarse graining in cholesterics If we average the director distribution in Fig. 2.11 about the axis pˆ , then the only preferred direction remaining is pˆ itself. We recognise that, on a length scale along pˆ greater than the pitch π /q, the variation of director can be coarse grained away leaving behind only the (non-centrosymmetric) cylindrical symmetry. In fact a coarse-grained cholesteric has a formal similarity to a smectic in that there is an underlying periodicity (equivalent to the smectic layers) along the pitch axis (equivalent to the layer normal). There is a modulus which resists compression and extension which derives from the Frank twist modulus (twist is changed when there is strain along the pitch axis), and likewise bending of the layers derives from the Frank bend and splay cost of bending

PIEZOELECTRICITY OF CHOLESTERIC ELASTOMERS

237

the pitch axis (de Gennes and Prost, 1994). A similar, cylindrically-symmetric elastic penalty will be found in distorting cholesteric elastomers. Coarse graining can sometimes be a useful view to take when calculating the piezoelectric response from a cholesteric solid. Imposed strains vary on a length scale far longer than the helical pitch and any resulting polarisation does not sense details of the spatial helicity. Exercise 2.5 about polarisation in the conical state illustrates the principle – the emergent polarisation direction is perpendicular to pˆ , varying with position along pˆ . It averages to zero on length scales longer than the pitch, a result that would have been obvious if one had considered the coarse-grained director distribution, neff = pˆ at the outset. Then with only one vector available and this being a pseudovector, no polarisation can result. Likewise, compression imposed along pˆ does not add a polar vector to the problem since the compression axis does not distinguish up from down either, and again no polarisation can result. Piezoelectric free energies can be constructed with terms like ni n j where directions i and j are perpendicular to pˆ . Averaging over the period of a helix one has: (tr)

ni n j  = 12 δi j (tr)

where δi j = δi j − pi p j is the Kronecker delta in the plane perpendicular to pˆ (the perpendicular projector for the director n used, e.g., in exercise 7.4). For instance nx nx  = cos2 φ  = 12 and nx ny  = sin φ cos φ  = 0. Cholesteric piezoelectric response Crosslinking a liquid crystal phase creates a solid matrix against which director rotations take on a physical significance. Furthermore, the matrix also offers the opportunity for symmetric strains to couple to the director. These two couplings are respectively governed by D1 and D2 relative rotation coefficients (recall secs. 7.1.1 and 7.1.2 sketching the physics behind D1 and D2 ). Such rotations and shears change the state of the elastomer and do work. They can also induce electric polarisation in chiral liquid crystal solids. A rotation has a (pseudo) vector associated with it, the rotation axis Ω = 12 curluu where, if the director is frozen, the relative rotation is simply the local rotation of the elastic matrix. The displacement of the matrix points is denoted by the vector u (rr ), as usual. Only the component of the rotation axis perpendicular to n generates any Ω · n )nn ≡ n × (Ω Ω × n ). Given a chiral meaningful relative rotation, that is Ω ⊥ = Ω − (Ω medium there will exist a pseudoscalar material constant, Γ2 say, with which we can E · Ω ⊥ ). By form a true scalar using Ω ⊥ and a true vector, E the electric field: Γ2 (E identification with the energy density in polarised systems, E · P , there is a polarisation generated by rotations relative to the director of: Ω × n) , P 2 = Γ2 Ω ⊥ = Γ2 n × (Ω

(9.14)

see Fig. 9.9(a). Recall that the product of a pseudoscalar and a pseudovector is a true vector, as the dielectric polarisation should be. Changing the sign of rotation, Ω ⊥ →

238

CHOLESTERIC ELASTOMERS

Ω⊥ will change P 2 → −P P2 . Changing from a right- to left-handed medium also re−Ω verses the piezoelectric polarisation (if the sign of the applied relative rotation is unchanged) since the sign of material constant Γ2 reverses. With a symmetric shear εi j , there are two associated, perpendicular directions i and j , but not a polar axis for polarisation. In a nematic medium, however, we have another direction, n . A pseudovector axis emerges if we take a cross-product, namely n × ε ·nn, or in diadic notation εi j [(nn · j )nn × i + (nn · i )nn × j ]. Given the elastic processes are different from before, another pseudoscalar material constant, Γ1 say, arises and makes possible E · [nn × ε · n ]). There is then another component the construction of the true scalar Γ1 (E of piezoelectric polarisation: P 1 = Γ1 [nn × ε · n ]

(9.15)

along the pseudovector, see Fig. 9.9(b). We can reverse P 1 by (separately) either reversing the strain or changing the handedness of the medium. To obtain a P 1 we require that n be in the plane of shear ε . The polarisation is maximal if n is in either the i or the j directions. The diadic form above easily gives the variation of the polarisation generated by symmetric shear, P 1 ∼ cos 2θ [ii × j ], as the angle θ between n and i is varied.

F IG . 9.9. The two underlying components of piezoelectric polarisation, P 2 and P 1 , generated by local rotations and symmetric shears respectively. These polarisations were first written down purely on symmetry grounds (Terentjev, 1993) with the notation Γ1 ≡ Q1 and Γ2 = Q2 − Q3 . A third component of polarisation, P 3 , also linear in rotations Ω , will emerge for cholesterics. However, in an untwisted chiral nematic medium it would be cubic in distortions due to the invariance with respect to an overall body rotation of the body. Since small distortions are involved, one does not distinguish between the original and current directors, n and n o , since to do so would involve terms ultimately in total of higher than first order. This phenomenological analysis is of course uninformative about the sizes or even signs of the Γs – they are simply symmetry-allowed in a chiral rubber. When we are interested in cholesteric piezoelectric rubber on length scales much greater than the pitch we can coarse grain. A free energy density can then be written down, again, purely on symmetry grounds (Pelcovits and Meyer, 1995): ? @ Fcg = γ1 qo E · pˆ × [δ (tr) · ε · pˆ ] + γ2 qo E · pˆ ( pˆ · Ω ) + γ3 qo E · δ (tr) · Ω ,

(9.16)

PIEZOELECTRICITY OF CHOLESTERIC ELASTOMERS

239

F IG . 9.10. The three polarisation components in eqn (9.16): P 1 generated by symmetric shears and those, P 2 and P 3 , produced by local rotations in a cholesteric rubber. with qo = π /po the helix wave number. This expression is written in terms of the pitch axis pˆ , which is the only vector surviving after the helix (a memory of the cholesteric origins of the material is visible in the qo factor that follows the coefficients γi ). The term γ1 coupling to symmetric shear is of the same form as before. Shears must span the helix axis pˆ and a direction perpendicular to it and the polarisation is perpendicular to this shear plane and to the helix axis. There cannot be shear-generated polarisation along the helix axis. By symmetry one expects two distinct couplings to the rotation, see Fig. 9.10. Thinking back to the cholesteric helix, a rotation Ω along pˆ has all directors perpendicular to it. Thus γ2 qo = Γ2 from eqn (9.14) for the underlying chiral nematic response. By contrast, a component of Ω perpendicular to pˆ has directors n at all angles to it. One can find γ3 qo , again in terms of Γ2 , by considering a rotation about the y-axis, that is Ω = Ωyy. Then the polarisation due to the local rotation component, eqn (9.14), is P 2 = Γ2 Ωnn × (yy × n ) where n = cos φ x + sin φ y . Recall the usual relations y × x = −zz etc., whence P 2 = Γ2 Ω(yy cos2 φ − x sin φ cos φ ). Averaging gives cos2 φ  = 12 , sin φ cos φ  = 0 and thus P 2 = Γ2 12 Ωyy = Γ2 12 Ω . Since, by comparison with the coarse-grained eqn (9.16) this is the third term, we conclude that γ3 qo = 12 Γ2 . The piezoelectric polarisation response to rotations parallel (γ2 ) and perpendicular (γ3 ) to pˆ thus differ in size but on average are along the local rotation axis in both cases. Again, this symmetry-based argument gives no clue as to the origins, sizes and signs of material constants γ1 , γ2 and γ3 . Flexoelectricity in cholesteric solids Perhaps the shear term has contributors that are nothing to do with intrinsic chirality (Pelcovits and Meyer, 1995). A cholesteric provides a non-uniform director distribution which, when acted upon by a shear field, can lead to polarisation by an essentially nonchiral mechanism, for instance flexoelectricity. One can thus model γ1 : Retreating from the coarse-grained approach, let us take the coordinate system of Fig. 2.11 and apply a shear εzx . It will modify the director field to give a pattern with a small component in the z-direction. The shear in the xz-plane convects the x-component of n toward z , thus the small z-component will scale with nx and with εzx , with a constant of proportionality β , say: n o (z) → n (z) ≈ x cos φ (z) + y sin φ (z) + z cos φ (z)β εzx .

(9.17)

240

CHOLESTERIC ELASTOMERS

The details of how great the director rotation is for a given imposed strain may be complex and all absorbed in the coefficient β for the sake of this argument. The director in eqn (9.17) remains a unit vector only at order β . Flexoelectricity, as the effect of dielectric polarisation induced in any non-chiral material subjected to non-uniform curvature deformations, was discussed in exercise 2.5. We must extract the splay and bend in the new director field, eqn (9.17). They are respectively d nz = −φ  β εzx sin φ (z) dz n × curlnn = φ  β εzx cos φ (sin φ , − cos φ , 0) divnn =

where φ  = d φ /dz = qo . Thus the local flexoelectric polarisation, eqn (2.49), P = e1 n div n + e2 [nn × curl n ], yields at order ε : P = −e1 qo β εzx (sin φ cos φ , sin2 φ , 0) + e2 qo β εzx (sin φ cos φ , − cos2 φ , 0) . Averaging this over the cholesteric helix period gives the macroscopic polarisation. By identification with eqn (9.16), γ1 is (Pelcovits and Meyer, 1995): P → − 12 (e1 + e2 )qo β εzx y ≡ γ1 qo εzx y P → γ1 = − 12 (e1 + e2 )β .

(9.18)

The other coefficients, γ2 and γ3 , govern the response due to local rotation of the rubber-elastic matrix, which seldom occurs in practical deformation situations except in the presence of simple shear. Applying hypothetical rotations allows one to relate these phenomenological constants to the underlying flexoelectric coefficients too. Exercise 9.2: Suppose rotations Ω of the matrix with respect to a clamped director can be achieved by suitable anchoring. Calculate the piezoelectric responses γ2 and γ3 in terms of the flexoelectric constants e1 and e2 . Solution: The rotation axis along pˆ causes a rotation of the helix by the angle Ω, see Fig. 9.10. Hence φ = qo z → φ = qo z + Ω, a uniform phase shift is generated by any change in twist, or the creation of splay and bend which would generate a P of flexoelectric origin. Thus γ2 = 0 in a model of piezoelectric response based on flexoelectrity of a cholesteric helical texture. The rotation axis along x , that is Ω = Ωxx, induces the y-component of n to change from sin φ y to sin φ cos ω y + sin φ sin ω z. Thus n and curl n are: n = (cos φ , sin φ cos Ω, sin φ sin Ω) curl n = −φ  (cos φ cos Ω, sin φ , 0) . Then the splay and bend present in the new director field are: divnn = qo cos φ sin Ω

PIEZOELECTRICITY OF CHOLESTERIC ELASTOMERS

241

∇ × curl n = −qo (sin2 φ sin Ω, sin φ cos φ sin Ω cos Ω, sin φ cos φ sin2 Ω) . From these we can deduce that the polarisation is P = 12 e1 qo sin Ω x − 12 e2 qo sin Ω x Ω ≈ 12 qo (e1 − e2 )Ω where we have averaged over a period of the helix (sin2 φ  = 12 ; sin φ cos φ  = 0) and have expanded sin Ω ∼ Ω for small rotations. Identifying with eqn (9.16) gives the third piezoelectric coefficient in terms of the flexoelectric coefficients:

γ3 = 12 (e1 − e2 ) (compare with the flexoelectric contribution to the symmetric shear response, eqn (9.18).

Piezoelectric experiments The first piezoelectric experiments on rubber (Meier and Finkelmann, 1990; Vallerien et al., 1990) were performed on polydomain cholesteric networks that became aligned after initial compressional strains were applied (we briefly discussed such alignment technique in Sect. 9.1.1 in connection with monodomain fabrication). For the first time stress-induced polarisation response was achieved in homogeneous soft solids, moreover in materials that are locally fluid-like and lack any of the positional order normally associated with piezoelectric solids. This positive response conclusively demonstrated a chiral mechanical coupling. The results at first appeared to register polarisation along the non-polar helix axis pˆ following compression along this axis, a response which however is not symmetryallowed (Pelcovits and Meyer, 1995). Further work (Meier and Finkelmann, 1993) concluded that inhomogeneities in the sample under compression resulted in the type of local shears that could generate piezoelectric response. Such shears can be generated in the setup of Fig. 9.11 which shows the displacement ux in the x direction of the upper surface of a piezoelectric cell (Chang et al., 1997b). It produces a simple shear uxz = dux /dz which follows from knowing the z-thickness of the sample. The corresponding symmetric shear is accordingly εxz = 12 dux /dz and the rotation is Ωy = 12 dux /dz; see Sect. 4.3 for a reminder of how infinitesimal simple shear can be broken down into symmetric (pure) shear and rotation in equal measure. The rotation and helix axes are perpendicular and thus the third (γ3 ) term of eqn (9.16) is active. Overall we have polarisation Py from a combination of shear and rotation: Py = (γ1 + γ3 )qo β εzx .

(9.19)

The measured dependence of polarisation on displacement amplitude (and thus on the shear/rotation) then directly gave the combination of constants (γ1 + γ3 )qo α . By identifying flexoelectric with piezoelectric coefficients as in eqn (9.18) and by taking representative values of the flexoelectric coefficients e1 + e2 from the literature, Chang et al. could estimate α = −2γ1 /(e1 + e2 ), ignoring in this estimate γ3 . They found α ∼ 0.1 for this coefficient connecting director rotation and shear.

242

CHOLESTERIC ELASTOMERS

F IG . 9.11. Schematic of simple shear applied to a cholesteric elastomer to generate piezoelectric polarisation (Chang et al., 1997b). A simple shear uxz generates a corresponding symmetric shear and local rotation in the xz-plane, both of which lead to a y-polarisation, P . The cholesteric pitch axis is z . 9.4

Imprinted cholesteric elastomers

We shall consider the physics of elastomers imprinted with chiral structure, but without chiral molecules, in the opposite way from intrinsic cholesteric elastomers. There we applied deformations to monodomain elastomers to induce their helical director fields to change – obtaining contractions, coarsening and conical states, all with non-classical mechanical responses. Here we take imprinted systems and see what changes in the director field can be driven from within. Director rotations are now not imposed mechanically, but by the imperative to reduce Frank elastic energy (which we previously ignored). We do not impose any distortions, though this is an interesting additional possibility. A simplest approach, avoiding compatibility problems generating expensive extra distortions, is not to allow any distortions at all, that is λ = δ . We can pose the problem simply. Crosslink nematic polymers in a chiral solvent to form a gel. The solvent, which causes a natural twist qo to the nematic, is then removed but nevertheless leaves behind a cholesteric elastomer (Hasson et al., 1998). There are two competing processes in the elastomer. • The twist φ  = qo of the gel, which gave the minimal energy at crosslinking, is now unfavourable since without chiral material the lowest energy twist is q = 0. The Frank energy density would be F = 12 K2 φ 2 if no twist is lost, see eqn (2.47) with qo = 0. • The director is anchored to the configuration in the matrix that pertained at the moment of crosslinking. In the absence of any mechanical distortions being allowed, this anchoring is given by the first de Gennes coupling, eqn (7.6): the energy density is 1 2 Ω n2 Ω 2 D1 [(Ω − ω )×n ] ∝ (Ω− ω ) , where (Ω − ω ) is the (small) rotation of the director relative to the solid. However, this nematic elastomer penalty must be described in a fully non-linear manner since the local rotation can be very large. The clamped,   large rotation 1 form was easily derived in Sect. 7.1.1 by inserting λ = δ into 2 µ Tr  o · λ T ·  −1 · λ   which becomes 12 µ Tr o · −1 , see eqn (7.1) and then finally eqn (7.5). This change in free energy density due to pure rotation is given by 12 D1 sin2 [φ (z) − qo z], where in the

IMPRINTED CHOLESTERIC ELASTOMERS

243

simplest model the coupling is D1 = µ (r − 1)2 /r, see eqn (7.4) and (7.6). The rotation is the difference between the current angle φ (z) of the director and its formation angle, φo = qo z. The effective pitch wavenumber qo hides in it any possible changes in the pitch length due to any solvent loss since gelation. The energy for an elastomer formed under a cholesteric solvent which is subsequently replaced with an achiral one is then: f = Axy



dz

1

2 K2

φ 2 + 12 D1 sin2 (φ − qo z) ,

(9.20)

where Axy is the sample area perpendicular to the helix. Two limits of the energy density are helpful: (i) the perfectly twisted cholesteric state has F = 12 K2 q2o where the current director points along its formation direction and thus pitch wavevector is unchanged from qo . (ii) the untwisted state with φ  = 0 has F = 14 D1 , the additional factor of 12 arising from the averaging of (sin2 ) over one period. Crudely, we expect the director to remain twisted if the twist cost is less than the anchoring penalty (strong imprinting): K2 q2o < D1 /2 .

(9.21)

The director untwists otherwise, because the demand of the nematic order to attain the uniform director conformation prevails over the local anchoring to the elastic network (weak imprinting). This balance is tuneable between the dominance of anchoring and of the Frank penalty since K2 , D1 and q2o vary relatively to each other with temperature, degree of crosslinking and swelling and the presence of additional chiral agents. Equation (9.21) anticipates the result of detailed analysis: initially highly twisted states (qo large) or systems with a large Frank constant K2 will pay a very high penalty on loss of spontaneous twist arising from the removal of the chiral solvent: a large combination K2 q2o will overcome the anchoring D1 and the imprinting will be lost. Weakly twisted elastomers with small Frank constants will not overcome the director anchoring and imprinting will remain.  The length ξ = K2 /D1 is essentially the nematic penetration depth (which we use here with the modulus D1 ). The estimate (9.21) for retention or loss of the helical q2o < 1/2. Recall that the initial cholesteric pitch is po = structure translates into ξ 2√ 2π /qo . Ignoring factors of 4π , we have:

ξ < po helix retention ;

ξ > po helix loss, unwinding .

If the effective director twist propagates against anchoring over a distance ξ greater than a period po of the helix, then the helix is destroyed. The spontaneous unwinding of helices in elastomers can be described exactly with elliptic functions (Mao and Warner, 2000) similarly to the classic case (Meyer, 1968; de Gennes, 1968) where it is a transverse electric field overcoming the tendency of an intrinsic cholesteric liquid to remain twisted. The combination ξ qo determines stability. It is a non-dimensional measure of the nematic length relative to the chiral pitch.

244

CHOLESTERIC ELASTOMERS

eo

xqO = 2/p

xq O

F IG . 9.12. Imprinting efficiency eo as a function of imprinting parameter ξ qo . The slope just above the critical point ξ qo = 2/π diverges and the efficiency decays rapidly on further increase in ξ qo . We expect imprinting to be lost at large ξ qo and retained at small ξ qo . It turns out for ξ qo < 2/π the cholesteric helix is stable. For ξ qo > 2/π the period lengthens and coarsens, that is, there are increasingly localised walls of twist separating largely untwisted regions. One can quantify this by defining an imprinting efficiency eo to be the fractional number of twists surviving in the whole helix. In other words, eo = q/qo , is the relative average wavenumber of the remaining periodicity. Figure 9.12 shows eo (α ). Nothing is lost (eo = 1) for ξ qo < 2/π . Twist is rapidly lost above the critical point ξ qo = 2/π , that is at ξ > po /π 2 . Practical ways of tuning ξ qo could be with light or solvent. If the chiral centres that induce twist are at or near photoisomerisable parts of a molecule, then absorption of light could alter this vital part of the molecular structure. The bent molecule could have an entirely different twisting power. We could speculate that such cholesterics could be radically altered by reversibly pushing them over the imprinting threshold. If the new natural pitch wavenumber becomes q, then the resulting effective ξ qo then becomes: ξ (qo − q). We sketched above the case of q = 0, assuming the complete actual removal of the chiral species, rather than the modification of chiral centres within an instrinsically chiral network (or an incomplete removal of chiral dopant). Cholesteric elastomers as chirality pumps The process of re-swelling of the imprinted elastomer can be subtle and lead to chiral separation of racemic mixtures (stereo-selection). Imagine an imprinted network with ξ qo > 2/π so that it has partially unwound from its originally right-twisted state. The elastic energy has risen because the Frank cost of remaining twisted was so high. The elastomer is now put in contact with a racemic mixture, that is, a solvent with equal number of right (R) and left (L) handed molecules. If R-molecules swell the network, they would restore the natural twist towards the imprinted value. The Frank energy is minimised, but now at a value of twist that also satisfies the elastic anchoring requirements since n is restored to n o . The collective re-winding of the rubber towards its initially crosslinked helical state is the reward for taking up one handedness of the solvent preferentially to the other (provided the gel is not so swollen that nematic order is lost altogether). Figure 9.13 shows calculations (Mao and Warner, 2001) of volume fraction

PHOTONICS OF CHOLESTERIC ELASTOMERS

245

r 1 0.8

r1

0.6 0.4 0.2 f 0

0.2

0.4

X

0.6

0.8

1

F IG . 9.13. Chiral separation by an imprinted cholesteric network. The reservoir of solvent has a volume fraction, ρo , of R-handed molecules. The R-volume fraction taken into the gel is ρ . A racemic example, ρo = 1/2, is illustrated, leading to ρ1 in the network. The gel anisotropy is taken as r = 2 and the interaction parameter between the different handednesses of solvent, χlr is assumed to be zero, that is R-L solvent demixing is purely due to the elastomer.

ρ of R-solvent in a gel, initially with ξ qo = 2/π , as a function of ρo , the volume fraction of R-species in the reservoir of solvent outside the gel. Figure 9.13 shows that φ > φo is generally satisfied: the solvent handedness which agrees with the network is favoured for absorption. Given the critical character of unwinding, it is plausible that the maximal resolving power is obtained when the imprinting power of the resulting network is near the transition point ξ qo = 2/π . We have taken this case in the illustration. We have also assumed for simplicity that the solvent is mesogenic in its own right, indeed that its absorption into the cholesteric gel does not change the underlying nematic order. This is a most restrictive assumption that complete theory will have to avoid. The process can then be reversed, at least conceptually. Mechanical stretching of the swollen gel can remove all twist, see Sect. 9.2. There being no intrinsically chiral material actually part of the network, the network has neither natural twist nor twist retained by nematic anchoring. Solvent expelled by the stretch will then simply have the enriched concentration, that is its concentration is not unbiased on expulsion because it is leaving a now achiral host. The process, we speculate, could be repeated cyclically leading to greater chiral enrichment. The first experiments (Hasson et al., 1998; Courty et al., 2003; Courty et al., 2006) have demonstrated the chiral imprinting, as well as the selective retention of one of the chiral enantiomers of a racemic solvent. 9.5

Photonics of cholesteric elastomers

Cholesteric elastomers are among the more unusual of such materials in that their photonic properties can be changed by strain and in that they have subtle polarisation effects. Photonic materials manipulate the properties of photons analogously to how semiconductors manipulate electrons. A periodic variation of potential interacts coherently with an electron wave to alter its dispersion relation connecting its angular frequency, ω (kk ), and wavevector, k . Equivalently the energy Ek = h¯ ω (kk ) is specified by the dispersion relation. Additionally gaps arise in this band structure where there are no allowed

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CHOLESTERIC ELASTOMERS

energy states at certain wavevectors. Photons too find their dispersion relation altered from their in vacuo condition, ω = cko where c is the speed of light and the wavevector is ko = 2π /Λo with Λo the wavelength in vacuo. The simplest modification of this by a medium is the development of a refractive index, m, so that ω = ck/m (we remind the reader of this unconventional notation for the refractive index, m, which we have to use to distinguish from the nematic director). Now the wavevector and wavelength in the medium are k = 2π /Λ and Λ. Since the angular frequency remains fixed on the passage between media, one deduces the usual ‘refraction’ result for the wavelength, Λ = Λo /m, and the speed of light d ω /dk = c/m: in the medium, the wavelength contracts and the speed of light diminishes. We shall find examples of where the group velocity of light will vanish and this will be of central importance to lasing. If there is a periodic modulation of dielectric properties, with a period close to the wavelength in the medium, then the coherent interaction of the optical field with the structure can lead to energy gaps and interesting photonics results. Most photonic materials have periodic variations of the magnitude of the dielectric constant. For example opal has periodically arranged inclusions, the scattering from which gives reflections coloured accordingly to which components of light satisfy the Bragg conditions at the angle of view. The reflection arises because states at that frequency are forbidden inside the medium – incident light is rejected because of a gap in the states available to it on entering the medium. Cholesteric structures are periodic too, but in the rotations of the principal directions of the dielectric tensor,
, rather than in variations in the magnitudes of the principal elements of
. A more subtle band structure arises since the medium then acts on optical waves also according to their polarisation. Control and sensing of polarisation is often in practice a preferred way of interacting with, steering and modulating light. We shall review the photonics of simple cholesterics first. A solution of the wave dispersion is possible (de Vries, 1951; Belyakov et al., 1979) and illuminates the rather different task we face with elastomers. We then sketch the solution to the photonics of elastomers and make contact with the band structure and lasing experiments. 9.5.1

Photonics of liquid cholesterics

The helical distribution of director in a cholesteric gives a periodically twisting dielectric tensor. It has a principal value
 along n and
⊥ in all directions perpendicular to n, neglecting biaxiality. The principal directions rotate as the director does. Typically rods are more polarisable along their length and their orientational order renders
 >
⊥ at the macroscopic scale. See Sect. 2.2, eqns (2.2) - (2.7), for the equivalent discussion of the closely related refractive index tensor, the optical indicatrix, and its connection with nematic order. In our later illustrations we adopt
 = 3.3 and
⊥ = 2.3, values common for liquid crystals. The√corresponding extraordinary () and ordinary (⊥) refractive indices, given by m =
, are me = 1.82 and mo = 1.52. The solution to the wave problem (de Vries, 1951) is easy in a coordinate system rotating with n since then these values of
appear fixed. At wavelengths long compared with the pitch, Λ  p, the waves only ‘see’ an averaged dielectric tensor and simply pick up an effective refractive index. When Λ ≈ p, the electric vectors of the eigenmodes coherently see either
 or

PHOTONICS OF CHOLESTERIC ELASTOMERS

247


⊥ and a gap develops. The reader can also consult the details of the de Vries analysis in a review (Belyakov et al., 1979), or a modern working of it in the context of photonic band structure (Bermel and Warner, 2002), for the details. Pictorially one sees (Kopp and Genack, 1999) why two waves of the same wavelength Λi can have different energies and thus how an energy gap can develop, see Fig. 9.14. At the very band edge,

~ w

2.5 RP

2 1.5

LP

1 0.5 0 0

0.1

0.2

0.3

0.4 ~ 0.5

(b)

(a)

k

F IG . 9.14. (a) The photonic band structure at normal incidence for a cholesteric. The gap arises because at a wavelength corresponding to the pitch, light can be polarised either along or perpendicular to the director (b), after Kopp and Genack. Arrows connect the two modes to the appropriate energy states in the band structure. the electromagnetic eigenstates of the cholesteric include plane polarised waves where the plane of polarisation rotates at the same rate as the helix. For the mode where the electric displacement vector D of the optical field always points along n , the field energy density 21 D2 /
 is low. When D is always perpendicular to n , the energy density   1 2 1 1 1 D2 , even though their 2 D /
⊥ is high. The two waves differ in energy by 2
−
⊥



wavelengths in the medium, Λi = p, are the same. The dispersion relation ω (k) has a gap in the allowed energies corresponding to the internal k = q. The figure uses reduced units for ω and k, to be detailed below. Right-circularly polarised light with a frequency in this gap, projected normally on the material as a ‘stop band’, will be totally reflected. The eigenstates just visualised are really standing circularly polarised waves with resulting linear polarisation (in an infinite medium) rotating with the cholesteric structure. Outside the sample, the interior eigenstates translate into right and left circularly polarised light. At the gap it is incident circularly polarised light of the same handedness (denoted by RP for concreteness) as the cholesteric that is reflected. The dispersion relation of Fig. 9.14 only has a gap for one mode (RP, rotating with the structure) and not for the other. Away from the gap, both modes have linear dispersion relations with the same slopes since they sense the same averaged refractive index as they rotate relative to the cholesteric helix. The ω (k) relation is plotted in the reduced band scheme common in electronic structures. Wave vectors with k > q are in a sense equivalent to those at (k − q) in the first by subtraction of a reciprocal lattice vector q since the system is periodic with period p = 2π /q. We thus fold back the band structure

248

CHOLESTERIC ELASTOMERS 2

~ w 3/2

1

1/2

(0, 0)

(0, 1/2)

(1/2, 1/2)

(0, 0)

~ ~ (kr , kz) (1/2, 0)

F IG . 9.15. The photonic band structure at oblique incidence for a cholesteric liquid crystal (Bermel and Warner, 2002). as it continues to higher k, and back again for k > 2q and so on. Note that there is only one gap, at k = q, and not any further gaps for higher k that one might have expected from electronic band structure. The ‘potential’, roughly equivalent here to the spatially dependent
(z), varies with only one spatial Fourier component. Oblique incidence enriches this picture. The local uniaxiality of cholesterics was established by oblique incidence analysis, in fact just at 45o (Berreman and Scheffer, 1970), but there are also more general approaches (Dreher and Meier, 1973; Belyakov et al., 1979). Figure 9.15 shows the dispersion relation for varying angles of incidence. As in electronic band structures, trajectories in k-space are chosen which can amount to varying angles. The normal wavevector is kz and that in the plane of the sample, perpendicular to the helix axis, is kρ . Thus the first segment represents normal incidence (kρ = 0) and is as in Fig. 9.14. The next goes from the zone centre at normal incidence to 45o at fixed magnitude of the z-component of wavevector in the medium, the next is fixed at 45o with reducing wavevector, and the last is grazing incidence, 90o . Small gaps in the eigenstate of the wrong handedness (LP) open up away from normal incidence. Still more complex band structures arise in liquid crystal blue phases (Hornreich et al., 1993). Liquid cholesterics can have their helical structure distorted by electric or magnetic fields applied perpendicular to the helix axis, case 1 in Sect. 2.8. The pitch lengthens and the helix coarsens, both factors changing the photonics (Chou et al., 1972; Dreher, 1973; Shtrikman and Tur, 1974). Lengthening the pitch, po → p, gives a smaller helix wavevector qo → q = qo · (po /p) and hence a shift of the gap to longer wavelengths. The helix coarsening distorts the initially purely sinusoidal variation of dielectric properties. Higher harmonics appear in the ‘potential’
(z) governing the band structure – more gaps open up, even at normal incidence. The rotating n is no longer static in the de Vries frame (which uniformly rotates with the same period as the now distorted helix). Coarsening means that it appears to rotate forward in the de Vries frame where the helix has sharpened up, and backwards where the helix is varying more slowly than before. Figure 9.5(b) gives an example of angular advancement in a deformed helix ( or 2) compared with the straight line φo = qo z in the undeformed case (◦). At first a deformed

PHOTONICS OF CHOLESTERIC ELASTOMERS

249

helix advances more slowly (relatively backwards) and then, near qo z ∼ π /2, more rapidly (relatively forwards). This variation with respect to the de Vries frame means that waves rotating oppositely with respect to the cholesteric also couple to the spatially varying
and gaps will open up for this handedness too. We shall find, theoretically and experimentally, all these effects in elastomers too. 9.5.2

Photonics of elastomers

An undeformed elastomer has the same helical structure and band structure as a simple liquid cholesteric, see Figs. 9.14 and 9.15. The large changes from simple to coarsened helices when elastomers are deformed suggest they will develop greatly modified band structures, colours (gaps) and lasing. Non-uniform rotation means we cannot simply transform to the de Vries rotating frame to solve for the single-gap band structure, but have to solve the general problem which we outline below. [In fact the helices of cholesteric elastomers can be coarsened even in the relaxed state if there is diffusion in an optical field during polymerisation (Broer et al., 1999)]. The Maxwell wave equation for the magnetic field in an optical wave of angular frequency ω is:

ω2 ∇ × H) = 2 H . ∇ ×
−1 (rr ) · (∇ c

(9.22)

The inverse dielectric tensor is:  1 1 1 n n + δ or, in components, −
−1 =

⊥
⊥    cos 2φ sin 2φ 0
−1 = b δ − α  sin 2φ − cos 2φ 0  ≡ b
˜ −1 , 0 0 −1

(9.23)

where
˜ −1 is an inverse reduced dielectric tensor. The related constants are, following the de Vries’ notation,
 −
⊥ α=
 +
⊥

and

b=

1 2

1 1 +

⊥

 .

(9.24)

The optical anisotropy α is not to be confused with the degree of semi-softness or with the factor by which volume changes during deformation, neither of which will appear in this section. The form of
−1 is reminiscent of −1 and likewise recognises that the vectors n and −nn are indistinguishable. Reduced variables (denoted by ˜) arise: (i) Extract the average measure of inverse dielectric constant, b from
−1 as in the definition (9.23). Likewise α is a reduced dielectric anisotropy. (ii) Lengths are reduced as z˜ = 2qz. Remember that the helix may be shifted by contractile deformations, for instance q = qo λ 2/7 in the limit of small strains; in general q = qo /λzz . (iii) Since lengths have been reduced, then so are wavevectors,

250

CHOLESTERIC ELASTOMERS

√ k˜ = k/2q. (iv) Frequency is reduced as ω˜ = ω /(2cq b) (which will also depend on deformation λ ). In reduced form, the wave equation becomes:   ˜ × H ) = ω˜ 2 H . ˜ ×
˜ −1 (˜z) · (∇ (9.25) ∇ Since
˜ −1 (˜z) is periodic in z˜ we can expand solutions to eqn (9.25) using Bloch’s theorem: ˜ ˜ (9.26) H (˜r ) = ∑ h G˜ ei(k+G)·˜r , ˜ G

˜ are reduced reciprocal lattice vectors and k˜ is a general wavevector, where the vectors G which however can be reduced to lie in the first Brillouin zone. One can choose a 2D ˜ · H = 0, that is H is transverse. ˜ since ∇ basis for the expansion functions h G˜ for each G We spare the reader the details of the band structure calculation and concentrate on representative results. Reduced units allow plots of photonic band structure universal to all elastomers. However the observed coloured reflections underscore that one actually observes the exterior eigenfunctions and wavelengths. These are matched to the interior wavefunctions and wavelengths we are describing here. Recall that the reduced frequency is √ ω˜ = ω /(2cq b). The angular frequency both outside and inside the elastomer is ω = c2π /Λo , where Λo is the in vacuo wavelength we associate with, for example, the observed colours. Eliminating the real angular frequency ω , using q = 2π /p and p = po /λ 2/7 , one obtains: √ (9.27) Λo = po /(ω˜ bλ 2/7 ) . The band structures have the reduced k˜ as an independent variable which produce a dependent reduced frequency ω˜ that eqn (9.27) translates into vacuum colours. The Brillouin zone centre is at k = q which translates into k˜ = 1/2 in reduced units. Normal and oblique dispersion relations are shown in Fig. 9.16 where the distortion is λ = 1.1 and the anisotropy r = 1.9. The deformation is less than λc and hence complete, albeit not uniform, twists still exist. Gaps in modes of the same handedness (RP) as the cholesteric open up at (reduced) wavevectors of 0 and 1/2 where there were none before, since harmonics have been introduced into
(z). A gap in modes of the opposite handedness (LP) opens up, inside what was the original de Vries gap. Where modes once crossed, they now anti-cross or repel each other (e.g. at k˜ ∼ 0.1), a phenomenon common in electronic band structures. When the modes meet and repel, they exchange some of their symmetry properties, for instance here the predominant sense of twist. The effective gaps move away from the zone centres and edges. The oblique structure at the same λ shows a similar complexity of new gaps, widening gaps and anti-crossings to which we return when discussing lasing. Figure 9.17 shows the equivalent structure for λ = 1.3 > λc . Now gaps are of sufficient width in both handednesses that gaps and anticrossings, in photonic bands are well away from the zone edges and centres. The effective backward rotations, as perceived in the de Vries frame, induce these new features. In fact, now that we have exceeded the

PHOTONICS OF CHOLESTERIC ELASTOMERS 2.5 ~ w 2

2 ~ w

RP

3/2

LP

1.5

1

1

1/2

0.5 0 0

251

0.1

0.2

0.3

0.4 ~ k

0.5 (0, 0)

(0, 1/2) (1/2 ,1/2) ~ ~ (0, 0) (kr , kz )

(1/2, 0)

F IG . 9.16. Normal and oblique photonic band structures for a cholesteric elastomer with anisotropy r = 1.9 stretched perpendicular to its helix axis by λ = 1.1 < λc . 2.5 ~ w 2 1.5

2 ~ w 1.5

RP

LP

1

1 0.5

0.5 0 0

0.1

0.2

0.3

0.4 ~ 0.5 k

(0,0)

(0,1/2)

(0,0) (1/2,1/2) (k ~,~ k) r z

(1/2,0)

F IG . 9.17. Normal and oblique photonic band structures for a cholesteric elastomer (r = 1.9) stretched by λ = 1.3, a deformation greater than that critical for unwinding. critical strain, there is no longer any net rotation, forwards or backwards. Both handednesses couple to essentially the same extent as each other to the varying dielectric tensor. 9.5.3

Experimental observations

We remind the reader that these universal plots using reduced variables hide the shifts with distortion in the characteristic features of the band structures. One recovers these from the affine contraction of the helical pitch with the elastomer thickness reduction, Λo ∼ λ −2/7 . Figure 9.18 illustrates experimentally the qualitative points we have discussed (Cicuta et al., 2002; Cicuta et al., 2004). The logarithm of attenuation of a through beam (by reflection at the incident surface) is plotted again vacuum wavelength Λo . There is some absorption in the UV by chemical bonds, which grows to dominance at about 350nm, that is not connected with the band structure. This can be seen in the left-circular (L∗ ) spectrum with no mechanical distortion which is devoid of any features. By contrast the right-circular (R∗ ) spectrum at λ = 1 has an edge about 800nm with a single stop band extending to about 650nm. There are no further gaps and attenuation only rises again at short wavelength because of chemical absorption, as in

252

CHOLESTERIC ELASTOMERS

the corresponding left-circular spectrum. On applying strain, the band moves to shorter wavelengths. The scaling of the selective reflection wavelength with distortion is one of the data sets shown earlier in Fig. 9.5(a). It indeed shifts close to λ −2/7 . The first shifted spectrum (at λ = 1.13) shows evidence of a second band gap starting at about half the wavelength of the de Vries gap, that is at Λ ∼ 400nm. The other edge of the gap is not seen since UV absorption intervenes at about 350nm, as we noted from the first spectrum. At the next spectrum, λ = 1.36, both edges have shifted further to the UV, the second edge coalescing with the chemical region. Future, true reflection experiments should be able to resolve this since the reflected beam does not suffer the chemical attenuation associated with passage through the sample, although there are modifications to the band structure associated with absorption. By λ = 1.72 it is clear that many gaps are active and a new band edge at lower wavelengths has intervened. One needs more complete data to map these results fully onto spectra such as in Figs. 9.16 and 9.17. Strain l

F IG . 9.18. Photonic band structures for right (R∗ ) and left (L∗ ) circular polarised light normally incident on cholesteric elastomers subjected to extensions (marked) perpendicular to the helix axis. In effect, reflection is plotted against vacuum wavelength (Cicuta et al., 2004).

PHOTONICS OF CHOLESTERIC ELASTOMERS

253

The L∗ spectra offer dramatic evidence for the mechanical generation of director twist that is reversed in parts of a period with respect to the uniform twist of a simple cholesteric, see Fig. 9.5(b). With such backward components of twist one can then couple to the left rotating eigenfunctions (LP) in the elastomer and induce gaps which are sensed by the left circularly polarised incident light in Fig. 9.18. For small strains there is little or no structure, which then develops at appreciable strains (note the dotted line showing the spectrum at the extension of 9%). Further, at λ = 1.36 one sees a LP gap inside what was the de Vries gap of the RP spectrum (as always, affinely shifted by λ −2/7 ). At large λ where there is essentially as much forward as backward twist, the two spectra resemble each other in their general features. In effect, the helix loses its chirality and turns into a sequence of aligned regions separated by twist defect walls (Cicuta et al., 2004). A new photonic gap associated with diffusion during photo-assisted cholesteric elastomer formation has also been detected in experiments of Broer et al (1999). The same group has also explored another very interesting route to modification of band structure – by the variation of pitch with distance into the sample (Broer et al., 1995; Broer et al., 1999). Gaps for different frequencies of light occur at different depths in the sample, crudely where the wavelength of light in the sample matches the local repeat distance of the helix. Thus a broad band of colours of light is completely reflected, see also theory (Kutter and Warner, 2003). Such materials are of great importance in liquid crystal displays. 9.5.4

Lasing in cholesterics

Low threshold lasing in cholesterics was proposed long time ago (Goldberg and Schnur, 1973; Yablonovitch, 1987) because of the recognition that the band structure can greatly modify the spontaneous emission from pumped molecules and can enhance the stimulated emission. Consider an excited dye molecule with its emission line in the gap of its cholesteric host. It will be difficult to find electromagnetic modes into which to emit. Suppression of fluorescence decay (spontaneous emission) from impurities in cholesterics is a well-known phenomenon. It has been discussed in the context of cholesteric elastomers (Schmidtke et al., 2003). The authors give pointers to the earlier literature. The extension of the gap to angles away from normal means that a cone of modes is forbidden to a molecule with its emission line in the gap, further suppressing emission. Thus pumping to excited states more easily creates a population inversion. Eventually there is emission into the allowed state closest in energy to the emission line, that is into a state at the edges of the bandgap enveloping the excited level. The band edges have very special properties since at that point the group velocity of light, d ω /dk, vanishes – see Figs. 9.14, 9.16 and 9.17. At this point the stimulated emission is greatly enhanced: The ‘dwell time’ that a photon spends in the medium will diverge as its velocity vanishes. Crudely speaking it is present long enough to stimulate excited molecules to emit at the same frequency, thereby building up a coherent wave. Such photonic materials are referred to in this context as ‘mirrorless cavities’. Alternatively, the rate of emission by excited molecules is, by Fermi’s Golden rule, proportional to the density of states at the resultant energy. The density of states at the band edges

254

CHOLESTERIC ELASTOMERS

F IG . 9.19. A phase jump of the director in a cholesteric structure arises when two sheets of rotated elastomer are glued together (sketch J. Schmidtke). is enhanced since it is proportional to the inverse of the group velocity, dk/d ω . The emitters, which are surrounded in energy by forbidden states, have a large coupling to the first states encountered after the gap. It is often observed that edge lasing occurs at the upper edge since the cone of forbidden states around the edge is greater than in the lower case [but see below]. Lasing in cholesterics has been subsequently observed (Kopp et al., 1998; Alvarez et al., 2001). There are complications – sometimes one has the edge lasing described above, at other times it is in defect states in the gap. The latter are localised, weakly coupled to the outside world and hence long-lived, and surrounded by forbidden extended states. Lasing has also been reported in cholesteric elastomers (Finkelmann et al., 2001b; Schmidtke et al., 2002). Initially the elastomers were biaxially stretched in the plane perpendicular to the helix axis. Such stretch increased the order by diminishing any wandering of the helix axis and thus sharpened up the band structure. As the elastomers were stretched, they were of course seen to change colour. It was also found that more tightly crosslinking the elastomer increased the order, thus reducing the lasing threshold, raising the quantum yield and improving the emission stability. Lasing observed at the lower edge (in frequency) was explained in terms of the polarisation of the relevant electromagnetic mode being parallel to the director and thus, in the case of the dyes used in these elastomers, being on average parallel to the dye’s transition moment. Elastomers offer more than just the advantage that their band structure and hence lasing can be mechanically deformed. Phase slippage of the cholesteric can be manipulated in the elastomer whereas it is seemingly impossible to do this in a localised way in the liquid state, see Fig. 9.19. Thus the theoretical proposal (Kopp and Genack, 2002) that photonic defects arising from such twist defects be used for lasing can be realised in practice (Schmidtke et al., 2003). Figure 9.20 gives a summary of the results. The main lasing emission occurs at the photonic band edge, with a light circular polarisation matching the cholesteric helix. However, this emission occurs at a substantial pumping threshold, Wp ≥ 50 nJ. At a much lower pumping level (threshold Wp ∼ 2.5 nJ, corresponding to the energy density ∼ 100 J/m2 ), one finds the lasing emission due to the defect mode. This occurs in the middle of the gap, with a much lower intensity and with the opposite circular polarisation. Undoubtedly more subtle forms of lasing and exploitation of the photonic band structure of cholesteric rubber await us in the future.

PHOTONICS OF CHOLESTERIC ELASTOMERS

L*

Intensity (a.u.)

Intensity (a.u.)

Wp =39 nJ

Defect mode

255

L* R*

R* Pump energy Wp (nJ)

L* R*

Band edge mode

Intensity (a.u.)

Intensity (a.u.)

Wp =91 nJ

L* R*

Pump energy Wp (nJ)

(a)

Wavelength (nm)

(b)

F IG . 9.20. Lasing in cholesteric networks. Plots (a) show the emission intensity at two different levels of pumping power Wp , in left- and right-handed circular polarisation. The low-threshold 591nm emission is due to the defect mode; the 613nm emission is at the lower band edge. Plots (b) show the emission intensity of both lasing modes as function of pump energy, indicating the lasing threshold and emission polarisation; from (Schmidtke et al., 2003).

10 CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

With the molecular theory of nematic networks, developed in the preceding chapters, many details of structure and properties of these materials have a clear explanation. The main success of this neo-classical theory of liquid crystalline rubber elasticity is in its ability to grasp a broad range of effects without much knowledge about the specific microscopic parameters. Therefore, many effects apparently have a truly universal nature – much sought after in physics in general. In spite of this seemingly fortunate situation, many partially explained phenomena depend of detailed molecular models, for instance the effects of molecular chirality on physical properties, random disorder introduced by quenched network crosslinks and its effect on macroscopic elastic response, additional constraints on polymer conformation in smectic phases of liquid crystalline elastomers and gels, and the practically important influence of crosslinking history and chemical composition of liquid crystalline networks. This specificity of behaviour is in contrast with the universality of the neo-classical theory of uniform nematic elastomers. While awaiting for such models to be developed and reconciled with each other, one is tempted to look at liquid crystalline elastomers and gels at a level of even greater universality, based on symmetry of the phases and corresponding degrees of freedom. In this chapter we shall also examine symmetry arguments behind soft elasticity. Constraints on non-linear deformations will determine the full form of soft deformation tensors and their associated rotations given within rubber theory in Chapter 7. Linear continuum theory based on the small deformation approximation so successful in general elasticity, has weaknesses from the very start when applied to rubbers. Rubber often involves large strains and, when applied to liquid crystalline systems, discontinuous orientational transitions. Both phenomena mean that an expansion-based phenomenological approach may be difficult to justify. For instance, a simple ‘antiFreedericks’ transition – reorientation of a uniform nematic elastomer under perpendicular mechanical extension, Sect. 6.7, cannot be fully described by a linearised continuum theory based on relative rotation couplings such as eqn (7.3). However, one should weigh this conceptual disadvantage against many possible benefits. Such possible benefits include the ability to predict a variety of new effects on the basis of phase symmetry only. In some situations, one can extend parts of the phenomenological description to the non-linear regime of large strains. A consistent non-linear continuum free energy is not yet available. In many cases the linearised continuum theory is fully sufficient, for example for the description of small fluctuations or small-amplitude acoustic waves. One may hope that the predictions of continuum theory show an informative trace of real physical effects. With this hope in mind, we shall proceed. 256

FROM MOLECULAR THEORY TO CONTINUUM ELASTICITY

257

10.1 From molecular theory to continuum elasticity The ideal nematic elastomer free energy density is a simple extension of the isotropic Gaussian elasticity of conventional rubbers:   (10.1) Fel = 12 µ Tr  o · λ T ·  −1 · λ . Recall that the polymer chain step-length tensor  o describes an average strand before the deformation has been applied (and is, therefore, a function of the initial director orientation n o ) and the tensor  describes the chain distribution in the deformed state of the elastomer (in which the director n may deviate from n o quite significantly). We ignore variation of the magnitude of the order parameter. We have seen in Chapter 6 how such variation can be taken up in the moduli. The elastic strain is defined in terms of the change in the position of a material point R o in the undeformed body to its position R in the deformed body (cf. Sect. 4.1):

λαβ =

∂ Rα . ∂ Roβ

(10.2)

We first examine how the small compressibility of nematic elastomers affects their response. We then reduce the Trace formula above to its continuum limit for small deformations, thereby identifying the seven elastic constants that appear in the elasticity of a uniaxial solid with an internal degree of freedom, here a nematic director. 10.1.1 Compressibility effects Compressibility issues have to be re-analysed for nematic rubber elasticity. Retracing the argument in Sect. 6.3, we first decompose the full strain into the initial spontaneous compression at formation and the subsequently imposed strain λ˜ and then separate this imposed deformation λ˜ into the additional small volume change and the remaining volume-preserving part λ ,

λ total = λ · (1 + α ) (1 + α (o) )δ .

(10.3)

Here λ˜ = λ · (1 + α )δ , see Sect. 4.4 for a systematic derivation, in particular eqn (4.48) (o) is the starting point. The network contracts at formation by an amount αeq. ≈ −µ /3B˜ with B˜ the bulk modulus of the rubber, and the resulting elastic free energy density is 2      µ 2  µ 3 T −1 ˜ 1 1# ˜ ˜ Tr  o · λ ·  · λ + 2 B 1 − Det λ − 1 (10.4) Fel = 2 µ 1 − 3B˜ 3B˜   with no a priori assumption on the value of Det λ˜ = (1 + α )3 . After minimisation of this expression, the equilibrium volume change in response to the subsequently imposed deformation is found to be, as in (4.50),   µ (10.5) αeq. ≈ − Tr  o · λ T ·  −1 · λ 9B˜ (o) [which includes the small spontaneous contraction αeq. = −µ /3B˜ at formation, checked as the limiting case of λ = δ in eqn (10.5)]. After substituting both α (λ ) and α (o) , the

258

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

effective nematic rubber-elastic energy arising in response to the volume-preserving part of strain λ becomes   Fel = 12 µ Tr  o · λ T ·  −1 · λ (10.6)   at µ /B˜  1 and with the condition Det λ = 1 satisfied exactly, by construction of eqn (10.3). As in the isotropic case of Sect. 4.4, the rubber actually is weakly compressible, changing its volume to preserve its constant pressure on deformation. However, for all practical purposes, this effect is small and the remaining relevant shear deformation λ is volume-preserving with an unchanged elastic energy. 10.1.2

The limit of linear elasticity

Let us now turn to the linearised limit of the theory when the elastic deformation and the potential director rotation are sufficiently small to justify Taylor expansions. Then changes in positions of material points Ro can be written as R = Ro + u, with the vector u(rr ) describing small distortions. This introduces the small deformation tensor uαβ :

λ˜ αβ = δαβ + uαβ ,

uαβ =

∂ uα . ∂ xβ

(10.7)

The tilde here signifies the fact that λ˜ is not yet made incompressible, cf. eqn (10.3). Usually, only the symmetric part of the deformation tensor, ε˜αβ = 12 (uαβ + uβ α ), plays a role in the elastic response. In liquid crystalline elastomers, with a possibility of local torques provided by the independent director rotation, we must not forget the antisym(a) metric part uαβ = 12 (uαβ − uβ α ). At small deformations this antisymmetric part, a rotation of the body, is often represented by a vector, Ωγ = 12
αβ γ uαβ , or Ω = 12 curl u , see the discussion of Sect. 4.3. In a particular coordinate frame, matrices representing small deformations and rotations are:         λxx λxy λxz εxx εxy εxz 100 0 Ωz −Ωy  λyx λyy λyz  ≈  0 1 0  +  εxy εyy εyz  +  −Ωz 0 Ωx  . (10.8) λzx λzy λzz εxz εyz εzz Ωy −Ωx 0 001 In fact the expression (4.29) for the matrix of finite rotations about z shows that sin Ω and cos Ω are involved, instead of just values of Ω, and in particular that there are diagonal entries of O(Ω2 ). Since we are deriving a free energy quadratic in distortions and rotations, we shall have to take greater care than in eqn (10.8), see Appendix E for details. Rotations of the solid matrix and of the director are depicted in Fig. 10.1. An infinitesimal rotation Ω of the solid matrix causes any vector, v say, to suffer a change δ v = Ω × v to give the new vector v  . The rotation vector Ω has to have a component perpendicular to v in order to have any effect on it. For simplicity Fig. 10.1 only shows the perpendicular case. Similarly, the director changes due to its infinitesimal rotation

FROM MOLECULAR THEORY TO CONTINUUM ELASTICITY






Æ
ª

Æ









ª



¼

ª


259




ª ¼




¼




¼

ª

ª
F IG . 10.1. Rotation of n about an axis ω (left), a body rotation Ω rotating body vectors v. In the frame attached to the body vectors v, the director n undergoes a relative rotation of Ω − ω (centre).The figure shows an extreme case, where the two rotations Ω and ω are around the same axis, but in opposite senses. will be written as n = n o + δ n with the small variation δ n being perpendicular to n o at linear order for unit vectors. The change in director is given by δ n = ω × n . We reserve ω for rotations of the director, while Ω is for the elastic matrix. Only rotations of the director relative to the solid matrix have any physical importance. A body rotation of both together leads to neither elastic nor nematic distortion. The final part of Fig. 10.1 illustrates the difference in directions of n and v after they have suffered rotations ω and Ω respectively. The rotation of n relative to the solid, Ω − ω , is accordingly Ω−ω)×n . δ n Ω − δ n ω = (Ω One should explicitly separate the traceless part of symmetric strain: εαβ = ε˜αβ − since the Trace, Tr[ε˜ ] expresses at lowest order the volume change associated with ε˜ . Up to quadratic order in strain, in the limit of full incompressibility, the result of this transformation is given by  (r − 1)2    2 1 (nn · ε · ε · n − (nn · ε · n ) ) (10.9) Fel = 2 µ 2Tr ε · ε + r  (r − 1)2 r2 − 1 Ω − ω ) × n ]2 + 2 Ω − ω ) × n] . + 21 µ [(Ω n · ε · [(Ω r r 1 ˜ 3 Tr[ε ]δαβ

Exercise 10.1: Derive the anisotropic elastic terms in the expression (10.9) for Fel Solution: Relative-rotation coupling terms, involving director variations, are discussed in the next section. Consider therefore just the elastic terms quadratic in symmetric small strain ε . The step-length tensors in the Trace formula deviate

260

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS from spherical as  o = δ + (r − 1)nnn and  −1 = δ + ( 1r − 1)nnn (we do not distinguish between n and n o since such a correction would be of higher order in small distortions). Putting these tensors into the Trace formula, one obtains:

Fel =

    1 Tr λ T · λ + (r − 1)nn · λ T · λ · n + ( − 1)Tr (λ T · n )(nn · λ ) r  (r − 1)2 (nn · λ T · n )(nn · λ · n ) − r 1 2µ

T Considering  only symmetric  strains, we can drop the transposition notation . Re(r−1)2 arrange Tr (λ · n )(nn · λ ) = n · λ · λ · n and regroup the middle terms as r n ·       λ · λ ·nn. Putting λ = δ + ε , then one obtains Tr λ · λ = 3+2Tr ε +Tr ε · ε =       3 + 2Tr ε · ε , because Tr ε = 12 Tr ε · ε + . . . by incompressibility. Gathering these and like terms from the expansion, gives the first line of eqn (10.9).

10.1.3

The role of nematic anisotropy

There are three terms in (10.9) that hinge upon the anisotropy of the nematic elastomers. They are proportional to (r − 1)2 and (r2 − 1). It will be helpful for the remainder of the chapter to establish pictures of anisotropic strain terms such as (nn · ε · n )2 − (nn · ε · ε · n ). The strain ε has indices which refer to directions, see Fig. 10.2. We can ignore director rotations in the second order symmetric strain terms in Fel since corrections for a changing director would introduce terms of still higher order. In these terms n is not different from n o . We shall sometimes take a particular coordinate system (x, y, z), which is indicated on the figure in addition to the frame-independent set of orthonormal vectors n , l and m . The elements of the tensor can be written out as

ε = εnn n n + εnm (nnm + m n ) + . . .

(10.10)

[We abandon the summation over repeated indices for l, m and n in this subsection.] Thus one can find out trivially that n · ε · n = εnn , only one diagonal element of ε , whereas n · ε · ε · n must involve a sum of three terms, εnm εmn + εnl εln + εnn εnn , where nn

n

n

nO

nO ll

nO

z

mm ml

nm

nl

C1

C4

C5

y

x

F IG . 10.2. Elements of strain illustrating their orientation with respect to the director n in a uniaxial medium, corresponding to the relevant elastic terms in eqn (10.12).

FROM MOLECULAR THEORY TO CONTINUUM ELASTICITY

261

the outside parts of ε · ε are forced to take n because of the dot products n · and ·nn. Overall the anisotropic strain term in Fel is 1 2µ

(r − 1)2 [εnm εmn + εnl εln ] . r

Figure 10.2 shows representative strain terms. Since there is symmetry of rotation about the director n , all orientations of the axes (l, m) are equivalent and all deformations εmm , εll and εlm are related by rotating the axes. The extensions εnn happen, for nematic elastomers without director rotation, to be equivalent to those such as εmm in the perpendicular plane. Examples of this are given in Chapter 6 and in particular in eqn (6.37) on page 140. This equivalence is a consequence of Gaussian elasticity and explains why for nematic elastomers there is only one anisotropy term, and not another describing 2 ) and in-plane (ε 2 etc.) distortions. When the equivalthe difference between axial (εnn ll ence is lost due to order parameter relaxation, there is an additional distinguished elastic constant, see Sect. 6.6 (page 143) where tensor relaxations of order are discussed. The only shears to retain a distinct character are then εnl and εnm which span the unique axis n and a transverse direction, see Fig. 10.2. Being different from the extensions along principal directions we discuss above, they are the only contributors with the (r − 1)2 factor. The last two terms in (10.9) are the relative rotation couplings phenomenologically introduced by de Gennes in 1982. They are unique to nematic networks because they require an independent, rotational degree of freedom (the director n and its rotation ω ) in the anisotropic elastic medium and thus involve antisymmetric components of shear strain. We discussed the full non-linear relative rotation coupling, from which the linearised expression D1 here derives, when exploring our first example of director rotation in the Trace formula, Sect. 7.1. The D2 term couples n to the symmetric part of shear in the plane that involves n (e.g. εzx if no = nz in equilibrium). These are the shears εnl and εnm in Fig. 10.2. Since infinitesimal, incompressible symmetric shear is equivalent to stretch along one diagonal and compression along another, it is reasonable that prolate molecules will reduce the cost of distortion by rotating their ordering direction to being as much as possible along the elongation diagonal. Figure 10.3 depicts this for both prolate and oblate elastomers, the latter rotating its director toward the compression diagonal. Oblate elastomers, having flattened, plate-like molecular distributions thereby arrange one of their extended directions to be along the elongation diagonal. Both D1 and D2 arise from anisotropy, r. Chain shape depends on the orientational order parameter, Q, through the connection r − 1 ∝ Q. The D1 coupling penalises rotation of the director relative to the solid matrix and is insensitive to the sign of order, prolate or oblate, and hence D1 ∝ (r − 1)2 as we see in eqn (10.9), whereas we show in Fig. 10.3 that D2 is sensitive to the sign of order. Indeed in eqn (10.9) this is so since D2 = 2µ (r2 − 1)/r ≡ 2µ (r − 1)(r + 1)/r ∝ Q. Figure 10.3 directly illustrates the geometry of n · ε · (ω × n ). The change in director δ n = n × ω is perpendicular to n (it is along m for the choice we have made in the figure). The only shear that can couple to this

262

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

nO dn

dn

nO

nl

nl

(a)

(b)

F IG . 10.3. Symmetric shear induces director rotation (a) in prolate elastomers toward the elongation diagonal (b) in oblate elastomers toward the compression direction. is εnm n m, by inspection of the vectors associated with εnm . Terms like m · εmm · (nn × ω ) are disallowed in Fel because they lack the necessary nematic invariance under n → −nn. A number of important physical effects in liquid crystalline elastomers arise from Ω − ω ) × n] to the extensional components of elastic the coupling of relative rotation [(Ω strain, for instance the director rotations discussed in Sect. 6.7 and Sect. 7.4.2. Such coupling turns out to be higher order in nominally small deformations δ n and uαβ , but in many physically relevant situations it will represent the leading effect, especially when an external deformation is imposed. The required additional terms, of symmetry differing from quadratics explored in the expression (10.9), are notionally cubic since they have two rotations and one strain:  r−1 Ω − ω ) × n ] · ε · [(Ω Ω − ω ) × n] [(Ω (10.11) −µ r Ω − ω ) × n ]2 (nn · ε · n ) +µ (r − 1)[(Ω Ω − ω ) is about the axis m (the y-axis in Fig. 10.2), then If the relative rotation (Ω Ω − ω ) × n is along l and the extension in the first cubic term is εll . In the δ n = (Ω second, the extension is clearly εnn , that is along n . This type of coupling leads to rotation of a director (initially, along, e.g. the z axis, no = nz ) in response to extension of the elastomer in the perpendicular direction, e.g. εxx . This was the cause of director re-orientation considered in Sect. 8.2 on stripe domains. One may also note that the effect of these terms is again opposite for oblate and prolate chains (r =  /⊥ less or greater than 1): stretching perpendicular to the director would cause a rotation of prolate-ellipsoid chains (see the negative coefficient in the first term), whereas oblate chains rotate their principal (short) axis away when stretch is along the initial director. 10.2

Phenomenological theory for small deformations

We have been fortunate in being able to actually derive  the linearisedcontinuum elastic energy density (10.9) from the molecular model Tr  o · λ T ·  −1 · λ . One might have taken a different, phenomenological route, obtaining the possible elastic terms in the expression (10.9) as scalar invariants involving the square powers of the two relevant physical fields, u ˜αβ and ω , allowed by symmetry transformations. Unusually, now the nematic medium with its essentially mobile director provides an energy penalty for its

PHENOMENOLOGICAL THEORY FOR SMALL DEFORMATIONS

263

uniform rotation, δ n , and for the rotation of the whole sample, 12 curl u , as long as they are not ‘in phase’. In other words, it is the relative rotation about any axis perpendicular Ω − ω )×nn], that is a valid physical variable of continuum to n , described by the vector [(Ω free energy – together with the usual translational and curvature deformation fields, εαβ and ∇ δ n , respectively giving Cauchy and Frank nematic elasticity. A model-independent linear continuum theory should be based on the symmetry of a uniaxial elastic medium and, thus, have five independent elastic moduli (three in the case of rigidly incompressible systems), plus two quadratic nematic-rubber coupling terms. Written in the frame-independent form analogous to (10.9), the continuum free energy density of a nematic elastomer is  2 F = C1 (nn · ε · n )2 + 2C2 Tr[ε˜ ](nn · ε · n ) +C3 Tr[ε˜ ] +2C4 [nn × ε × n ]2 + 4C5 ([nn × ε ] · n )2 plus : the relative rotation coupling terms Ω − ω ) × n ]2 + D2 n · ε · [(Ω Ω − ω ) × n] + 12 D1 [(Ω

(10.12)

plus : higher order (cubic) relative rotation terms Ω − ω ) × n ] · ε · [(Ω Ω − ω ) × n] + 12 D22 [(Ω Ω − ω ) × n ]2 (nn · ε · n ) + 12 D33 [(Ω plus : the Frank elasticity part involving ∇(δ n ) + 12 K1 (div δ n)2 + 12 K2 (nn · curl δ n)2 + 12 K3 [nn × curl δ n]2 ˜ As before, ε is the traceless, symmetric part of strain The linear bulk modulus C3 ≡ 12 B. tensor. Note that most of the terms in the free energy density (10.12) contain the traceless ε , which is not always the case with analogous expressions in the literature. The difference is trivial, but establishing the direct connection between terms in these two representations may be quite tricky. The directors n and n o are not distinguished for small director rotation, apart from in the relative rotation coupling terms via the variable ω . We have also included the Frank elastic energy describing non-uniform directors, where div and curl are shown acting on δ n , the part of n that varies, and reminds us that n is a unit vector with variation perpendicular to itself for small changes. With this, the quadratic-level continuum theory of nematic elastomers is complete. The coupling terms D22 and D33 are nominally of the third order in small deformations – they are proportional to (δ n)2 ε . However, we must keep them (while discarding many other third-order terms in F) because in some cases, in particular when an extensional strain is applied to the sample, this is the leading coupling between elastic strains and director rotations. With the aid of Fig. 10.2 and eqn (10.10) one can easily form a geometric picture of each strain energy term. In eqn (10.10) one sees that the tensor ε has vector ‘legs’ which then have to form vector or scalar products with the vector n in terms like n × ε × n . In this particular example the vectors associated with ε cannot be n , but must be l and m , that is they must be in the plane perpendicular to n . The relevant shears are then in this plane, that is εmm , εml and εll , examples of which are shown in Fig. 10.2. Another

264

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

important example is n × ε · n for which one leg of ε has clearly to be n and the other must be perpendicular to n , that is m or l . These are the shears in planes encompassing n , for instance εnl . One can re-write the free energy density F in a specific coordinate frame where the z-axis is chosen along the initial nematic director n o . The new director defines δ n via n(rr ) = no + δ n(rr ) and no = (0, 0, 1) We then have a more familiar elastic expression: F

 2 = C1 εzz 2 + 2C2 Tr[ε˜ ] εzz +C3 Tr[ε˜ ] +

(10.13)

+2C4 (εxx 2 + 2εxy 2 + εyy 2 ) + 4C5 (εxz 2 + εyz 2 ) plus : the terms involving δ n   (a) (a) + 12 D1 (uxz − δ nx )2 + (uyz − δ ny )2   (a) (a) −D2 (uxz − δ nx ) εxz + (uyz − δ ny ) εyz ,  (a) (a) + 12 D22 (uxz − δ nx )2 εxx + (uyz − δ ny )2 εyy  (a) (a) +2(uxz − δ nx )(uyz − δ ny ) εxy   (a) (a) + 12 D33 (uxz − δ nx )2 + (uyz − δ ny )2 εzz + 12 K1 (∇x δ nx + ∇y δ ny )2 + 12 K2 (∇x δ ny − ∇y δ nx )2 + 12 K3 [(∇z δ nx )2 + (∇z δ ny )2 ] , (a)

where ui j = 12 (ui j − u ji ) is the antisymmetric part of the small deformation. Such a symmetry-based phenomenological analysis should also give estimates for constants: In nematic elastomers the three shear moduli C1 , C4 and C5 must all be of the same order of magnitude as the ordinary rubber modulus, µ = ns kB T . The differences between the Ci in the nematic state, for example the elastic anisotropy C5 −C4 , must be weak functions of the nematic order parameter Q ∝ (r − 1). Of the constants involving compressibility, one can only say the coefficient C3 ∼ B˜ is very large. Strictly, there is no physical reason for the other constant, the anisotropy of compressibility C2 , to be that the ideal anything but the rubber shear modulus µ . However,   we have seen already Trace formula for the free energy density 12 µ Tr  o · λ T ·  −1 · λ captures the main features of nematic rubber elastic response. This perhaps gives us confidence in the result it produces, namely that C2 = 0, see eqn (10.9). There are many sources of small corrections to the Trace formula, e.g. the semi-soft elasticity discussed in sections 7.4, 7.6.2 and 7.6, or the non-Gaussian corrections in Sect. 6.8, which may contribute to a non-zero C2 which we should still expect to be small. More importantly, the coupling constants D1 , D2 , as well as D22 and D33 , must also vanish in the isotropic phase, at Q = 0, while at Q → 1 their magnitude should be of the order µ again. Therefore, in the isotropic limit, r → 1, we should have, for any system, C2 → 0, D1 → 0, D2 → 0, D22 → 0, D33 → 0,

STRAIN-INDUCED ROTATION

265

and C1 → µ , C4 → 12 µ , C5 → 12 µ The molecular model of nematic rubber elasticity, eqn (10.9), gives for these phenomenological coefficients: C2 = 0, C1 = 2C4 = µ , C5 = 18 µ

(r + 1)2 r

(10.14)

(r − 1)2 r2 − 1 , D2 = −µ , r r r−1 and D33 = 2µ (r − 1), D22 = −2µ r

D1 = µ

where the chain anisotropy parameter remains r =  /⊥ . These expressions behave correctly when the isotropic limit is reached, r → 1. The main conclusion that arises from linear elastic theory, eqn (10.12), is that the free energy depends on the asymmetric part of the strain tensor uαβ , through the relativeΩ − ω ) × n . This, of course, has been already seen rotation coupling terms involving (Ω from the results of molecular theory of nematic elastomers in the previous chapters: the presence of an independent degree of freedom, the director orientation n coupled to elastic deformations, makes invalid the standard argument of classical elasticity that it can only depend on symmetric combinations of strains. The results of this change in continuum symmetry are profound and we shall examine them in the next sections. There have been attempts to develop a fully phenomenological theory of high (nonlinear) elasticity of nematic rubbers (Lubensky et al., 2002), employing the fully nonlinear form of the symmetric strain tensor ε = 12 (λ T · λ − δ ), eqn (4.13). Such an approach is highly non-trivial because at large deformations the rotational invariance is required with respect to the two separate states of the system, before and after the deformation, cf. Sect. 4.1. In addition, the director n that appears in the five elastic terms in an expression like eqn (10.12) is now capable of rotation. Changes in n can make terms take on an order higher than apparent at first inspection. Much remains to be done to incorporate all these factors and consistently generate expressions for higher-order elastic energy. 10.3

Strain-induced rotation

Let us re-examine the experiment already discussed in Sect. 7.4.2, the stripe instability, in the light of continuum small-deformation theory of nematic elastomers. Obviously, we should not expect to describe the whole effect because the end of this transition corresponds to director rotation by up to 90o and large elastic strains – both beyond the applicability of eqns (10.12) and (10.13). However, we would like to detect and correctly describe the onset of instability. Thus, we shall be able to judge the degree of usefulness of such a linear continuum elastic theory for externally imposed strains or stresses. Consider the nematic rubber uniformly aligned along the z axis, as on the left in Fig. 10.4, and apply a uniaxial extension, measured by the fixed strain component

266

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

F IG . 10.4. Diagram of the stripe-instability. The uniformly aligned nematic elastomer is stretched along the perpendicular axis. This causes the director rotation and associated shears, the dominant one of which, uxz , is sketched.

εxx = ε . If, as in the experiment by Kundler and Finkelmann, the sample is actually a thin strip of rubber (in the xz-plane of the diagram) we could reasonably assume that the director will only rotate in this plane and thus only δ nx ≈ θ is present. Similarly, no shear encompassing the y direction should be expected. If we also assume full incompressibility, a reduced form of the full free energy density (10.13) takes the form F = C1 εzz 2 + 2C4 (εxx 2 + εyy 2 ) + 4C5 εxz 2     (a) (a) + 12 D1 (uxz − δ nx )2 − D2 (uxz − δ nx ) εxz     (a) (a) + 12 D22 (uxz − δ nx )2 εxx + 12 D33 (uxz − δ nx )2 εzz

(10.15)

+ 12 K1 (∇x δ nx )2 + 12 K3 (∇z δ nx )2 ] , where εxx = ε is an imposed (small) constant, εyy = −ε − εzz due to incompressibility and we shall adopt the one-constant limit for the Frank elasticity, K1 = K3 = K. Finally, as a first approximation, we may assume that the main elastic shear occurs in the x direction with the gradient along z, that is uxz  uzx and thus εzx ≈ εxz ≈ 12 uxz . We have then a much simpler problem with a free energy density: F ≈ 4C4 ε 2 + (C1 + 2C4 ) εzz 2 + 4C4 ε εzz +C5 uxz 2 + 12 D1 ( 12 uxz − θ )2 − 12 D2 ( 12 uxz − θ ) uxz

(10.16)

+ 12 D22 ( 12 uxz − θ )2 ε + 12 D33 ( 12 uxz − θ )2 εzz + 12 K(∇z θ )2 . The transverse contraction εzz is obtained immediately, by minimisation of eqn (10.16) which contains the quadratic and linear terms in εzz (the latter – from the square of εyy and from the D33 coupling):

εzz = −

2C4 D33 ε− ( 1 uxz − θ )2 . C1 + 2C4 4C1 + 8C4 2

(10.17)

Returning this to F, keeping only the relevant quadratic terms and minimising over uxz , one then finds the expression for shear strain

STRAIN-INDUCED ROTATION

uxz ≈

2(D1 + D2 ) θ + O(ε θ ) . D1 + 2D2 + 8C5

267

(10.18)

The resulting free energy density, which is now a function only of the director angle θ , becomes     2C4 C1 +C4 2 1 2 − D22 ε ε + 2 θ D1 − D22 /(8C5 ) − 14 D33 F ≈ 4C4 C1 + 2C4 C1 + 2C4 + 12 K(∇θ )2

(10.19)

If the deformation ε is imposed, the first term is a constant. If a stress σxx is imposed instead, we would need to add −σxx ε to eqn (10.16) or (10.19) and, after minimising over the now free parameter ε , would obtain the corresponding coupling between θ 2 and σ . In the second term, the coefficient of θ 2 will change sign at a threshold value εt . In the absence of gradient terms, the limit to stability of the unrotated phase is reached. The extent of rotation is determined by higher order terms in F, one must consult the full solution to the problem in Sect. 7.4.2. The uniform threshold strain is:

εt ≈ 4

D1 − D22 /(8C5 )   . 4 D33 C12C +2C4 − D22

(10.20)

The combination of rubber elastic constants D1 − D22 /(8C5 ) is critical in determining the point of instability. Let us, for concreteness, put the specific values of material constants given by the ideal molecular model, eqn (10.14). We then have instead of eqn (10.19): (r − 1)(r + 2) 2 1 ε θ + 2 K(∇θ )2 . (10.21) r There is no threshold for the director rotation! External strain ε would induce uniform director rotation from the beginning of deformation, via the negative coupling −ε θ 2 . The threshold expression (10.20) of continuum theory and vanishing of the threshold in eqn (10.21) suggest that the combination D1 − D22 /(8C5 ) is indeed the continuum measure of semi-softness of nematic elastomers. For ideal elastomers this combination of continuum elastic constants will vanish. Obviously, when instead oblate nematic polymers are considered, (r − 1) < 0, the perpendicular strain does not induce director instability: an attempt to stretch a ‘pancake’ sideways makes it only more stable – as opposed to the prolate ‘cigar’ shape with r > 1. The experiment of Fig. 10.4 resembles the Freedericks effect in an ordinary nematic. We concentrate on boundary conditions for the director, θ = 0, and for the strain, (a) εxz = uxz = 0 at all four edges of the strip. (The details of resulting stripe texture and role of constraining clamps are quite complex, see Chapter 8.) Assuming a harmonically oscillating director with θ = θo sin (kz) sin(π x/L), the average free energy density (10.21) becomes  (r − 1)(r + 2) 1 K 3 ε + (k2 + [π /L]2 ) (10.22) F ≈ µ ε 2 + µ θo2 − 2 8 r µ F ≈ 32 µ ε 2 − 12 µ

268

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

(an additional factor of 1/4 arises from the average over sin2 (kx) sin2 (π z/L) in the macroscopic sample). It is unstable with respect to the director rotation θo (exactly as in the Freedericks effect) at

εc ≈

K  π 2 r µ L (r − 1)(r + 2)

(10.23)

where L is the sample’s z-dimension where the principal director variation takes place.  The characteristic nematic penetration depth ξ = K/µ ∼ 10−8 m, is small compared with macroscopic lengths. Even for a very narrow strip of elastomer, say L ∼ 1 mm, such a threshold purely due to Frank effects is very small indeed (εc ∼ 10−12 ). Despite the apparent similarities with Freedericks transitions, strain-induced nematic transitions are of very different physical character. The threshold is not due to the Frank elastic resistance to the applied field that arises from gradients induced by boundary conditions. It results from the semi-soft anchoring of the director to the bulk of the material. Only a perfectly soft elastomer would have its Freedericks transition governed by Frank elasticity. There are two conclusions. The continuum description of nematic elastomers at small deformations provides a self-consistent way of modeling both the elastic strains and the director rotation modes. However, in order to describe real physical effects involving imposed deformations and director rotations, we had to resort to complex arguments and considerations, for instance the need for third-order coupling terms D22 and D33 . There is no surprise here – most rubber-elastic effects occur in the non-linear regime. Having in our possession a full molecular model, which describes the rubber elasticity and director variations at arbitrary deformations λ , we should perhaps argue that all problems in nematic elastomers that involve imposed deformations or stress should   T −1 be addressed from the standpoint of the Trace formula Tr  o · λ ·  · λ . Only special groups of physical effects, such as acoustic waves or thermal fluctuations (which do not involve externally imposed strains), are adequate candidates for the continuum small-deformation theory developed here. 10.4

Soft elasticity

Golubovic and Lubensky (1989) first demonstrated that some solids must, on general symmetry grounds, possess soft elastic modes. To understand this argument, it is particularly important to examine the reference and the deformed states of such an elastic continuum. First, we apply symmetry conditions to non-linear strains which turns out to give the full functional form of large amplitude soft modes. We analysed the thermomechanical history dependence of soft elasticity in sections 7.4 and 7.6 using a molecular approach. We found there too that it is in the reference states that the symmetry conditions can be invalidated and the residual semi-softness results. Here we then instead examine linear continuum elasticity to illustrate the mechanism by which the soft and semi-soft elastic effects are attained and also violated. One of the elastic constants of the general uniaxial harmonic elastic free energy will be shown to vanish.

SOFT ELASTICITY

10.4.1

269

Symmetry arguments

Imagine an initially isotropic elastic body, where all material points are labelled by a vector X in the laboratory frame. The reference state possesses full rotational symmetry, that is, the state described by the set of points X is completely equivalent to the set U φ · X which have been rotated by φ (with U φ a rotation matrix). On transition into a uniform single-domain nematic these material points undergo a spontaneous deformation, see secs. 5.4.1 and 6.2. They are now R = λ m · U φ · X . Another nematic state may be obtained from the isotropic reference state without such a rotation, R o = λ m · X .

Inverting this we find an expression for the reference point, X = λ −1 · R o , which we m insert in the expression for R . The latter now gives a matrix relation between the states, U φ · λ −1 · R o . The deformation gradient tensor that connects R and R o , that is: R = λ m ·U m these two states is therefore U φ · λ −1 λ = ∂ R /∂ R o = λ m ·U . m When the intermediate state is truely isotropic, then the two nematic states must be physically equivalent and the deformation λ between them costs no energy. See Fig 10.5 for the routes to these equivalent deformations and for a picture of λ (Warner, 1999; Lubensky et al., 2002). The equivalence of these very general soft modes to those emerging from an explicit theory of nematic rubbers, Chapter 7, has already been suggested by Fig. 7.4 in which the states connected by a soft deformation can be thought of as having an isotropic intermediate state. (Some theoretical methods, for instance those of Lubensky, DeSimone, = r1/6 λ m , see Fried and others, use this isotropic state as a reference state.) Since  1/2 φ eqn (7.14), then the soft modes emerging above can be equivalently written as U φ · λ −1 U Tφ · λ m ·U U φ · λ −1 λ = λ m ·U = U φ ·U m m = U φ ·  1/2 ·  −1/2 −φ o

(10.24)

U Tφ . The regrouped comIn the second term, unity has been inserted from the left as U φ ·U

bination U Tφ · λ m · U φ represents a backward rotated spontaneous distortion λ m (−φ ): care must be taken with the definition and the sense of rotations. In the final term, the leftmost rotation by φ restores the deformed matrix to the orientation seen on Fig. 10.5. ·  −1/2 . Comparison To within rotations, such soft deformations reduce to the form  1/2 φ o of Figs. 7.4 and 10.5 confirms indeed that the current soft modes are directly related to · −1/2 . (There we were concerned with rotathose arising from the Trace formula, 1/2 θ o tions θ of the director rather than φ of the solid matrix. The two are equivalent since it is relative rotations between the two that are significant.) Thus this symmetry argument yields the detailed soft modes just as before and explains thereby their universality. One can also find the corresponding soft symmetric strain tensor in its fully nonlinear form (Lubensky et al., 2002)

270

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

εi j = 12 (λ T · λ − δ )i j =

1 2



 −1 T T T U U (λ −1 .(10.25) ·U · λ ) ( λ ·U · λ ) − δ i j ik k j m m φ m m φ

Take the spontaneous deformation on entering the nematic √ phase to have a principal extension value λm along n o and principal contractions 1/ λm perpendicular to n o , as in Sect. 6.2. (The extension depends on the chain anisotropy as r = λm3 .) In the principal frame, shown in Fig. 10.5, the strain tensor can be explicitly derived by multiplying the matrices in (10.25) (with the initial director along z):   3/2 −3/2 3 − 1)[1 − cos 2φ ] ( λ 0 −( λ − λ ) sin 2 φ m m m 1  ε =  0 1 0 4 3/2 −3/2 −3 −(λm − λm ) sin 2φ 0 −(1 − λm )[1 − cos 2φ ]   − √1r sin 2φ (1 − cos 2φ ) 0 r−1  . 0 1 0 (10.26) ≡ 4 1 1 √ − sin 2φ 0 − (1 − cos 2φ ) r

r

Since ε has been symmetrised, it is not so immediately evident as in Fig. 7.3 what the division between body rotation and actual body shape is as the soft deformation proceeds – this information can however be extracted. Recognising that these deformations are soft means that one has a connection between the components of strain and rotations along a soft trajectory. To this connection one can add boundary conditions (for instance

X

lm

o

R (X)

l

Uf

lm

f Uf.X

R(Uf.X)

F IG . 10.5. The alternative paths to two nematic states differ only by a body rotation, φ , of the isotropic reference state from which a spontaneous distortion has occurred. The nematic states, although equivalent in energy, differ in shape (see the cartoon Fig. 7.3). They are evidently connected by a deformation, λ that is soft and is parameterised by rotations of the isotropic reference state.

SOFT ELASTICITY

271

the suppression of certain shears, for example by clamps, as we saw in Sect. 7.4.2) and then derive precise expressions for the components of soft shear in terms of, say, one imposed extension. Explicit examples are given by Lubensky et al (2000) which demonstrate that in the purely soft case the universal, non-linear soft response curves, Fig. 7.10, are the result of symmetry and not of any particular form chosen for the free energy. That these strain-rotation curves remain universal even in the semi-soft case is rather more remarkable. 10.4.2

The mechanism of soft deformation

We now demonstrate explicitly why it is the shear modulus C5 of linear theory that must vanish due to this additional symmetry (Olmsted, 1994; Warner, 1999). Consider just the elastic terms of the continuum free energy (10.12), that is we simply take a uniaxial elastic body but do not follow the evolution of any internal rotational degrees of freedom it might possess. If we further ignore the rubber compressibility, the remaining part of the energy density becomes: ∆F = C1 (nn · ε · n )2 + 2C4 (nn × ε × n )2 + 4C5 (nn · ε × n )2 .

(10.27)

As in eqn (10.12), the director n is here identified with the initial principal axis direction, n o . Other terms in eqn (10.12) account for the energy cost, at harmonic order, of director rotation. If we restrict our attention to the C4 and C5 terms above and ignore the internal director degrees of freedom, i.e. allow the director to adopt any orientation it prefers to lower the total free energy, we cannot assume the elastic constants in eqn (10.27) have the same values as in the full eqn (10.12). Indeed we shall find that one constant, C5 , is renormalised by the evolution of the director modes. If the angle of rotation φ is small, the deformation gradient tensor connecting the nematic states in Fig. 10.5 reduces to   −3/2 1 0 −φ λm   U φ · λ −1 0 λ ≡ λ m ·U ⇒ 0 1 (10.28) , m 3/2 φ λm 0 1 and will be relevant for linear elasticity. It acts in the plane of rotation. The linearised strain tensor obtains from the symmetrising expression εi j = 12 [(λi j − δi j ) + (λ ji − δ ji )] = 12 [λi j + λ ji ] − δi j and is   −1 U φ · λ −1 U − δi j εi j = 12 (λ m ·U ) + ( λ ·U · λ ) (10.29) i j ji m m m φ    0 0 1 φ  3/2 −3/2  λm − λm = 0 0 0 (10.30) 2 1 0 0 the latter expression being in a principal frame with the z-axis along the initial director n o . It is obvious that the strain (10.30) does not have a diagonal extension component, that is, (nn · ε · n ) = 0, in the elastic energy density (10.27). The in-plane shear is not

272

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

represented either: (nn × ε × n ) = 0, since εyz , εyy and εzz do not appear in (10.30). However, the shear component (nn · ε × n ) of eqn (10.30) does not vanish (the εxz elements) and thus contributes to the part of the elastic energy determined by C5 . We then obtain a non-zero energy difference between the two states, characterised by R and R o and connected by ε :     3/2 −3/2 2 ∆FGL = C5 φ 2 λm − λm = C5 φ 2 λm3 + λm−3 − 2

(10.31)

∝ C5 φ 2 Q2 where Q is the nematic order parameter and φ is the arbitrary angle of the rotation matrix U φ . This energy must be zero since the φ -rotations are simply of an isotropic reference state and cannot be of physical significance. Hence rotational invariance under U φ demands that (10.31) vanish. This is only possible if the effective shear modulus also vanishes, C5 = 0. When this modulus has been eliminated, the response of the elastic system to any strain εi j related to C5 (with n · ε ×nn = 0) is soft. Such deformations are shears in a plane encompassing the director, which we have already discussed in previous chapters. The continuum analysis above simply compared equivalent undeformed states at which the nematic elastomer arrived in two different ways. The conclusion, that ‘the effective shear modulus C5 must be zero’, is correct but perhaps puzzling. One needs to understand the meaning of effective in this context: certainly, the familiar Trace formula for nematic rubber elasticity does not give C5 = 0, see eqn (10.14)! By effective shear modulus we mean the response to an imposed elastic strain, when the nematic director is free to adopt its optimal possible configuration and we are not interested in it. We simply stretch the rubber and measure the force, allowing the internal degrees of freedom to evolve as they choose. (This is not the philosophy of eqn (10.12) and the equations that immediately follow it.) In this way, by optimising the nematic director evolution one can explicitly find modifications to the effective elastic modulus C5 that, by the symmetry argument, must exactly give no overall energy cost (Olmsted, 1994). Conversely, when the conditions for the Golubovi´c-Lubensky (GL) theorem to hold are violated, we show in the next section how the limited shear modulus renormalisation defines the degree of hardening, that is of semi-softness. If the local rotation of the deforming elastic matrix Ω and that of the director are coaxial, that is Ω and ω are parallel and uniform, the argument is simple. Minimising the relative-rotation part of the energy density (10.12), 1 Ω − ω ) × n ]2 + D2 n · ε · [(Ω Ω − ω ) × n ], D1 [(Ω 2 Ω − ω ) × n one obtains the optimal relative rotation for a given shear with respect to (Ω strain ε Ω − ω ) × n] = − [(Ω

D2 (nn · ε ) D1

or

Ω−ω) = − (Ω

D2 [nn · ε × n ] . D1

(10.32)

CONTINUUM REPRESENTATION OF SEMI-SOFTNESS

273

The difference between the two expressions (10.32) is a vector ∼ (nn · ε · n ) n , parallel to n , involving the strain εzz – something that does not appear in the relevant vector product [(. . .) × n ]. We now substitute this back into the energy density and obtain 2 1 Ω 2 D1 [(Ω − ω ) × n ]

Ω − ω ) × n] + D2 n · ε · [(Ω ⇒ −

(10.33)

 D2  D22 [nn · ε × n ]2 = − 2 εxz 2 + εyz 2 8D1 8D1

The last expression is written in a coordinate frame where the initial director n o is parallel to z-axis. We can now unite this expression with the rest of the elastic energy density, eqns (10.12) or (10.13) and thus obtain the effective rubber-elastic energy which depends on strains but not on the director, for example in the specific coordinates:  2 F = C1 εzz 2 + 2C2 Tr[ε˜ ] εzz +C3 Tr[ε˜ ]  D22 2 2 2 (εxz 2 + εyz 2 ) . +2C4 (εxx + 2εxy + εyy ) + 4 C5 − 8D1

(10.34)

The renormalisation of C5 to an effective value C5 − 18 D22 /D1 can produce a remarkable cancellation: taking the molecular-model values from eqn (10.14) we have (r + 1)2 (r − 1)2 r2 − 1 , D1 = µ , D2 = −µ , r r r D2 C5R = C5 − 2 = 0 ! 8D1

for C5 = 18 µ

(10.35)

as required above on general symmetry grounds. We saw also the cancellation of the C5 and D22 /(8D1 ) in the continuum analysis of the stripe instability (10.20) where the threshold disappeared in the ideal case. Olmsted (1994) proposed this mechanism: shape depends on the orientation of an internal (nematic) degree of freedom, the rotation of which causes a natural shape change at zero cost for suitable solids, leading to the relation C5 = D22 /(8D1 ) for systems that meet the GL symmetry criterion. 10.5

Continuum representation of semi-softness

For systems which cannot attain the isotropic reference state, required for the GL argument, the rotations U φ of the reference state generate physical strains with real ener-

gies; thus C5R > 0, or C5 > D22 /8D1 . We called this effect semi-softness in the previous chapters. There exist several mechanisms for retaining anisotropy in all reference states and hence for losing ideal softness. K¨upfer and Finkelmann (1991) have shown experimentally that if crosslinks are rod-like, then crosslinking in the nematic state pins the crosslinks’ orientation even at high temperatures, above the nominal Tni , and the network is always at least weakly nematic. Theoretically, the study of quenched random fields of like origin dictate a residual nematic order at high temperatures and polydomains

274

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

at lower temperatures in networks crosslinked in the isotropic state (Fridrikh and Terentjev, 1997). These and other mechanisms give quadratic (λ 2 ) additions to the basic Trace formula (10.1) and are discussed in sections 7.6 and 7.6.2. We illustrate continuum semi-softness by considering a particular mechanism studied in some detail in Sect. 7.4.1 – fluctuations in chemical composition and hence strand anisotropy in the copolymers commonly used for nematic elastomers. The physical idea is that the soft mode (7.35) depends on the chain anisotropy  /⊥ . It cannot be simultaneously soft for every individual network strand if there are random variations in local chemical composition. The mean soft mode does however provide only a low-energy elastic response, thus providing the reason for the term ‘semi-soft’. We now see how semi-softness can arise in the continuum picture. Taking the ideal molecular expressions for elastic and coupling constants (10.14), we have seen explicitly that the effective (renormalised) shear modulus, C5R vanishes. If now chains fluctuate in composition and therefore in anisotropy, we replace r → r and 1/r → 1/r in all expressions for Ci and Di , the averaging . . . being over polymer strands of a rubbery network. Now instead of r meeting 1/r, it is r meeting 1/r and there is no longer a cancellation. The elastic constants are amended and there is no longer a cancellation in the renormalised C5R as in (10.35). Instead we obtain C5R = C5 −

D22 r1/r − 1 . → 4µ 8D1 r + 1/r − 2

(10.36)

Convexity arguments on the distribution of r and 1/r show that the renormalised modulus (10.36) of the statistical mechanical model does indeed satisfy C5 ≥ D22 /8D1 , and hence C5R ≥ 0. This is a necessary condition for stability: the arguments leading to eqns (10.31) and (10.33) involve deformations unrelated to C1 and C4 which thus do not enter the above inequality for stability. The extent to which the renormalised C5R is positive is the continuum measure of how semi-soft a nematic elastomer becomes in response to fluctuations in its structure or other sources of non-ideality. Our example of continuum soft deformations was the stripe instability in the director orientation when a nematic elastomer strip is strained by εxx = ε across the axis of n . The free energy density obtained in eqn (10.19) involving director rotation θ was   C1 +C4 2 1 2 C5 D1 − 18 D22 2D33C4 − D22 (C1 + 2C4 ) F ≈ 4C4 ε + θ − εxx C1 + 2C4 xx 2 C5 C1 + 2C4 + 12 K(∇z θ )2 .

(10.37)

The coefficient of θ 2 changes sign at a certain threshold value of strain. The combination C5 D1 − 18 D22 = C5R D1 , a measure of semi-softness, occurs in this coefficient. In an ideally soft case, such as the one provided by the coefficients (10.14) of the ideal Trace formula, any imposed strain resulted in a torque on the director without a threshold, apart from the tiny effect of Frank penalty on director gradients, eqn (10.23). If, on the other hand, arbitrary values of elastic and coupling constants are adopted, the combination (C5 D1 − 18 D22 ) in eqn (10.37) does not vanish. Re-writing it in terms of renormalised modulus C5R , we now obtain

CONTINUUM REPRESENTATION OF SEMI-SOFTNESS

F(θ ) ≈

1 2

µ θo2

C5R D1 2D33C4 − D22 (C1 + 2C4 ) − ε R 2 C1 + 2C4 C5 + D2 /8D1  K + (k2 + [π /L]2 ) . µ

275

(10.38)

giving the actual threshold strain

εth ∼

K  π 2 C5R + . µ L µ

(10.39)

We can safely ignore the small correction due to the Frank and boundary effects. We assume above that all moduli are of the same order of magnitude, D1 ∼ D22 ∼ µ , as in the particular model (10.14), and that the semi-softness is small C5R  C5 ∼ µ . From this one can estimate the actual value of semi-soft parameter α , discussed in Sect. 7.4, from the measured threshold strain:

α  (C5R /µ ) ∼ εth

(10.40)

This gives a small factor, of order 0.1, in the two stripe experiments discussed in Chapters 7 and 8. Semi-softness and the nematic order Let us now discuss the mechanisms for softness and non-ideality as the temperature T increases and the nematic order becmes small, Q → 0. (The same can be achieved by increasing the concentration of isotropic solvent in a gel.) The phenomenon of elastic softness occurs because anisotropic network chains can accommodate a macroscopic shape change by rotating their distribution of shapes at constant entropy. When chains tend to isotropy, Q → 0 and r → 1, shape change can only be accommodated by chains distorting their distribution, thereby decreasing their entropy and increasing their free energy. The rubber modulus is then µ . We must see how this classical limit is achieved; after all the cancellation C5R = C5 − D22 /8D1 = 0 based on the ideal values (10.14) appears to hold for all r including r → 1. Semi-softness is expected when there are fluctuations of composition, crosslinks of rod-like nature, quenched sources of random nematic field, and any other form of non-ideality that prohibits the finding of an isotropic reference state. Then there are additions to the Trace formula (6.5) which are of the form α sin2 θ for simple shears, see for instance (Verwey et al., 1996). The degree of semi-softness, α , can be calculated directly from various models of non-ideality, or measured from experimentally detected thresholds. For continuum theory such additions appear in eqn (10.14) for the Ci and Di , and upset the cancellations yielding a C5R = 0. However, the additions inspired by the semi-soft factor α clearly must vanish as Q → 0 and r → 1 and the delicate limit question must still be resolved, in particular, the limit C5R → C5 → 21 µ as r → 1. In the ideal case there is no high temperature order, softness is perfect and for T < Tni the modulus C5R is identically zero. Several systems, depending on their thermomechanical history, are close to this situation. Since ideally Q jumps to zero discontinuously,

276

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

on heating to T = Tni , there is no limit problem: D1 and D2 cease to exist and C5 is not renormalised. In the semi-soft case, the condition for C5R places bounds on the form the non-ideal additions to D1 can take. Since D1 ∝ Q2 and D2 ∝ Q, in the ideal case the limit D22 /D1 is finite as Q → 0. If however D1 has additions D1 ∝ (Q2 + a1 Q), then the limit of D22 /D1 vanishes as Q → 0, eliminating any renormalisation. Explicit model calculations agree with these symmetry arguments: the semi-soft additions to D1 are indeed ∼ Q and reflect the thermomechanical history of the material. For instance fluctuations (due to compositional fluctuations in the polymers that make up the network) in the effective order felt by a chain, Qeff = (1 + δ )Q (with δ  = 0), yield for the modulus:   2  (r − 1) + a1 Q (10.41) D1 = µ r   where µ  = Det[ f ]/Det[] µ . The constant a1 is a1 = 3δ 2 

Q f (3 + 2Q f ) . (1 + 2Q f )(1 − Q f )

Here formation conditions for the network are denoted by a subscript f : the step-length tensor at formation is  f with order parameter Q f , see Sect. 6.3. The determinant factors reflect spontaneous shape changes since formation, which can be very large. The sign of the coefficient a1 is that of the formation order parameter Q f . It is the order at formation that induces the residual order Q at high temperatures and the sign of Q follows that of Q f . Thus the combination a1 Q is always positive and the non-ideal additions to D1 in (10.41) are really proportional to the absolute value ∼ |Q|. 10.6

Unconstrained director fluctuations

So far we assumed that only uniform director rotations are present in nematic elastomers, or as above in sections 10.3 and 10.5, non-uniformity is energetically insignificant. However, non-uniformity is what an optical experiment first notices, as a large scale rotation of the birefringence axis, or change in transparency due to scattering. Complicated, non-uniform director deformations can arise provided they satisfy boundary conditions and lower the total free energy of the material. There are continuous sets of possible soft deformations at zero (or small) free energy cost. Perhaps, in a particular set-up, boundary conditions do not allow uniform soft deformations to develop. An example is again the configuration leading to stripe domains, Fig. 10.4, where the rigid clamps prevent uniform shear strain. The best way for the material to lower the total free energy was then to break into a non-uniform texture, with a corresponding low elastic response during the transition. In general we have to analyse an arbitrary non-uniform configuration of strains and director rotations and find the so-called effective rubber elastic free energy. This would describe the linear response of nematic elastomer to mechanical deformations when one ignores the nematic director completely, permitting it to adopt the optimal configuration

UNCONSTRAINED DIRECTOR FLUCTUATIONS

277

consistent with the strains and allowed by boundary conditions. Here we shall consider the simplest case when no elastic strain is externally imposed on the sample so that the ‘ground state’ of our system is δ n = 0 and ui j = 0 and all deformations with respect to it are small. This assumption allows us to retain only the leading, quadratic form of the free energy density, without the additional higher-order terms discussed in Sect. 10.1. To find an effective rubber elastic energy one needs to minimise the full free energy, for example, eqn (10.13), without the nominally third-order coupling terms D22 and D33 , with respect to all possible non-uniform variations δ n(rr ). Because, by assumption, boundary conditions preserve the director no , or δ n(rr ) = 0, we should be looking at a class of oscillating functions with nodes at boundaries and, therefore, zero mean value. This is usually done in Fourier space, with

δ nq =



drr eiqq·rr δ n (rr ) ;

uq =



drr eiqq·rr u (rr ) ⇒ uαβ (qq) = (−iqβ )uα .

In other words, one may regard the process of optimisation with respect to an arbitrary non-uniform director modes with zero mean δ n (rr ) as integrating out the director fluctuations. Consider a block of nematic rubber: inside it one would find all possible modes of non-uniform fluctuations of the director field. So, when we are simply stretching this rubber and measuring the elastic force, the effective response function is the average over all particular director modes, as long as they comply with boundary conditions. This is the idea behind the original GL argument for soft elasticity. Equilibrium is determined by the partition function       1 1 H (nn, u ) ≡ Duu exp − Heff (uu) Z = Dnn Duu exp − kB T kB T with H given by the volume integral of eqn (10.12). The second form above is the result of having integrated by steepest descent over director modes, δ n q , in Fourier space. Only the elastic displacement modes then remain and one has an ‘Effective Hamiltonian’ depending only on these modes:    1 H (nn, u ) . Heff (uu) = −kB T ln Dnn exp − kB T The two methods, minimisation with respect to Fourier modes δ n q , and evaluation of the corresponding path integral over thermal fluctuations, are equivalent in the limit of small deformations because the steepest-descent treatment of the path integral requires the minimisation of the exponent, δ H /δ n = 0. The relevant parts of the full nematic rubber energy density (10.12), which involve the variation of nematic director δ n , are the relative-rotation coupling terms and the Frank elasticity: 2 1 Ω Ω 2 D1 [(Ω − ω ) × n ] + D2 n · ε · [(Ω − ω ) × n ] 2 1 1 + 2 K1 (div δ n ) + 2 K2 (nn curl δ n )2 + 12 K3 [nn × curl δ n ]2

(10.42) ,

278

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

Performing the Fourier transformation of this expression, we obtain the full energy of  q δ F(δ n , u ) with its Fourier-transformed the fluctuating director modes δ H = (2dq π )3 density     δ F = 12 D1 + K1 q2⊥ + K3 q2z |δ n ⊥ |2 + 21 D1 + K2 q2⊥ + K3 q2z |δ nt |2 1 + (D1 + D2 ) [iq⊥ (δ n ⊥ u∗z − δ n ∗⊥ uz )] (10.43) 4

1 − (D1 − D2 ) iqz (δ n ⊥ u∗⊥ + δ nt ut∗ − δ n ∗⊥ u⊥ − δ nt∗ ut ) , 4 where we chose the principal axis of initial director n o z and the notation for perpendicular ()⊥ and transverse ()t components of vector fields in the plane perpendicular to z as explained in Fig. 10.6. This geometry is similar to that used in the continuum analysis of liquid crystals (Chaikin and Lubensky, 1995). Essentially, because there is no preferred direction in the xy-plane, we choose one axis along the projection of the wave vector q onto this plane, q ⊥ ; the transverse direction in the plane is then perpendicular to q ⊥ . Accordingly, the director component δ n ⊥ (qq) describes a mixture of splay and bend deformations, while δ nt (qq) corresponds to a combination of twist and bend. Finding the optimal values for two components δ n ⊥ and δ nt is now straightforward. We obtain from the quadratic forms in (10.43): i (D2 − D1 ) qz ut 2 D1 + K2 q2⊥ + K3 q2z i (D2 − D1 ) qz u⊥ + (D2 + D1 ) q⊥ uz δ n⊥ = . 2 D1 + K1 q2⊥ + K3 q2z

δ nt =

(10.44)

The effective elastic response of a nematic rubber, which is allowed to freely fluctuate its director or choose any particular director conformation in reaction to small elastic deformation, is obtained by substituting the optimal director modes (10.44) back into z (n ) n q

x( )

y(t) dn

F IG . 10.6. Geometry of relevant vectors in the problem. The axis z is chosen along n o , with the director deviation δ n is perpendicular to n o . Two arbitrary axes in the xy-plane are chosen along the projection of the wave vector q ⊥ (axis ⊥) and perpendicular to it (the transverse axis t).

UNCONSTRAINED PHONONS

279

the full free energy. Neglecting the corrections higher order in (small) wave vector q , the corresponding minimal value of the energy density associated with a given Fourier mode becomes

δ F∗ ≈ −

D22 2 q⊥ |uz |2 + q2z (|u⊥ |2 + |ut |2 ) + q⊥ qz (u⊥ u∗z + u∗⊥ uz ) . 32D1

(10.45)

(The corrections to this and subsequent expressions are of the order ∼ Kq4 |u|2 , i.e. represent the gradient elasticity that one usually neglects in a basic elastic continuum analysis.) Transforming δ F ∗ back to real-space variables and gradients, we find that

δ F∗ ≈ −

D22 2 εxz + εyz 2 . 8D1

Finally, bringing it back to the full rubber-elastic energy density eqn (10.13), we recover the effective uniaxial response to small strains:  2 (10.46) Feff = C1 εzz 2 + 2C2 Tr[ε˜ ] εzz +C3 Tr[ε˜ ] +2C4 (εxx 2 + 2εxy 2 + εyy 2 ) + 4C5R (εxz 2 + εyz 2 ) with the now familiar renormalised value for the shear modulus C5R = C5 − D22 /8D1 . In ideal soft elastomers C5R → 0 and shear strains in the plane including the uniform director (xz or yz) are penalised only by higher-order gradient terms, e.g.     (D2 − 2D1 )2 ∂ εzx 2 ∂ εzx 2 K1 . + K3 ∂x ∂z 8D21 However, even when a small semi-soft correction is present, C5R = 0, we should expect it to prevail over the gradient elasticity and, in the first approximation, neglect the gradient contribution to the effective rubber elasticity. 10.7

Unconstrained phonons

One may now ask the opposite question: what is the nematic director configuration in a sample that is not mechanically constrained and free to adopt any shape. In a liquid nematic, the only contribution to the continuum free energy is the Frank curvature elasticity. When the director is coupled to the rubbery network, we also include the coupling terms and the elastic energy of the network itself, arriving at the elastic energy expression (10.12), or (10.13). For instance, consider the case when there are no deformations in the network: λ = δ or ui j = 0. Then the full free energy density (10.12) reduces to F|λ =1 = 12 K(∇ δ n )2 + 12 D1 (δ n )2

(10.47)

(in the simplifying one-constant approximation for Frank elasticity). The D1 term imposes a large penalty on any director rotation, uniform or non-uniform, because the

280

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

director axis is ‘frozen’ into the network that we assumed to be immobile. The result is an almost complete prohibition of long-wavelength director fluctuations, manifesting itself in the high transparency of monodomain elastomers. Another effect of the non-deformable rubbery network is its resistance to external electric and magnetic fields. In order to overcome the local resistance of elastic network, the external field has to exceed the relative rotation coupling, e.g. for electric fields
o ∆
E 2 ≥ D1 . These are very high fields, as the discussion in Sect.8.1 illustrated, unless there are physical reasons for an exceptionally small coupling constant D1 (Terentjev et al., 1994; Urayama et al., 2005). This argument, however, was built on the assumption that the rotating director has to meet the full resistance of the relative rotation coupling. In reality, the network may generate any non-uniform strain to lower its total energy, with the only demand that strains are modulated in a way to comply with boundary conditions. In other words, the rubbery network is free to adopt its minimum-energy conformation corresponding to a given director field n (rr ). This will effectively reduce the penalty on director variations. To find out what strain modes are optimal, one needs to minimise the free energy density (10.12) with respect to the elastic variables u ∼ eiqq·rr with a non-zero wave vector, q , (thus preserving the overall shape of an undeformed sample). This corresponds to the ‘integrating over phonon modes’ and finding the ‘Effective Hamiltonian’ for the nematic director field n (rr ), for which the elastic network has found a corresponding lowestenergy strain configuration. Technically, the correct way of finding the minimum for a non-uniform field variable u (rr ) is to examine its Fourier modes (each of which satisfies the boundary condition u = 0 at the nodes of oscillation). The steepest-descent integration over phonon modes, u q in Fourier space, is equivalent to minimisation δ H (nn, u )/δ u = 0 at a fixed n q . As in the Sect. 10.6, taking for convenience the specific frame n o ||z, eqn (10.13) without the nominally third-order terms D22 and D33 , and writing the Fourier components of elastic and rotational modes as  1 2 2 F ≈ C3 qz + (D1 + 2D2 + 8C5 ) q⊥ |uz (q)|2 8  1 + C3 q2⊥ + (D1 − 2D2 + 8C5 ) q2z |u⊥ (q)|2 +C3 qz q⊥ (uz u∗⊥ + u∗z u⊥ ) 8  1 (D1 − 2D2 + 8C5 ) q2z +C4 q2⊥ |ut (q)|2 (10.48) + 8 1 + (D1 + D2 ) [iq⊥ (δ n ⊥ u∗z − δ n ∗⊥ uz )] 4

1 − (D1 − D2 ) iqz (δ n ⊥ u∗⊥ + δ nt ut∗ − δ n ∗⊥ u⊥ − δ nt∗ ut ) 4    + 12 D1 + K1 q2⊥ + K3 q2z |δ n ⊥ |2 + 12 D1 + K2 q2⊥ + K3 q2z |δ nt |2 . (c.c. denotes ‘complex conjugate’ of a preceding expression). In this expression we have suppressed some unnecessary information: certain modes u (qq) are penalised by the bulk elastic response and the effect of other elastic constants is negligible on its background.

UNCONSTRAINED PHONONS

281

Eqn 10.48 has the limit C3  Ci implemented. The notation for perpendicular (..)⊥ and transverse (..)t vector components is the same as in Fig. 10.6, based on an instantaneous orientation of the wave vector q . When all network deformations are prohibited (uuq = 0), the remaining nematic director is described by the last two terms, identical to eqn (10.47). The ability of the network to find a lower energy state with an optimal distribution of strains u q , for each mode q , leads to the reduction of the effective nematic energy. Although we are dealing with vector fields u (q) and δ n (q) with complex components, the minimisation of the quadratic form (10.43) is straightforward, albeit algebraically tedious. The optimal network deformation modes (again, using the fact that the bulk modulus C3 is large) are given by u¯z (q) = −2iq⊥ δ n ⊥ (q)

(D1 + D2 )q2⊥ + (D1 − D2 )q2z (D1 + 2D2 + 8C5 )q4⊥ + (D1 − 2D2 + 8C5 )q4z

(D1 + D2 )q2⊥ + (D1 − D2 )q2z (D1 + 2D2 + 8C5 )q4⊥ + (D1 − 2D2 + 8C5 )q4z D1 − D2 . u¯t (q) = 2iqz δ nt (q) 8C4 q2⊥ + (D1 − 2D2 + 8C5 )q2z

u¯⊥ (q) = 2iqz δ n ⊥ (q)

(10.49)

The effective nematic free energy density of a macroscopically undeformed, but freely fluctuating elastomer is then   (10.50) Feff |u =u¯ = 12 M⊥ (qq) + K1 q2⊥ + K3 q2z |δ n ⊥ |2   2 2 2 1 + 2 Mt (qq) + K2 q⊥ + K3 qz |δ nt | , with the corresponding ‘effective masses’ M⊥ (q) = Mt (q) =

[8D1C5 − D22 ](q2⊥ + q2z )2 − [2D21 + 16D1C5 − 4D22 ]q2⊥ q2z (D1 + 2D2 + 8C5 )q4⊥ + (D1 − 2D2 + 8C5 )q4z 8D1C4 q2⊥ − [8D1C5 − D22 ]q2z . 8C4 q2⊥ + (D1 − 2D2 + 8C5 )q2z

(10.51)

Although these expressions appear quite complicated, one thing is clear: the effect of rubbery network on the nematic director depends on the mutual orientation of three relevant vectors: the initial director n o , its variation δ n and the spatial gradient, q . Figure 10.6 identifies the relevant angles describing the orientation of wave vector q , θ and ψ . The effective masses may then be written in a different format, which is valid even at |qq| → 0: M⊥ (q) = Mt (q) =

[8D1C5 − D22 ] − [2D21 + 16D1C5 − 4D22 ] tan2 θ (D1 + 2D2 + 8C5 ) tan4 θ + (D1 − 2D2 + 8C5 ) 8D1C4 tan2 θ − [8D1C5 − D22 ] . 8C4 tan2 θ + (D1 − 2D2 + 8C5 )

(10.52)

The familiar combination of constants C5R = C5 − D22 /8D1 is dominating these expres-

282

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

(b)

(a)

(c)

F IG . 10.7. Three-dimensional polar diagrams of the effective masses: Mt (a), and M⊥ (b), as functions of the wave vector q orientation with respect to the undistorted nematic director n (vertical in (a) and (b)). Plot (c) brings the two graphs together (Mt - solid line, M⊥ - dashed), in a meridional plane, to show their relative magnitudes. The plots are made for the chain anisotropy r = 2 and large semi-softness α ∼ C5R /C5 = 0.2. sions again. For the particular values of elastic and coupling constants provided by the Trace formula, eqn (10.14), the principal effective masses become M⊥ (q) = 2µ

(r − 1)2 tan2 θ (r − 1)2 tan2 θ ; M . (q) = µ t r2 tan4 θ + 1 r tan2 θ + 1

(10.53)

Both expressions are thus proportional to the square of nematic order parameter, via the backbone chain anisotropy factor r =  /⊥ . Figure 10.7 shows, in contrast, the role of semisoftness in eliminating the degeneracy of θ = 0 case, when q is parallel to n o . 10.8

Light scattering from director fluctuations

Light passing through a birefringent material is scattered by the fluctuations in the uniaxial dielectric tensor (10.54)
i j =
⊥ δi j + (
 − ε⊥ ) ni n j which has its principal value
 along the local nematic director n. As an experimental technique, polarised light scattering has been used to study liquid crystalline and polymer morphology for some time and excellent reviews are available (Haudin, 1986; Hashimoto et al., 1989). Much of the interpretation of scattering results is based on various modifications of swarm theory, when the nematic director is assumed to be constant within certain small volumes (with the size of order of light wavelength) and abruptly changing its orientation between such domains. The scattering amplitude from each such domain is then calculated. The most commonly used are the Rayleigh-GansDebye (RGD) or the anomalous diffraction (AD) approximations. The RGD approximation is discussed in the literature in great detail, for instance, in (Haudin, 1986; Rhodes and Stein, 1969). This approach is widely used in the interpretation of light scattering results, very often in the context of liquid crystals (de Gennes and Prost, 1994). The approximation is based on direct superposition of the electric

LIGHT SCATTERING FROM DIRECTOR FLUCTUATIONS

283

  field radiated by point dipoles P (rr ) =
o
(rr ) − δ · E o (rr ) induced by the incident light wave. The careful analysis of its validity, e.g. (van der Hulst, 1991), shows that it is only applicable in the cases of small birefringence and the size of scattering objects (e.g. director fluctuation wavelength) much less than the wave length of light. Both requirements could be questioned in ordinary liquid crystals, where the fluctuations are the strongest at long wavelengths and the anisotropy refractive indices could be m − m⊥ ≥ 0.2. However, the pinning of thermal fluctuations of the nematic director in elastomers make RGD the excellent approximation for analysing the static light scattering in monodomain, single-crystal liquid crystalline elastomers. The AD approximation for birefringent materials is originally described in (Meeten, 1982) and adapted for the case of disordered nematic elastomer in (Clarke et al., 1998). In contrast to the RGDh, the AD approximation is most suitable when the size of than the scattering objects (e.g. characteristic size of director fluctuations  ξ ) is larger  wavelength of light λ and the birefringence is substantial, i.e.
 /
⊥ − 1 ξ  λ . The method requires that the main scattering goes into small angles, thus allowing an effective treatment of a sample as a plane two-dimensional array of scatterers. On the balance of evidence, it appears that the AD approximation has to be used instead of the traditional RGD in the cases of disordered, polydomain liquid crystalline elastomers and, possibly, ordinary liquid crystals too. The results of two methods become identical in the case of very small birefringence. Damping of director fluctuations Using the expression (10.54) of the dielectric tensor through nematic director n = n o + δ n (rr ) we should discard the constant, non-fluctuating parts of it as they vanish on integration with ei q·rr . We only need the linear part (
 − ε⊥ ) [nno δ n + δ n n o ] of ε in this expression. It is convenient to use the geometry expressed in Fig. 10.6, decomposing the fluctuation δ n into perpendicular and transverse components with respect to the scattering vector q . In a typical geometry with crossed polars, Fig. 10.8, the scattering intensity (often called IHV , probably after ‘horizontal–vertical’ orientation of polars) is

F IG . 10.8. Diagram of the optical geometry, indicating the plane of polarisation of the incident light, Eo , that on the detector, Ea , and the relevant wave vectors.

284

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

the squared ratio of electric field components allowed through by the polariser and analyser. The result for the dimensionless scattering intensity in the RGD approximation is: IHV =

π (
 − ε⊥ ) m2s λ 2 R

2

[(eep )⊥ (eea · n o ) + (eea )⊥ (eep · n o )]2 |δ n ⊥ |2 

(10.55)

+[(eep )t (eea · n o ) + (eea )t (eep · n o )]2 |δ nt |2 

where (ee)⊥ is the projection of a polarisation vector on the direction q − (qq · n o )nno and (ee)t is its projection on [nno × q ], see Fig. 10.6. The mean square average of director fluctuations for each mode q is conveniently obtained from the Fourier transform of the free energy, using the equipartition theorem of thermodynamics. The average of energy for each mode (i.e. per degree of freedom) is 12 kB T . Therefore, for an ordinary nematic liquid crystal with free energy density F

1 V

∑ 21 Kq2 |nn(qq)|2

i.e. |δ n q |2  =

q

V kB T , K q2

which leads to strong scattering of light by fluctuations of orientation at small q, seen as the turbidity of a nematic (as opposed to the low scattering from density fluctuations in an isotropic liquid). When elastic modes adopt their optimal values, the effective nematic free energy (10.50) produces the following expressions for the mean-square director fluctuations |δ n q |2  = |δ n ⊥ |2  + |δ nt |2  =

(10.56)

V kB T sin ψ V kB T cos2 ψ + 2 2 M⊥ + K1 q⊥ + K3 qz Mt + K2 q2⊥ + K3 q2z 2

[see Fig. 10.6 for definition of the azimuthal angle ψ and eqns (10.52) and (10.53) for Mα (qq) expressions]. In general, the conclusion is that the coupling to the rubbery network still prevents the strong scattering of light on director fluctuations (|δ n |2  ∼ 1/q2 in an ordinary nematic liquid crystal). The uniform single domain nematic elastomer is optically transparent, in contrast to the characteristically turbid nematic liquids. The effect of unconstrained elastic modes (phonons in a rubbery network) reduces such a penalty, especially in the ideal soft case with D1C5 − 18 D22 → 0. The effective masses M⊥ and Mt strongly depend on the orientation of the scattering vector with respect to the nematic director and its fluctuations and may even vanish in some scattering geometries. Dynamic light scattering Dynamic light scattering is an important method for studying the viscoelastic properties of optically inhomogeneous media, see for example (Geissler, 1993). It is one of the traditional experimental tools in the physics of liquid crystals, with a large dedicated literature. The method (also called photon correlation spectroscopy) uses the same

LIGHT SCATTERING FROM DIRECTOR FLUCTUATIONS

285

optical geometry as shown in Fig. 10.8, with an added autocorrelator which measures the correlation of the scattering intensity I(qq) at different times, g2 (t) =

Iq (0)Iq (t) − 1. Iq (0)2

At long times the correlation is gradually lost due to the internal viscous relaxation g2 → 0, while the short-time limit depends on the proportion between the true dynamic (fluctuating) component and the static scattering component in the total intensity I(q,t). A so-called generalised Siegert relation (Geissler, 1993) gives g2 (t) ≈ [α 2 g21 (t)+ 2α (1 − α )g1 (t)], where g1 = exp[−Dq2t] and α = Ifluct /Itotal  is the ratio of the dynamic to the total scattering intensity. When the system has low static scattering, in the homodyne regime, as is for instance the case in colloids where Itotal  ≈ Ifluct  provided by the moving particles, the correlation function g2 (t) ≈ exp(−2Dq2t) ≡ e−2t/τ approaching g2 → 1 at short times. If a sample strongly scatters light on static inhomogeneities, which is called the heterodyne regime, then Ifluct   Itotal  and the correlation function g2 (t) ≈ 2α exp(−Dq2t) ∝ e−t/τ , having only a very small increment over zero at t → 0. In both limits the relaxation time τ is determined by the scattering vector q and an appropriate diffusion constant for the given physical process. In a nematic liquid, scattering light on inhomogeneous modes of director fluctuations, δ n⊥ (qq) and δ nt (qq), has two corresponding relaxation times depending on the scattering geometry (de Gennes and Prost, 1994):

τ⊥−1 =

K3 q2 + K1 q2⊥

η1 (qq)

;

τt−1 =

K3 q2 + K2 q2⊥

η2 (qq)

(10.57)

with the effective viscosities η1,2 being functions of scattering geometry. In experiments (Sch¨onstein et al., 2001) particular configurations of n , k and the scattering vector q were chosen such that the ideal nematic relaxation times are simply −1 τsplay =

K1 q2 ; γ1 − α32 /ηb

−1 τbend =

K3 q2 γ1 − α22 /ηc

(10.58)

(with ηb,c the corresponding Miesowicz coefficients). This method, after plotting the relaxation time τν (q) against q2 , is one of the main methods of finding the viscous coefficients of nematic liquid crystals (provided an independent measurement of a corresponding Frank constant Kν is available). Generally, the same applies to liquid crystalline polymers as well – only the viscosities are much larger and relaxation times longer, as one would expect. However, these results (Sch¨onstein et al., 2001) indicate that in a crosslinked nematic elastomer many differences arise in comparison with the corresponding uncrosslinked melt. First of all, the short-time results have been noticeably more noisy than it is usually expected. Then, characteristically, the scattering is clearly in the heterodyne regime, with a large static component – in marked contrast to the behaviour in nematic liquids.

CONTINUUM DESCRIPTION OF NEMATIC ELASTOMERS

g 2 (t)

286 1

0.05

0.8

0.04

0.6

0.03

0.4

0.02

0.2

0.01

0 -5 10

(a)

10

-4

10

-3

-2

10

10

-1

10

0

10

1

2

10

Log(t /s)

0 -5 10

10

(b)

-4

10

-3

-2

10

10

-1

10

0

10

1

2

10

Log(t /s)

F IG . 10.9. Characteristic output of photon correlation spectroscopy, g2 (t), for an ordinary liquid crystal polymer, (a) – the homodyne regime, and for a crosslinked nematic elastomer, (b) – strongly heterodyne regime. These plots are typical of experimental results in the literature (Schmidtke et al., 2000; Sch¨onstein et al., 2001). And finally, the attempts to fit the existing decay of g2 (t) were only successful with a stretched exponential function ∼ exp[−(t/τ )0.45 ], indicating a broad distribution of relaxation times throughout the system. Plotting the relaxation time τ (q) against the scattering wave vector, Fig. 10.10, is a direct way of measuring the effective diffusion constant, or in the nematic case - the ratio K/η , eqn (10.58). However, when the average relaxation time is extracted from the intensity correlation function for a crosslinked nematic network, its plot against q, Fig. 10.10(b), shows no wave-vector dependence at all (Sch¨onstein et al., 2001). Such a result is, of course, expected from the earlier analysis of static light scattering. Equation (10.56) gives the spectrum of nematic director fluctuations modified by the coupling to network elasticity. In full similarity, when all elastic deformations are prohibited in a nematic elastomer (which may be exactly the case at a characteristic frequency of scattering experiment), the estimate of the relaxation times should read −1 τeff =

Kq2 + D1 . ηeff

(10.59)

A full calculation, allowing for the network phonons and the corresponding reduction of elastic penalty on certain (soft) director modes, results in the estimates (in the geometry used in experiments of Stille et al.) −1 τsplay =

K1 q2 + M⊥ (q) ; γ1 − α32 /ηb

−1 τbend =

K3 q2 + Mt (q) . γ1 − α22 /ηc

(10.60)

It is unclear whether the elastic modes u¯ (qq), eqn (10.49), can be excited in a polymer network at a frequency corresponding to the light scattering experiment. Chapter 11 elaborates on this point of dynamics, but no definitive answer exists at this stage. Perhaps the safest assumption is to regard the underlying elastic network as a rigid medium, corresponding to the glass plateau of Figs. 11.9 and 11.10. In that case the estimate

LIGHT SCATTERING FROM DIRECTOR FLUCTUATIONS 700

120 52.1C 72.5C 82.9C 96.3C

600 500

t -1(s -1)

287

63C 50C

100 80

400 60 300 40

200

20

100 0 0

(a)

1

2 4 3 q 2 (10 -14 m-2 )

5

0

0

10

20

30

40

50

60

q 2 (10 -12 m-2 )

(b)

F IG . 10.10. Relaxation time τ (q), for an ordinary liquid crystal polymer, (a), and for a crosslinked elastomer, (b), following (Schmidtke et al., 2000; Sch¨onstein et al., 2001). The temperature-dependent slopes in (a) give the effective diffusion constant D = K/η . In a crosslinked elastomer one finds very little q-dependence in the average relaxation time, confirming the expectation that τ −1 ∼ D1 /η . (10.59) applies. For typical values of the wavelength of light, the scattering vector is of order q ∼ 5 × 106 m−1 . Taking the Frank constant K ∼ 10−11 N and D1 ≈ µ ∼ 105 Pa, we find that D1  Kq2 with a very large margin. So it is not surprising that the average relaxation time of nematic elastomer does not depend on q. Instead, using an independent knowledge of D1 values, one can estimate the effective rotational viscosity ηeff (T ) and compare its values with the previous measurements in un-crosslinked liquid crystalline polymers, as well as the ordinary low-molecular mass nematics (where γ1 ∼ 10−1 −10−2 Pa s). The outcome of this crude analysis gives a very high value for the effective rotational viscosity of director in nematic elastomers: ηeff ∼ γ1 (LCE) ∼ 105 Pa s (Schmidtke et al., 2000; Sch¨onstein et al., 2001).

11 DYNAMICS OF LIQUID CRYSTAL ELASTOMERS Among various dynamical properties of polymeric materials, an important and most frequently encountered is their mechanics. The equilibrium response of liquid crystalline elastomers can be soft or hard depending on the relation between the imposed strains and the nematic director, in particular, if the director is able to respond by rotating. With such unusual equilibrium elasticity one might expect dynamic mechanical response to be equally unusual. If the elastic forces are small, then the return to equilibrium is driven more weakly than in conventional systems. How does the dynamics of internal director rotation, and the corresponding time-dependent softening of rubber-elastic response, determine the dynamic mechanical response of a nematic rubber to a small amplitude oscillatory shear? There are two main aspects of dynamical response in a nematic elastomer or gel, which we shall concentrate on in this chapter. The first question is about the liquid crystalline (internal) degree of freedom: how would the classical Leslie-Ericksen picture of anisotropic nematic viscosity be modified by the coupling to the underlying rubbery network? The second issue is the converse of the first - what would be the effect of internal nematic relaxation on the effective dynamic-mechanical response. We shall see that there is indeed a dynamical analogue of soft elasticity! A linear continuum description in the hydrodynamic limit (the same level of approximation as in the Leslie-Ericksen approach) will be described below. This analysis provides insight into the problem and addresses the first experiments conducted, by groups in Freiburg (Schmidtke et al., 2000), in Bordeaux/Strasbourg (Stein et al., 2001) and in Cambridge (Clarke et al., 2001). However, the full dynamical problem, combining the role of entanglements and the Rouse dynamics of polymer strands with nematic director motion, is far from being solved theoretically. This problem pervades all of this chapter and the attempts in the literature to describe nematic network dynamics: in the hydrodynamic limit of any viscoelastic system one finds stresses and torques proportional to rates of strain and rates of director rotation. Symbolically, in the frequency domain this corresponds to

σ ∼ Cε + iωηεε

h ∼ Dδ n + iωγδ n ,

(11.1)

where C, D are elastic constants and η , γ are the corresponding viscous coefficients. This results in exponential decays and associated characteristic timescales η /C and γ /D. Exercise 11.1 gives the classical illustration of when the effects can be simply combined. Unfortunately, in most polymer systems the effects of elasticity and viscous flow are not simply additive since they have a spectrum of internal modes and generate power-law dependencies on time and frequency, σ ∝ ω y ε . Networks are even more 288

CLASSICAL RUBBER DYNAMICS

289

subtle since there are made of even longer (effectively infinite) connected molecular paths to convey and correlate disturbance and thus create even longer timescales. Experimental results offer some clarification and guidance despite the above complexities. We shall review how the response at low frequency is radically different in the isotropic phase and in the nematic phase (provided one has a distortion geometry that provokes director rotation). Strain rates slow enough to allow director rotation generate a low stress whereas faster strain rates give a classical rubber response which can indeed be confirmed over wide ranges by time-temperature superposition. Thus nematic and network effects to an extent can be separated. We shall thus make the crude separation implied by considering (11.1) as an initial attempt to describe dynamics. First, we shall review the ideas and the results of classical (isotropic) polymer dynamics (Doi and Edwards, 1986) which give a picture of the real difficulties this area must confront. 11.1

Classical rubber dynamics

Rubber technology heavily depends on understanding of linear response of this complex viscoelastic material to small and large deformations (Roberts, 1988). In spite of our every day experience of highly stretchable elastic bands, the overwhelmingly large proportion of practical applications of rubbers explores their response to very small local strains: from automobile tyres and anti-vibration suspensions to seals and noise insulators. There are two ways the mechanical response of rubbers can be invoked: by imposing an external strain (a specific component of shear, ε ) and measuring the stress response, or by applying a force (stress, σ ) and detecting the resulting deformation. There are also two main ways these external influences can be exerted - in a steady, stepwise fashion, thus exploring the time dependence of the dynamical response, or in an oscillatory fashion, varying frequencies in the Fourier domain. In the special case that the external perturbation is small, one arrives at the linear-response constitutive relationships summarised in Table 11.1 and Fig. 11.1 (Ferry, 1980). Such relationships are based on the assumption of additivity of sequential changes in the external input and include only one material-dependent function – a shear modulus or a creep compliance. Considerations for the tensile (Young) modulus E(t) and the corresponding tensile compliance D(t) are completely analogous. Note that, perhaps a little unfortunately, the definition of dynamic modulus G∗ (ω ) (and the compliance alike) is not a direct Fourier transform of the retarded relaxation modulus G(t − t  ), denoted by F T [G]. Recall that, by causality, the value of G(t < t  ) has to be exactly zero to prevent us discovering the future (at t  > t) from the current response. In that case one can alter the limits of integration and find that the convolution in the time-domain becomes the product of the corresponding Fourier transforms in the frequency domain,

σ (t) =

 ∞ −∞

G(t − t  )

d ε (t  ) dt  ; dt 

σ (ω ) = F T [G(t)] i ω ε (ω ) .  ! "

(11.2)

The underlined expression is, of course, what we called the dynamic modulus, which, accordingly, is defined as G∗ (ω ) = i ω F T [G(t)]. In exactly the same way one defines

290

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

Table 11.1 Linear mechanical response relations and corresponding material functions. Steady, stepwise input ε = ε (t) t σ (t) = −∞ G(t − t  ) dtd  ε (t  ) dt  Shear relaxation modulus

Imposed strain:

σ = σ (t) t ε = −∞ J(t − t  )σ (t  ) dt  Creep compliance t    0 G(t − t )J(t )dt = 1

Imposed stress:

Dynamic, oscillating input ε = εo ei ω t σ = G∗ (ω ) εo ei ω t Dynamic modulus G∗ = G (ω ) + iG (ω ) σ = σo ei ω t ε (t) = J ∗ (ω ) σo ei ω t Dynamic compliance J ∗ (ω ) = 1/G∗ (ω )

the other dynamic constants: the compliance J ∗ = i ω F T [J(t)], the extension modulus E ∗ = i ω F T [E(t)], and so on. Another direct consequence of causality is the Kramers-Kronig relations between the real and imaginary parts of the complex linear response coefficient. In our current notation the components of dynamic modulus G∗ (ω ) satisfy: G (ω ) = −P

 ∞ ω G (ω1 ) d ω1 −∞

ω1 ω − ω1 π

and

G (ω ) = P

 ∞ ω G (ω1 ) d ω1 −∞

ω1 ω − ω1 π

(,11.3)



where the integrals are regarded as their principal values (denoted by P ). These are very basic properties of linear response functions found in all areas of physics. In order for Kramers-Kronig relations to be valid one must have a response function [G(t − t  ) here] genuinely independent of the input [ε here]. However, we know from the earlier chapters of this book that even the equilibrium stress-strain relations in nematic elastomers are often highly non-linear due to various aspects of soft elasticity and corresponding stress plateaux. In dynamics too, we shall find that linearity of stress with strain is not always found, even at small strains. e(t)

e(t)

e(t)

s(t)

t

t

t

(a)

t

t

t

s(t)

s(t)

(b)

(c)

F IG . 11.1. Schematic representation of mechanical linear response: (a) steady stress relaxation in response to a stepwise constant shear, (b) oscillatory shear response, (c) creep on application of constant stress.

CLASSICAL RUBBER DYNAMICS

291

The classical retarded linear constitutive relationships between stress and strain have a simple meaning if one considers two obvious limiting cases. An ideal Newtonian liquid is characterised by the linear relation σ (t) = η ε˙ (t), with η the viscosity, corresponding to G(t − t  ) = ηδ (t − t  ). In Fourier space, according to the definition above, we have G∗ (ω ) = i ω η , a purely imaginary modulus corresponding to G = ω η . The viscous dissipation is directly related to the imaginary part, the ‘loss modulus’ G . Now consider an ideal elastic solid: clearly σ (t) = Go ε (t), with Go the shear modulus, corresponding to the constant value G(t − t  ) = Go . As a result, in the frequency domain, G = Go – the ‘storage modulus’ defining the amount of elastic energy stored in the material. Rouse model and entanglements

11.1.1

So far in this book we have dealt with equilibrium properties of polymer chains and their networks, starting with the simplest Gaussian model of random walks in space, representing connected sequences of monomers. To describe the corresponding dynamical phenomena one can follow a similar approach, however, a new concept has to be added: that of time-dependent thermal noise, the source of chain fluctuations, and local friction which provides damping and the overall conservation of energy (fluctuationdissipation balance). If the polymer chain, the monomers (segments) of which experience these two influences, is regarded as an ideal Gaussian then the resulting dynamical description carries the name of Rouse, after (Rouse, 1953). Consider an ideal polymer as a sequence of monomers (beads) connected in a sequence via a harmonic potential (springs) (Doi and Edwards, 1986). Assuming the motion of each monomer in the dense system is overdamped (that is, the inertia and instantaneous acceleration are irrelevant), the remaining dynamic equation is a balance of all forces acting on the selected segment (with the current position in space r s ):

ζ

drr s = κ (rr s+1 − r s ) − κ (rr s − r s−1 ) + A s (t) ; dt

ζ

∂r ∂ 2r = κ 2 + A (t) , (11.4) ∂t ∂s

the second equation being the continuum representation for a chain coordinate r (s,t). On the left hand side one finds the friction force, with a local friction constant ζ , obviously a material parameter. On the right hand side, the two spring forces due to the connectivity with monomers (s + 1) and (s − 1) are determined by the effective spring constant κ . It is easy to verify that in order for the corresponding ‘beads and springs’ model to be compatible with the Gaussian distribution (3.3), one must have κ = 3kB T /2o . The last term, A (t), represents the ‘white noise’ – the stochastic force from thermal collisions of the chosen monomer with the environment. The fluctuation-dissipation theorem relates its second moment to the friction constant because both, in essence, describe the effect of interaction with the same local environment: As (t  ) = 2ζ kB T δss δ (t − t  ). As (t)A A Equations (11.4) represent what is known as the Langevin equation in stochastic physics (Doi and Edwards, 1986; Mazo, 2002).

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DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

The important next step is to transform this microscopic dynamical model to collective coordinates describing the modes of chain deformation as a whole. In essence, this is a Fourier transformation with respect to the index s numbering the segments along the chain (or the equivalent continuum variable s ∈ (0, N)). Calling the reciprocal variable p, the index of the corresponding collective mode, and the Fourier transform of the chain coordinate x p (t), we can write the definitions: xp =

1 N

 N 0

ds r (s,t) cos

 π sp  N

∞

;

r s = x0 + 2 ∑ x p (t) cos p=1

 π sp  N

.

(11.5)

The Langevin equation for the collective mode takes the form

ζ

∂xp π 2 p2 = − 2 κ x p + A p (t) , ∂t N

(11.6)

where the Fourier transform of the stochastic force is exactly the same as in eqn (11.5) A p A q  = N1 ζ kB T δ pq δ (t − t  ). and its second moment is A  Note that the ‘zero mode’, x 0 , describes the chain centre of mass, x 0 = N1 0N ds r (s) = R cm . Its mean square fluctuation describes the self-diffusion of an ideal polymer chain: Rcm (t) − R cm (0)]2  = [R

6kB T t. Nζ

The non-zero collective modes are described by eqn (11.6), which is exactly the problem of Brownian motion of a ‘particle’ with coordinate x p in a harmonic potential with a spring constant κ (π 2 p2 /N 2 ). The solution of this stochastic problem is well-known and it results in the expression for the second moment xx p (t)xxq (0) = δ pq

3NkB T −t/τ p e , 2π 2 p2 κ

with τ p =

1 ζ N2 . p2 π 2 κ

(11.7)

Returning back to the real space and chain coordinates, we can obtain the chain end-toend distance R = r N − r 0 = 2 ∑ p x p [cos(π p) − 1] = −4 ∑ p : odd x p (t). The mean square fluctuation of the chain size is now represented by the sum of odd-numbered collective modes:   t p2 3NkB T −t/τ p 1 2 R(t)R R(0) = 8 ∑ , (11.8) e = 8 N exp − R ∑ 2 2 o 2 2 τR p : odd 2π p κ p : odd π p where we have used κ = 3kB T /2o and identified the so-called Rouse time as the relaxation time of the longest-living, p = 1, mode:

τR =

ζ 2o N 2 . 3π 2 kB T

Essentially, the Rouse time measures how long it takes for a signal (e.g. a thermal fluctuation) to propagate along the whole length of the polymer chain. Note that the algebraic

CLASSICAL RUBBER DYNAMICS

293

identity ∑ p : odd 1/p2 = π 2 /8 ensures that the instantaneous (at t = 0) mean square chain R2  = 2o N ≡ o L. size is R The famous Doi-Edwards expression for the microscopic stress tensor of a polymeric material, 3 2  3kB T 3kB T N ∂ ri (s) ∂ rk (s) 2π 2 p2 =c 2 ∑ xi (p)xk (p), σik = c 2 dn (11.9) o N 0 ∂s ∂s o N p N with c the concentration of chain segments, allows an immediate calculation of the relaxation modulus (Doi and Edwards, 1986). Within Rouse model, the result is G(t) =

c kB T N

∞

∑ e−t/τ p ≈

p=1

c kB T  τR 1/2 . 2N t

(11.10)

The power-law decay with time, G ∝ t −1/2 , is the result of summation over all modes of chain collective vibrations. The exponent of −0.5 is the consequence of the ideal Rouse model, which ignores, e.g., the chain excluded-volume effects. Corrections, most notably due to hydrodynamic interactions when solvent is present as in the Zimm model (Zimm, 1956), result in a different power-law exponent but do not change the basic underlying physics of internal relaxation of chain segments. For instance, for a chain in a good solvent, the Zimm model gives G ∝ t −5/9 , while in the Θ-conditions the relaxation is faster, G ∝ t −2/3 . However, the screening of excluded volume interactions in a dense melt conditions and, more generally, the adequate performance of Gaussian models in polymer physics make the Rouse model and its characteristic time scale an important starting point in understanding polymer dynamics. The main difficulty of this model and deviations from it are determined by chain entanglements, which are especially pronounced in a crosslinked network. Figure 11.2 sketches the linear response of an entangled polymer, showing the three plots for chains of increasing molecular weight. At very short times the material cannot relax in any way and the resulting response modulus saturates at the ‘glass plateau’. The corresponding time is of the order of shortest lifetime of the highest collective mode: for pmax  π /N the time τ p  ζ 2o /kB T ∼ τR /N 2 . If the distance between entanglements on the chain is Ne < N, so that the average number of entanglements per chain is M = N/Ne , the characteristic entanglement time is τe = τR /M 2 . For very long chains one expects τ p  τe  τR . Both these characteristic times do not depend on the overall length of the chain L = o N. In this book we deal with crosslinked networks so that the chains cannot undergo reptation diffusion, which is the mechanism of disengagement from entanglements (Doi and Edwards, 1986) with a characteristic time scale of τd  M τR ∝ N 3 , very strongly dependent on the chain length. 11.1.2

Dynamical response of entangled networks

After early work (Green and Tobolsky, 1946) attempting to describe relaxation processes in crosslinked polymer networks, the Lodge-Yamamoto theory of rubber-like fluids was developed (Lodge, 1956) and (Yamamoto, 1956) It treats the network crosslinks

294

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

ln G(t)

ln G’(w)

glass

1/2

te

(a)

N3

rubber

N2

td

N3

ln t

(b)

N2

N1

te-1

N1

ln w

F IG . 11.2. Schematic representation of mechanical linear response: (a) steady stress relaxation on stepwise shear, (b) oscillatory shear response, showing the reciprocal picture. Shortest times and highest frequencies do not allow the material to relax in any way and result in the ‘glass plateau’. The power-law decay (shown with the Rouse exponent of 12 ) ends at the entanglement time τe after which the polymer responds as a rubber, until the reptation motion allows disentanglement. The disentanglement time τd ∝ N 3 is different for chains of different lengths, as shown in the sketch. as temporary physical junctions. Several extensions of this model have been developed over the years, in particular in the context of transient networks. However, as we have discussed in earlier chapters dealing with the equilibrium static response, the key factor that needs to be considered is the effect of entanglements of chains crosslinked into a network and thus unable to disengage. The tube model and reptation theory (Edwards, 1967; de Gennes, 1971) were developed to describe dynamics of polymers in the melt. Its application to equilibrium rubber elasticity of polymer networks has been summarised in the literature (Edwards and Vilgis, 1988). Here we sketch its application (Edwards et al., 2000) to the microscopic dynamical description of an ideal network. One needs a combination of the tube model ideas, sketched in Sect. 3.5, and the Rouse-like concepts based on fluctuation and dissipation of individual chain segments. The segment coordinate, r (s,t) is now confined within a narrow tube around the axis of the primitive path, defined by the coordinate R (m,t). Here the index m labels the steps of the primitive path. An essential variable in the problem is the number of chain segments contained within each step of the primitive path; the relation between the position s ∈ (0, N) on the polymer and the corresponding position m ∈ (0, M) is given then given by the function m = m(s,t). Note that, while r (s,t) and m(s,t) are the dynamical variables of the problem, the tube conformation itself, R (m,t), is given externally by the surrounding constraints. After geometric and algebraic manipulations with the dynamical equations one finally obtains the time-dependent correction to the equilibrium relation mo = (M/N)s, which is determined by the condition    d dRi dRk N ∂ m 3kB T ∂ 2 m − = εik . (11.11) ∂t ζ2 2o ∂ s2 2o M ds ds ds The microscopic friction constant ζ2 in this equation is different from that controlling the segmental motion (ζ in the Rouse model) as a different physical process affects

CLASSICAL RUBBER DYNAMICS

295

the sliding motion of chain segments re-distributing between the portions of primitive path. In particular, one expects ζ2 to be proportional to the square of the tube diameter. The right-hand side of eqn (11.11) represents the source of perturbations, in this case proportional to the externally imposed strain. Solving for the new m(s,t) as function of εik and relating the chain coordinate to it, via Taylor expansion around the tube axis (Edwards et al., 2000) r (s,t) ≈ R o [mo (s)] + ε · R o +

∂ Ro [m(s,t) − mo (s)], ∂ mo

we can once again compute the stress response, using the definition (11.9). The results can be summarised as following. The fast fluctuating modes of chain transverse excursions perpendicular to the axis of primitive path are responsible for the α -relaxation during the dynamic glass transition, but contribute little to the rubbery response (similar to the static equilibrium case of Sect. 3.5). In the absence of reptation diffusion, the main relaxation is the sliding motion of chain segments along the tube, eventually finding a new equilibrium distribution along the affinely deformed primitive path, see eqn (3.36). When this happens, as t → ∞ or as ω → 0, the rubber shear modulus reaches its equilibrium value, increasing above the ideal rubber modulus µ = ns kB T by the measure of entanglement density. The tube model (Kutter and Terentjev, 2001) and the hoop model (Higgs and Ball, 1989) produce similar results, coincident in the linear regime:  2M + 1 1 11M + 5 + 5 (M − 1) , Go ≈ µ 43 3M + 1 3M + 1 The slip link model (Ball et al., 1981) gives, in our current notation,   M−1 Go ≈ µ 1 +

2 . 3 2 1 + 25 ns (1 + M)2 /M 2 The higher frequency regime sets in when τe2 ω  1, with the ‘second entanglement time’ given as by an expression similar to τe = τR /M 2 , but involving the different friction constant: τe2 = (N/M)2 ζ2 2o /kB T . In this regime the effective rubber plateau modulus reaches the limit  M−1  −1/2 (τe ω ) ; G ≈ 15 µ (M − 1)(τe ω )−1/2 . G ≈ µ M 1− 5M At frequencies above the segmental entanglement scale, τe ω  1, the ‘usual’ Rouse-like relaxation should dominate the final rise of the response modulus to the glass plateau, see the sketch in Fig. 11.3. Theoretical estimates of the characteristic time scales involved in this analysis are difficult, mainly because little is known about the microscopic friction constants ζ and ζ2 . Recent NMR experiments (Komlosh and Callaghan, 1998) once again confirmed that these times are short, e.g. the chain length-independent segmental entanglement time is τe ∼ 10−7 − 10−8 s. The longitudinal sliding motion is a collective process and

296

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS ln G’(w)

glass

1/2

M3

te-12

M2 M1

te-1

M1 M 2

M3

ln w

F IG . 11.3. The dynamic response of an entangled rubber, based on the predictions of (Edwards et al., 2000). The high frequency (short time) dynamic glass transition is the same as for the corresponding melt, whether of long or short chains (shown by dashed lines, distinguished at low frequencies). The lowest curve of rubber plateau is for a network of chains with no entanglements (M = 1). On increasing the entanglement density one finds an additional frequency dependence accompanying the increase in both the apparent rubber plateau and the zero-frequency equilibrium rubber modulus. one could expect it to be slower, τe2 > τe . It is not known whether or not these two time scales are separated widely enough to unambiguously observe the second rubber plateau. Another observation about the time scales could be made. In a crosslinked elastomer, the average number of monomers on each network strand is usually not high. In all nematic elastomer materials mentioned so far in this book this number, N, is seldom above 20 and is frequently only about 10. Polymer strands in a dense network are highly entangled since one does not need a topological knot, the main mechanism of entanglement of free chains to have an effect. In a network, any non-parallel neighbouring chain represents a lateral constraint. The number M should accordingly be high. Then the ratio N/M could be of order one and the only intrinsic time scale of relaxation of polymer network is the Rouse time scale. 11.1.3

Long time stress relaxation

The above analysis of the dynamic mechanical response of a rubbery polymer network assumed an ideal network. This means all chains are of the same nature and are connected by their ends. In effect one assumes highly functional crosslinks as well (to suppress junction point fluctuations). How good are these assumptions, how well do the real rubbery networks follow theoretical predictions? It turns out that the high-frequency (or short-time) response is described adequately, although only qualitatively because the current understanding of the dynamic glass transition is even less developed than that of thermal glass transitions. However, powerlaw decays (with Rouse-like, or similar exponents) and the attainment of the rubber plateau are found in all rubbers. Significant deviations from ideal network behaviour become evident in the low-frequency regime, or during the long stress relaxation (Gaylord and DiMarzio, 1984). It was discovered long ago (Chasset and Thirion, 1965), and since confirmed in

CLASSICAL RUBBER DYNAMICS

297 retracting chain

G(t) T1 T2

l

T3

(a)

ln t

equilibrium conformation

(b)

F IG . 11.4. (a) A long stress relaxation experiment at different temperatures. The logarithmic time scale compacts the long-time region and reveals the continuing decay of the modulus, where one might have expected a constant equilibrium value. (b) The scheme of a free dangling end retraction in a deformed network, providing the additional relaxing stress (Curro and Pincus model). many different materials and experiments (Roberts, 1988), that ordinary rubbers do not seem to ever reach a constant equilibrium value of the relaxing modulus. In a typical steady stress relaxation after a stepwise application of constant strain, one could wait hours, or days, or months (which are not too far apart on the logarithmic time scale) – and still observe a continuous decay of the response. Performing the same experiment at a higher temperature speeds up the relaxation, but does not change the qualitative feature of this long stress relaxation, Fig. 11.4(a). There could be many reasons for such a behaviour, all demanding major deviations from the ideal network concept. For instance, in natural rubbers (polyisoprene or polybutadiene) the traditional method of crosslinking is sulfur vulcanisation; it has been discovered that sulfur bridges linking the network together can, in fact, isomerise (Pond, 1989) and effectively slide along chains, slowly relieving the internal stress. Another mechanism of very long relaxation in non-ideal networks considers the possibility of free dangling chain ends interspersed inside the crosslinked network (Curro and Pincus, 1983; Curro et al., 1985). In equilibrium, before deformation, each such chain adopts an optimal (highest entropy) conformation. When the network is deformed, one end of the chain is pulled away, but the rest of it remains confined in the dense (tube-like) environment, see Fig. 11.4(b). Reptation theory (de Gennes, 1979) tells that the time required for the complete relaxation, by retraction of the chain along its confining tube to reach the new equilibrium conformation, is very long – an exponential function of the length of the dangling end, t = τo eaL/o ∝ eN with a a material constant. In other words, the length of a chain that has retracted in time t is L(t) = (o /a) ln[t/τo ]. Clearly, while the chain is still retracting, it provides an additional resistance to deformation (local stress), acting as an anchor attached to a crosslinking point. By calculating the (binomial) probability distribution of lengths of free dangling ends in a network of chains formed by radical polymerisation, P(N)  q(1 ˜ − q) ˜ N−1 (with q˜  ns /c  1 the probability of any individual monomer to be crosslinked), Curro et al. were able to find the average relaxation stress due to the free dangling ends in a rubber, which is proportional to the total fraction of chains still in the process of retraction:

298

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

σ (t) − σeq = const

∞

∑

˜ (N − 1a ln[t/τo ])P(N) = const (1 − q) ˜ L(t)/o ∝ (t/t ∗ )−q/a ,

N=L(t)/o

for q˜  1. This interesting new power law relaxation has an exponent, determined by the molecular properties of polymers forming the network. In all cases the estimates give a very small value of this non-universal exponent, of the order q/a ˜ ∼ 0.1. This fits very well to the classical results on long stress relaxation by Chasset and Thirion. 11.2

Nematohydrodynamics of elastic solids

In order to study the dynamics of the mechanical response of nematic networks, one needs to describe viscous dissipation in a system of anisotropic chains moving towards its equilibrium in both translational and orientational spaces. We started this chapter by identifying the conflict between hydrodynamic and polymer problems. At present, there is no microscopic model that would even approximately describe the dynamics of anisotropic rubbery networks, although interesting attempts have been made to address the viscosity of nematic polymers (Long and Morse, 2000). However, as in the theory of viscosity of ordinary liquid crystals, briefly described in Sect. 2.7, much progress can be made on a phenomenological level using the symmetries of variables contributing to the physical effects in question. In the first attempt of such combining of simple hydrodynamic and elastic degrees of freedom (Brand and Pleiner, 1994) wrote down equilibrium and dissipation terms according to their underlying symmetry. Later, nematic viscous terms were regrouped to match the symmetries of the elastic terms (Clarke et al., 2001) and the subtleties of nematic rubber response in the dynamical context addressed in a concrete manner. We briefly examine this hydrodynamic approach here. Under the assumption of constant magnitude of local nematic order Q, two physical fields describe the state of motion of a viscoelastic nematic solid: the local variation of director orientation δ n(rr ,t) = n − no with respect to the equilibrium no and the local fluid velocity v(rr ,t), which is the time derivative of the corresponding elastic displacement u(rr ,t). After the extensive discussion of relevant relative rotations in Chapter 10, it is useful introduce a compact notation, Θ = Ω − [nn × δ n], which is the independent rotational variable in both the equilibrium and the relaxation problems. The dynamic equations are then obtained by adding the forces deriving from the classical Lagrangian density, L , and those from the Rayleigh dissipation function, T s. ˙ Let us, as usual in this book, neglect in the first instance the effects of Frank elasticity that arise from director gradients. Adding the kinetic energy density to the familiar expression for the elastic potential energy density F (e.g., eqn (10.12) of Sect. 10.2), one obtains L = 12 ρ (∂t u )2 − F(uu, Θ ). The dynamic equations result from the variational principle applied to the Lagrangian and the Rayleigh dissipation function, separately, with the variables strain and strain rate, respectively:

ρ∂t2 u = ∂k

  ∂F ∂ T s˙ + ∂k ∂ [∇k u ] ∂ [∇k u˙ ]

(11.12)

NEMATOHYDRODYNAMICS OF ELASTIC SOLIDS

0=

∂ F ∂ T s˙ . + ˙ ∂Θ ∂Θ

299

(11.13)

The first equation describes the balance of forces acting on a material point, while the second represents the balance of torques, both of elastic and viscous origin. Strictly, to obtain the proper torque, the derivatives here should be taken with respect to the nematic director rotation vector [nn × δ n ] (and its time derivative), which is equivalent to (11.13) due to the linearity of the relative rotation Θ . The local moments of inertia are neglected in eqn (11.13): at low frequencies and flow rates this is well-justified in classical hydrodynamics, see (de Gennes and Prost, 1994), while in the high-frequency regime the effects of the polymer glass transition would overwhelm the angular inertia. We now develop the form of the dissipation function, describing the linear viscous effects in the nematic fluid, by a direct analogy with the Leslie-Ericksen theory of liquid nematic liquid crystals. However, after the earlier detailed analysis of equilibrium continuum elasticity, it is more natural to use the same symmetry-grouping of terms as in (10.12), rather than as in (2.33) as is usual for classical nematics. Neglecting the effects of heat convection, the total energy dissipation (the entropy production) in a uniaxial anisotropic medium is expressed by the volume integral of the conjugate forces and fluxes, eqn (2.36). Let us now re-write the density of dissipation function in a form to match the elastic energy density (10.12): T s˙ = A1 (nn · ε˙ · n )2 + 2A4 [nn × ε˙ × n ]2 + 4A5 ([nn × ε˙ · n ])2 + 12 γ1 N 2 + γ2 n · ε˙ · N

(11.14)

in the fully incompressible case. The Leslie notation denotes the relative rotation rate ˙ × n ]. In the regrouped form the new viscosity coefficients Ai are linear comas: N = [Θ binations of the classical Leslie coefficients (applied to the nematic elastomer medium) A1 = 12 (α1 + α4 + α5 + α6 ),

(11.15)

A4 = 14 α4 , A5 = 18 (2α4 + α5 + α6 ). In the isotropic limit, one finds A1 = 2A4 = 2A5 → η , reminiscent of the isotropic Lam´e limit of the corresponding elastic constants. By differentiation of (11.14), respectively by the symmetric strain rate ε˙i j and by ˙ i , one obtains a representation of the symmetric viscous stress tensor and the nematic Θ molecular field, contributing to the local viscous torque, analogous to the parallel expressions (2.33) and (2.35) for simple nematics:       ˙ ˙ n · n × n × σisym = 2A ε · n n n + 4A ε × n × n i j 1 4 j ij        n × ε˙ · n × n n j + n × ε˙ · n × n ni + 4A5 i j   ˙ j +Θ ˙ in j ; (11.16) + 12 γ2 ni Θ

300

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

˙ i + γ2 n j ε˙i j , hi = γ1 Θ sym

where σi j is the symmetric part of an expression like (2.33), which emerges naturally since we express T s˙ in terms of symmetric and anti-symmetric variables separately. The molecular field h, expressing the torques of viscous origin, cf. eqn (2.35). In Sect. 2.7 we have already mentioned two important non-dimensional numbers controlling the regimes of fluid dynamics in nematic liquids, the Reynolds number Re and the Ericksen number Er. In elastomers and gels, one is much more concerned with the balance between flow-induced torques, scaling as ∼ η ∇vv, and those of the rubbery matrix, expressed by ∂ F/∂ Θ ∼ D1 Θ. This yields a new dimensionless group of parameters characterised by the number Ne = η v/(LD1 ) (the abbreviation Ne chosen to stand for ‘Nematic Elastomers’). Note that the assumed dominance of rubber-elastic over Frank effects essentially means that D1  K/L2 , that is, Ne  Er. 11.2.1

Viscous coefficients and relaxation times

There is a clear parallel between the form of equilibrium potential energy density F(uu, Θ ), ˙ ), eqn (11.14): eqns (10.12), and the viscous coefficients of the dissipation function T s( ˙ u˙ , Θ Ai → Ci ; γ1 → D1 and γ2 → D2 . Of course, this is not a surprise since elastic deformations and viscous flows in a uniaxial continuum share the same symmetry. Equally, there is a direct correspondence in the dependence of the various coefficients on nematic order parameter Q, see eqn (2.33) for viscosities and eqn (10.14) for elastic constants (where the network strand anisotropy r ≈ 1 + β Q). Thus in this hydrodynamic model the viscous coefficients are related to their corresponding elastic coefficients by a certain time scale: Ai = Ci τR ; γ1 = D1 τ1 ; γ2 = D2 τ2 .

(11.17)

These relaxation times τR do not have to be equal for each pair of coefficients, however, one can expect them all to be of the same order of magnitude - of the order of the only relevant time in the problem, the Rouse time for the corresponding polymer backbone, see the discussion in Sect. 11.1.2. Two orientational relaxation times, τ1,2 , describe the rotational dynamics of the nematic director. One expects, in many polymer systems, to find τR ∼ 10−4 − 10−6 s (Doi and Edwards, 1986), while the nematic director relaxation time is τ1 ∼ 10−2 s (Sch¨onstein et al., 2001). However, as we have discussed earlier in this chapter, in practice very little is known about the intrinsic time scales of polymer chains forming the rubbery network and one must not attribute too much importance to the value of such estimates of, e.g., a Rouse-like time scale in a nematic elastomer. The hydrodynamic approach combines linear viscous stresses (11.16) with the corresponding elastic stress producing effective frequency-dependent moduli in the form (Ci + iω Ai ). This is the low-frequency limit of a general complex modulus G∗ (ω ), see Sect. 11.1, showing the rubber plateau modulus G = Ci and the initial rise in the loss modulus with frequency, describing a viscous dissipation. In the classical picture of network dynamics, the next characteristic regime is at a Rouse-like frequency, when

NEMATOHYDRODYNAMICS OF ELASTIC SOLIDS

301

the signal cannot propagate along the polymer chain length and mechanical response is provided by individual segments, thus causing a climb of G∗ (ω ) towards the glass plateau. Therefore, the characteristic time scale τR in the relations (11.17) should indeed be of the order of the Rouse time. Could there be viscous softness analogous to the elastic softness of ideal elastomers? Physical arguments suggest that this may not be possible. The viscous combination A5 − γ22 /8γ1 , analogous to the renormalised elastic constant C5R = C5 − D22 /8D1 , can be rewritten with the benefit of timescales (11.17) as:  τ22 D22 2 . (11.18) A5 − γ2 /8γ1 = τR C5 − τR τ1 8D1 Only if the three relaxation times combine in such a way that τ1 τR = τ22 , one has a possibility of renormalising eqn (11.18) to zero, if C5R → 0. Elastic softness arises because an anisotropic distribution of chains can be rotated undistorted and thus at constant entropy. However, individual chains will be distorted and there should reasonably be dissipation associated with their flow relative to the matrix. Thus elastically soft distortions should nevertheless have associated dissipation, which could only accidentally vanish. Another aspect of the relationship between elastic constants and relaxation times, eqn (11.18), is the requirement of positive-definiteness of the entropy production. It is very easy to check that T s˙ is non-negative only when the combination A5 − γ22 /8γ1 ≥ 0. This imposes a constraint on the values of relaxation times (assuming τR is an average ‘Rouse-like’ time between different modes of anisotropic deformations and flows, Ci and Ai ): D22 2 τ 1 τR ≥ τ . (11.19) 8C5 D1 2 In an ideally soft elastomer, with C5 = D22 /8D1 , this reduces simply to τ1 τR ≥ τ22 . For instance, if one assumes (based on certain compelling experimental evidence to be discussed below) that τ1 ∼ 10−2 s and τR ∼ 10−4 s, this requires that τ2 ≤ 10−3 s, at least an order of magnitude shorter than τ1 . If one takes the Rouse time as τR ∼ 10−6 s, then τ2 ≤ 10−4 s. 11.2.2

Balance of forces and torques

To complete the general phenomenological analysis of nematic rubber dynamics, we briefly discuss the stresses and torques that follow from the combination of elastic and dissipation functions. The set of dynamic equations now takes the ‘standard’ form of balancing local forces and local torques:   ∂ F ∂ (T s) ˙ 2 sym (11.20) ρ∂t u = ∇ · σ ≡ ∇· + ∂ε ∂ ε˙   ˙ × n ] + D2 (nn · ε ) + γ2 (nn · ε˙ ) . Θ × n ] + γ1 [Θ (11.21) 0 = n × D1 [Θ

302

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

If one assumes that the nematic elastomer penetration length is not negligibly small, K/µ L2 ∼ 1, then the torque balance will be controlled by the full molecular field with contributions from Frank elasticity of the nematic, rubber-elastic coupling and the viscous terms: ˙ × n + γ2 n · ε˙ , h full = K∇2 δ n + D1 Θ × n + D2 n · ε + γ1 Θ

(11.22)

in one-constant approximation for simplicity. Such a situation may occur in highly diluted gels where the rubber elastic energy scale µ is very low – or indeed when the relevant sample is very thin (L small) and is affected by boundary conditions. Here we continue to discard the contribution of the Frank elasticity to the torque balance (11.21), assuming it is negligibly small in the presence of relative rotation coupling to the underlying rubber-elastic network. In the limit of isotropic rubber, Q → 0, the only relevant equation is that for the symmetric stress, which in this case reduces to

σ = C4 ε + A4 ε˙ . This is a simplest viscoelastic approximation at lowest frequencies, or shear rates, of a  general linear-response expression σ (t) = G(t − t  )ε˙ (t  )dt  . The other limiting cases are those of uncrosslinked, nematic liquid crystals. The equilibrium (zero-frequency) elastic moduli Ci and Di are zero and the only contribution to the stress and torque balance are the Leslie-Ericksen eqns (11.16). In a low molecular weight nematic, this is the full description of an anisotropic Newtonian liquid. However in a polymer nematic, one again expects a complex viscoelastic response function, G(t − t  ), with several characteristic time scales, from the shortest α -relaxation time, to Rouse, entanglement and disengagement times (if applicable). The Leslie-Ericksen equations (11.16) are thus the low-frequency limit of such complex nematic viscoelasticity. Experiment (Figs. 11.8 and 11.9 below) will show there is a range of low frequencies where viscoelastic stress is subordinate to quasi-equilibrium nematic rubber and Ericksen-Leslie hydrodynamic stresses and torques. The proof is that shear geometries sensing director rotation are much softer than that which does not, the distinction being lost at higher frequencies where viscoelastic effects common to all geometries and to the isotropic state are dominant. The balance of local torques, [nn × h ] = 0 in the absence of external fields, eqn (11.21), is thus achieved by two groups of terms which are manifestly of the same order of magnitude in the regime of frequencies characterising director rotations. The condition of zero net torque allows one to obtain, for instance, the rate of re˙ In a fixed coordinate system, for instance with the initial director n lative rotation Θ. along the z-axis, and aiming to describe oscillating solutions giving dtd ⇒ iω , this is a straightforward procedure. The formal solution of (11.21) reads   D2 + iωγ2 −εyz Θx = . (11.23) Θy εxz D1 + iωγ1 Substituting it into the corresponding components of stress σ sym , one obtains the effective viscoelastic response. This operation, known as the integration out of internal

NEMATOHYDRODYNAMICS OF ELASTIC SOLIDS

303

degrees of freedom, is precisely what has been done in Chapter 10 in the analysis of static soft elasticity. Here it is complicated by the viscous terms, but the essence remains the same: the effective shear modulus is renormalised C5R (ω ) = C5 −

D22 (1 + iωτ2 )2 . 8D1 (1 + iωτR )(1 + iωτ1 )

(11.24)

Clearly, at any non-zero frequency there will no longer be a complete cancellation – however, the reduction of the effective elastic modulus will nevertheless occur. 11.2.3

Symmetries and order parameter

We have seen that the anisotropic rubber moduli and the Leslie viscous coefficients depend on the underlying nematic order parameter Q, usually a function of temperature or solvent concentration. In particular, note that the coupling constant D2 depends on the linear power of Q. As a result, when there is anisotropy, Q = 0 and r = 1, the sign of the elastic constant D2 depends on whether r > 1 (prolate order, D2 < 0) or r < 1 (oblate order, D2 > 0). Accordingly, the sign of director rotation relative to the matrix, ω − Ω , varies according to whether chains are prolate or oblate. The former align along the extension direction associated with the shear, the latter along the compression direction. In both of these two cases, a term such as D2 εnk ω , bilinear in εnk ω , always reduces the elastic energy, irrespective of the sign of D2 , and indeed this is the actual mechanism whereby a shape change can ideally be achieved with no energy cost, see Sect. 10.4.2. There is an appealing analogy between this behaviour and the Leslie coefficient γ2 . This viscous coefficient is usually negative for a nematic liquid of classical rod-like molecules. Indeed, it is known (Volovik, 1980; Carlsson, 1983) that disk-like molecular shape leads to the opposite sign of γ2 in discotic liquid crystals, with the according consequences for the flow alignment properties. The ideal rubber-elastic moduli (10.14) suggest that the coupling constant D1 is proportional to (r − 1)2 ∼ Q2 . Continuing the analogy with liquid nematic viscosity, we should recall the estimates of Sect. 2.7 (Imura and Okano, 1972), but also the following qualitative consideration. The rotational viscosity γ1 is determined by the antisymmetric part of viscous stress tensor σij , which cannot be proportional to the linear power of the symmetric tensor nematic order parameter Qi j , but at least its square. However, there is a delicate problem arising in the analysis of soft elasticity. The renormalised shear modulus C5R = C5 − D22 /8D1 should reduce to the bare C5 → 12 µ in the isotropic phase at Q → 0. However, the ideal values of D1 and D2 , scaling as Q2 and Q respectively, result in a finite renormalisation of D22 /8D1 = 0 in the limit Q → 0! The answer to this paradox is thought to be in the phenomenon of semi-softness, however small it might be. This point is discussed around eqn (10.41). In fact, D1 has a additional small ‘semi-soft’ correction, which is proportional to a lower power of order parameter, ∼ Q. This resolves the problem of making the renormalisation D22 /8D1 vanish at Q → 0, but raises a question about the symmetry consideration that γ1 (and also D1 ) cannot be linear functions of Qi j . The problem is safely resolved when one recognises that the semi-soft addition a1 , e.g. in eqn (10.41), is a linear function of Qf , the order at network formation. Therefore, in fact, both D1 and γ1 in nematic elastomers

304

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

depend on the bilinear combination Qfi j Qi j , which means overall a variation ∝ |Q| (since Qf is a constant provided by sample history), and thus no symmetry problem arises. In an ordinary liquid nematic there is no issue of formation order being frozen by crosslinking since Qf = 0, and thus γ1 ∝ Q2 as expected. Strengths and weaknesses of the hydrodynamic approach In the face of complex polymeric influences on the dynamics we have preliminary adopted a hydrodynamic approach. A strength of this is that it is symmetry based. We can group simple viscous stresses in the same way that we have done for the equilibrium elastic stresses, that is, based on the geometry of deformations. The elastic and viscous coefficients that emerge in correspondence to each other, the Cs and As, are related in an internally consistent way via a certain timescale, C/A = 1/τ . On the other hand we have, optimistically, used equilibrium values of the rubber elastic moduli, C, even though we are not at equilibrium. Experimentally there is a range of low frequencies where nematic-viscous plus equilibrium elastic effects dominate over visco-elastic effects, even though one is still distinctly away from equilibrium. Of course in reality we cannot separate and then add viscous, elastic and nematic effects as in eqns (11.12) and (11.13). One should instead have complex responses A∗i (ω ) where neither the real nor imaginary parts can be simply identified as above or where the ω dependence is so simple as above. In the absence of a more complete approach, we proceed in the spirit of hydrodynamic approach. 11.3

Response to oscillating strains

The three principal shear geometries of classical dynamic mechanical experiments on a nematic elastomer are shown in Fig. 11.5. The three cases are labelled according to the orientation of the director n with respect to the shear directions: G when n is along the shear gradient, D when it is pointing along the displacement direction and V when n is along the vorticity. Let us assume that deformations are sufficiently small and the sample is big so we can neglect director variation and therefore also neglect Frank elasticity. The strain is the same for each director orientation. It is a simple shear and the incompressibility constraint is satisfied automatically. We have then the symmetric part εzx = 12 ε and the antisymmetric (body rotation) part is Ωy = 12 ε . Note that the three principal shear geometries in Fig. 11.5 are the same as in the classical setting for Miesowicz viscosity experiments, cf. (de Gennes and Prost, 1994). That is, if the director is kept immobile (e.g. by a strong external field, which incidentally would be hardly possible in elastomeric network under strain), then the viscous coefficients in the flow orientations V, D and G are the Miesowicz viscosities ηa , ηb and ηc , respectively. Our purpose is to examine the effect on viscoelastic response of the director’s freedom to rotate. The elastic free energy density, eqn (10.12), takes the form, in the three cases of Fig. 11.5:   FG = C5 + 18 [D1 − 2D2 ] ε 2 − 12 (D1 − D2 )ε θ + 12 D1 θ 2   FD = C5 + 18 [D1 + 2D2 ] ε 2 − 12 (D1 + D2 )ε θ + 12 D1 θ 2

RESPONSE TO OSCILLATING STRAINS

305

F IG . 11.5. The geometry of simple shear experiment with three principal orientations of the nematic director n 0 . The small-amplitude shear ε ∼ eiω t is applied to the elastomer and the measured stress σ (ω ) provides the linear response modulus in each of the three configurations G , D and V . FV = C4 ε 2 ,

(11.25)

where the small change in director orientation, δ n , is taken equal to the angle θ . Clearly, one does not expect director rotation to occur in the ‘log-rolling’ geometry V . The bilinear term ∼ ε θ reduces the elastic energy: if a strain ε is imposed then the director responds by adjusting θ . Given sufficient time to reach equilibrium, θ adopts its optimal value for a given deformation ε . Returning this minimised value θG,D from eqns (11.26) back into (11.25), the free energy density at a given strain is also optimal: D1 ∓ D2 θG,D = ε 2D1

⇒

FG,D

 D22 → C5 − ε2 8D1

(11.26)

If the nematic elastomer is ideally soft, then the free energies in the cases G and D vanish because their geometry allows the director to respond to the shear and internally relax. Case V remains elastically hard. In fact, because of the chosen restricted strain geometry, the response of even ideal elastomers in geometries G and D is actually quartic, rather than completely soft (true softness requires some unconstrained extension as well as shear). This is formally the same as in the Freedericks transition, Sect. 8.1. The molecular model discussed in that section yields a quartic penalty F = 12 µ

r2 ε 4, (r − 1)2

which one must neglect in linear-response analysis. One might well refer to quartic response as soft, since in linear elasticity there is no energy cost to such strains. (To see the correspondence with Sect. 8.1, recall there one was interested in the angle of the corresponding director rotation which was linearly related to the amplitude of simple shear. The energy being ideally quartic in the angle implies a quartic dependence in the corresponding shear.)

306

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

In the dynamical case, the evolution of the director is given by the torque-balance eqn (11.21); for the two geometries where director rotation is present it takes the form (G :) (D :)

γ1 θ˙ = −D1 θ + 12 (D1 − D2 ) ε (t) + 12 (γ1 − γ2 ) ε˙ (t) γ1 θ˙ = −D1 θ + 12 (D1 + D2 ) ε (t) + 12 (γ1 + γ2 ) ε˙ (t) ,

(11.27)

with θ the angle of relevant director rotation. These linear, inhomogeneous differential equations are easily solved. After the transient relaxation θ  θ0 e−(D1 /γ1 )t associated with starting the strain oscillations has completely relaxed, the steady-state response is given by the particular solutions with the oscillating driving shear ε ∼ ei ω t :   t γ2 1  −(D1 /γ1 )[t−t  ] D1 ∓ D2   θG,D (t) = dt e ε (t ) + (1 ∓ ) ε˙ (t ) (11.28) 2γ1 2 γ1 −∞ where the signs − or + correspond to the G or D geometry, respectively. The solutions depend on a characteristic time for director rotation, τ1 = γ1 /D1 . The linear ‘nominal stress’ in response to the imposed simple shear deformation ε (t) is given by the sum of elastic and viscous contributions, that is

σ=

∂ F ∂ (T s) ˙ + . ∂ε ∂ ε˙

Substituting the optimal value for the director orientation angle (11.28), at a given frequency of imposed strain, one obtains the form σ (ω ) = G∗ (ω )ε (ω ) with the linear complex shear modulus G∗ = G + iG . Remarkably, although perhaps predictably, the response in the two geometries, G and D , involving n -rotation is exactly the same: σG = σD , despite the difference in the rotation angles θG and θD . The corresponding storage and loss moduli have a single-relaxation time behaviour with a characteristic frequency ω1 = D1 /γ1 and can then be written in a universal form: (D2 γ1 − D1 γ2 )2 (11.29) 4D1 γ12 1 + (ωτ1 )2   ωτ1 (D2 γ1 − D1 γ2 )2 + ωτ1 12 D1 (γ2 /γ1 )2 − 2A5 /τ1 . G (ω ) = 2 2 4D1 γ1 1 + (ωτ1 ) G (ω ) = 2(C5 − D22 /8D1 ) +

(ωτ1 )2

The ‘non-soft’ V -geometry returns a trivial sum of the elastic and viscous parts: GV∗ (ω ) = 2C4 + 2iω A4 ≡ 2C4 (1 + i ωτR ) ,

(11.30)

which is the essence of hydrodynamic approximation. Figure 11.6(b) shows schematically the storage modulus (the real part, G ) of an ordinary polymer network. To make the simple conceptual point, this plot ignores more complicated effects, such as entanglements. Thus, at ω → 0 the isotropic modulus approaches the constant value G = 2C4 ≡ µ , which is of course the equilibrium rubber modulus. At ωτR  1 one finds the dynamic glass transition, the α -relaxation. The shear modulus saturates at a very high (glassy) value at high frequencies.

Storage modulus G’/m

RESPONSE TO OSCILLATING STRAINS

1

G’ =m

0.1 -3 10

G’/m (Log-scale)

Rubber plateau -2

10

-1

0 101

10

1

10

Reduced frequency wt 1

Glass plateau

2

10

1

1.5

Loss modulus G’’/m

307

t -1 R Frequency (Log-scale)

1

(b) 0.5

0

0

2

4

6

8

10

Reduced frequency wt 1

12

(a)

F IG . 11.6. (a) The frequency dependence of storage (G , top graph, log-scale) and loss (G , bottom graph, linear scale) moduli, in units of rubber modulus µ . At ω → 0 both G → 2C5R . The families of curves in both plots correspond to the increasing nematic order (e.g. by lowering the temperature), indicated by arrows; at Q = 0 one finds G = µ as in the classical rubber. (b) The sketch of a simplified ‘classical rubber response’, a variation of G with frequency. Theoretical plots (a) represent the region of low frequencies, modifying the characteristics of the rubber plateau. The main content of Fig. 11.6 is given in the two plots (a), which show the frequency dependence of G and G , expressed by eqns (11.29) in the two director-rotating geometries, G and D (remember, the linear complex modulus is the same in these two cases). One finds the classical single-relaxation behaviour with the characteristic time scale τ1 = γ1 /D1 . At ωτ1  1 (assuming this is still much lower than the dynamic glass transition), the storage modulus saturates at the constant level of modified rubber plateau, given by eqn (11.29), while the loss modulus begins its linear rise towards the glass transition peak at ω  τR−1 . More interesting is the behaviour at ωτ1  1. Here we see a substantial drop in the storage modulus due to the internal director relaxation - the effect of dynamic soft elasticity. The family of curves G (ω ) in Fig. 11.6(a) is drawn for the increasing nematic order parameter Q, which enters into the elastic constants (via r ≈ 1+ β Q) and the Leslie coefficients. At Q = 0, in the isotropic phase, G = µ and G ∝ ω in this hydrodynamic limit we take as ω → 0. As the nematic order increases, we see an increasing drop in G and also the rise of a new loss peak in G at ωτ1 ∼ 1. To emphasize the role of nematic order, Fig. 11.7 shows the temperature dependence of G and the loss factor tan δ = G /G , at several frequencies around ωτ1 ∼ 1. To plot this, we need some model for the dependence Q(T ), or alternatively – take interpolated

308

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

F IG . 11.7. (a) Plots of the reduced storage modulus G /µ against reduced temperature for a number of increasing frequencies: ωτ1 = 0.1 (circles), 0.5 (squares), 1 (triangles), 2 (diamonds) and 10 (stars). An apparent critical behaviour near the transition point Tni is due to the nematic order Q(T ). (b) Plots of the loss factor tan δ = G /G against reduced temperature for the same set of increasing frequencies (the direction shown by arrows). experimental values from, for example Fig. 6.1. At low temperature and ω → 0 the storage shear modulus approaches the semi-soft constant Go = 2(C5 − D22 /8D1 ), cf. Fig. 11.6(a) and eqn (11.29). 11.4 Experimental observations As we have discussed in Sect. 11.1, there are two main ways one can examine internal dynamics and relaxation mechanisms. One method involves an application of a dynamic oscillating perturbation (either force or displacement) and monitoring the (complex) linear response function. The other approach is to monitor the static relaxation a fixed constant deformation is applied to the sample (stress relaxation) or a constant force is imposed (creep). At a basic level, the two approaches should provide the same information (Ferry, 1980) contained in the appropriate time- or frequency-dependent linear response coefficient, defined in Table 11.1. We illustrate these two methods in a concrete manner below: Exercise 11.1: Find the frequency-dependent complex shear modulus for a Debye relaxation process. Solution: The Debye process is the viscous relaxation when the returning force is proportional to the distance to equilibrium, that is x˙ = −x/τ , resulting in the simple exponential decay. In our case, this corresponds to the exponential decay  of ‘memory’, G(t − t  ) = Go e−(t−t )/τ . So, if a constant strain εo is applied as an instant step at t = 0, its time-derivative ε˙ = εo δ (t) and the stress in the Debye model would relax as

σ (t) = Go εo e−t/τ . In the frequency domain one looks for the coefficient connecting σ (ω ) and ε (ω ). The arguments leading to eqn (11.29) follow the direct approach of imposing a

EXPERIMENTAL OBSERVATIONS

309

single-frequency mode ε (ω ) and simply calculating the stress response. The other possibility is to examine the Fourier transformation of (11.2):    F T Go e−(t−t )/τ = −

iτ Go , i + ωτ ωτ Go ε (ω ) hence σ (ω ) = i + ωτ

The coefficient is, by definition, the linear complex modulus of a single relaxation time process: (ωτ )2 iωτ Go − Go . G∗ (ω ) = (ωτ )2 + 1 (ωτ )2 + 1 Exercise 11.2: Find the complex shear modulus for a power-law relaxation. Solution: Many complex systems experience slow relaxation described by a stretched exponential, or even a power law (usually this is a signal of cooperative processes and interactions between different relaxation modes). Assume the linear response function decays as a power law, at long times G(t) ≈ Go t −y . Calculating the corresponding Fourier integral and multiplying it by iω , we can obtain an estimate of the complex shear modulus:   1 3 (11.31) G∗ (ω ) ≈ Go ω y Γ(1 − y) e 2 iyπ − e− 2 iyπ , where the constant complex factor depends on the relaxation exponent y, but the most important feature is the power-law frequency dependence, G ∝ ω y .

11.4.1 Oscillating shear Dynamic-mechanical thermal analysis (DMTA) is a traditional experimental technique when oscillating strain is applied to the sample and the response force is interpreted in terms of the complex modulus G∗ (ω ) = G + iG . In practice, whether using commercial or purpose-made DMTA devices, the range of accessible frequencies seldom stretches beyond the 10−2 -103 Hz range. An analytical technique traditional in polymer dynamics is the method of time-temperature (t-T) superposition. Described in classical monographs (Ferry, 1980; Struik, 1978), this method allows one to build what is called the Master Curve G(ω ) by shifting along the frequency (or time) axis and superposing the data obtained at different temperatures. There is no rigorous theory underpinning t-T superposition, but the empirical arguments of Williams, Landel and Ferry (WLF) (Dickie and Ferry, 1966) are based, in essence, on ideas about polymer glass transition. So, by choosing an arbitrary reference temperature, Tref , for the modulus G(ω ), one shifts to lower frequencies the data obtained at T > Tref and to higher frequencies the data at T < Tref . This presumes that the available experimental window of frequencies explores the same physical process in portions, depending on the working temperature in relation to the glass transition Tg . In other words, at low temperatures (especially at T < Tg ) practically no viscoelastic relaxation takes place in the polymer material, which is equivalent to examining a high-frequency response; at high temperature the internal relaxation is accelerated and this shows as an effectively low-frequency response.

310

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS 9

Storage modulus G’ (Pa)

10

G' (100Hz) G' (50Hz) G' (5Hz) G' (0.5Hz) G' (0.05Hz)

8

10

D 50 space 50Hz V 50 D5 5Hz V5 D 05 0.5Hz V 05

7

10

6

7

10

10

T ni

T ni

6

10

5

10 5

10

Tg

4

4

10

-50

0

50

100

Temperature T ( C)

10

20

40

60

80

100

Temperature T ( C)

O

O

(a)

(b)

F IG . 11.8. (a) Plots of the storage shear modulus G (for the shear in D -geometry) against temperature for a number of frequencies, showing the full evolution of the system from isotropic to nematic and further to a polymer glass. (b) Similar data, focusing on the vicinity of Tni , and comparing the storage modulus in the D - and V -geometry, 0.5, 5 and 50 Hz. Clearly, the drop in the modulus is related to the director’s ability to rotate. The variation of the storage modulus of nematic elastomer with temperature at a number of fixed frequency values is shown in Fig. 11.8(a). This example (Clarke et al., 2002) uses a siloxane side-chain material with a glass transition temperature Tg ∼ 0o C and its nematic-isotropic transition Tni ≈ 90o C. It is evident that the low-frequency shear modulus experiences a pronounced drop just below Tni – a result comparable with the theoretical model of Fig. 11.7(a). Figure 11.8(b) shows the transition region in greater detail; more importantly, it gives the comparison of the measured shear modulus in the D -geometry, when the director rotation causes dynamic softness, and in the V geometry, when there is no director rotation and no drop in G . Figure 11.9 shows the parallel result – the frequency dependence of storage modulus at a number of different temperatures. Characteristic frequency scans are obtained by applying small amplitude oscillating shear to a nematic elastomer. Let us use this example to illustrate the t-T superposition. Choose the reference temperature to be the ambient Tref = 22o C (295K), and shift the frequency along the logarithmic temperaturedependent scale for each data set until they match on a continuous curve. The Master Curve is thus obtained, Fig. 11.10(a,b), for all the data below Tni , or more precisely – for the temperature up to the minimal value of G (T ) in Fig. 11.8. The shift factors, aT , defined as ωscaled = aT ω , strongly depend on temperature (Ferry, 1980): Fig. 11.11 shows that an adequate fit of the low-temperature region is achieved by the empirical WLF relation −C1 (T − Tref ) (11.32) log aT ≈ T − Tref +C2 with the WLF coefficients C1 = 17.4 and C2 = 102, not dissimilar to ordinary isotropic polymer systems. This is perhaps not surprising when one recalls that the empirical

EXPERIMENTAL OBSERVATIONS

311

9

Storage modulus G’ (Pa)

10

-10C 8

10

50C

60C 70C 80C 90C 100C 130C

70C

Tni =90C

5C 10C

7

10

105

20C

Isotropic rubber Nematic plateau

}

6

10

5

10

4

10 0.01

0.1

1

10

100

0.01

0.1

1

10

100

freq Frequency (Hz)

Frequency (Hz)

(a)

(b)

F IG . 11.9. Plots of the storage shear modulus G (for the shear in D -geometry) against frequency for a number of temperatures, showing the full evolution of the system from the rubbery plateau to the glass state. (a) All temperatures are below Tni , so the low-frequency end shows the semi-soft value G0 ; the frequency dependence invites the time-temperature superposition. (b) Higher temperatures, below and above Tni , show the reduction of the low-frequency modulus from its isotropic value G = µ to the nematic (semi-soft) plateau value G0 . WLF relationship is ‘designed’ to describe the dynamic glass transition region. Note that both elastomers illustrated in Figs. 11.11(a) and (b) are analysed with nearly the same reference temperature, chosen around 20o C. In contrast, the shift factors deviate from the ‘classical’ WLF behavior very strongly as soon as higher temperatures (and, correspondingly, long relaxation times) are considered. Here, different materials clearly behave differently: the same side-chain siloxane elastomer as shown in Figs. 11.9 and 11.8 only allows the t-T superposition with much smaller frequency shifts aT , while the Master Curve for the highly anisotropic main-chain nematic elastomer demands much greater shifts. This difference is also seen in the Master Curves for these two contrasting nematic elastomers. One should note that, in practice, the t-T superposition (shifting of plots along the frequency axis) is very sensitive to the value of each shift factor aT and, accordingly, the errors in Fig. 11.11 are very small. It is more likely that the noise in the aT data is due to uncertainties in temperature. The experimental Master Curves, constructed by t-T superposition, show the key characteristic features of polymer dynamics. Generally, the high frequency regime of α relaxation and the dynamic glass transition is very similar to what ordinary elastomers and polymer melts show. However, the Rouse time scale at a reference Tref ∼ 20o C, determined by the onset of the power-law rise towards the glass transition, appears to be at a level τR ∼ 10−2 -10−3 s for the two very different materials. This is a most unusually low characteristic frequency (most polymers and networks have τR ∼ 10−5 -10−6 s). One could question whether the traditional way of estimating τR from the position of the peak in the parallel data for G (or the ratio tan δ ) is valid in nematic elastomers where we expect the dissipation to be dominated by nematic relaxation effects. Still,

312

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS 9

9

10

Storage modulus G’ (Pa)

10

0.6 T

10

6

5

10

Isotropic Nematic

(oC) -10 5 15 22** 50 85 130

7

10

T (oC) -5 5 20 ** 35 50 60 65 70 75 130

0.25

8

10

t -1 R

8

10

7

10

Isotropic Nematic

6

t -1 R

10

0.07 4

10 -7 -5 -3 -1 1 3 5 7 9 10 10 10 10 10 10 10 10 10

Scaled frequency (Hz)

(a)

5

10

-20

-18

-16

-14

-12

-10

-8

-6

4

6

10 10 10 -15 10 10 10 -10 10 10 0.0001 0.01 1 100 210 10 10 -4

10

8

10 10 10 8 10 Scaled frequency (Hz)

(b)

F IG . 11.10. The master curves G (ω ) obtained by the time-temperature superposition of individual experiments, cf. Fig. 11.9. Plot (a) – the same side-chain siloxane elastomer as shown in Figs. 11.9 and 11.8, for the reference T = 22o C; plot (b) – a different material, a highly anisotropic main-chain nematic elastomer with mesogenic siloxane crosslinks, for the reference temperature T = 20o C. All superposed data are for temperatures below Tni . The rubber modulus in the isotropic phase (at 130o C in both cases) is much higher than the ‘nematic plateau’. The expected Rouse frequency at 22o C could be of the order τR−1 ∼ 102 -103 Hz. The lines indicate power laws with the corresponding exponent. the comparison of experimental data at low frequencies in Fig. 11.10(a) with the qualitative theoretical argument expressed in Fig. 11.6 reveals no distinct nematic transition at ωτ1 ∼ 1. It is clear that, in these materials, the Rouse frequency is not sufficiently separated from the nematic relaxation time τ1 for us to see the ‘secondary rubber plateau’ at τ1−1  ω  τR−1 suggested by the hydrodynamic prediction in Fig. 11.6(a)top. In fact the main-chain nematic elastomer, the Master Curve of which is shown in Fig. 11.10(b), could be an example of a polymer network where the relation between τR and τ1 even reverses – the nematic director becomes the faster relaxation mode and one has τR−1 < τ1−1 . Nevertheless, in all cases, at low frequencies the dynamic softening of the D and G -shears remains prominent. The drop to a much lower value of G due to dynamic softness is evident, from comparison with the correspondingly shifted data for the high-temperature isotropic phase, but instead of finding a characteristic singlerelaxation sigmoidal shape, the experimental G (ω ) follows a gradual slope reduction. Dynamic-mechanical analysis of liquid crystalline polymers and the issues of t-T superposition and power-law frequency dependence G (ω ) have been actively discussed in the literature (Colby et al., 1993; Gallani et al., 1996; Fourmaux-Demange et al., 1998). Generally, no universality was found, although the region just before the glass transition has an unambiguous power law behaviour. The two different materials characterised in Fig. 11.10 have G ∝ ω 0.6 and ω 0.25 during the main α -transition (note that the Rouse model predicts G ∝ ω 0.5 ). The side-chain nematic rubber has a faster internal relaxation and at a scaled frequency of ∼ 10−4 Hz the modulus has clearly reached an equilibrium rubber plateau. The relaxation is very much slower in the main-chain nematic rubber

Shift factor logarithm

EXPERIMENTAL OBSERVATIONS 8

10

6

5

4

0 -5

2

-10

0

T ni

-2 -4 260

313

-20

T ref 280

300

320

340

360

Abs. Temperature (K)

380

T ni

-15

-25 260

T ref 280

300

320

340

360

380

Abs. Temperature (K)

(a)

(b)

F IG . 11.11. A typical WLF plot of the temperature dependence of shift factor logarithm, log aT : (a) for the same side-chain siloxane elastomer as in Figs. 11.9 and 11.8, for the reference T = 22o C, and (b) for the main-chain nematic elastomer with mesogenic siloxane crosslinks, for the reference T = 20o C. The solid line in both graphs is the attempted fit to eqn (11.32). – it appears that even at ω ∼ 10−19 the material is still far from its equilibrium response. Note that the traditional approach of simply studying the DMTA response at an even higher temperature, thus effectively exploring lower frequencies, is not possible here: at Tni = 105o C the elastomer becomes isotropic! There are indications (Elias et al., 1999) why the relaxation in main-chain nematic polymers may be frozen by hairpin folds (de Gennes, 1982; Meyer, 1982). However, both Master Curves in Fig. 11.10, for side-chain as well as for the main-chain network, indicate that complex, cooperative dynamic processes take place in nematic elastomers at low frequencies, when the underlying director rotation is involved. Finally, Fig. 11.12 makes two didactic points about the dynamic mechanical response in general. One of the plots illustrates the simultaneous t-T superposition of the storage and the loss moduli, G and G . Exactly the same shift factors, cf. Fig. 11.11(a) for this material, produce both Master Curves. Note the low-frequency variation of G ∝ ω 0.6 and the wide range of frequencies where G ≥ G . The other plot compares the storage modulus of (the same) nematic elastomer and its corresponding undercrosslinked version (the crosslinking density below the network percolation limit). Clearly, the latter system has no rubber plateau, G ∝ ω 0.6 all the way to ω → 0, but the higherfrequency response is notably the same as in the crosslinked rubber. 11.4.2

Steady stress relaxation

Oscillating dynamic-mechanical studies described above are usually conducted in simple shear geometry. Steady stress relaxation is more relevant in situations when a fixed extensional strain is applied (or, conversely, when a uniaxial extensional stress is applied and the creep is measured). In the high-frequency regime, ωτR  1 (or, equivalently, at very short times) the glassy shear modulus is not very different from the compression modulus and the condition of rubber incompressibility is no longer valid. However, at

314

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS

low frequencies, or at long times of continuous relaxation, the shear modulus G(t) is directly proportional to the extensional Young modulus E(t) (in an isotropic rubber it is simply E = 3G, but with nematic order there are some effects of anisotropy). Applying an instantaneous strain step of a constant value, and monitoring the relaxation of the stress, one can reformulate it in terms of the general linear response analysis of eqn (11.2), with ε (t) = εo Θ(t) which involves the step-function at t = 0. Then the stress σ (t) = G(t) εo and one can expect to be measuring the Fourier-transformed function of G∗ (ω ). Therefore, we could expect a parallel between the results of DMTA analysis and those of steady stress relaxation. Continuous stress relaxation offers a different angle to the complex problem of polymer dynamics exploring the time, rather than the frequency domain, see Sect. 11.1.3. The classical study (Chasset and Thirion, 1965), supported by many subsequent experiments on stress relaxation performed by different groups, described a very slow power law decay in ordinary crosslinked polymer networks such as an isoprene or butadienestyrene copolymer rubber, σ ∝ 1/t y at very long times, with y taking values in the range 0.1-0.15. In liquid crystalline networks, early results showed a different response. The problem in such experiments is always the need to access very long times, when the main transient processes in the polymer network have relaxed and only the slowest mode remains. A peculiar dual regime of long-time relaxation was found in polydomain siloxane nematic rubbers (Clarke and Terentjev, 1998). A characteristic time t ∗ ∼3000 s separated a regime of fast power-law relaxation, σ ∝ t −0.5 or higher, and a very slow relaxation at the later stage, attributed to an unusual logarithmic law σ ∼[1 + ln(t/t ∗ )]−1 . 9

10

9

10

8

10

7

10

6

10

5

4

10

4

3

10

3

10

2

Storage and Loss moduli

10

G' (Pa) G'' (Pa)

8

10

0.6

7

10

6

10

5

10 10 10

2

10

-7

-5

10 10

-3

-1

10 10

1

10

3

5

7

10 10 10

Scaled frequency (Hz)

9

10

(a)

G'(10%) G'(5%)

-7

-5

10 10

-3

-1

10 10

1

10

3

5

7

10 10 10

Scaled frequency (Hz)

9

10

(b)

F IG . 11.12. (a) The Master Curves for both storage and loss moduli, G and G , for the same side-chain siloxane elastomer as in Figs. 11.9 and 11.10(a). The power-law slope of the approach to dynamic glass transition is clearly the same for both functions. (b) The comparison of storage modulus G (ω ) of a ‘normal’ elastomer (the same data as in the previous plots, labelled here as 10% for its molar density of crosslinks) and the same material with number of crosslinks below percolation point (that is, nominally a polymer melt, labelled as 5% in the plot). The absence of rubber plateau at low frequency is reminiscent of sketches in Fig. 11.2, while the approach to the dynamic glass transition is evidently exactly the same.

EXPERIMENTAL OBSERVATIONS

315

A similar dual regime was reported by (Hotta and Terentjev, 2001) in the study of acrylic nematic networks, with σ ∝ t −0.67 in the early stage of stress relaxation. At very long times, the very slow relaxation regime takes over, in this case attributed to a power law decay with the exponent ∼ 0.15, similar to the one observed in natural rubbers by Chasset and Thirion. More interesting is when director rotation is involved in the response of mechanically deformed samples (this is always the case in a polydomain nematic network, but the response here would be less clean, being diffused by the quenched averaging over the sample). Accordingly, we should compare the results of stress relaxation when a monodomain nematic rubber is stretched perpendicular to the director, cf. Figs. 5.18 and 7.7, with the DMTA response of the same material in the non-trivial D- or G-geometry, Fig. 11.10. Figure 11.13 shows typical stress relaxation curves (represented by a time-dependent shear modulus G). This example is for a composite siloxane network containing mainchain nematic polymer, cf. Figs. 11.11(b) and 11.10(b), in response to a small extension of ∼ 9% applied at t = 0, for several temperatures. The double-logarithmic scale plot 11.13(b) reveals the extent of relaxation and also the power-law nature of stress decay. One can successfully attempt the time-temperature superposition of stress relaxation curves, e.g. (Hotta and Terentjev, 2001) for acrylic nematic rubbers, but more interesting is the observation that, at least in some cases, one can make a direct match between the DMTA and the relaxation data. Figure 11.14 illustrates this point, first showing the ‘inverse’ DMTA Master Curve G (ω ) plotted against the ‘effective time’ ω −1 and the relaxation data sets superposed on top of it by shifting along the time-axis for the reference temperature Tref = 20o C. This means that the respective power laws for the time and the frequency domain are exactly matching in the corresponding time or temper6

Modulus G (Pa)

4 10

31 o C

6

3 10

6

10 6

2 10

51 o C

6

1 10

83 o C

5

10

0

1

2

3

4 4

Time (10 s)

5

6

10

100

1000

4

10

Time (s)

F IG . 11.13. Stress relaxation curves, represented in (a) linear and (b) double-logarithmic scales, for the main-chain nematic elastomer after a small extension is applied instantly at t = 0, perpendicular to the nematic director. The three data sets are for different temperatures in the nematic phase, labelled on plot (b). It is evident that the extent of relaxation decreases on the approach to the isotropic phase, above Tni ≈ 90o C.

316

DYNAMICS OF LIQUID CRYSTAL ELASTOMERS 9

10

Shift factor logarithm

Shear modulus G (Pa)

10

8

10

7

10

6

10

5

10

10 -8

10 -2

10 4

10

10

10 15

10 20

5 0 -5 -10 -15 -20 -25 260

280

300

320

340

360

Abs. Temperature (K)

Inverse frequency (s)

(a)

(b)

F IG . 11.14. (a) The data for the Master Curve from Fig. 11.10(b), connected line, plotted against the inverse frequency ω −1 , representing the effective time in seconds. On top of this Master curve one can see the three superimposed data sets for the stress relaxation, from Fig. 11.13. (b) The logarithm of shift factor for this Master Curve, bullets, from Fig. 11.11(b), and the three shift factors required to superpose the stress relaxation data sets onto the Master Curve, squares. ature ranges (cf. Exercise 11.2). The plot (b) of Fig. 11.14 shows that the respective shift factors required to superpose the dynamic or steady relaxation data are the same as well. Such a conclusion indicates that if one stretches this particular nematic elastomer at 22o C, the stress would continue to decay and the mechanical equilibrium would not be reached even at the times of order ∼ 1020 s. All these, and many other theoretical and experimental paradoxes and difficulties, clearly demonstrate that much remains to be discovered in the dynamic mechanical properties of liquid crystalline polymer networks.

12 SMECTIC ELASTOMERS When positional ordering, in the form of layering, is added to the orientational order of nematics, we have the smectic phase. Chapter 2.9 briefly describes SmA and SmC liquid crystals. They offer possibilities analogous to those of nematic liquids studied so far. When made into networks, they remain locally liquid-like, can suffer large extensions and have mobile internal degrees of freedom. This freedom, which gave spontaneous distortions and soft elasticity in nematics, is restricted in smectics by the layers to which the matrix is often strongly coupled. The mechanics of smectic elastomers is decidedly more complex than that of nematic elastomers. Indeed, the effect of layers means we must construct the non-linear mechanics of a 2-D rubber where there can be mechanical escape into the third dimension. In this chapter we sketch the materials that make such elastomers and review their remarkable physical properties. We then build a non-linear, molecularly-based model of the phenomena that occur in smectic elastomers and gels. Such a model, as was the case with the Trace formula for the nematic networks, can naturally describe the very large strains liquid crystal elastomers can sustain. SmA is treated in greater detail since it is a paradigm for how rubber elasticity is constrained by rigid layers. In the following chapter, we develop continuum theory for smectic elastomers. 12.1 Materials and preparation Several classes of synthetic polymers exhibit smectic A order (McArdle, 1989). Historically, the most common are the side-chain mesogenic polymers based on siloxane [-O-Si-] and aliphatic carbon [-C-C-] chains, with rod-like molecular moieties attached to one of the reacting bonds of Si or C atoms. The synthetic strategy is the same as that for obtaining nematic polymers – briefly mentioned in Sect. 3.2 and described in review articles (Finkelmann, 1984; Zentel, 1989). Exploiting naturally smectic polymers was the first route to smectic elastomers (Finkelmann et al., 1981). The recognised method to obtain smectic rather than nematic materials is to increase the length of flexible spacer between the mesogenic side groups and the backbone. Long spacers (note (CH2 )11 in Fig. 12.1) provide sufficient mobility for the backbone to aggregate in a microphase separated way between the lamellar regions with higher concentration of rod-like mesogenic groups. This, combined with the orientational order of the rods, leads to smectic layer formation. In contrast, shorter spacers with 3-5 carbon atoms usually result in a nematic phase – as we have seen in examples of Figs. 3.4, 5.1 and 5.2. Smectic elastomer preparation also has to take into account crosslink size and its relation to layers. In fact, the sketch in Fig. 2.14(c) illustrates that one needs to be careful in selecting the size of crosslinking molecular groups. From the geometric point of view, their length must equal an integer multiple of the smectic layer spacing do . A 317

318

SMECTIC ELASTOMERS

O

O

O

(CH2)

11

(CH2)

(CH2)

O

O O

O

O

CH3

C2 H 5

Iso 60 Nem 44 SmA 20 Cr

Si O

CH3

H3C C

(CH2)11

O (CH2) 2 O Iso 35 Cr

CH3 Si O 0.9

(CH2)11

O

O

O O C

O O C

0.1

CH C H3

Cl

Iso 89 SmA 53 SmC* 29 Sx

Iso 70 Nem 59 SmA 28 Cr

(a)

2.7

O

O

O

CH3

Si O

2

11

O

CH3

(CH2)5 NH

C O

(b)

F IG . 12.1. Typical molecular structures of smectic A side-chain polymers. (a) Mesogenic vinyl ethers (1,2) and the crosslinking group with two reacting ends (3): the carbon backbone is established by polymerisation across terminal vinyl groups (Andersson et al., 1994). A similar chemical construction could be achieved by acrylate-terminated molecules. (b) A typical polysiloxane elastomer, where the crosslinking group is mesogenic on its own (Pakula and Zentel, 1991). ‘point-like’ crosslink (or a flexible chain of the same nature as the backbone – which in some sense accounts for two ‘point-like’ crosslinks) would bind backbones confined within one interlayer plane. A rod-like crosslink such as bi-functional groups shown in Figs. 2.14(c) and 12.1(a) would bind backbones across one layer if its length is ≈ do . The crosslinking group shown in Fig. 12.1(b) reacts with a vinyl bond of another such group from a different chain – the resulting length of the binding aggregate becomes ≈ 2do . Anything in between would create a strong distortion of the local smectic order near an incommensurate crosslink and, therefore, would depress the existence of the phase. In any event, crosslinking polymers in the smectic phase creates a local dependence between the crosslink and the layer position and their relative movement along the layer normal should be difficult. We shall soon return to this effect, crucial to understanding of smectic rubber elasticity. Note that the position of crosslinks has no effect in a homogeneous nematic phase: one needs to break the translational symmetry of the mesophase in order to experience such a coupling. Having briefly discussed the preparation of elastomeric networks from side-chain liquid crystalline polymers, we should not forget that many other materials possess the same class of one-dimensional, lamellar symmetry breaking. The Lα phases of nearsymmetric block copolymers are portrayed in Fig. 2.15(a) and can be crosslinked into a rubber network. They will be treated identically to the usual smectic elastomers deriving from conventional melts. Similarly, the layered phases formed from the aggregation of amphiphilic molecules, Fig. 2.15(b), when polymerised by linking the heads or tails together, can further be crosslinked to form networks (Meier, 1998). Such an amphiphilic system was prepared with non-ionic hydrophilic ethyleneoxide derivative heads and hydrophobic (CH2 )n tails attached to an (equally hydrophobic) long polysiloxane back-

MATERIALS AND PREPARATION

(a)

319

(b)

F IG . 12.2. (a) Polydomain (opaque) and (b) monodomain (transparent, crosslinked in the aligned state) smectic elastomers (Semmler and Finkelmann, 1994). The optical perfection of the monodomain means that it can only be seen where there is slight curling of the sample! See also (Nishikawa et al., 2004). bone (Fischer et al., 1995). The lamellar phase formed in water solution and the bilayers were tethered via crosslinking the siloxane backbones with an appropriately small proportion of bi-functional groups, Fig. 2.15(b). Again, lengths were carefully matched. In crude terms, for a {=(CH2 )9 O(CH2 CH2 O)6 CH3 } reacting amphiphile one needs a {=(CH2 )9 O(CH2 CH2 O)6 CH2 (CH2 )9 =} bi-functional group. As a result, a SmA water gel was prepared, with many interesting hydroelastic properties and highly anisotropic swelling behaviour. All such lamellar gels equally form the subject of this chapter and the next. Spherical micelles, hexagonally packed cylinders and a number of complex diamond and cubic phases exist in addition to lamellar structures in phase diagrams of many complex fluids. Crosslinking of these phases into rubbery networks can also result in a variety of elastomers and gels with complex symmetry and, as a result, with many of the unusual physical properties we shall explore. However, the study of such non-lamellar systems goes far beyond our scope. As with nematic networks, the conditions of crosslinking define the texture of the resulting mesophase. When the network is formed in the isotropic phase, or no special aligning procedure is applied, the liquid crystalline elastomer invariably forms polydomains with a very small characteristic texture. Polydomain nematic and smectic elastomers and gels strongly scatter light and, thus, appear opaque. However, when the final crosslinking of the network is performed in an aligned state (whether in a mesophase or in the isotropic state, for instance, by applying a stress or a strong magnetic field, which is then frozen in by crosslinking) a monodomain liquid crystalline phase results, see Fig. 12.2. As in nematic elastomers, smectic monodomains are transparent since they do not scatter light due to long wavelength director fluctuations being suppressed by the polymer network. 12.1.1

Smectic A elastomers

Prolate nematic polymer chains are best aligned by uniaxial extension. By contrast smectic or lamellar polymers usually align their layers on uniaxial compression. One can understand this from a sketch of backbone conformation, for example in Fig. 12.3(a). Elastic deformation of a network is transmitted via this backbone. The uniaxial compression, accompanied by symmetric biaxial extension in the perpendicular direction due to incompressibility, results in the appropriate oblate configuration promoting a

320

SMECTIC ELASTOMERS

F IG . 12.3. The effects of deforming the smectic A elastomer network: (a) uniaxial compression leads to the symmetric extension of backbones in the layer plane; (b) uniaxial extension in the smectic phase leads to the alignment of layer planes, but not their normals (only the layer plane positions are shown in this sketch); (c) uniaxial stretching during crosslinking in the nematic phase aligns the director n , on subsequent transition into the smectic A this orientation is preserved and the layers are formed by re-arranging (partial flattening of) the backbone. Associated uniaxial contraction occurs at the N-A transition, see Fig. 12.5. (d) the macroscopic arrangements of layers with respect to sample shape arising in case (a). The bookshelf geometry (e) arises from case (c). monodomain smectic A alignment. The resulting monodomain sample has its smectic layers parallel to its flat surface, Fig. 12.3(d). An effective way of extending a weak, partially crosslinked gel is to deswell it while it is stuck to a substrate. Contraction is only possible in film thickness, while the other two dimensions, in not shrinking, are effectively stretched. When crosslinking and solvent removal are complete, a totally transparent monodomain SmA elastomer results (Nishikawa et al., 2004). Uniaxial compression is the same in effect as biaxial extension in an incompressible system. Polydomain systems can be compression-aligned subsequent to their formation – as in the water-polysurfactant lamellar system (Fischer et al., 1995), but the effect is not permanent unless second-stage crosslinking is used to freeze in the established alignment. Application of uniaxial extension in that case resulted in layers oriented along the stretching direction, but with the layer normal randomly oriented in the perpendicular plane, Fig. 12.3(b). However, in this situation one would still encounter internally quenched disorder from the random crosslinks of the first stage. When the liquid crystalline polymer which will form the network also possesses a prolate nematic phase at higher temperatures, aligning this phase by uniaxial extension produces a monodomain nematic. Subsequent cooling into the smectic A preserves

MATERIALS AND PREPARATION

321

this director orientation, with layers spontaneously forming in the plane perpendicular to the stretching axis, Fig. 12.3(c), thus providing an equally monodomain smectic A elastomer. In a thin film the layers are in the bookshelf geometry with respect to the flat and long dimensions of the sample, as shown in Fig. 12.3(e). It has to be noted that, in some cases, no equilibrium nematic phase exists between the smectic and the isotropic states. However, if the crosslinking has been performed under a uniaxial load (which is one of the very few way of producing monodomain textures), prolate nematic order may be induced in the resulting elastomer network just above the smectic transition point (Nishikawa and Finkelmann, 1999). 12.1.2

Smectic C and ferroelectric C∗ elastomers

The synthesis of smectic C∗ polymers and their crosslinked elastomer networks was greatly motivated by their liquid analogues displaying ferroelectricity. Some of the earliest work in liquid crystalline elastomers was focused on this particular aim (Finkelmann et al., 1981; Kapitza and Zentel, 1988) and many interesting results have been reported. However, as with nematic elastomers, direct synthesis will always result in a highly non-uniform texture with sub-micron size domains in equilibrium. Systematic studies of most important properties require well oriented samples, ideally with an untwisted helical structure. The traditional alignment techniques of surface treatment, electric field or shear flow often fail for polymers and are practically hopeless when applied to elastomers when already crosslinked. Rapid progress in discovery and studies of novel physical properties of smectic C∗ elastomers was achieved after effective methods of preparing monodomain aligned structures were developed. Finkelmann et al. utilised experience in nematic and smectic A networks and crosslinked under selective mechanical deformations. To obtain monodomains, one mechanical stretching is not enough since the SmC phases is clearly biaxial. After an initial uniaxial stretch aligns the nematic director, the second stretch at an appropriate angle to n is applied to align the layers, Fig. 12.4(a). All this is performed while network crosslinking is in process, which then permanently fixes a well-aligned, untwisted bookshelf C∗ -smectic structure (Benn´e et al., 1994). In a novel method, monodomain smectic C elastomers were made by blowing a bubble of a smectic melt which is thereby aligned by the accompanying biaxial elongational flow in the layer planes. Crosslinking of the bubble then follows (Schuring et al., 2001). The resulting balloons have most unusual elasticity. A third mechanical method (Hiraoka et al., 2005) first obtains a monodomain SmA by stretching as in Fig. 12.3(c). A second crosslinking while imposing an in-plane shear on the still SmA elastomer imposes a preferred inplane direction. Indeed, the SmA in Fig. 1.9(a) shows a residual sheared shape. Cooling to the SmC* state gives rise to a spontaneous shear, confirming mechanically the X-ray and optical evidence that a monodomain state has been achieved. A perhaps more natural method of preparing monodomain ferroelectric elastomers is by aligning the polymer melt (in the C∗ phase) in a cell subjected to a high external electric field, Fig. 12.4(b). This creates the perfect bookshelf geometry, which is then crosslinked by a UV-initiated reaction (a non-invasive method of crosslinking at a fixed temperature) to form a permanent defect-free elastomer network (Brehmer et al., 1994).

322

SMECTIC ELASTOMERS 1st deformation (a) q

2nd deformation

Electric field

(b)

F IG . 12.4. Aligning a smectic C (or C∗ ) system before crosslinking, leading to the bookshelf geometry of the final elastomer. Method (a): a sequence of two mechanical deformations applied to a partially crosslinked gel at a prescribed angle to each other, followed by the final crosslinking. Method (b): aligning of a polymer melt by an external electric field, followed by UV-initiated crosslinking (evidently, this would only be applicable to ferroelectric C∗ -smectics). In this approach the electric field aligns a relatively thick layer of material, but one may need to overcome the problem of removing the electrode plates from the surfaces (usually achieved by coating the electrodes by a sacrificial, e.g. water-soluble, layer). An alternative to electric alignment has been to use free-standing films of the corresponding polymer melt stretched on a rigid frame (Gebhard and Zentel, 1998). In this case a (very thin) film of a smectic polymer shows a very strong alignment of layers in the film plane, in the geometry of Fig. 12.3(d). If the crosslinking is performed in the smectic A phase, the resulting film has been demonstrated to have a giant electrostriction due to the electroclinic effect, the field-induced A-C transformation (Lehmann et al., 2001). When, after the drawing of a free-standing film in the C∗ phase, a lateral electric field is applied between two electrodes in the frame, a well-aligned molecular tilt is established and, after crosslinking, one obtains a ferroelectric film with polarisation in the plane (Zentel et al., 2000). Elastomers prepared from ferroelectric liquid crystalline polymers combine the elasticity of rubbers with the ordered structure and mobility of liquid crystalline phases, thus offering a chance to manipulate optical and electric polarisation states by mechanical fields. 12.2 12.2.1

Physical properties of smectic elastomers Smectic-A elastomers

As smectic order develops, it changes the chain shape distribution and thus the macroscopic shape of the elastomer. Figure 12.5 shows the uniaxial thermal expansion on cooling the elastomer (Nishikawa and Finkelmann, 1999) into such a prolate ‘inducednematic’ state (iN), which is followed by contraction in the smectic A phase due to the flattening by layers of some portions of the otherwise stretched backbones. This system

PHYSICAL PROPERTIES OF SMECTIC ELASTOMERS

smA

(iN)

323

Iso

F IG . 12.5. Thermal expansion of a smectic A elastomer with an associated prolate nematic phase at a higher temperature (Nishikawa and Finkelmann, 1999). The approximate smectic phase transition temperature (62o C) is marked by the arrow on the plot; above it one finds a prolate ‘induced-nematic’ state, which eventually terminates at a true isotropic phase. Different data sets correspond to the same sample cooled or heated under an increasing load. was twice-crosslinked under uniaxial extension, thereby remembering a preferred direction of elongation. But the elastomer has no instrinsic nematic phase. As incipient layer formation and orientation develops, chains extend to a prolate shape, finally shrinking at a characteristic temperature where layer order becomes very high. Stress, σ , accompanies elongational strain, Fig. 1.8, from which a (Young) linear elastic modulus E could be extracted: σ = E ε . Firstly, for εzz imposed parallel to the layer normal, a modulus of E = 1.03 × 107 N/m2 was measured. For εxx imposed inplane, a linear modulus E⊥ = 1.4 × 105 N/m2 emerged – about 75 times smaller than E . The physical reason for this one high modulus is clear: stretch along the layer normal attempts to alter the layer spacing which is resisted by the smectic ordering. This ordering acts on each side-chain monomer, rather than on chains as a whole as in rubber elasticity, and correspondingly dominates. Such high elastic anisotropy is unparalleled even in strongly ordered nematic networks, where both Young moduli arising from the nematic elastomer free energy, E and E⊥ , are of the same magnitude: of the order of an ordinary rubber modulus µ . In fact, one may recall that an ideal Gaussian nematic elastomer (satisfying the Trace formula) must have exactly the same rubber modulus in response to small deformation along and across its director – see Sect. 6.7. Any observed anisotropy in a monodomain nematic rubber is due to secondary effects, such as non-Gaussian corrections for shorter chains, semi-softness induced by aligned rigid-rod crosslinks, or strain-induced changes in nematic order parameter (see Sect. 6.6). In any case, such anisotropy is known to be small in nematic elastomers. With such a high elastic anisotropy, the shape of monodomain smectic rubber on being stretched (εxx ) in the layer plane, Fig. 12.6, does not look as surprising as it might

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F IG . 12.6. A smectic A elastomer showing no transverse contraction in the z-direction on being strained by ∼ 80% along x, in the layer plane. The sample is prepared with layers uniformly aligned with their normal along z (Nishikawa et al., 1997).

e zz

e xx

x

z z

(a)

(b)

x

F IG . 12.7. Uniformly stretching a monodomain smectic elastomer along the layers (a), strain εxx is imposed (see Fig. 12.6) and along the layer normal, (b), when εzz imposed. In the first case the high-modulus resistance to layer compression prevents the sample from changing its perpendicular dimension. In the second case there is no such resistance and a familiar transverse contraction occurs, except at the clamps, see Fig. 12.8(a). However, at non-infinitesimal strains εzz , the smectic elastomer finds a lower energy deformation path, the layer buckling instability Sects. 12.4 and 13.5 and exercise 2.6. have. We know that all rubbers are physically incompressible, because the bulk modulus is many orders of magnitude greater than any of the rubber moduli. This usually means that a rubber strip narrows on stretching; with rigid clamps it results in a most characteristic shape, see Chapter 8. However, the sample in the picture, although stretched by 80%, apparently did not reduce its width at all. The reason, of course, is the elastic anisotropy E  E⊥ . The contraction of the sample width in the direction visible in the figure would mean the deformation along the layer normal z, which is penalised much more strongly. So, instead, the elastomer preserves its volume by reducing the thickness of the sample in Fig. 1.8, which is another ‘weak’, quasi-fluid direction in the layer plane. The high modulus against extension along the layer normal in both liquid and rubbery smectics can be avoided by rotating the layers and thereby presenting along the extension direction a longer repeat interval, Fig. 2.20(a). This is the Clark-Meyer-Helfrich-

PHYSICAL PROPERTIES OF SMECTIC ELASTOMERS

325

150 125

snom (kPa)

100 75 50 25

(a)

(b)

0 1.00

1.05

1.10

1.20

1.15

lzz

1.25

F IG . 12.8. Layer buckling instability on stretching the monodomain smectic A elastomer along the layer normal. Images (a) show that the strip experiences the ordinary transverse contraction, but also becomes optically opaque. The stress-strain plateau (b) shows how internal layer rotations reduce the effective elastic modulus and also indicates the strain threshold εth ∼ 0.05, from (Nishikawa and Finkelmann, 1999). Dotted line, theory. Herault (CMHH) effect, see Sect. 2.9 of liquid smectics. In rubber the elongation can then proceed by in-plane shear that we will see below in molecular and in continuum theory is cheaper than layer distortions. Rotation can, however be incompatible with boundary constraints and layers have to have a periodic reversal in the sense of their rotation, that is they must buckle, see Figs. 12.7 and 2.20(b). The buckled layers now scatter light and are no longer transparent. Nishikawa and Finkelmann (1999) have thoroughly investigated this effect of layer buckling under extension of monodomain smectic rubber along its principal axis – the layer normal, Fig. 12.7(b). At strains below threshold, the elastomer responded as a classical rubber, albeit with anomalously high Young modulus, cf. Fig. 1.8 and exercise 13.1. The onset of instability is easy to detect, both optically (due to strong light scattering on birefringence axis modulations, making the material opaque) and from the 50

f

40

f 30 20 10

(b)

(a) 0 0

0.1

0.2

0.3

0.4

0.5

0.6

e

F IG . 12.9. (a) An X-ray scattering image illustrating the zig-zag coarsened buckling of smectic layers, while the nematic alignment is left largely unchanged – vertical, as indicated by wide-angle arcs. Measuring the angle φ between the two pairs of smectic reflexes allows plotting the angle of local layer rotation φ (ε ) against strain, (b), from (Nishikawa and Finkelmann, 1999). Dotted line, theory.

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stress-strain variation (which develops a ‘soft’ plateau due to internal relaxation of layer orientation), see Fig. 12.8 and the original literature (Nishikawa et al., 1997; Nishikawa and Finkelmann, 1999). Chiral SmA* elastomers exhibit an electroclinic effect where an in-plane electric field induces a tilt of the director from the layer normal to the in-plane direction orthogonal to the field (Terentjev and Warner, 1994a). Where an associated mechanical distortion arises we have an inverse piezoelectric effect, possible in liquid crystal elastomers because now there is an elastic rigidity. We return to both these effects in reviewing SmC* elastomers where the deformations and polarisation are natural rather than induced. We have concentrated on bulk phases of strongly coupled elastomers. It is also possible to make free-standing, submicron thick elastomeric films on which one can conduct mechanical and electro-optical experiments. For instance one can induce compression of the smectic layers by applying an in-plane strain (Stannarius et al., 2002; Stannarius et al., 2004; Stannarius et al., 2006). Evidently these elastomers do not have the strong coupling of the elastomers we have described thus far. Further, manipulating the micron-scale roughness of free-standing smectic films by mechanical stretching has also been reported (Brodowsky et al., 2002). 12.2.2

Smectic-C elastomers

Liquid SmC has the director tilted by an angle θ from the layer normal. The solid, elastomeric analogues are also interesting because in their chiral forms they too are (improper) ferroelectrics (see Chapter 2.9). The are perhaps unique among solid ferroelectrics in that they are locally fluid-like, capable of huge distortions and with low moduli (about 104 times smaller than crystalline materials or ceramics). Additionally their extra azimuthal degree of freedom of the tilted director, combined with their solidity, gives them the shear equivalent of the spontaneous distortion we saw in nematic elastomers. Shear associated with the SmA to SmC transition (Hiraoka et al., 2005) is shown in Fig. 1.9. These authors characterise shear by the angle θE that the sample mechanically tilts through: tan θE = λxz where x is the in-plane tilt direction and z that of the layer normal. They denote the molecular tilt that they measure by X-ray scattering by θX (we simply use θ ). Its evolution as the C-phase is entered is shown in Fig. 12.10(a), while Fig. 12.10(b) plots the dependence θE (θX ). It shows that mechanical tilt (i.e. shear λxz ) is not identical to the molecular tilt (Hiraoka et al., 2002). Indeed there is a residual shear in the SmA, untilted phase from the two-step crosslinking, see Sect. 12.1.2. The second stage was performed under an imposed shear, a memory of which is imprinted in the matrix, both in the mechanical shape and in the residual director tilt. In SmC, director tilt has created a second special direction in addition to the layer normal. It was explained in Sect. 2.9 that, for a chiral system, the existence of two non-colinear axes allows us to find a third polar vector, in this case the polarisation, P , in the orthogonal direction. Thus changes in the tilt magnitude or tilt direction should correspondingly change the magnitude or direction of P . We can now understand how an electric field, E , applied in the layer planes of

PHYSICAL PROPERTIES OF SMECTIC ELASTOMERS 30

327

30

SmC*

SmA

High temperature

Iso

25

Heating Cooling

25

Tilt angle q X

Shear angle qE

SmX* 20 15 10

20

SmC*-SmX* transition point

15 10

T’ at 80ºC Heating Cooling

5

5

0 20

40

60

80

Temperature

100

120

140

TCA at 105ºC SmA

0 0

5

10

SmC* 15

SmX* 20

25

30

Tilt angle q X

F IG . 12.10. (a) Increase of molecular tilt θX on entering the SmC phase, according to (Hiraoka et al., 2005) (b) Shear angle θE against molecular tilt. chiral SmA* can induce a polarisation and, by the inverse of the above connection between tilt and polarisation, the field also induces a tilt. In effect a SmC* phase is being induced; this is the electroclinic effect (Garoff and Meyer, 1977; K¨ohler et al., 2005). The electroclinic coefficient is defined as α = ∂ θ /∂ E and the values obtained in in SmA* elastomers were in the range α = 0.045m/MV. Since tilt and mechanical shear are related, then E can as well induce shear λck (in a frame-independent form; c is the c-director, the director projection on the layer plane, and k the layer normal). This is the inverse piezoelectric effect, the direct effect being where an imposed λck induces a P in the orthogonal c × k direction. The normal and inverse effects can be found in both SmA* and SmC* phases and tend to be largest near the A-C transition where the order is least rigid against changes of tilt (K¨ohler et al., 2005). When orientationally ordered rods tilt, they also effect the layer spacing. Naively one might expect in a tilted system that d(θ ) = d0 cos θ . One can directly measure d by X-rays as tilt is altered, for instance by electric fields. Alternatively, one can deduce there has been a layer spacing change by measuring possible deformation λkk along the layer normal, assuming strong coupling between the layers and the rubber matrix, that is λkk = d(θ )/d0 . In SmA* there is initially no tilt; the flatness of the cosine function around θ = 0 means that λkk ∼ 1 − θ 2 /2, that is strain, ε , is second order in θ and in this mode is contractile (Terentjev and Warner, 1994b). In SmA* therefore, the electrostriction resulting from field-induced layer tilt has a coefficient, a, defined by
= λkk − 1 = −aE 2 , that is connected to the electroclinic coefficient by a = α 2 /2 ∼ 10−15 m2 /V2 (K¨ohler et al., 2005). To summarise, one can induce λck and λkk at linear order in E in SmC*, but in SmA* λkk is second order in tilt. This contrast is to be expected since for electrical effects one requires two directions, naturally provided by either shear or the tilted SmC* environment, but the two directions are not provided by compression or by a SmA* environment. We have met this distinction between shear and compression in polarisation effects in cholesterics, see the discussion of Sect. 9.3. Second order effects are also

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discussed in Sect. 13.7, as are experiments. In SmC* where there is a pre-existing tilt, θ0 say, then the connection between δ d/d0 and δ θ is linear, that is δ d/d0 ∼ − sin θ0 δ θ . Now we have a piezoelectric response to compression along the layer normal, or inversely a contraction along the layer normal in response to an in-plane field. Thus we have two piezoelectric responses – to compression along the layer normal and to shear λck , the latter being associated with much lower stresses since layer separation is not effected. A final interesting possibility arises from analogous liquid smectic experiments, namely that of photo-ferroelectricity (Langhoff and Giesselmann, 2002a; Langhoff and Giesselmann, 2002b; Saipa et al., 2006) where a SmC* phase has photoisomerisable elements that on photon absorption reduce the molecular tilt and hence also reduce the electric polarisation. The first steps to synthesising photo-switchable SmC elastomers has been taken (Beyer and Zentel, 2005) where light can induce both mechanical tilt and an electrical response. Further recent related work examined the solidification of smectic liquid crystal phases with photosensitive gel forming agents (Deindorfer et al., 2006) where irradiation causes the network to lose memory of its formation state – the crosslinks are essentially dissolved and then remade on de-excitation. 12.3 A molecular model of Smectic-A rubber elasticity Since smectic elastomers are rubbery, that is capable of large, reversible deformations, their chains locally retain a liquid-like mobility and readily explore configurations. Being long, the chains have a Gaussian distribution of shapes that is not necessarily isotropic. They are both orientationally ordered and subjected to spatial localisation by layers. Being Gaussian, their conformations are characterised by a second moment proportional to an  tensor. There is thus an entropic part to their free energy density given   by 21 µ Tr  o · λ T ·  −1 · λ . We took an analogous view with cholesteric elastomers. Because the underlying order is locally nematic, then the basic rubber free energy is also locally nematic, but with constraints offered by the new phase. We now take this approach to smectic elastomers. In smectic elastomers the layers have a profound effect on physical properties, Sect. 12.2, and chains are constrained on an energy scale far greater than µ . Extensions along the layer normal that would require crosslink translation with respect to the layers are penalised by a modulus B comparable to or larger than that of the layer spacing modulus of smectic liquids. We shall assume that, in strongly coupled smectics, layers rigidly localise crosslinks. Layers must then move affinely with the rubber matrix, that is, their orientation and spacing must deform as their local matrix does. We must then add to the rubber elastic deformation cost a penalty 12 B(d/d0 − 1)2 where d and d0 are the current and original layer spacings. Thus the overall smectic-A rubber energy density is (Adams and Warner, 2005a)   FA = 12 µ Tr  o · λ T ·  −1 · λ + 12 B(d/d0 − 1)2 (12.1) The second part of this energy may not be relevant to weakly coupled smectic elastomers where there is apparently no elastic signature of the smectic layer system (Stan-

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329

R2

V

R1

F IG . 12.11. A microscopic model of a smectic elastomer. Crosslinks sit in a periodic potential V = gs cos(qqR i ) resulting from the smectic ordering. For clarity, the mesogenic side-groups are not shown. narius et al., 2002). Such elastomers could be adequately described by classical isotropic rubber elasticity as is appropriate for a nematic elastic matrix where the director cannot rotate (biaxial strains in the layer planes were examined). They have been prepared as very thin samples (a few hundreds of molecular layers) and it seems that layer numbers are not necessarily preserved under deformations. In contrast, there is evidence (Stannarius et al., 2004) that homopolymer networks, where the smectogens are not diluted, experience a strong potential (suggested, as we will see, by their extreme Poisson ratios (0, 1)). The interaction of crosslinks with a smectic potential and their induction of randomness has been studied (Olmsted and Terentjev, 1996) in determining the character of the nematic to SmA transition in elastomers, see Sect. 13.6. It is possible to derive eqn (12.1) from the statistical mechanics of a chain with its Ri ) = end points (crosslinks) strongly confined to the minima of a periodic potential, V (R gx cos(qqR i ), Fig. 12.11. Each crosslinking point pins the positions of φ monomers (with φ the crosslink functionality), probably further enhanced by short-range correlation along the chain. The barrier for such a composite object to ‘tunnel’ through a smectic layer is a molecular characteristic of the material describing the degree of miscibility of backbone and the mesogenic side-groups, and is roughly of order φ χ , where χ is the Flory-χ parameter describing interactions between mesogenic groups and the backbone. In molecular theory (Adams and Warner, 2005a) the additivity of the energy of rubbery and layer effects emerges naturally. It shows the polymer network contribution to the layer compression modulus to be Bx = (2π )2 gx (R /d0 )2 ns where the amplitude of the localising potential gx is certainly proportional to the smectic order parameter |ψ |. R2 = L /3 is the mean square dimension of chains parallel to the director. Bx is a component to the layer compression modulus B in eqn (12.1), the other part being a usual smectic contribution discussed in eqn (2.54) and estimated as ∼ K/do2 . The magnitude of the overall B in strongly coupled systems turns out to be much larger than that encountered in liquid smectics. Polymeric smectics evidently have a mechanism to give

330

SMECTIC ELASTOMERS

coherence to the movement of connected rods with respect to the layer potential and hence give more resistance to layer-spacing change, and then there is the above addition due to crosslinking. Overall, in Fig. 1.8 one sees the very much greater resistance to elongations along the layer normal than the rubbery response in-plane, the modulus for the former being around, or even greater than 107 Pa. We revisit with continuum mechanics in Sect. 13.1.1 the question of the great rigidity of matrix-layer coupling. The geometry of the rotation and deformation of layers under λ must be solved in order to give explicit form to the innocent-looking last term in theenergy (12.1). The  energy FA is highly non-linear because of incompressibility, Det λ = 1, and from evaluating the distorted layer spacing d(λ ), as we now demonstrate. 12.3.1

The geometry of affine layer deformations

The layer normal (equivalent to the director in SmA) and the layer spacing transform as a result of layers affinely following the deformation λ . Since the layer normal is a unit vector and must remain so, the process is non-linear. The reader not interested in the details need only refer to eqns (12.3) and (12.4) for these two quantities, and then look at the explicit, frame-dependent examples we give. Two non-colinear vectors span a plane. It suffices then to take two perpendicular unit vectors, k and m say, since their cross product is the layer normal n 0 = k × m that defines the plane, Fig. 12.12. If k and m deform affinely with λ , they become λ · k and λ · m in the deformed plane. They yield the new normal by again taking the cross product: n=

(λ · k ) × (λ · m ) . |(λ · k ) × (λ · m)|

(12.2)

m are no longer necessarily unit vectors or even still perpendicular In general λ ·kk and λ ·m to each other. Thus there is the need for the normalising denominator in eqn (12.2). −1 −T
ijk = λpk
ijk one can readily show that Using the matrix identity
αβ p λα i λβ j = λkp the normal, of layers deforming affinely with an incompressible medium, itself deforms according to λ −T , the inverse transpose of λ : n=

k

k

m m

λ −T · n 0 |λ −T · n 0 |

λ

.

λ. k

(12.3)

λ

.m λ

.m

λ. k

F IG . 12.12. The normal to the layer deforms when the vectors in the layer deform according to x → λ · x .

A MOLECULAR MODEL OF SMECTIC-A RUBBER ELASTICITY

331

The new layer spacing then also follows: corresponding points in adjacent layers before deformation have a separation d0 n 0 which then deforms to λ · (d0 n 0 ). The separation vector is not necessarily still along the normal of the deformed system of layers. But the perpendicular separation between the layers, d, is the projection of λ · (d0 n 0 ) onto the new (unit) normal, n , we have just derived: d = d0 (λ · n0 ) · n = d0 (λ · n0 ) · d/d0 =

n 0 · λ −1 · λ · n 0 |λ −T · n 0 |

≡

λ −T · n0 |λ −T · n 0 |

1 |λ −T · n 0 |

.

(12.4)

A frame-dependent illustration is when k = x and m = y , and hence n 0 = z . Then 1 d ; = d0 |
ijk λjx λky |

ni =

λiz−T . |
ijk λjx λky |

(12.5)

The full non-linearity of a set of affinely deforming unit vectors remaining unit vectors in an incompressible medium is exposed when eqn (12.3) is inserted in the second part of the free energy (12.1):  2   1 T −1 1 1 FA = 2 µ Tr  o · λ ·  · λ + 2 B (12.6) −1 . |λ −T · n 0 | 12.3.2 Response to principal deformations Despite the apparently highly tensorial and highly non-linear character of FA , one can analyse fundamental deformations, those of Fig. 12.13. None of these deformations are soft since director rotation is associated with deformations highly constrained by layers. We now give specific forms of the minimum energy deformation tensors and layer rotations for each of Figs. 12.13. (a) Imposed in-plane extension λxx We impose an extension λ = λxx along one (x) of the equivalent in-plane directions, see Fig. 12.13(a). As in earlier chapters, our notation is that the externally imposed component of λ is written without its subscript index. The deformation gradient matrix and its inverse transpose take the form:  1    0 0 λ 0 λxz λ 1   λ =  0 λyy 0  ; λ −T =  0 λyy 0  . (12.7) λxz 1 0 0 λzz − λ λzz 0 λzz As expected, the director is unchanged and the layer spacing changes simply with λzz :

λ −T · n 0 = (0, 0, 1/λzz ) ⇒ n = (0, 0, 1) and d/d0 = λzz .

(12.8)

Volume conservation is simply expressed by 1 = λ λyy λzz because λ is upper triangular and the determinant trivial. This is because the shear λzx is absent since it introduces

332

SMECTIC ELASTOMERS

torques from the change of shape in the presence of an x component of force (from the xx-stress associated with imposing λxx ) which tends to eliminate the distortion. We shall see that the λxz shear vanishes too, as expected by symmetry of this deformation. Inserting λ (with λzz → 1/λ λyy ) into the free energy results in: A 2 B 1 1 2 2 , (12.9) + rλxz +λ2 + 2 2 +b −1 FA = 12 µ λyy λyy λ λyy λ with the non-dimensional layer compression ratio b = B/µ . The shear is uncoupled to other deformations, hence minimisation yields λxz = 0. Minimization of this free energy with respect to λyy gives: 4 λ 2 λyy − 1 = b(1 − λ λyy ) . (12.10) Our illustrations below are for b = 5, r = 2. Material properties such as Poisson ratios, strain thresholds and moduli depend on this coupling and anisotropy; small b amplifies thresholds in our illustrations. Although a quartic in λyy , eqn (12.10) can be solved by treating it as a quadratic in λ (λyy ) and plotting the result parametrically, Fig. 12.14. It is clear that the limits (a)

(b)

(c)

(d)

z x l xx

l xz

l zx

l zz

F IG . 12.13. Imposed deformations: (a) stretching perpendicular to the layer normal, (b) shearing the layers in their plane, (c) stretching parallel to the layer normal, and (d) shear out of the planes. 1.0

lyy 0.8

0.6

0.4

1.5

2.0

2.5

lxx

3

F IG . 12.14. In-plane contraction λyy in response to an in-plane stretch √ λxx . r = 2 and b = 5. The dashed lines are yy-contractions 1/λxx (lower) and 1/ λxx (upper) shown for comparison.

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333

(dashed lines) of large and small b correspond respectively to the bounds √ λyy = 1/λ (with no contraction along the layer normal, λzz = 1) and λyy = λzz = 1/ λ (the classical limit with equal perpendicular contractions). The elastically isotropic material with a small ratio B/µ is still a SmA phase in the sense that the director is constrained to lie along the layer normal and could still be in the strong coupling limit in that layer-matrix coupling is rigid; the layer strength is simply lower, as in (Stannarius et al., 2002) or near the SmA-N transition. The Poisson ratios for b → 0 and b  1 reveal the isotropy and dimensional reductions implied in the limits of weak and strong layers. For small strains λyy = 1 +
and λ = 1 + ω , Eqn (12.10) becomes at first order 4
+ 2ω + b(
+ ω ) = 0. The Poisson ratio in the y direction, νy = −
/ω , and that in the layer direction, νz , are:

νy =

2+b ; 4+b

νz =

2 . 4+b

(12.11)

The crossover from Poisson ratios (z, y) = (0, 1) to (1/2, 1/2) is thus relatively slow. However, it is clear that for b ∼ 60, as found in some experimental situations (Nishikawa and Finkelmann, 1999) the material is firmly in the (0, 1) class. In this regime of large b, where layer compression is suppressed (λzz → 1), the elastomer is like a classical 2-D rubber. Since there is no director rotation, r does not enter into the elastic free energy density (12.9), which reduces to 6 7 1 6 2 7 2 = 2 µ λ + 1/λ 2 (12.12) FA = 12 µ λ 2 + λyy as a consequence of higher layer stiffness and volume conservation. The Young modulus derived from eqn (12.12) is * ∂ 2 F ** = 4µ , (12.13) E⊥ = ∂ λ 2 *λ =1 rather than the value 3µ that obtains for a classical isotropic rubber: dimensional constraints have increased the effective response modulus. (b) Imposed in-plane shear λxz Consider the in-plane shear with deformation gradient matrix:   λxx 0 λ λ =  0 λyy 0  . 0 0 λzz

(12.14)

We suppress λzx since typically the application of λxz is with parallel plates that constrain the sample. Again, volume conservation is simply expressed 1 = λxx λyy λzz and eliminates λyy . The free energy is: 4 5 1 2 2 FA = 12 µ λxx (12.15) + 2 2 + λzz + rλ 2 + b (λzz − 1)2 . λxx λzz

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SMECTIC ELASTOMERS

The shears λ and λzz do not couple and so the imposed shear cannot affect λzz , as expected in this geometry. Minimizing over λxx and λzz yields λzz = λxx = 1 and the free energy density becomes 6 7 (12.16) FA = 12 µ 3 + rλ 2 . The result is the same as for the simple shear in a nematic elastomer with unrotating director. The linear shear modulus is r µ . The anisotropy enters in the classical way – for large r the modulus becomes large because chains extend across many molecular shear planes. It will turn out that this modulus is identical to that obtaining for  1 after the instability on stretching along the layer normal, because then deformation is largely via shears in the rotated planes, a case to which we now turn. (c) Imposed out of plane shear λzx The two principal deformations analysed above have, characteristically, preserved both the layer spacing d0 and the orientation of the layer normal intact. We now examine the other two deformations of Fig. 12.13 which, with increasing complexity, affect these two smectic variables. Let us now impose the deformation of Fig. 12.13(c) by placing the lower triangular element in the matrix λ :   λxx 0 0 (12.17) λ =  0 λyy 0  . λ 0 λzz As in the previous cases we must suppress the opposite shear, here λxz , because of the mechanical constraints implied in imposing λzx . From the geometry of this shear one expects the layers to be forced to rotate by an angle we denote as φ (see Sect. 13.1.1 for a more detailed discussion of the consequences of such a uniform rotation). Volume conservation gives λyy = 1/λxx λzz . The director evolves as: n=

1 (−λ , 0, λxx ) . 2 λ 2 + λxx

(12.18)

1

d/d0 1.2

(a)

1 0.8 0.6 0

(b)

0.8 0.6

lxx lyy lzz

0.4

cosf

1

lzx 2

0.2

0

1

2

lzx

F IG . 12.15. (a) The relaxing components of the deformation tensor on imposing λzx for b = 5 and r = 2; The associated director (layer normal) rotation is given by cos φ . (b) The associated change in layer spacing.

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The free energy density is:   2    2 (λ 2 + r λ 2 ) 1 λ λ λ xx zz 2 2 + 2 2 + λ 2 + rλzz + zz 2xx 2 +b  −1 FA = 12 µ λxx 2 +λ2   λxx λzz λxx + λ λxx which can be numerically minimized (Adams and Warner, 2005a). A typical solution for the components of λ and the corresponding layer spacing changes is illustrated in Figs. 12.15. In this zx-deformation we are thus effectively compressing, as well as rotating the layers, which is now penalised by the large modulus B. Even for very large ratio b, the layer spacing eventually yields and begins to decrease. (d) Imposed extension λzz along the layer normal The smectic planes are known experimentally to at first resist such extension with a high modulus and then complex rotational and shear instabilities arise that reduce the modulus, Fig. 12.8. In this case, Fig. 12.13(d), the deformation tensor is:     λxx 0 0 λyy λ 0 0 −T (12.19) λ =  0 λyy 0  with λ =  −λzx λyy λxx λ 0  . λzx 0 λ 0 0 λxx λyy The shear λzx induces layer rotation by an angle φ which, for purely geometrical reasons, presents a longer distance d/ cos φ along z than the simple value d. This is the the origin of CMHH instability for elongation along the layer normal in smectic liquid crystals, see Fig. 2.20(a). The shear λxz in the presence of a z-force of extension would lead to torques and is not observed in smectic elastomers (Nishikawa and Finkelmann, 1999) or in the analogous nematic elastomer geometries that involve simultaneous director rotation and shear, see Sects. 7.3.2 and 7.4.2. The volume constraint simply gives λyy = 1/λxx λ . Since λ −T · n 0 = (−λzx λyy , 0, λxx λyy ), then the layer spacing and the director become respectively:  (12.20) d/d0 = √ 12 2 and n = − √ λ2zx 2 , 0, √ λ2xx 2 . λyy

λxx +λzx

λxx +λzx

λxx +λzx

The elastic free energy density is then:   2    2 2 2 1 (λ + rλzx )λ λ λxx 2 2  + 2 2 + λzx + xx 2 + b − 1 (12.21) FA = 12 µ λxx 2 2 +λ2   λxx λ λxx + λzx λxx zx Minimizing this free energy fixes the free components of the deformation tensor for a particular b value; they are shown in Fig. 12.16(a). There is a critical, threshold value of the elongation λ = λc (b, r) when the layer rotation starts to occur. It is rather large because a relatively small ratio b has been adopted for illustration. The threshold is to a uniform state given by the strain (12.19), see the sketch included in Fig. 12.16(b).

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SMECTIC ELASTOMERS

1.0

snom/m

4

0.8

(a)

3

(b)

0.6

2 0.4

lzx lyy lxx

0.2 0.0 1.0

1.5

2.0

1

lzz 2.5

0 1.0

1.5

lzz 2.0

F IG . 12.16. (a) Deformation tensor components for an imposed stretch λzz = λ parallel to √ the layer normal, for b = 5 and r = 2. The shear λzx relaxes to an asymptote of 1/ λc for large λzz . (b) The nominal zz stress. It exists independently of other possible causes of threshold, namely clamp constraints and microstructure which arise in practical cases, see Sect. 12.4. Frank elasticity, that partly determines the fineness of microstructure, has not yet been invoked and will turn out to be largely irrelevant for the threshold. Shear λzx starts with a singular edge at the threshold and the transverse contraction λyy thereafter remains constant. This reminds the analogous response in nematic elastomers, e.g. Fig. 7.8, where the λyy =const was the key signature of soft (or semi-soft) deformations. The accompanying stress also divides into two distinct regimes with a much higher modulus before the transition, see Fig. 12.16(b) and the fit to experiment in Fig. 12.8(b) below. The layer rotation, given by n , eqn (12.20), has the same singular edge at λcr . It is plotted against
= λ − 1 in Fig. 12.9(b) along with experiment (Nishikawa and Finkelmann, 1999) where λcr ∼ 1.04. The geometric reason why the shear λzx , the layer rotation and softer shear do not start immediately, but only onsets after a threshold, is addressed in Sect. 12.3.3. Analytically, the solution to this model splits into two parts: before and after the discontinuity. Before the layers start to rotate, λzx = 0. The free energy density is: 4 5 1 2 2 2 1 (12.22) FA = 2 µ λxx + 2 2 + λ + b(λ − 1) λxx λ 2 = λ 2 = 1/λ . Thus, for λ < λ , the two perpendicular dirand has a minimum when λxx c yy 1 1 ections are equivalent and the material has Poisson ratios ( 2 , 2 ) in the (x, y) directions. The free energy density and nominal stress, σnom = ∂ FA /∂ λ , are: 6 7 (12.23) FA = 12 µ λ 2 + 2λ + b(λ − 1)2 6 7 2 σnom = µ λ − 1/λ + b(λ − 1) . (12.24)

Layer rotation starts at a threshold λc that we show below is set by the tensile nominal stress in the unrotated and rotated phases being the same. As λyy is a constant, √ volume conservation then requires λxx = λc /λ . If deformations were small, then one

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337

could say that the material would now have Poisson ratios (1, 0), to which we return when comparing with experiment. 2 , after some simplification we obtain: Minimizing the free energy w.r.t. λzx  √ (r − 1)λc λc λc , (12.25) 0 = 1+ + b − p4 p3 p4 2 + λ /λ 2 . From eqn (12.25), where the shear λzx only appears in the combination p2 = λzx c p is clearly a function only of r − 1, b and λc , but not a function of λ . Thus p can be fixed at any convenient value of λ , for instance at the critical extension λc where λzx = 0. Hence p2 = 1/λc , and thus from the definition of p2 the induced shear after the instability is then: 0  2 1 1 1 λc λc . (12.26) λzx = ± √ 1− ; λyy = √ ; λxx = √ λ λc λc λc λ

The shear displays the singular edge seen in Fig. 12.16(a). Both signs of shear give the same dilation along z at constant layer spacing. Both shears, and indeed all directions perpendicular to z (not just x), are equivalently possibly and thus required in a description of any induced microstructure, see Sect. 12.4. The director (and thus layer) rotation can be derived from the explicit expression (12.20) for n . For instance the first component gives:

(12.27) sin φ = 1 − (λc /λ )2 with the singular edge and distinctive form also seen in experiment, Fig. 12.9(b). All components of induced deformation and the rotation of layers depend solely on (the imposed) λ and (the observable) λc , not in any separate or detailed way on the smectic potential b or the anisotropy r. Theory is very tightly constrained since there are no free parameters. It requires all relaxation-strain and rotation-strain relations, (12.26) and (12.27), to be of the same form for all systems when λ is reduced by λcr . We substitute λzx back into (12.21) to obtain the free energy and thus the nominal stress for λ > λc : 4 5 2 2 2 2 2 1 (12.28) + λc + r(λ − λc ) + b(λc − 1) FA = 2 µ λc σnom = µ rλ . (12.29) Notice that the energy associated with layer spacing, the b term, is now constant. The gradients of the nominal stresses eqns (12.24) and (12.29) give the effective Young moduli in each regime: 4 3 µ + B λ < λc . (12.30) E = µr λ > λc The layer spacing of the system as a function of the imposed stretch is:

338

SMECTIC ELASTOMERS

1 d λ λxx  = = = 2 2 2 +λ2 d0 λyy λxx + λzx λxx zx

4

λ for λ < λc . λcr for λ > λc

(12.31)

Before layer rotation starts λzx = 0 and so the layer spacing increases as d/d0 = λ . After layer rotation starts the layer spacing remains fixed; the only cost of deforming the system is that of shearing the rubber. This is because, as the layers rotate, the component of the force along the layer normals remains constant. Explicitly this component of nominal stress is σnom cos φ which, by eqns (12.27) and (12.29), above the transition is µ rλcr and thus constant. Further, because shear (as opposed to extension and contraction along principal directions) involves the chain anisotropy and it is the cost of this shear that is to be compared with that of layer dilation, we can understand that r (as well as the relative modulus B/µ ) enters the expression for λc . Finally we determine the threshold elongation: the continuity of the nominal stress with λ , that is equating eqns (12.24) and (12.29), yields a cubic equation for λc :

λc3 (r − b − 1) + bλc2 + 1 = 0 .

(12.32)

To obtain a threshold at all we require b > r − 1. Below this value there is no instability: layer dilation is not significantly more costly than matrix distortion and it is no longer avoided by the intercession of an instability. Instead of solving eqn (12.32) analytically, it is more useful to analyse it in the physically important large b limit. The first few terms in an expansion yield the variation of critical extension with layer modulus:  1 1 r (12.33) λc = 1 + + r(r − 3) 2 + O 3 . b b b The choice of r and b in Figs. 12.16(a) and (b) is clearly outside the asymptotic limit of eqn (12.33) but allows for an exaggerated clarity in Fig. 12.16. The ratio of the linear response moduli of eqn (12.30) before and after the transition, can be related to λc , which provides another stringent constraint on theory. For large b their ratio is: (12.34) r µ /B ≈ λc − 1. Experimentally one could obtain µ (actually 4µ , see (12.12) and below) from stretching the rubber in the layer plane, and the anisotropy of the polymers, r, from the Young modulus after the threshold (12.30). Thus the value of the threshold itself gives a direct measure of the layer compression constant B, which can then be compared with the modulus before threshold in (12.30). The large B/µ expression (12.34) for the threshold strain essentially agrees with the result of linear continuum mechanics (Weilepp and Brand, 1998) if one neglects Frank effects, which have a small effect on the threshold strain (although do influence the length scales of the subsequent microstructure). This conclusion follows for the same reasons as explored for stripe domain microstructure in nematic elastomers, Sect. 8.2 and the discussion below eqn (8.13). If one ignores the anisotropy of the underlying nematic network, then the factor r is of course absent from an expression for the threshold (Weilepp and Brand, 1998).

A MOLECULAR MODEL OF SMECTIC-A RUBBER ELASTICITY

12.3.3

339

General deformations of a SmA elastomer

We have seen that deformations of SmA elastomers are very complex because of the rigidity of layers. One shear, λxz (and the y-equivalent), causes no layer spacing changes and is rubbery, i.e. penalised by the entropic effects, while the other, λzx rotates layers and compresses them and is therefore non-rubbery, ultimately controlled by the potential forces of the lamellar system. Elongations λzz are initially non-rubbery until an instability at which point they generate rotations, softer rubbery shear stress and no further increase in layer spacing. The post-instability combinations of deformations and rotations subtly conspire to give d/d0 =const. They can be understood via their decompositions. Armed with this we can then answer the question why there is a threshold if shear deformation costs so much less energy. Exercise 12.1: Decompose a general distortion of a SmA elastomer into its minimal, natural set of three component deformations in its own frame – an imposed λxx , λxz and λzz – plus a rotation. These are natural since λzz by itself describes layer compression or extension, λxx along with constancy of volume describes shape changes of the smectic layers, and λxz describes in-plane shears at constant layer spacing. Solution: zx deformations are complex since they induce layer compression and rotations. We attack these since all other deformations are trivial non-linear combinations of the fundamental components λij . Let us break up a deformation gradient in the LAB frame Λ , consisting of zx shear and simple stretches/contractions (to be determined), into a deformation D in the frame of the elastomer followed by a rotation W : 

Λxx Λ = W ·D =  0 Λ

0 1 Λxx Λzz

0

 0 0  Λzz

(12.35)

(the imposed zx component as usual losing its index). W rotates by an angle γ about the y axis (and hence no shears involve y). The compounded deformation D consists first of diagonal stretches/contractions, followed by a simple xz shear, all in the elastomer frame (and hence denoted by lower case λij ). Explicitly: 

 cos γ 0 sin γ W = 0 1 0  − sin γ 0 cos γ



 λxx 1 0 λxz D =  0 1 0 · 0 00 1 0

0 1 λzz λxx

0

 0 0  (12.36) λzz

Taking the product W · D , the overall deformation gradient tensor is thus: 

λxx cos γ 0 Λ= −λxx sin γ

0 1 λxx λzz

0

 λzz (λxz cos γ + sin γ )  0 λzz (cos γ − λxz sin γ )

(12.37)

which can be cast into the required form (12.35) by taking tan γ = −λxz to eliminate the Λxz element. Explicitly, the LAB stretches/compressions and shear are:

340

SMECTIC ELASTOMERS



λxx λxz 2 ; Λ =λ 2 . Λxx = λxx / 1 + λxz zz zz 1 + λxz ; Λ =  2 1 + λxz

(12.38)

The director is straightforwardly rotated with the layers, that is n = (sin γ , 0, cos γ ), which can be confirmed by use of eqn (12.2) and Λ .

By resolving Λ in the LAB frame into D of (12.36) in the elastomer frame, we have shown that there are in fact only three different deformations that the sample undergoes, apart from a trivial rotation. These are the deformations sketched as (a), (b) and (d) in Fig. 12.13, with all other possible strains decomposing into a combination of them. This situation is, again, due to the stringent layer constraints and is dramatically different from nematic elastomers. 12.4

Instability and CMHH microstructure

If shears are cheaper than layer dilations by a factor of 1/b = µ /B, why is there any threshold at all for the CMHH deformations in response to an imposed Λzz , especially for high b? Leaving aside the usual gradient terms that are relatively small in elastomers but which provide the threshold in liquids, such a question has led to the complicated concept of semi-soft elasticity in nematic elastomers. The situation in smectic elastomers is much simpler and is due to geometric constraints (which are also the reason why there is no soft elasticity in SmA rubber). At first sight eqn (12.19) and Fig. 12.16(b) would seem to suggest that there is simply a Λzx shear after the instability. In the LAB frame this is indeed true, but it is clearly accompanied by other distortions required to keep the layer spacing constant (see the insets to Fig. 12.16(b)). Imposing a Λzx shear alone causes the layer spacing to contract. λzz deformation A decomposition as in Ex. 12.1 can be performed on the imposed  2 , eqn (12.38). and gives a geometric reason for the threshold: identify Λzz = λzz 1 + λxz Firstly, suppose that the elastomer deforms with only the λzz component; there is no shear λxz . The free energy density and nominal stress are then: (1)

F1 ≈ 12 B(λzz − 1)2 ≡ 12 B(Λzz − 1)2 ; σzz ≈ B(Λzz − 1),

(12.39)

in the limit B  µ where layer spacing constraint dominates. Alternatively the sample could deform by a shear λxz , which leaves the layer spacing unchanged, and then rotate to accomplish the same Λzz value. In this case the free energy density is: (2)

2 = 12 µ r(Λ2zz − 1) ; σzz = µ rΛzz . F2 = 12 µ rλxz

(12.40)

For small
= Λzz − 1, we find F1 ∼ 12 B
2 and F2 ∼ µ r
. The latter energy is first order rather than second order in the strain and explains why shear and layer rotation is initially more costly. The equality of nominal stress and hence the transition occurs when (1) (2) σzz = σzz , that is when
c = µ r/B. When extension along the layer normal, λzz becomes too costly we have seen that layers rotate instead of dilating further, as illustrated in the cartoon inset of Fig. 12.16(b). However the sample must be clamped in order to apply a z-extensional force and thus

INSTABILITY AND CMHH MICROSTRUCTURE

341

F IG . 12.17. The microstructure of a sample loaded past the threshold stress. Stripes (dotted) of width h are shown coarsened, the layer normals (arrows) being at ±φ with respect to the extension axis, z. the rotation cannot occur uniformly throughout the sample. It must vanish at z = 0 and z = Lz , that is at the clamps, and must alternate in the x-direction between the values ±φ , eqn (12.27), sufficiently frequently that large layer translations are not built up which would then cost large elastic energies to satisfy the clamp constraints, see Fig. 12.17. On the other hand very frequent alternation between ±φ leads to the creation of many stripe interfaces (where a Frank elastic energy cost is paid). The resulting x-length scale and overall energy cost arises from optimising the sum of these two energies. There will turn out to be significant differences from the standard CMHH instability to stripes, Sect. 2.9, where the thresholds are small and are set by the interplay between Frank and layer elasticity. Here it is primarily set by the interplay between layer and rubber elasticity, discussed in greater detail in Sect. 13.5. Here we give a short analysis to produce a first estimate. Length scales emerge naturally from layer and matrix elastic moduli B and µ competing with Frank elastic energies which, for simplicity, we represent by a single  constant K. One obtains geometric quotients from Euler-Lagrange analysis: ξ = K/µ ∼ 10−8 m for the nematic penetration depth. It is a measure of how deeply a director variation can penetrate into the depth of a material while acting against the rubber elastic penalty for director rotation, see Sect. 8.1. It determines stripe interfacial width and the seemingly instant coarsening in the analogous strain-induced microstructure of stripe domains in nematic elastomers, Sect. 8.2. Analogously, one defines the usual smectic  penetration depth ξsm = K/B  d0 ≤ 10−9 m which determines the penetration of distortion into a smectic structure. Note this length is shorter than layer spacing in elastomers, due to the fact that the layer compression  constant B is larger than in liquid smectic and lamellar phases, where naturally K/B ≈ d. The length scale ξsm is independent of rubber elasticity and its small value compared with that of nematic elastomers suggests that smectic microstructure should also be instantly coarsened. The geometric mean of the smectic and Frank scales gives a √ surface tension of interfaces, as the energy cost per unit area of stripe formation: γsm = KB. As in the analysis of stripe domains in nematic elastomers, the threshold λzz = λc of a uniform system will be shifted very slightly by the addition of Frank effects to a higher λc  λc at which point there will be a small jump to a finite φ > 0. Microstructure to accommodate clamp constraints means there are spatial variations and thus a (small) Frank contribution to the energy. A little more strain must be imposed to overcome this additional cost. The stripe period in the x-direction can be estimated as

342

SMECTIC ELASTOMERS

0 h∼

Lz ξsm B 1 . r µ (λ − λc )1/4

(12.41)

The period never diverges, since λ  λc  λc (instant coarsening), and rapidly saturates to the value of a few µ m for a sample of length Lz ∼ 10−2 m. Note that, in contrast to liquid smectics, in elastomers the long sample dimension is in the direction of the stretch, that is the original layer normal. The resulting length scale would give the strong light scattering that is actually observed, such as in Fig. 12.8(a). See however the end of Sect. 8.2 where the possibility of a 2/3 power scaling of microstructure with sample length is raised (Kohn and M¨uller, 1992). 12.5

Comparison with experiment

Three different types of experiments give insight into SmA elastomers – strain response, stress-strain and rotation-strain measurements. We compare theory with relevant experiment (Nishikawa and Finkelmann, 1999), principally the λzz and λxx distortions. Strain response The Poisson ratios for imposed λzz are isotropic and volume preserving, (1/2, 1/2), until√the layer √ instability is reached. Thereafter the transverse relaxation is apparently (1/ λ , 1/ λ ) from closely inspecting the snapshot of large strain shown in Fig. 12.8(a). The authors do not give a functional dependence, but the figure rules out the predicted monodomain post-threshold xx and yy responses of eqn (12.26) which, if strains were small would correspond to Poisson ratios (1, 0) respectively. This discrepancy is not surprising given that there is clearly not a monodomain after λc . Either layers are being destroyed or they are rotating to all possible directions in the xy plane and not just to the x-direction as in our monodomain analysis. We return to this question of developing microstructure shortly. 2 Imposed in-plane stretches λxx give predicted Poisson ratios (νy , νz ) = ( 2+b 4+b , 4+b ). Experiment apparently gives (1, 0) corresponding to B/µ  1, see Fig. 12.6. These Poisson ratios agree with the stress results that show that smectic order is much more rigid than rubber elastic effects. Stress-strain relations Nominal stress against imposed strain (λzz − 1) along the initial layer normal direction in Fig. 12.8(b) is fitted to eqns (12.24) and (12.29). The ratio of the slopes is 4.1 × 10−2 . Theory gives µ r/B for this ratio whereupon one deduces b ≈ 25r, which is evidently large. In this limit eqn (12.34) predicts the direct connection r/b = λc − 1 for the threshold. This gives for the threshold strain
c = λc − 1 ≈ 4% which is extremely close to that observed in Fig. 12.8(b). In-plane stress and moduli in response to imposed λxx were not reported by Nishikawa and Finkelmann. The in-plane Young modulus is in theory E⊥ = 4µ , eqn (12.13). It is known from other work to be comparable to the post-threshold modulus against stretches parallel to the layer normal, Eafter = r µ , in accord with theory, see eqn (12.29). However, the situation is potentially more subtle:

COMPARISON WITH EXPERIMENT

343

Many SmA elastomers that have been investigated are suspected to be de Vries phases, that is where there is incipient SmC ordering. In this state the director tilt is not long-range correlated in its azimuthal direction. The signature of such local order is that the transition to the SmC state, with long-ranged order of tilting molecules, is not accompanied by a layer spacing change as expected from the transition from a standard SmA. Only small layer spacing changes have been observed even for induced tilts as large as 31 degrees (Spector et al., 2000) and also in ferroelectric SmC* examples (R¨ossle et al., 2004). Applied strain λxx in one in-plane direction could extend the correlation in tilt alignment and direct it along the strain, allowing the rubber to extend along x at lower energy cost than 4µ . Tests of this type of response would be: (i) The observation of in-plane induced optical birefringence. While an untilting director remains anchored along the layer normal, the response should be that of a classical elastomer where stress induces very small birefringence compared with that in any liquid crystal system. In comparison a de Vries elastomer would have a huge birefringence response. (ii) The ratio Eafter /E⊥ is predicted to be r µ /4µ = r/4. Departures from this ratio could be due to a low E⊥ because of de Vries phase effects. However to some extent de Vries effects should also intervene in Eafter since there is an element of in-plane stretch in the now rotated planes. The balance between the intervention of de Vries effects in the two moduli is not trivial since the component of stretch in-plane and the degree of shear acting both change with strain beyond the threshold for the Eafter case. One should also draw a distinction between the structure of a layer in classical liquid crystals and in side-chain polymeric systems. In the latter, a strong tendency for microphase separation (demixing between the backbone and the mesogenic groups, especially prominent when the backbone is polysiloxane) would lead to the backbone preferentially concentrating in the space between the mesogenic groups. Thus the total smectic layer periodicity is made up from adding the contributions of aligned mesogenic rods and the amorphous polymer backbone, in analogy with, for example, the structure of starch (Daniels and Donald, 2004). In smectic polymer melts, very flat oblate backbone conformations are well-established (Noirez et al., 1992). The overall distribution of end-to-end distances in an elastomer may still be prolate if the chains were crosslinked under uniaxial stretching in isotropic or nematic phase and only subsequently cooled into SmA, as is the case in Fig. 12.5. However, in the SmA state the backbone is flattening in the inter-layer spacing while straightening further on traversing the mesogenic region to minimise its exposure. On deformations the relative thickness of the amorphous inter-layer region may be slightly adjusted, a factor not encompassed by the above model. Strain-layer rotation and X-ray scattering Layer rotation against strain starts in a singular manner at a threshold λc both in theory, eqn (12.27), and in an X-ray determination of layer orientation, Fig. 12.9(b). Agreement with φ (λ ) is good, but a major problem of interpretation remains. As strain increases, the X-ray intensity associated with the rotating layer lines diminishes sharply. Nishikawa and Finkelmann proposed that above λc a diminishing fraction of the sample

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rotates while an increasing fraction of the layer system melts to a nematic state. The entropy change for the smectic-isotropic phase transition was ∆S = 2.4 × 10−2 JK−1 g−1 . Thus the cost of melting at 300K for a sample with density ρ ∼ 1g cm−3 is T ∆Sρ ∼ 7.2 × 106 J m−3 . To pay the cost of melting, a mechanical energy density 12 B(λc − 1)2 ∼ 8 × 104 J/m3 is available and is clearly rather little. In other words, the smectic energy scale is high compared with the rubber elastic scale. Further work is required to confirm this possibility. A more direct explanation of the reduced intensity of X-ray scattering is that layers rotate their normals towards all directions perpendicular to the stretch along the original layer normal n0 . Section 12.3.2 calculates the monodomain contraction and shear in the x direction perpendicular to original layer normal z, see eqn. (12.26), and the rotation of the normal toward x, see eqn (12.27). But no direction perpendicular to the original director is privileged, in contrast to stripe formation in nematic elastomers. We must consider all other axes perpendicular to n0 . This break up of the sample into a microstructure of regions of tilted domains is cylindrically symmetric around the stretch axis. The regions that are tilted toward the X-ray beam no longer meet the Bragg condition for diffraction, and as a result do not contribute intensity to the observed scattered beam. One could suggest that the drop in X-ray intensity is simply a result of polydomain formation. Additionally, the overall Poisson ratios observed in the two, now equivalent directions perpendicular to the original layer normal are ( 12 , 12 ) rather than the monodomain values (1, 0). Again, further work is required, for instance rotation of the sample in the X-ray beam about the stretch axis; see Adams and Warner (2005) for further discussion. 12.6

Smectic-C rubber elasticity

Molecular tilt leads to spontaneous shear in making the transition from the SmA to SmC phase, Sect. 12.2.2. The development of such shear has been described within nonlinear continuum theory (Stenull and Lubensky, 2005) and within the molecular theory of this book (Adams and Warner, 2006). Both methods draw a distinction between the molecular tilt and the shear angle, as required in experiment. The other challenge for theory is to describe the shear offset and the dependence of the spontaneous shear on the crosslinking history, that is the dependence on the shear that was applied during the second-stage crosslinking and then allowed to relax when crosslinking was complete. Further details are available in the original papers. Recall that the most remarkable aspects of nematic rubber elasticity were the spontaneous distortions on entering the nematic phase and shape change at minimal energy cost – soft elasticity. The two phenomena are intimately related: soft elasticity arises because of the degeneracy in the problem, that one can rotate the director independently of the body; the extent of the soft modes is given by the magnitude of the spontaneous distortion arising when symmetry is broken. In SmA networks the constraint of director perpendicular to the layers removed this freedom. In smectic C we have this possibility – now the director is tilted one can move it independently of the body by rotating it on a cone about the layer normal, and with this rotation must come elastic distortion at no, or low elastic energetic cost. With such director rotations and deformations we

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345

are not altering the layer spacings and thus in these deformations not transgressing the rigid constraints of SmA. In fact biaxial SmA elastomers could also rotate their in-plane director about the layer normal and change shape without effecting layers. This route to elastic softness has been recognised theoretically (Adams and Warner, 2005b; Stenull and Lubensky, 2005). It is equivalent to the simplest biaxial soft modes considered in Appendix B treating the soft deformations of biaxial nematics. We again consider strongly coupled smectic elastomers, that is where the layer compression modulus B is large and all deformations λ affinely convect the layer system while respecting their spacing. In general the tilt angle, θ , might be induced to change by stresses, but we take it to be fixed here. Firstly, the rigidity of the smectic order we have seen can be very high. Secondly, we shall be interested in soft deformations (Adams and Warner, 2005b) in which case changes of θ would cause an increase of thermodynamic energy and vitiate softness. In a recent non-linear continuum theory of SmC elasticity (Stenull and Lubensky, 2005) the absolute rigidity of the tilt angle has been relaxed. In both approaches considerable care is needed to record the elements of body rotation this is generated and thus determining the final orientation of the sample, information that can be extracted from the λ tensor. The difficulty arises in respecting the strong layer spacing constraint, that is initially unit layer normal k 0 transforms to new k which remains a unit vector: k = λ −T · k 0 with k T0 · λ −1 · λ −T · k 0 = 1 .

(12.42)

The transformation is the equivalent of eqn (12.3) for SmA rubber where there the director and the layer normal were identical. In addition to not incurring any smectic penalty, the last part of the free energy eqn (12.1), the deformations must also incur no nematic energy cost, the first (Trace) part. Such deformations are the soft modes of eqn (7.10), that is: W ·  −1/2 λ =  1/2 ·W , n 0 with the rotation W being able to be chosen arbitrarily. In Appendix F we sketch schemes for finding k , n and λ in the general case, which has considerable geometric complexity. In particular we make contact between methods akin to those we have used in nematic soft elasticity, and those which take a virtual, isotropic reference state. 12.6.1

SmC soft deformations

We illustrate soft deformations here with an important set of modes analogous to the soft modes of a nematic in response to an extension imposed perpendicularly to the director. For SmC elastomers we impose extension in the layer planes perpendicular to the initial director which is thereby induced to rotate without layer rotation or contraction of the layer spacing. The director rotation can then only be azimuthal, on surface of a cone about the unchanging layer normal. Let us consider the specific coordinates shown in Fig. 12.18(a): An extension in the y direction is imposed; we take a deformation matrix of the form:

346

SMECTIC ELASTOMERS k0 n 0

z (a)

y

y x

(b)

x

F IG . 12.18. (a) The undeformed SmC elastomer with layer normal k 0 = z and with n 0 tilted toward the x direction. (b) The shape changes of the elastomer viewed along the layer normal as the c -director advances (right to left) through φ = 0, ±π /2, ±2π /3 and ±π . In this example the anisotropy and tilt are r = 8, θ = π /6.   λxx 0 λxz (12.43) λ =  λyx λyy λyz  . 0 0 λzz The components λxy and λzy are excluded as in the nematic case – they deform the sample by translating the y faces of the sample in the ±xx and ±zz directions. Any small y forces associated with the yy elongation would generate counter torques and quickly eliminate them. The λzx component and (again) the λzy component are excluded because, as we shall see, they rotate the layer normal, which we want to remain fixed. In our coordinates, the director and the (unchanging) layer normal are: k 0 = (0, 0, 1) ; n 0 = (sin θ , 0, cos θ ) → n = (sin θ cos φ , sin θ sin φ , cos θ ) where the tilt angle θ is typically around 20◦ . The layer normal, k 0 = z , cannot be moved by deformation tensors of the form eqn (12.43) since it derives from k = λ −T · k 0 – the element ki is thus the cofactor of the element λiz , all of which vanish except for that of λzz , as can be seen by inspection of (12.43). The soft mode, not rotating the layer normal and with rotation φ of the director about k 0 , is:   (r−1) sin 2θ 0 a(φ ) 2ρ  (−a(φ ) + cos φ )   2φ 1 (r−1) sin 2θ  2φ    (12.44)  1 − ρr sin . sin φ − 1 − ρr sin 2ρ 2a(φ ) a(φ ) 2a(φ ) 0 0 1 Anisotropy and tilt are encoded in the mode in the parameter ρ = 1 + (r − 1) cos2 θ ≤ r. 1/2  2 2 ≤ 1. The deformation is The rotation angle appears in a(φ ) = 1 − r−1 r sin θ sin φ pictured in Fig. 12.18(b) for a sequence of rotations φ . One clearly sees that yx shear deforms the initially square xy section shown on the right (φ = 0) and that xz and yz shears give the lean of the layer stack seen at all other rotations. The final state (φ = π ) on the left is purely (negative) xz shear resulting from a reversal of the tilted director. It is the maximum extent of soft shear and one thereby sees a connection to the spontaneous shear: consider Fig. 12.19 which starts (a to b) with a SmA elastomer spontaneously

SMECTIC-C RUBBER ELASTICITY

k

n0

L

q

k

c0

c^ p (a)

n k0

q n0

k0

n

347

z

p0

(b)

(c)

-2L -c0 p

x

(d)

F IG . 12.19. (a) A SmA elastomer spontaneously shearing by Λ on achieving the SmC state (b). A regular shape (c) is cut out of this sheared elastomer and becomes the reference state for subsequent distortion. (d) has had its tilt direction reversed with an associated shear reversal of −2Λ. For the three shapes the polarisation p arising in the chiral (SmC*) case is shown in or out of the plane. 0.4

f

0.2 0 -0.2 -0.4 1

1.02

1.04

l yy

1.06

1.08

1

lxz lyz lyx 1.02

1.04

l yy1.06

1.08

F IG . 12.20. (a) The director rotation in response to an imposed stretch λyy perpendicular to the initial director. (b) The soft yx, yz and xz shears thereby induced by the imposed yy extension. The anisotropy and tilt are r = 2, θ = π /6. shearing by Λ to the SmC state (Hiraoka et al., 2005), see Fig. 1.9. The maximum extent of possible soft shear in going from (c) to (d) is 2Λ, that is twice the spontaneous shear on entering the SmC state. State (d) corresponds to the left-most state in Fig. 12.18(b). The director rotation and the shears that develop as λyy is imposed are shown in Figs. 12.20. The relation λyy = 1/a(φ ) ≥ 1 between extension and rotation φ can be read off the appropriate element of λ in (12.44) and, using the definition of a(φ ), can easily be inverted. For small φ and for φ → π it is clearly a singular relationship,

2r / sin θ , in common with theory (eqn (7.44) with λxx in that exφ ∼ (λyy − 1)1/2 r−1 ample) and experiment (Fig. 7.10) for the rotation associated with the soft modes of  nematics. The maximum soft extension occurs at φ = π /2 and is λyy = λm = r/ρ ≡  −1/2 2 1 − r−1 . The other components of λ can equally easily be read off. A very r sin θ subtle aspect of non-linear elasticity is apparent: a maximal yz shear of magnitude Λ must develop when the director is tilted purely in the y direction, that is when φ = π /2. Inspection of Fig. 12.20(b) shows that in fact this particular shear is apparently maximal for λyy ∼ 1.045, and thus for a rotation φ  π /4. It is a consequence of geometrical nonlinearity of finite deformations, which must be compounded rather than added. At the time of writing no very large strain experiments have yet been performed

348

SMECTIC ELASTOMERS

and this area remains one of exciting future studies and applications now that large SmC monodomains are accessible. Applications are especially attractive when the chiral SmC* elastomer is considered. In the above discussion of soft c -director modes, the associated electric polarisation was originally originally along the y direction. On deformation, its y-projection varies as

(12.45) Py = P0 cos φ = P0 ((λm /λyy )2 − 1)/(λm2 − 1). It is not singular at the start of the extension, at λyy = 1, but does vary rapidly around the maximum λm which may be of practical significance. Experiments probing the electrical response to this particular (extensional) mode have not yet been performed, but rather shear has been investigated. We now look at imposed shear and address the related question of deformation in the presence of constraints, that is the induction of stripe microstructure during the low energy deformations of SmC elastomers. 12.6.2

SmC deformations with microstructure

A deformation more natural than a y-extension to apply to a SmC elastomer might be an xz in-plane shear. One could view the deformations as being simply those of the +lyz

±p/3

+lyz

y

k0 c

z x

lxz

f=0

lxz

s

_ _ __ _ _ __ ___ _ _

±2p/3

+

+_ __

+

+

_

P ±p

P

z y x

+lyz

(a)

+lyz

(b) P

P

F IG . 12.21. (a) The top and bottom plates are moved relatively in the x direction to create a λxz while perpendicular (y) displacement of the plates is forbidden. Alternating bands of ±λyz shears ensure the macroscopic boundary conditions are respected. (b) When all but xz shear is macroscopically suppressed, an initially cubic sample deforms softly with laminates of director rotation and of yx and yz shears of alternating sign and with xz shear advancing from 0 to −2Λ. It is shown in steps of director rotation φ = ±π /3. The laminate normal s  is shown for the example of φ = π /3. It starts and finishes parallel to n but in general makes a different angle with the layer normal. The deformations away from the initial shape (light reference frame) reveal the slight contraction λxx along x and a compensatory lengthening along y. The ±π /2 case would be like (a), but with λxx , λyy = 1.

SMECTIC-C RUBBER ELASTICITY

349

previous section, but using λxz as a variable instead of λyy or φ which played somewhat equivalent roles. The soft λ would be identical and the accompanying polarisation also. However, depending upon whether one has a slab or thin sheet geometry, that is whether z or y is the smallest dimension of the sample in Fig. 12.18(a), one might have one or more elements of macroscopic shear suppressed by clamps, used to hold the sample, or by electrodes used to measure the electric polarisation changes. In this event, the system will try to deform as softly as possible, albeit by the development of microstructure. Take as an example an in-plane shear λxz imposed by gripping a slab-like sample using rigid plates in the in the xy planes bounding the top and bottom of the sample in the z direction, and then moving the plates relatively in the zx direction. We have seen that, in order to deform softly, the sample must rotate the director around on a cone, ideally from tilting in the +x direction to the −x direction, and thereby develops a λyz , among other shears. Suppose that a constraint on the sample is that the top plate may not move in the y direction relative to the bottom plate. In that case the average λyz shear must be suppressed; in other words an oscillating λyz must develop preserving the zero y-displacement on the two plates. Figure 12.21(a) proposes one scheme as to how this can be done by alternating displacements (and hence shears) in the y direction. There are in fact also other shears in the soft mode and all of these except the xz shear must be involved in laminates to avoid macroscopic distortion suppressed by clamps. Fig. 12.21(b) shows a possible set of laminates of shears of alternating sign that could develop in the sample, the normal to the laminates rotating in the zx plane as the director φ rotates about the layer normal (the latter remaining along the z direction). It is known from parallel studies in Martensitic alloys that many different microstructures are possible, and the same is true in Smectic C elastomers. In Appendix F we show how textures may be calculated, using the soft modes we have already discussed, and how the requirement of rank-one connectedness between the deformation gradients of neighbouring parts of the system of laminates can be satisfied. Since the particular laminates shown above are oblique to the z axis and thus cut through the layers, as the polarisation rotates the laminates will also cut the polarisation and the internal interfaces between laminates will be charged. An external yz surface is shown decorated with surface charges of sign that alternates from laminate to laminate since the x component of polarisation that terminates of starts on this surface reverses too. Other laminates are known which do not cut the polarisation and hence do not lead to internal charging. At the time of writing little is known of the structure of the laminates and, as yet, no large amplitude experiments have been performed. Small shear piezoelectric experiments do show changes of polarisation – see Sect. 12.2.2.

13 CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS The continuum theory of liquid smectics was discussed in Sect. 2.9 in order to have a basis from which to construct the free energy of smectic rubbers. As we have seen, smectic elastomers differ from their corresponding liquids in two important regards. Just as in nematic elastomers, internal rotations are now significant. In addition, the spatial position of layers is also no longer arbitrary. Their translation with respect to the rubbery matrix is penalised by a periodic potential provided by the lamellar order. Having set up the continuum free energy we pursue two separate routes. Eliminating the director and the layers by letting them optimally relax, one is left with the elasticity of the effective medium that has unusual properties: it is rubber-elastic in two dimensions and a conventional solid in the third. Alternatively, we can eliminate the rubbery matrix (by letting it relax in a mechanically unconstrained sample) and concentrate on the layer structure. We then find the behaviour of the smectic layers radically altered from the liquid case. The layers interact via the now-invisible effective background provided by the elastomer network. They no longer display the celebrated Landau-Peierls loss of long range order. The Clark-Meyer-Helfrich-Hurault (CMHH) layer buckling instabilities are different from those of liquid smectics. They also display a new mechanism for quenched disorder in the smectic layers. We conclude this chapter by considering the continuum elasticity of smectic C elastomers. This phase has the director tilted in the layer planes. Their chiral variants are ferroelectric solids, but soft, greatly extensible and non-crystalline and thus of great technological significance. We make further contact with the growing body of experiment. 13.1 Continuum description of smectic A elastomers To understand the physical properties of smectic (lamellar) elastomer networks, we reexamine in continuum theory the variables describing different aspects of their behaviour. Nematic order underpins smectic layering. Nematic and relative-rotation effects arise in smectic elastomers again, but are modified by the constraints offered by layers. The continuum free energy of smectic A elastomers has a considerable complexity of couplings (Terentjev and Warner, 1994a), especially if one considers also polarisation effects. The most important effect is the layer-matrix relative displacement coupling (Lubensky et al., 1994). A simplified description (Weilepp and Brand, 1998) has assumed from the outset a rigid identification of layer displacements with those of the matrix, which is appropriate for some problems, for instance the CMHH instability (Terentjev and Warner, 1994a; Weilepp and Brand, 1998). However, we shall use the more general layer-matrix elastic coupling which allows for fluctuations which can be important in questions of the Landau-Peierls instability and long range layer order generally that are of interest in smectics. Indeed, a new length scale will emerge in Sect. 13.4 350

CONTINUUM DESCRIPTION OF SMECTIC A ELASTOMERS

351

on effective layer elasticity and order. It arises from the competition between fluctuations of layers from their locations in the matrix and the effects of incompressibility governing strains of the matrix. 13.1.1 Relative translations in smectic networks revisited We saw in the first sections of Chapter 12 that the formation of a rubbery network in a smectic or lamellar phase results in preferential placement of crosslinks with respect to the layers. In the geometry of Fig. 2.14(c) the crosslinks are locked within the layer, as the real mesogenic side-chain polymers in Fig. 12.1 would indeed do. The sketches in Figs. 12.3(c) and 13.1 show the crosslinks within the backbone, preferentially concentrated in the interlayer space. In synthetic work (Zentel et al., 2000) and (Zubarev et al., 1998) the crosslinking was of this type, but there are also many examples of effectively rod-like crosslinks spanning the mesogenic layer. In any case there clearly is a barrier for a crosslinking ‘point’ to cross the neighbouring layer. Of course, they can freely migrate within their chosen layer plane. Accordingly, only the z-component of the crosslink displacement vector u participates in the relative translation coupling, shown in Fig. 13.1 and eqn (13.1). As a result, any attempt to deform the rubbery network (which amounts to the relative movement of crosslinks along the smectic layer normal, described by the displacement vector u ) causes an energy penalty, if the smectic layers do not move with the matrix, see Fig. 13.1. Thus it is the relative displacement uz − v that is penalised. Unusually, such a coupling does not have to involve spatial gradients of the displacement fields u and v (see the remark above eqn (2.54) about the choice of variables for the matrix and the layer displacement). In a classical elastic continuum with only one deformation field, u or v, the free energy cannot depend on such a field alone, otherwise a constant displacement of the whole sample (amounting, e.g. to a uniform vector u ) would cost energy. That is why we usually only look at the derivatives, such as ∇ v or the strain ui j = ∂ ui /∂ x j . Nematic elastomers conformed to this irrelevance of the network translation vector u because the second deformation field, the director rotation δ n , was of different symmetry (note, however, that director rotation and local matrix rotation were of the same symmetry and their difference did prominently enter the free energy). Now this situation has changed. With the deformation variables u and v both

F IG . 13.1. Coupling between the elastic matrix and the layer system through relative translation. Because network crosslinks are locked with respect to the local smectic layer position, their displacement along the layer normal described by the component uz has to be the same as the layer displacement v at this point.

352

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

corresponding to translations, we have a possible energy penalty on their local relative magnitude along the layer normal. For small relative displacements between the layers and the elastic medium the penalty must be harmonic with a free energy density of Λ [V (rr ) −Uz (rr )]2 .

(13.1)

We deliberately write the capital letters in the notation for local layer displacement V (rr ) and the corresponding component of network displacement Uz (rr ). Because no spatial derivatives are involved in eqn (13.1), it is essential to separate the displacement fields V and Uz into their constant components, uniform (average) strains, and also fluctuations with respect to these average values, in both layer and elastomer sub-systems. Displacements in eqn (13.1) can be decomposed as following: U (rr ) = U (0) + u · r + u (rr );

V (rr ) = V (0) + r · ∇V0 + v(rr )

(13.2)

where, for the elastic medium, U (0) is the displacement of the origin in the body, the constant matrix ui j describes the uniform part of strain, and u (rr ) represents the phonons and other non-uniform distortions with zero average over the sample. A similar decomposition applies to the layer displacement field V : V (0) describes the uniform movement of layers along their normal, the constant vector (∂xV0 , ∂yV0 , ∂zV0 ) is the uniform layer rotation and compression, and v(rr ) represents the layer fluctuations, which average to zero over large distances. The decomposition into uniform and fluctuating parts makes it apparent that uniform layer rotations (arising from ∇ ⊥V ) and uniform compression (arising from ∇zV ) are rigidly locked with the corresponding components of uniform strain, uz j , of the rubbery ∇⊥ denotes variations in the layer planes.) Let us consider the effect of such network. (∇ decomposition on the full free energy (over the whole sample volume ∼ L3 , where L here is the linear size of the system) resulting from the coupling (13.1):

u z= e x

u x= e z

(a)

(b)

L

F IG . 13.2. The effects of relative translation: (a) Displacement in the layer plane Ux causes no effect in eqn (13.1) [but, of course, the corresponding symmetric and antisymmetric components of shear ui j will contribute to the relative rotation terms in eqn (13.5)]. (b) The displacement along layer normal Uz is relevant. A uniform simple shear uzx causes an enormous displacement over the sample dimensions L and rigidly demands the affine rotation of layers.

CONTINUUM DESCRIPTION OF SMECTIC A ELASTOMERS

F =Λ



[V (rr ) −Uz (rr )]2 drr

353

(13.3)

∼ Λ L3 [V (0) −Uz (0)]2 + Λ L5 ∑[∇ jV0 − uz j ]2 + Λ



[v(rr ) − uz (rr )]2 drr .

j

The first term, proportional to the system volume, implies that there can be no relative uniform displacements, that is, there is no Goldstone mode on relative translation in our system. The second term is extremely large with a coefficient L3 (L2 Λ) and hence an effective elastic constant proportional to Λ times a factor corresponding to the 23 -power of the system volume. It constrains uniform layer rotations to exactly follow the corresponding uniform shear strains of the rubbery network ( j = x and y in eqn (13.3)), and similarly with compressions ( j = z). The effect of such a constraint is shown in Fig. 13.2(b). Finally, the third term penalises layer fluctuations with respect to the matrix, or conversely elastic phonon type displacements with respect to the layers. These couplings are additional to those acting in conventional smectics or rubbers and correlate layer fluctuations with phonons in rubbery, layered networks. Let us now give an estimate of the new coupling constant Λ. In the original work (Lubensky et al., 1994), eqn (13.1) was regarded as a purely entropic effect of the network strands. With the characteristic energy scale kB T and the number of strands per unit volume ns , we can write on dimensional grounds Λ ≈ α ns kB T /R2o ≡ α µ /R2o (using the notation for the rubber modulus µ ). Here α is a coefficient of order unity, which must depend on the smectic order parameter ψ tending to zero near a transition. Hence Λ ∼ |ψ |µ /R2o and, in a material with a rubber modulus µ ∼ 105 J/m3 and network span Ro ∼ 1 − 10 nm, the constant is around Λ ∼ 1021 − 1022 J/m5 . However, a molecular model (Adams and Warner, 2005a) gives this coupling constant explicitly, depending on the smectic layer potential strength gs and the layer spacing (see Fig.12.11 in Sect.12.3): Λ = 16π 2 ns gs /d 2 . Since the layer compression modulus B of SmA elastomers is also determined by these parameters, and is independently measured by mechanical experiments, we deduce a much higher value Λ = 4B/R2o ∼ 1024 − 1025 J/m5 , as indeed expected for significant enthalpic contribution of smectic order. 13.1.2

Nematic -strain, -rotation and -smectic couplings

We now introduce the nematic effects into the smectic elastomer explicitly and will naturally find an energy that involves the fluctuating nematic director δ n. Therefore, the possible deviations of the director from the layer normal would be allowed, although naturally penalised by the corresponding coupling energy, see eqn (13.8) below. Having assembled the full smectic elastomer energy in this section, in the next we shall integrate out the director modes δ n q , i.e. allow the system to equilibrate to its optimal director fluctuation modes. As usual, this means that we minimise the free energy w.r.t. n , after which the nematic director will no longer appear as an independent variable. This gives the effective Hamiltonian, or free energy density that depends only on rubber-elastic and smectic degrees of freedom.

354

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

Uniaxially anisotropic elasticity Since a nematic phase underpins the smectic elastomer, we have the same kind of uniaxial elastic energy as in a nematic elastomer. Recall that only the symmetric part ε˜αβ = 12 (uαβ + uβ α ) of the strain tensor contributes to the ordinary elastic response of a uniaxial material, with the free energy density in the frame-independent form  2 Felastic = C1 (nn · ε · n )2 + 2C2 Tr[ε˜ ](nn · ε · n ) +C3 Tr[ε˜ ]

(13.4)

+ 2C4 [nn × ε × n ]2 + 4C5 ([nn × ε · n ])2 , with n the local axis of anisotropy and εαβ = ε˜αβ − 13 Tr[ε˜ ] δαβ being the traceless part of strain, exactly as in eqn (10.12). We again expect that the bulk modulus C3 is very large, ∼ 109 -1010 J/m3 , making the elastomer physically incompressible, that the three moduli C1 , C4 and C5 are of the same order of magnitude as the rubber shear modulus µ ∼ 104 -106 J/m3 , and that the anisotropic compression correction C2 is much smaller, determined by secondary effects in networks. However, the rigidity of layers will mean that C5 will suffer a rather different renormalisation from that arising due to softness in nematic elastomers. Relative rotation and Frank coupling Frank elastic terms play a significant role in short wavelength director fluctuations. However, several examples in Chapter 10 show that in the first approximation their effect could always be neglected at low q or long wavelength deformations, the only physically relevant situation. We would have a similar case here, if it were not for the specifically smectic effect of layer curvature: the second term in the smectic elastic energy eqn (2.54) represents a Frank splay contribution. Thus, taking care not to neglect the weak Frank nematic elasticity too early, we write the main terms in eqn (10.12) that involve the director fluctuations δ n , perpendicular to the initial director n which we take to be along z :   (a) (a) Fnem (δ n ) = 12 D1 (uxz − δ nx )2 + (uyz − δ ny )2   (a) (a) −D2 (uxz − δ nx ) εxz + (uyz − δ ny ) εyz

(13.5)

+ 12 K1 (div δ n )2 + 12 K2 (curlz δ n )2 + 12 K3 (∂z δ n )2 , where the constants D1 and D2 are given, for instance, by the expressions (10.14). When there is no underlying elastic network (crosslinking density is zero), only the third line remains in the expression for Fnem . Nematic-smectic coupling The free energy density describing the N-A phase transition is written, as in Sect. 2.9, in terms of the smectic order parameter ψ (rr ) = |ψ (T )|e−iqo v(rr ) , which describes the departure of the local density ρ (r) from a uniform value ρo , in a form of a single wavelength

EFFECTIVE SMECTIC ELASTICITY OF ELASTOMERS

355

modulation. This free energy and the N-A transition are thoroughly discussed in the literature (Lubensky, 1983) and summarised in monographs (de Gennes and Prost, 1994; Chaikin and Lubensky, 1995):   (13.6) FsmA = 12 τ |ψ |2 + 12 β |ψ |4 + g |∂z ψ (rr )|2 + g⊥ |(∇⊥ − iqo δ n )ψ (rr )|2 where, as usual in Landau theory, τ = τo (T /Tc − 1) and the constants governing the gradient terms differ, g = g⊥ because of the uniaxial nematic anisotropy. The last gradient term expresses the correlations between fluctuations of a forming layer (both the magnitude and phase of ψ are varying). In a system away from the N-A transition, the fluctuations of amplitude |ψ | are small. Mainly ∇⊥ acts on the phase of ψ and the layer normal is well defined. Director fluctuations δ n away from the normal are then penalised by the last gradient term. Another way of re-writing both gradient terms in FsmA is 2 2 2 2 2 2 1 1 2 g qo |ψ | (∂z v) + 2 g⊥ qo |ψ | (∇⊥ v + δ n )

.

(13.7)

Note that the first term has the meaning of a layer compression penalty and the corresponding constant in eqn (2.54) for liquid smectics is, therefore, B = g q2o |ψ |2 . The g⊥ term expresses the nematic-layer coupling and we express the combination of terms as the single constant b⊥ = g⊥ q2o |ψ |2

(13.8)

governing the deviation of the director away from the layer normal. Later, in considering the effective matrix-layer coupling, eqn (13.16), we shall explore the limits of strong (b⊥  D1 ) and weak (b⊥  D1 ) rigidity compared with the usual rotational coupling in the nematic state. 13.2 Effective smectic elasticity of elastomers We shall not be interested to follow the director fluctuations δ n in eqns (13.5)-(13.6) and will simply let them adopt their optimal values, leaving behind an energy that only then depends on the layer and matrix distortions. As before, in the analogous calculation for nematic elastomers, we choose the notation for perpendicular ()⊥ and transverse ()t components of vector fields in the plane perpendicular to z, as explained in Fig. 10.6. Performing the Fourier transformation of the relevant parts of free energy density (13.5), that is those involving δ n , one obtains a free energy density in q-space:   (13.9) F = 12 D1 + b⊥ + K1 q2⊥ + K3 q2z |δ n ⊥ |2    2  2 2 2 2 2 1 1 + 2 D1 + b⊥ + K2 q⊥ + K3 qz |δ nt | + 2 Bqz + b⊥ q⊥ |v|  ∗ ∗ 1 + 4 iq⊥ [D1 + D2 ] δ n ⊥ uz + 4iq⊥ b⊥ δ n ⊥ v + c.c.

− 14 (D1 − D2 ) iqz (δ n ⊥ u∗⊥ + δ nt ut∗ ) + c.c. . The choice of perpendicular and transverse has left the corresponding modes, δ n ⊥ and δ nt , decoupled from each other. Note that in an ordinary smectic liquid (with no background rubbery network) the relative-rotation constants D1 and D2 are zero and the only addition to the Frank elastic terms is proportional to b⊥ .

356

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

The energy eqn (13.9) has terms linear and quadratic in the components δ n ⊥ and δ nt and thus the system can always lower its energy by adopting optimal spontaneous director distortions:

δ nt = δ n⊥ =

i (D2 − D1 ) qz ut 2 D1 + b⊥ + K2 q2⊥ + K3 q2z

(13.10)

i (D2 − D1 ) qz u⊥ + (D2 + D1 ) q⊥ uz + 4b⊥ q⊥ v . 2 D1 + b⊥ + K1 q2⊥ + K3 q2z

These expressions are very similar to the director rotations in the soft nematic elastomer case, eqn (10.44), except for the presence of the smectic order-dependent coupling b⊥ which shifts combinations of elastic constants in the denominators of the above expressions, and also directly couples δ n ⊥ with v. As a check, when the crosslinking density is zero and D1 = D2 = 0, eqn (13.10) reduce to δ n ≈ −∇⊥ v, or in separate components:

δ nt = 0;

δ n⊥ = i q⊥

 K 2 b⊥ v  i q⊥ 1 − q v. b⊥ b⊥ + K1 q2⊥ + K3 q2z

In this case the effective smectic elastic free energy density takes the form FsmA =

1 2

Bq2z + b⊥ q2⊥ −

Expanding at small wave vectors q ( only the leading terms, this reduces to 1 2

b2⊥ q2⊥ (b⊥ + K1 q2⊥ + K3 q2z )2

 |v|2 .

(13.11)

 b⊥ /K below the N-A transition) and retaining



 Bq2z + K1 q4⊥ |v|2 ,

which is the Fourier transform of eqn (2.54), the elastic energy of a lamellar phase. We see that it is indeed the splay Frank constant of the underlying nematic that controls the layer curvature term, proportional to the fourth-order gradient q4⊥ . We shall see below that the coupling to a rubbery network removes this degeneracy. Returning to the elastomer case, we can now obtain the effective elasticity energy density of the layers, the matrix and their coupling. Substituting the optimal director modes eqn (13.10) into eqn (13.9) and bringing the result together with the ordinary uniaxial nematic rubber-elastic terms eqn (13.4), one obtains the overall smectic elastomer energy: F = Felastic + FsmA + Fcoupling .

(13.12)

We now examine the different contributions in turn. The technical details of how to obtain each effective energy by eliminating δ n are tedious but straightforward (Osborne and Terentjev, 2000).

EFFECTIVE SMECTIC ELASTICITY OF ELASTOMERS

357

Elasticity of the rubbery matrix The rubber-elastic part of F derives from the derivative terms of u . It is determined by the same five-constant expression, eqn (13.4), dictated by the uniaxial symmetry. As in Sect. 10.6 for nematic rubber elasticity, we minimise over the director modes so that they are no longer visible in the elastic free energy. Again, the modulus against shears in the plane of rotation of the director, C5 , is renormalised by the removal of the director degree of freedom: C5smA = C5 −

b⊥ D22 D22 D2 becomes if C5 = 2 . 8(b⊥ + D1 ) 8D1 (b⊥ + D1 ) 8D1

(13.13)

Note the familiar soft combination C5 − D22 /8D1 → 0 for elastically soft nematics. Clearly, the coupling in this strain mode to smectic layers sufficiently constrains director response to prevent complete softness, even if the underlying nematic is ideally soft (which has been assumed to obtain the last part of eqn (13.13)). Not surprisingly, the surviving modulus is then proportional to the degree of smectic order, via the order parameter dependence b⊥ ∼ |ψ |2 . Figure 13.3 shows why C5 is renormalised by layer effects while C4 , which controls the in-plane shears, is not. Smectic elasticity The part of F depending only on v yields the smectic elastic part which is modified as well. Coupling of layers to the rotational elastic influence (D1 ) of the network increases the penalty on layer deformations. We obtain in eqn (13.12), after expansion at small wave vectors of deformation,  b2⊥ b⊥ D1 2 2 4 1 q + Kq |v|2 (13.14) FsmA (v) = 2 Bqz + b⊥ + D1 ⊥ (b⊥ + D1 )2 ⊥ z

x x

y

F IG . 13.3. Shearing in the xz-plane (governed by C5 ) distorts layers and carries an additional energy cost, while shear in the xy-plane perpendicular to the layers and director (C4 ) has no effect on layers in a smectic A.

358

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

≡ 12 B

∂v ∂z

2 +

 2 2 b2⊥ K b⊥ D1 (∇⊥ v)2 + ∇⊥ v . 2 2(b⊥ + D1 ) 2(b⊥ + D1 )

This differs greatly from the ordinary smectic expression eqn (2.54). The network constant D1 , which penalises relative rotation in nematic elastomers, now acts to penalise uniform layer rotations as well, the term in (∇⊥ v)2 which was formerly absent, see Sect. 2.9 and in particular Fig. 2.16. As a result, the familiar degeneracy (the leading q4⊥ dependence associated with layer bending) no longer dominates the corresponding lamellar rubbery network. Thus the smectic layer-rotation and layer-bending constants become, respectively: 0 →

b2⊥ K b⊥ D1 ; K → . (b⊥ + D1 ) (b⊥ + D1 )2

(13.15)

Of course, when the network is not crosslinked (i.e. D1 → 0) the new quadratic layerrotation penalty disappears and the bending constant becomes the usual nematic splay constant K. Layer coupling to the matrix Finally, the coupling between rubber-elastic and smectic layer deformations in (13.12) emerges when δ n in eqn (13.10) is returned to eqn (13.9) and terms in v2 , u2 and uv, with factors of q giving the derivatives as appropriate, are extracted. The layer-matrix coupling takes the explicit form Fcoupling = Λ[v(rr ) − uz (rr )]2 Ω − ω A ) × k ]2 + ∆2 k · ε · [(Ω Ω − ω A) × n] + 12 ∆1 [(Ω

(13.16) .

The layer normal is k and is along the original director. One might expect to differentiate between the original and current layer normal, that is perhaps to use k o . The difference between the current and original layer normals introduces another small difference, the account of which would give us small terms, higher than second order – these are therefore neglected in linear continuum elasticity. We concentrate on the fluctuating parts of the total displacement variables V and U since we have already discussed below eqn (13.3) how the uniform parts of the matrix and layer displacements and shears are ∇⊥ v)]. The associated rigidly coupled. Layer rotation is expressed as ω A = [kk × (−∇ elastic constants are given by combinations of the underlying nematic relative rotation couplings, D1 and D2 , and the smectic coupling b⊥ ∝ |ψ |2 : ∆1 =

b⊥ D1 b⊥ D2 and ∆2 = . b⊥ + D1 b⊥ + D1

(13.17)

When the nematic director is very strongly anchored along the layer normal, the nematic relative rotation is really equivalent to that of a smectic. Accordingly, if b⊥  D1 , such an equivalence emerges from expressions (13.17), ∆1,2 → D1,2 . Some authors (Weilepp and Brand, 1998) assume from the outset this limit of rigid identification of

EFFECTIVE SMECTIC ELASTICITY OF ELASTOMERS

359

the director with the layer normal and use D1 and D2 directly as the layer-matrix relative rotation and shear coupling moduli. In the opposite limiting case, when the network is rigid and smectic layers impose a weaker influence, D1  b⊥ (which should also be the case near the N-A phase transition when the order parameter |ψ | diminishes), one obtains a correspondingly weaker coupling, ∆1 = b⊥ ; ∆2 = b⊥ (D2 /D1 ). Molecular estimates of constants This section has thus far discussed the continuum elastic theory of elasticity in rubber networks consistent with their possessing smectic (lamellar) order. All expressions have been written phenomenologically, on symmetry grounds, because no microscopic theory as yet exists for smectic rubbers. However, to complete this description, we present the elastic and coupling constants that emerge from the underlying nematic elastomer – for which we do have a molecular-level description. This allows predictions of the magnitude of these constants in the smectic phase as well. Using the ideal Trace formula (that is, with softness and with the initial C5 = D22 /8D1 ), the constants follow from eqns (10.14) and from the layer-inspired renormalisation of C5 , eqn (13.13). We have for the uniaxial smectic rubber moduli in eqn (13.4): C2 = 0, C1 = 2C4 = µ , C5smA =

1 8

b⊥ µ (r + 1)2 , µ (r − 1)2 + b⊥ r

(13.18)

where µ = ns kB T is the shear modulus of the rubber in its isotropic phase. Again, as in eqn (13.13), one discovers that the effective shear modulus C5smA is only non-zero by virtue of coupling to smectic layers b⊥ . Near the nematic-smectic phase transition, when b⊥ = g⊥ qo |ψ |2 decreases, the renormalised shear modulus C5smA ≈ 1 2 2 8 b⊥ (r + 1) /(r − 1) . It vanishes on b⊥ → 0, that is, losing the layer structure as one leaves the smectic state and enters the soft nematic state above it in temperature. On the other hand, the factor (r − 1)2 in denominator could make this crossover very near the transition point or even altogether absent for materials with very low effective chain anisotropy, r ≈ 1. Deep in the smectic phase, we can assume that the magnitude of b⊥ is similar to the liquid-crystalline (not the much larger elastomer) contribution to the layer compression constant Bo = g qo |ψ |2 ∼ 106 J/m3 , which is usually greater than µ . In the limit b⊥  µ the shear modulus recovers its ‘bare’ value, as given by the original Trace formula for a 2 nematic elastomer: C5smA ≈ 18 µ (r+1) r . Similarly, the model expressions for the smectic-rubber rotation and shear-rotation coupling constants (13.17) become: ∆1 =

b⊥ µ (r − 1)2 b⊥ µ (1 − r2 ) and ∆ , = 2 µ (r − 1)2 + b⊥ r µ (r − 1)2 + b⊥ r

(13.19)

with the analogous set of limiting cases at |ψ | → 0 and |ψ | → 1. The final remark due here is about smectic or lamellar elastomeric networks that do not have any intervening nematic phase. This is not such a rare case even for ‘classical’

360

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

thermotropic smectic systems, many of which have a direct transition from isotropic to the smectic phase. Lyotropic lamellar systems based on microphase separated layers, whether in the presence of solvent or in a block-copolymer melt, have no means of forming a nematic phase at all. In such physical systems we would not need the procedure used in the beginning of Sect. 13.2, cf. eqns (13.9) and (13.10), and start directly from the smectic elastomer free energy density (13.12). Nor would we have the underlying molecular estimates for the constants, leading to eqns (13.18) and (13.19). Further reducing the effective elastic energy In the sections above we have set up the free energy of smectic elastomers and have left open the degree of coupling of the director to the layer normal. We then removed the explicit appearance of the director and emerged with an energy that depends on the elastic strain and layer displacement variables and their coupling. We have seen that layer rotation is now penalised and that layer bend is no longer one of the dominant effects as it was in the liquid smectic case. Several further new effects will emerge from this continuum description. To obtain these we shall take the reduction further by eliminating either of the remaining variables, strain or layer displacement, depending upon what we then want to focus on. • Eliminating layers, one is left with a homogeneous medium with effective elastic properties. We thus examine in depth (in Sect. 13.3) the effect on shear and extensional deformation elasticity of the relative translation coupling. Indeed, the decomposition of deformations into their average uniform parts and fluctuations with zero mean, eqn (13.3) and Fig. 13.2(b), has indicated an anomalously large – essentially rigid locking between uniform shears. Extreme mechanical anisotropy will correspondingly emerge when we eliminate the layers. • Eliminating strains and focussing on the smectic layers alone, we shall see that the effect of the elastomer network is to produce many changes to the effective smectic layer elasticity and ordering. The Landau-Peierls effect in liquids is lost to long range order in elastomers, see Sect. 13.4 where the properties of this effective smectic are derived. Random disorder couples to layers in ways not accessible to liquids, Sect. 13.6. The CMHH undulation layer instability from uniform stretching deformation along the layer normal in ordinary liquid crystals is too changed fundamentally in elastomers due to the coupling of layers to strains, Sect. 13.5. 13.3

Effective rubber elasticity of smectic elastomers

We determine the effective mechanical response of a smectic rubber when its layers are allowed to fluctuate freely and adopt the optimal conformation that lowers the total free energy (13.12). The effective equilibrium rubber-elastic energy obtains on eliminating the layer fluctuations, v(qq). Strictly, the result of such optimisation depends on the magnitude and geometry of imposed strains ui j , externally applied to the smectic elastomer sample. We shall have to examine several such cases below. Here, however, we are interested in effective rubber-elastic moduli only. These are determined by the effective equilibrium energy with no externally imposed deformations. Mechanical experiments unconcerned about layers per se (Nishikawa et al., 1997) explore just this elasticity, see

EFFECTIVE RUBBER ELASTICITY OF SMECTIC ELASTOMERS

361

the discussion in Sect. 12.2 and Fig. 1.8 where a monodomain smectic elastomer displayed a remarkable in-plane fluidity with rigidity perpendicular to the smectic layers. Smectic A elastomers were shown to form two-dimensional rubbers and to be solids in the third dimension! The physical reason for such response is clearly that the matrix is coupled to the layers that are fluid in-plane (and only the matrix elasticity is then significant) while the layers are solid along their normal and overwhelm the rubbery matrix in this direction. One derives this analogously to Sect. 10.6 for the nematic case. Write the free energy density in Fourier space and then minimise it with respect to the smectic layer fluctuation modes v(qq). At leading order of expansion at small wave vectors q, the resulting optimal modes are given in terms of uz (qq) as B 2 (∆1 − ∆2 ) q uz − q⊥ (q⊥ uz + qz u⊥ ) + ... 2Λ z 4Λ or in real space as   (∆1 − ∆2 ) ∂ εxz ∂ εyz B ∂ εzz + + v(rr ) ≈ uz + 2Λ ∂ z 2Λ ∂x ∂y

v(qq) = uz −

(13.20)

The expansion parameters, as one can see from the ratio of terms in eqn (13.20), are (B/Λ)q2z  1 and (µ /Λ)q2⊥  1, assuming all coupling constants have the magnitude of the rubber modulus µ . Recall that Λ ∼ B/R2o where Ro is the average chain span between crosslink points in the network. For the first condition it is then straightforward to check that the series expansion is valid when q⊥  1/Ro , which is easily satisfied in any practical situation. The relation is much stronger enforced since we expect B  µ . Return the optimal value for the layer fluctuation mode to eqn (13.12) and re-arrange terms. One obtains the effective elastic distortion energy density of a smectic elastomer, now without layers, in the form of Felastic of eqn (13.4) for a uniaxial elastic medium but with elastic constants. In a coordinate dependent form, with the initial layer normal along z, one obtains Felastic = C1eff εzz 2 +C3eff (Tr (ε˜ ))2 + 2C2eff Tr (ε˜ ) εzz +2C4eff (εxx 2 + 2εxy 2 + εyy 2 ) + 4C5eff (εxz 2 + εyz 2 ),

(13.21)

where, as usual, εi j represents the symmetric traceless part of the deformation gradient ui j . The linear elastic moduli are renormalised by smectic fluctuations and acquire the effective values: 1 B, C2eff = C2 + 16 B, C3eff = C3 + 18

C1eff

= C1 +

1 2 B,

C4eff

= C4 ,

C5eff

(13.22)

= C5smA + 18 (∆1 − 2∆2 ) ,

where, in turn, C5smA is given by the expression in eqn (13.18) and, of course, one can safely neglect the correction to the main bulk modulus C3 . See Table 13.1 for comparative estimates of values. To complete, and make a connection with the molecular theory of nematic elastomers, let us present the effective rubber moduli of a smectic elastomer through the

362

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

Table 13.1 The values and estimates of the smectic-A elastomer constants discussed in this section. The numerical values are obtained for |ψ | ∼ 1, using µ = ns kB T = 105 Pa, r = 3, and the average network span Ro = 10 nm. The factors of order 1 are retained so the relative sizes of coefficients are showing. C1

µ ∼ 105 Pa

b⊥

g⊥ q2o |ψ |2 ∼ 106 Pa

C2 C3

0

B K

g q2o |ψ |2 + gx ns ( d0 )2 ≥ 107 Pa ∼ 10−11 N

D1

µ (r−1) ∼ 1.33 × 105 Pa r

C4 C5 C5SmA C5eff C1eff

1010 Pa 1 5 2 µ ∼ 0.5 × 10 Pa 2 5 µ (1+r) 8r ∼ 0.67 × 10 Pa 2 D2 C5 − 8(b +D1) ∼ 0.61 × 105 Pa ⊥ C5SmA + 18 (∆1 − 2∆2 ) ∼ 0.68 × 105 Pa C1 + 12 B ∼ 0.7 × 106 Pa

D2

R

2

−µ r

2 −1

5 r ∼ −2.67 × 10 Pa b⊥ D1 5 b⊥ +D1 ∼ 1.2 × 10 Pa b⊥ D2 5 b⊥ +D1 ∼ −2.1 × 10 Pa ∼ µ /R2o ∼ 1021 Nm−4

∆1 ∆2 Λ

model microscopic expressions given by the ideal trace formula extended to smectics, eqns (12.1) and (12.6), and the renormalisations (13.17) and (13.22). We thus obtain (cf. Table 13.1): C2eff = 16 B, C1eff = 12 B + µ , C4eff = 12 µ , C5eff =

1 2

(13.23)

µ r2

b⊥ . µ (r − 1)2 + b⊥ r

As before, near the smectic phase transition the shear modulus C5eff ∼ b⊥ → 0. Deep in the smectic phase C5eff → 12 µ r, leaving no room for softness. Comparisons with the results of molecular model of Sect. 12.3 are subtle to make because eqn (13.21) is necessarily written in terms of traceless εi j . The strong smectic limit for C5eff is relatively simple since the trace is not involved in the zx shears. The limit of εxz = 12 λ → 0 of FA in Sect. 12.3.2 can be taken (remembering that λxx and λzz differ from 1 only by small quantities that can be neglected. The most important effect is the asymmetric renormalisation of the moduli C1 and C4 , as well as the effect of layer compression. Even taking a typical value for smectic liquids, B ∼ 106 J/m3 , we would find it much greater than a typical µ ∼ 104 − 106 J/m3 . However, the discussion in Sect. 12.3 indicates that in an elastomer network the additional contribution to this modulus could be at least an order of magnitude greater, Bx ≥ 107 Pa. On stretching the smectic elastomer in the layer plane (penalised by the modulus ∼ C4eff ) the incompressibility-driven contraction along the layer normal z is resisted by the high effective modulus C1eff  B, insisting that the layer spacing do remains constant. Because the ratio of the effective moduli is large, no noticeable transverse contraction along the layer normal would take place. Since the deformation is at constant volume, the other relaxation must also be anomalous for a rubber: εyy = −εxx . In the reverse situation, when the sample is stretched along z and the measured effective modulus is E ∼ C1eff , both perpendicular directions in the layer plane experience the usual incompressibility-driven contraction. Here, however, the material will be able

EFFECTIVE RUBBER ELASTICITY OF SMECTIC ELASTOMERS

363

to find a lower-energy deformation mode than the uniform strain εzz with its high effective modulus C1eff : the CMHH type of layer instability intervenes. It is discussed below in Sect. 13.5; see also the sketch in Fig. 12.7 and the experiment in Fig. 12.8 . Exercise 13.1: Find the anisotropy of Young moduli in a SmA elastomer. Solution: The effective linear elasticity of a smectic rubber is highly anisotropic and thus the effect of a uniform extension applied to a monodomain smectic rubber in the two principal orientations of layers will be very different, see sketch in Fig. 12.7 and illustrations in Figs. 12.6 and 12.8. Consider infinitesimally small strains so that no effects of director or layer rotations are relevant. Both geometries prevent macroscopic shear strains and the only relevant components of εi j are 

 εxx 0 0 ε =  0 εyy 0  . 0 0 εzz   Section 10.1.1 shows that deviation of Tr ε˜ from zero occurs at order µ /C3 , the ratio of a shear modulus  to  the bulk modulus, which is small and which we shall ignore. We then set Tr ε˜ = 0 and also do not need to distinguish between ε˜ and ε . Further one can then write εyy = −(εxx + εzz ). The effective elastic energy, e.g. (13.21), explicitly written in terms of these components of strain, is then:   2 2 (13.24) + 2C4 εxx + (εxx + εzz )2 Felastic = C1 εzz where C1 takes its renormalised, effective value, eqn (13.22). Case (a), in-layer stretching: the strain εxx ≡ ε is imposed. The other independent component, εzz , obtains by minimisation of the energy (13.24). We have

εzz ≈ −

2C4 C1 ε → 0 ; εyy ≈ − ε → −ε ; C1 + 2C4 C1 + 2C4

(13.25)

where the limits obtain when the renormalised C1  C4 and are as suggested by Fig. 12.7(a). C1 is the elastic modulus most heavily renormalised by its coupling to the layer compression mode. At these optimal values of volume-conserving strains, the rise in elastic energy in response to the imposed strain ε is given by F≈

1 2

8C4 (C1 +C4 ) 2 8C (C +C4 ) ε ⇒ E⊥ = 4 1 → 8C4 ∼ 4µ , C1 + 2C4 C1 + 2C4

i.e. the perpendicular Young modulus we have already seen emerging in molecular theory, eqn (12.13) Case (b), stretching along the layer normal z, Fig. 12.7(b): the strain εzz ≡ ε is imposed. Proceeding as above gives

εxx ≈ − 21 ε ; εyy ≈ − 21 ε ,

(13.26)

the symmetric transverse contraction, as one finds in any isotropic incompressible

364

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS material, such as a conventional rubber. The energy response to the imposed ε is now F ≈ 12 2(C1 +C4 ) ε 2 ⇒ E = 2(C1 +C4 ) → C1eff ∼ B, the parallel Young modulus corresponding to the large-B limit of the below-threshold molecular result (12.30).

We have assumed that the elastic moduli not involving volume compression are small compared with the bulk modulus. This is true even for C1eff ∼ B which itself has become large with respect to the rubbery moduli, that is those of the order of µ of molecular theory. The assumption, however, would break down for solvent-based lamellar gels deformed slowly enough that solvent can be lost – that is where the effective bulk modulus becomes an osmotic modulus and reduces to order µ . 13.4

Layer elasticity and fluctuations in smectic A elastomers

We have examined the effective rubber elasticity, with moduli renormalised by the freely fluctuating smectic layers whose only constraint was the coupling to the polymer network. Let us now examine the opposite question and eliminate strains to leave behind the layers. This smectic degree of freedom can be easily monitored. The interest, apart from the practical attraction in manipulating the birefringent optical properties or the anomalous anisotropic viscosity (Mazenko et al., 1982), is the fundamental problem of the thermodynamics of a one-dimensional crystalline lattice. This problem carries names of Landau and Peierls (Landau and Lifshitz, 1970) and is treated below. A translational symmetry breaking in one dimension only (i.e. a system of parallel equidistant layers with no intra-layer structure – which we call a smectic A here) has, by symmetry, a degenerate elastic energy density expressed by eqns (2.54) or (13.11) in real and Fourier space, respectively. Assuming reasonably long-wavelength deformations, the equipartition theorem of thermodynamics tells that each mode of layer fluctuation v(qq) is determined by the size of its mean square average |v(qq)|2  =

V kB T B q2z + K q4

(liquid smectics)

(13.27)

where V is the sample volume. Because layer fluctuations in the plane are penalised by a higher-power of displacement gradient, the fluctuation modes have a much higher amplitude at q → 0 than in a usual three-dimensional solid (where the analogous mean square average is proportional to kB T /G q2 , with G an elastic modulus). The purely in-plane modes have qz = 0 and thus behave instead like 1/q4 at small q. In thermodynamics, the quality of order in a lattice (or indeed any system with broken symmetry) is determined by the correlation of these fluctuations in real space. When at each point in space lattice particles fluctuate with respect to their assigned equilibrium positions independently from each other, the overall lattice is well defined as a set of such equilibrium positions. In this case the correlation function v(0)v(rr ) ≈ v(0)v(rr ) → 0 at long distances. If, on the other hand, the sites far away from each other fluctuate in a correlated manner, it is hard to identify the equilibrium lattice and the order is not considered long-range. This is represented by a correlation function actually increasing with the distance, v(0)v(rr ) → ∞ as r → ∞.

LAYER ELASTICITY AND FLUCTUATIONS IN SMECTIC A ELASTOMERS

365

Accordingly, in an ordinary solid, the real space correlation function is obtained as a Fourier transformation11   1 kB T kB T −i(qq·rr ) d 3 q →0 (13.28) v(0)v(rr ) = e = G q2 (2π )3 G 4π r In contrast, in a smectic one-dimensional lattice the fluctuations correlate over large distances. A direct integration gives v(0)v(rr ) =



 3 kB T r kB T −i(qq·rr ) d q e ≈ ln (2π )3 do B q2z + Kq4⊥ 4π (B K)1/2

(13.29)

This weak (logarithmic) divergence is known as the Landau-Peierls effect and was first applied to the smectic-A system in (de Gennes, 1969b). This corresponds to the slow, power-law decay of the structure factor, S (r), and signals ‘quasi-long range order’ – the marginal case between true crystalline or bond-orientational order (with the structure factor remaining a constant) and the short-range order of liquids with rapidly decaying of density fluctuS (r) ∼ e−r/ξ . (The structure factor is defined here as the correlation

ations, S (r) = exp [iqo [v(r) − v(0)]], which is equal to exp − 12 q2o [v(r) − v(0)]2  in the harmonic (Gaussian) approximation.) Such an effect is directly seen in X-ray scattering experiments, where the diffraction from smectic layer density modulation generates a peak in reciprocal space at a wave vector qo = 2π /do . The scattering intensity I(q) is proportional to the Fourier transform of structure factor S (r) and thus reflects the nature of correlations in the system. In a crystalline lattice with long range order, the Bragg reflections are nominally delta-functions, I(q) ∼ δ (q − qo ) at each reciprocal lattice vector (Cullity, 1978), which are broadened by the diffusive scattering and modulated by the Debye-Waller factor ∼ exp −π 2 kB T /do3 G , a measure of thermal fluctuations in eqn (13.28). Quasi-long range order in a one-dimensional smectic lattice results in the famous polynomial decay of order (Caille, 1972) as seen in reciprocal space: I(qz , q⊥ = 0) ∼

1 (qz − qo )2−η

and I(qz = 0, q⊥ ) ∼

1 η q4−2 ⊥

.

(13.30)

The power-law decay of intensity around the peak position is also anisotropic, with √ 2 the factor η contributing to the two separate exponents being η (T ) = 8qπo kB T / B K ∼ 1/|ψ | near the nematic-smectic transition. Precision X-ray scattering experiments (AlsNielsen et al., 1980; Safinya et al., 1986; Bouwman and de Jeu, 1992) have confirmed this famous prediction and the consequent critical behaviour. The profound effect of long wavelength, transverse fluctuations of smectic layers has been studied over the years and led to several important theoretical discoveries. 11 A

useful Fourier transformation formula reads:

 e−ar i(qq·rr ) 3 e d r= r

4π , q2 +a2

 −i(qq·rr) 3 reverse: eq2 +a2 (2dπ q)3 = 4π1 r e−ar recovers the ‘screened Coulomb’ form.

resulting in the Lorentzian. In

366

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

It was found (Grinsten and Pelcovits, 1981) that renormalisations of the smectic constants B and K are, in fact, more serious than the leading harmonic approximation suggests. The renormalisation-group treatment taking into account two relevant higherorder terms (∂z v)(∇⊥ v)2 and (∇⊥ v)4 concludes that the renormalised compression constant BR vanishes and the curvature constant K R diverges logarithmically for deformations with wavelength longer than a certain characteristic length scale l ∗ . In practice, such an effect may be difficult to notice because l ∗ turns out to be quite large [≥ 100 µ m (Grinsten and Pelcovits, 1981)]. However, in principle, it is very important, in particular for the case of uniform strains. Another classical result expressing the role of anomalous thermal fluctuations in a smectic system is the divergence of certain viscous coefficients (Mazenko et al., 1982). In particular, the shear viscosity controlling the undulation mode behaves as 1/ω at low frequency, which is just another representation of long wavelength anomalies in a one dimensional lattice. Description of all such effects could be a subject of a separate book and well beyond the scope of this one. Eliminating the rubber-elastic deformation variables In nematic elastomers the coupling to the underlying rubbery network depresses thermal director fluctuations, although particular fluctuation modes corresponding to soft geometries are not penalised (or penalised weakly in a semi-soft material). However, in smectic or lamellar networks the soft deformation conditions are never geometrically fully satisfied and, therefore, one should expect that thermal fluctuations in a onedimensional (smectic) lattice would play a minor role, see for instance Fig. 13.3 where the potentially soft condition for the effective C5 is thwarted by layer coupling. This depression of fluctuations is somewhat at odds with a general thermodynamic expectation about the smectic class of order, where the in-layer elastic degeneracy (∼ q4⊥ ) is seemingly dictated by symmetry. As always in liquid crystalline elastomers, the answer to this lies in having two independent deformation fields, or degrees of freedom, acting in the system, here v and uz . We again need to find an effective Hamiltonian of the smectic system, this time described by the layer phase variable v(rr ) or its Fourier modes v(qq). To do this, we need to integrate out the unconstrained fluctuations of the elastic deformation field u (qq) – the collective phonon modes in the rubbery network. Technical details, including the relation to the Landau-Peierls problem are available in the literature (Terentjev et al., 1995; Osborne and Terentjev, 2000). The Fourier transform of the full free energy density eqn (13.12), which depends on both u (qq) and v(qq), can be arranged as a quadratic form: (13.31) F = 12 u · G (qq) · u ∗ − Γ (qq) · (uu v∗ + u ∗ v) + 12 M (qq)|v|2 , where, from the result of Sect. 13.1.2, the elements of the elasticity matrix, G , are presented for the components (uz , u⊥ , ut ) in a coordinate frame (z, ⊥,t) based on the layer normal z and the chosen direction of the wave vector projection q⊥ (cf. Fig. 10.6): G⊥⊥ ≈ 2C3 q2⊥ + 14 (8C5smA + ∆1 + 2∆2 )q2z ; Gtt = 2C4 q2⊥ + 14 (8C5smA + ∆1 − 2∆2 )q2z ; G⊥t = Gt ⊥ = 0;

(13.32)

LAYER ELASTICITY AND FLUCTUATIONS IN SMECTIC A ELASTOMERS

367

Gzz ≈ 2Λ + 2C3 q2z + 14 (8C5smA + ∆1 + 2∆2 )q2z ; G⊥z = Gz⊥ ≈ 2C3 qz q⊥ ; Gzt = Gtz = 0, where constants of order µ are neglected when they appear together with the bulk modulus C3 , and the shear modulus C5smA is already renormalised by nematic director fluctuation modes. In Gzz the layer-matrix relative displacement coupling Λ and the bulk modulus C3 appear together, the latter involving a gradient of displacement (a strain) squared. two terms become of equal influence, Λ = C3 q2z , at a new length  Hence the scale C3 /Λ = Ro C3 /µ . Given that the bulk modulus is much greater than the shear modulus, this length can be significantly longer than the extent of the network polymers. In the same frame, the elements of the vector Γ = (Γz , Γ⊥ , Γt ), that is the coefficients Γ · u )v, take the form of the cross term (Γ Γ⊥ = − 14 (∆1 − ∆2 )qz q⊥ ; Γz = Λ + 14 (∆1 + ∆2 )q2⊥ ; Γt = 0.

(13.33)

Finally, the coefficient penalising the ‘bare’ layer fluctuations v(qq) is given by M = 2Λ + Bq2z + ∆1 q2⊥ +

b2⊥ Kq4 , (b⊥ + D1 )2 ⊥

(13.34)

although we expect it to be changed significantly when the rubber-elastic phonon modes are integrated out, or in other words, the variables u are minimised over). The further calculation is straightforward and was already performed on other occasions: minimising the quadratic form (13.31) we obtain the optimal modes of network G−1 · Γ ) v(qq) and the effective free energy density bedeformations, e.g. here u = 2(G comes   2 − 2G Γ Γ + G 2 G Γ Γ zz z z⊥ ⊥ ⊥⊥ z ⊥ |v(qq)|2 . (13.35) F eff = 12 M (qq) − 4 Gzz G⊥⊥ − G2z⊥ The full expression, with matrix elements explicitly put in, is tedious but there are possible simplifying approximations due to the large bulk modulus C3 (as always, negsmA lecting terms of order  C5 /C3 ) and the constraint of small wave vectors (explicitly estimated as q  Λ/µ  1/Ro , the network span). We shall also find out that the traditional smectic term ∝ Kq4⊥ will no longer be relevant for the layer fluctuation spectrum. After all these considerations are implemented, we have the effective free energy density  eff 4  ∗ 4 2 2 eff 2 1 8C5 qz+ 8C5 q⊥− 4(∆1 − ∆2 )qzq⊥ + Bqz |v(qq)|2 (13.36) F ≈2 4q2⊥ + 4C5eff /C3 q2z + 4C5eff /Λ q4z where the twice renormalised effective constants C5 are [compare with eqn (13.22)] C5eff = C5smA + 18 (∆1 − 2∆2 ) and C5∗ = C5smA − 18 (3∆1 + 2∆2 ) .

(13.37)

Note that the term ∼ (4C5eff /C3 )q2z in the denominator of the kernel in eqn (13.36) cannot be neglected in spite of the nominal incompressibility. One may encounter a situation

368

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

when q⊥ = 0 and this term will become the leading effect. The expression eqn (13.36) may still appear cumbersome, but it is easy to study its consequences in the two limiting cases, qz → 0 and q⊥ → 0 separately. These two limiting cases, in which the results can be presented in a simple intuitive way, are also the focus of the classical theories and scattering experiments on liquid A-smectics. The mean square average of layer fluctuations, remaining after the elastomer penalty has been implemented, follows directly from eqn (13.36). The result depends on the orientation of fluctuation modes: In the most common case q2⊥ /q2z  C5eff /C3 , the bulk modulus cancels: |v(qq)|2  ≈ =

V kB T ∗ ∗ 2 B qz + 2C5 q2⊥ + 2C5eff [q4z /q2⊥ ]

(13.38)

sin2 θ V kB T 2 2 q 2C5eff + (B∗ − 4C5eff ) sin θ − (B∗ − 2C5eff − 2C5∗ ) sin4 θ

where the new, only slightly modified layer compression modulus is B∗ = B − ∆1 + ∆2 .

and θ  C5eff /C3 is the angle between q and the layer normal z. One can verify that the kernel of eqn (13.38) is positive for all directions of q , even though some combinations of B, C5smA , ∆1 and ∆2 may be negative. In the case when the wave vector is almost parallel to the layer normal, q2⊥ /q2z  eff C5 /C3  1, the bulk modulus dominates the fluctuation spectrum: 1 + (C3 /Λ)q2z V kB T → at |qq| ≈ qz → 0 C3 q2z C3 q2z  V kB T = const at still very small qz  Λ/C3 → 2Λ

|v(qq)|2 z ≈ V kB T

(13.39) (13.40)

The constant (non-diverging) value of the average mean square layer displacement is an indication of true long-range order! The regime (13.39) occurs inside the narrow

cone of wave vector directions, given by the condition θ  C5eff /C3 . For a typical elastomer this amounts to θ  0.6o , or even less when the smectic order is weak and C5smA ∼ C5eff ∼ ψ 2 . This essentially means a pure compression modes along the layer normal. The crossover between the bulk compressibility response to a purely longitudinal distortion and the ‘mass-like’ response of eqn (13.40) lies at the wave vector q∗z being  the inverse of the new characteristic length scale, q∗z = Λ/C3 ∼ [10Ro ]−1 . The latter response is more unusual in this context. In the case of weak smectic order q∗z ∼ ψ and, therefore, both types of behaviour can be readily accessed by a detailed experiment on layer compression-extension. Physical reasons for having the effective smectic-A elastic energy eqn (13.36) proportional to an overall square power ∼ q2 are quite transparent and follow directly from the main feature of smectic order in a rubbery network – the relative translations coupling (13.1). In a liquid smectic, only layer curvature ∇2⊥ v is elastically penalized. When

LAYER ELASTICITY AND FLUCTUATIONS IN SMECTIC A ELASTOMERS

369

there is an underlying elastic network, coupled to the layer displacement, the smectic degeneracy with respect to uniform layer rotations ∇⊥ u is lost and we obtain the solidlike elastic energy, eqn (13.36). Naturally, the renormalisation is determined solely by rubber-elastic parameters: the shear modulus C5 and coupling constants ∆. The special case is the longitudinal layer deformation of an incompressible smectic elastomer, when the system becomes very rigid indeed, controlled by bulk modulus effects. Equations (13.38) and (13.39) also tell us that there is no longer a Landau-Peierls instability of smectic fluctuations. The logarithmic divergence of the correlation function v(0)v(rr ) is suppressed by the network elasticity. Therefore the X-ray scattering on smectic layers will no longer be purely diffusive, but will take a usual form of the Bragg peak with a thermal Debye-Waller factor. The scattering intensity in reciprocal space will take the form I(qq) =



d 3 r eiqq·rr ρ (0)ρ (rr ) =



d 3 r ei(qq−qqo )·rr S (rr )

(13.41)

for the first-order reflection peak in an ideally infinite sample (so that no additional finite-size peak broadening occurs). In an ordinary smectic, one obtains from the logarithmic divergence (13.29), (Caille, 1972) πk T

− 2 √B do BK

S (z)  |z|

and the predicted scattering intensity (13.30) is observed experimentally. In contrast, in an ideal, defect-free smectic elastomer the diffusive scattering is far less important. (In Sect. 13.6 we shall return to the effect of X-ray peak broadening by diffusion scattering on quenched defects and distortions in real smectic elastomers.) The primary effect in the structure factor is a Debye-Waller constant given by direct integration of eqns (13.38) and (13.39) (Terentjev et al., 1995). Far from the smecticnematic transition, when the layer compression constant is large, B  C5eff , one obtains 



2(1 − cos[qq · r ]) d 3 q 2q⊥ dq⊥ dqz |v(qq)|2   |v(qq)|2  3 V (2π ) V (2π )2 2kB T kB T v2  ≈

(13.42)   ∗ ∗. do 8C5∗ (B∗ − 2C5eff − 2C5∗ ) do 2C5 B

[v(rr ) − v(0)]2  =

Near the nematic-smectic transition in an ideally soft elastomer, when the constants B, C5smA and the ∆s are all proportional to |ψ |2 and thus vanish with the smectic order parameter, the mean square fluctuation becomes √ 2 2kB T 2 . (13.43) v   do (C5∗ )3/4 (C5eff )1/4 Note that these simplified mean-square fluctuations grow as the smectic order diminishes near the nematic-smectic (or isotropic-smectic) phase transition, |ψ | → 0.

370

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

The opposite limiting case, C5eff /B  1, could be achieved near the smectic transition point in a highly non-soft elastomer, where the effective shear modulus C5eff is not proportional to |ψ |2 , but the layer compression constant is vanishing. Then, in a network with a relatively high degree of crosslinking, near the transition point where B → 0, the mean square fluctuation becomes v2  

kB T . 4π doC5

(13.44)

Accordingly, the X-ray scattering intensity (13.41) becomes modulated by the Debye2 Waller factor ∼ exp[− 12 4dπ2 v2 ], where the mean square fluctuation of layers is given o by eqns (13.42)- (13.44). For each consecutive n-order peak one obtains  1 4π 2 n2 2 − |v | 2π n e 2 do2 . (13.45) I(qq)  ∑ δ qz − d o n Qualitatively, neglecting all effects of potential softness, uniaxial anisotropy and layering, all rubber moduli are of the order of µ . We have then for the Debye-Waller  2 factor e−n D in (13.45): D  (4π 2 /do2 ) a/ns ∼ (aRo /do2 ) (where a is a characteristic size of a mesogenic monomer). We see that at µ ∝ ns → 0, that is when the crosslinking density of the matrix vanishes, then the Debye-Waller exponent diverges and thus suppresses the Bragg peak in the X-ray scattering intensity. The usual smectic-A diffusive scattering (Caille, 1972; Als-Nielsen et al., 1980) would then prevail as the effect of the rubbery matrix is switched off. Precision X-ray scattering is not a common experimental technique because it requires a very narrow dispersion of the incident beam I(q) in order to resolve the lineshape of a smectic diffraction peak. This is usually achieved by multiple-Bragg reflection in the monochromator and analyser crystals – a technique used by (Als-Nielsen et al., 1980) in their pioneer study of smectic structure and in many following publications. Long-range order in a monodomain side-chain polymer smectic-A system, capable of forming an elastomer by photocrosslinking, has been first studied by the group of de Jeu (Wong et al., 1997). Free-standing films of SmA polymer were aligned by a magnetic field, after which the network crosslinking has been performed. The measurements have achieved a sufficiently narrow incident beam dispersion to resolve the difference between the Bragg and the Caille diffraction. Figure 13.4 shows an example lineshape for a smectic elastomer I(qz ) ∼ 1/q2.4 (shown on plot) in a broad range of wave vectors down to the resolution limit of several inverse micrometers. In contrast, the same polymer material, not crosslinked into the network, under the same conditions shows the intensity profile I(qz ) ∼ 1/q1.85 , as expected, somewhat less than the theoretical limit of 1/q2 for the Landau-Peierls quasi-long range order, see eqn (13.30). In addition many higher-order reflections of the layered structure were observed, which is another characteristic of Bragg scattering as opposed to the suppressed high-order peaks in ‘liquid’ fluctuating smectics. Equation (13.45) describes the reduction in each consecutive Bragg peak intensity arising from thermal fluctuations controlled by the effective Hamiltonian (13.36). In

LAYER BUCKLING INSTABILITIES: THE CMHH EFFECT

371

(a)

uncrosslinked crosslinked

Intensity (Arb.Units)

Intensity (Arb.Units)

(b)

-1.85 -2.4

F IG . 13.4. Precision X-ray scattering results of (Wong et al., 1997) on monodomain smectic polymer melt, and its weakly crosslinked elastomer; (a) linear and (b) log-log scale [reproduced with permission from Nature]. The comparison of intensity scans I(qz ) across the first-order diffraction peak illustrates the high degree of smectic (lamellar) order in the elastomer: the slope of decay is much steeper than the normal Caille lineshape (expected to be somewhat less than the theoretical limit of −2 owing to the finite mosaic width of the sample). Multiple-order reflections in small angles also confirm Bragg-like scattering of smectic elastomer. real smectic elastomers, there are quenched random undulations of the smectic layers that result from the random nature of the crosslinking of the underlying network. This effect is most pronounced when the network crosslinking is not performed in the wellaligned smectic state, but in any homogeneous phase: nematic or isotropic. In that case, on establishing the smectic order, crosslinks will find themselves out of register with the layers and will exert a local force. These random forces will lead to randomly quenched layer undulations and further reductions in Bragg peak intensity, and its diffuse broadening. Section 13.6 describes these effects of quenched disorder in SmA elastomers.

13.5

Layer buckling instabilities: the CMHH effect

In the previous section, we looked at the role played by the rubber-elastic network and its spontaneous thermal excitations u (rr ), with respect to its equilibrium undistorted state. Now suppose that a small uniform extensional strain ε is applied to the SmA elastomer along the (uniformly aligned) layer normal. Taking the z-axis as the layer normal, the expressions for displacement fields take the form V = v(rr ) + z ε Uz = uz (rr ) + z ε ;

(13.46) Ux = ux (rr ) − 12 x ε ;

Uy = uy (rr ) − 12 y ε

372

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

Here, in the components of the full elastic displacement vector U (rr ), the material incompressibility is taken into account; the components along x and y axes are equivalent by symmetry, cf. eqn (13.26). The corresponding affine expression for the full layer displacement vector V (rr ) is the same as in the classical description of the CMHH effect in liquid A-smectics (Clark and Meyer, 1973), see Sect. 2.9, page 43. The original and subsequent authors (de Gennes and Prost, 1994; Singer, 1993) give details of this generic layer-buckling instability and how the smectic layer-compression term should be additionally modified in the presence of non-zero layer tilting ∇⊥ v (see the geometrical basis for how tilt is involved before exercise 2.6): 1 2B

∂v + ε − 12 (∇⊥ v)2 ∂z

2 .

(13.47)

The subsequent analysis in our case is completely analogous to that leading to the full free energy density of smectic A elastomer, eqn (13.12), and the integration out of rubber-elastic modes, eqns (13.31)-(13.35). The difference is in the presence of an additional constant strain in the corresponding diagonal components, εzz  = ε , εxx  = − 12 ε , εyy  = − 12 ε as well as the effect of eqn (13.47). As a result, the effective free energy density of smectic layer system takes the form, instead of eqn (13.36),  eff 4 ∗ 4 2 2 eff 1 8C5 qz+ 8C5 q⊥− 4(∆1 − ∆2 )qzq⊥ (13.48) F ≈ 2 4q2⊥ + 4C5eff /C3 q2z + 4C5eff /Λ q4z    b2⊥ 4 |v(qq)|2 Kq +B q2z − ε q2⊥ + (b⊥ + D1 )2 ⊥ The cross term involving the imposed strain, ε (∇⊥ v)2 , is nominally third order in deformation. However, one retains it as the main source of instability since ε is an externally imposed parameter, unlike the (small) spontaneous layer fluctuation v(rr ) (exercise 2.6). The non-linear analysis of CMHH buckling at high amplitudes (Clark and Meyer, 1973; Singer, 1993) would require high-order contributions, such as the fourthorder term arising from (13.47), 12 B(∇⊥ v)4 , as the amplitude stabilisation. Here we shall only discuss the threshold region of instability, for illustration. The classical threshold analysis assumes that the fluctuations v(rr ) away from the affine imposed strain ε are described by a single mode v = vo sin qz z cos qx x , where the wave vectors are constrained by boundary conditions to be multiples of π /Lz and π /Lx , respectively (Lz,x are the sample dimensions along and perpendicular to the layer normal direction), see Fig. 2.9(b). In practice, one finds that the in-layer modulation (layer buckling) starts at a finite wave vector qx , while in the linear regime qz = π /Lz , the longest wavelength available. Then the limiting case of qz /qx  1, cf.

LAYER BUCKLING INSTABILITIES: THE CMHH EFFECT

373

eqn (13.38), can be implemented. The average free energy density, obtained by substituting v(x, z) into (13.48) and integrating over the sample volume, takes the form:   2  2 2 b2⊥ ∂v 2 ∗ ∗ 1 + 2C5 (∇⊥ v) + K ∇⊥ v (13.49) F ≈ 2 (B − B ε ) ∂z (b⊥ + D1 )2   # x 4 v2o (13.50) = 18 B∗ q2z − B ε q2x + 2C5∗ q2x + Kq # is used for the modified layer-curvature constant). (in the second line, the abbreviation K The periodic undulations v(x, z) will develop when the above average free energy density is less than zero, in other words – when the coefficient at the squared amplitude v2o becomes negative: # x 4 − B (ε − 2C5∗ /B) q2x + B∗ (π /Lz )2 ≤ 0. Kq

(13.51)

Clearly, the increasing imposed strain ε is the driving force for this instability. The threshold condition occurs when this bi-quadratic equation for the wavevector qx first has a real solution, that is, when its determinant equal to zero, giving  # ∗ 2C5∗ 2π KB + ; (13.52) εth = Lz B B 1 π B∗ at threshold : q2x = , (13.53) # Lz K (recall the definition (13.37) of C5∗ ). The first part in the right-hand side of the threshold strain εth is the classical ClarkMeyer-Helfrich-Hurault (CMHH) result (Clark and Meyer, 1973; Singer, 1993), expressing the role of boundary conditions for the layers stack. In a liquid smectic A, the only reason for non-zero threshold ε (and the non-zero wave vector qx at the start of layer buckling modulation) is the demand that the modulation amplitude is zero at both boundaries and modulated as a single half-wave, sin π z/Lz . In smectic elastomers, due to the coupling to the underlying rubber-elastic network, there is additional resistance to spontaneous layer modulation, expressed by the second term in εth , (13.52); this was first recognised by Weilepp and Brand (1998). Even in an infinitely long sample, the imposed strain has to overcome a barrier 2C5∗ /B to allow any non-affine layer modulation (which in this case, at Lz → ∞, starts with the zero wave vector qx ). Let us try estimating the threshold and critical modulation wavelength values, using the values of constants already discussed in the previous sections. Taking the values in Table 13.1, and expressions for C5∗ and B∗ from (13.36), we could approximate that # ≈ K which gives, very crudely, B∗ ≈ B and K 0.2π do µ 2r − 1 ∼ 10−6 + 0.02 ; + Lz B r  π /qx ≈ 0.1π Lz do ∼ 2 µ m,

εth ≈

(13.54) (13.55)

takingLz ∼ 1cm and assuming that in elastomers with an additionally enhanced B the ˚ about an order of magnitude smaller than the smectic layer ratio K/B ≈ 0.1do ∼ 4A,

374

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

spacing. The values for the two parts of εth show that Frank effects are indeed minor in determining the transition in elastomers. Clearly the estimate of the threshold (totally dominated by the elastomeric contribution to the layer compression constant Bx ) is very close to the observations, Fig. 12.8. The fact that light scattering is strong, even near the threshold, indicates that the modulation period is in the range of microns, also close to the estimate (13.55). These estimates would, however, change in the vicinity of smectic phase transition, where both b⊥ and B → 0. More importantly, as the external strain increases above εth , the non-linear regime of layer buckling sets in. The sinusoidal modulation at the threshold rapidly becomes coarsened, in a zig-zag fashion, and the modulation period saturates at a constant value independent on the sample size. In the case of smectic elastomers, the analysis is almost identical to the case of coarsened stripe domains in nematic elastomers – Appendix C. 13.6

Quenched layer disorder and the N-A phase transition

In Sect. 13.4, we have seen that the suppression of thermal fluctuations of smectic layers due to the coupling to the rubber-elastic matrix leads to effectively long-range order in a one-dimensional stack of layers. A confirmation of this was seen in a narrow Bragg peaks of X-ray diffraction (Wong et al., 1997). The aligned monodomain smectic-A elastomer used in that study has been prepared by crosslinking of polymer chains in the smectic phase, subjected to an external magnetic field for alignment. In this case, one expects that crosslinks are formed in register with already present layers and, indeed, provide an enhancement of thermodynamic order. In contrast, a similar X-ray scattering experiment (Nishikawa and Finkelmann, 1998) on smectic rubber, which was aligned and crosslinked in the nematic phase and later cooled down to the smectic-A state, has reported the opposite trend. The scattering from the layer stack was very diffuse, with very broad peaks and clearly frustrated order. In this section we shall examine the role of randomly quenched disorder in the layer system, imposed by network crosslinks that are established not in full register with the smectic density modulation, see Fig. 13.5. In the conventional liquid N-A transition, the degrees of freedom are the smectic order parameter ψ (r) and the director fluctuations δ n . As we have seen in Sect. 13.1.2, ψ (r) describes the departure of the local density from its uniform average value in a form of single wavelength modulation |ψ |eiqqo ·rr . While the N-A transition should be continuous according to group-representation arguments, director fluctuations may change this picture: the coupling to the fluctuating gauge field δ n should induce a first-order phase transition in type-I smectics (and the analogous superconducting system). This important result is due to Halperin, Lubensky and Ma (HLM) (Halperin et al., 1974). In corresponding ‘type-I’ smectics the characteristic length for penetration of the director twist into the ordered smectic state is much smaller that the correlation length for the smectic order parameter, so that director fluctuations may be treated at a mean field level. Type-II smectics (and superconductors), in which the gauge field fluctuations are correlated on length scales comparable or larger than that of the order parameter ψ (rr ), require a more sophisticated analysis. The beliefs about the nature of this transition have varied over the years. A renormalization group
-expansion (Halperin et al., 1974) yielded no fixed point for physical order parameters, suggesting that the

QUENCHED LAYER DISORDER AND THE N-A PHASE TRANSITION

375

F IG . 13.5. A sketch of a stack of smectic layers having to avoid the randomly positioned quenched crosslinks and, as a result, generating a frustrated texture. transition is weakly first order. On the basis of a dislocation-melting theory (Toner, 1982) the transition should be continuous. Critical exponents were calculated within self-consistent perturbation theory (Andereck and Patton, 1994); a similar calculation (Radzihovsky, 1995) for the type-II superconducting transition, demonstrated the existence of a fixed point not found in
-expansion of (Halperin et al., 1974). Experiments on the N-A transition have generally yielded a continuous transition of the non-inverted three-dimensional XY class (Bouwman and de Jeu, 1992; Garland et al., 1993), but there are differing views on this (Anisimov et al., 1990). Perhaps the source of most reliable information on critical behaviour during the smectic phase transition still remains the review article (Lubensky, 1983). The N-A transition in an elastomer network is expected to be qualitatively very different from the same transition in melts because of the presence of elastic strain degrees of freedom and the local effects of random crosslinks (Olmsted and Terentjev, 1996). As the sketches in Figs. 12.11 and 13.1 suggest – there is a preferential positioning of crosslinks with respect to the smectic layers. In that case, if the crosslinking is performed not in the perfectly aligned smectic state,12 then on cooling into the smectic the layers will end up frustrated because their preferred straight passage will frequently be incompatible with a crosslink position, Fig. 13.5. This quenched pinning of the smectic phase to network inhomogeneities, which we represent by a random distribution of crosslinks, can be represented by the sum of such coupling for each crosslink (labelled by the index i) in the smectic potential and gives the free energy, FRF , the form FRF = ∑ grf |ψ (rr i )| cos [qo (zi − v(rr i ))] ,

(13.56)

i

where grf is the strength of coupling and r i is the position of the ith crosslink, in which we explicitly separated the order-parameter dependence, cf. Sect.12.3. Let us now introduce the continuum density of crosslinks, as in Sect. 8.4: by definition ρ (rr ) = ∑i δ (rr − r i ), so that under an elastic deformation r i → r i + u they distort by ρ (rr ) → ρ (rr − u ). One then transforms (13.56) to collective variables. After changing variables r  = r − u (rr ) we have 12 If the crosslinks are set in the aligned smectic state, then one expects a modulated density of them to result, with other interesting physical implications for rubber elasticity.

376

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

FRF =



grf ρ (rr )|ψ (rr )| cos [qo (z − v(rr ) + uz (rr ))] d 3 r.

(13.57)

Changing variables in the argument of v(rr ) introduces higher-order gradient corrections which are irrelevant in a mean-field treatment. The probability distribution function of crosslinking points is assumed Gaussian, cf. eqn (8.28). For this random distribution of crosslinks the characteristic moments are ρ (rr ) = ρo and ρ (rr )ρ (rr  ) = ρo δ (rr − r  ). To find the effects on the N-A transition in an elastomer network and, eventually, on the nature of the smectic state, one can use the replica trick, averaging over the quenched random disorder associated with the crosslinks (Olmsted and Terentjev, 1996). To compute the values of observable physical quantities one needs to write the full free energy of the system as 1 − Z n P , n→0 n

F  = −kB T ln ZP = −kB T lim

(13.58)

introducing n identical replicas of the system, initially described by the partition function   7 1 6 Z = D ψ DuuD δ n exp − Felastic + FsmA + Fcoupling + FRF , (13.59) kB T cf. eqn (13.12) for the rest of the terms entering the exponent. The analogous analysis for the case of orientational quenched disorder in nematic elastomers, Sect. 8.4 and references therein, provides a description of equilibrium glass-like textures and corresponding properties, which are also seen in experiment. In frustrated smectic systems one also finds characteristic glass structures (Bellini et al., 1992; Clarke et al., 1997; Bellini et al., 2001). Different theoretical approaches have been applied to the basic model of quenched sources of random disorder, in particular the methods of renormalisation group (Radzihovsky and Toner, 1999; Saunders et al., 2000). This, as well as the method of replica symmetry breaking, see (Giamarchi and Doussal, 1995) and references therein, describe the correlations of smectic order (or nematic, in the orientational case) and make detailed predictions for the nature of ‘Bragg glass’ textures. The qualitative mean-field results for the N-A phase transition in the network with quenched sources of random disorder can be summarised in the following way: 1. Near the putatively continuous N-A transition, the relative-translation coupling, eqn (13.1) may be ignored, since it represents a primarily dynamic effect at |ψ | → 0. At low enough temperatures (when the characteristic timescale of the crosslink ‘climbing’ over the barrier presented by the layer becomes essentially infinite) this term must be considered. 2. At the mean-field level, nematic director fluctuations no longer induce the HLM first-order smectic phase transition: fluctuations acquire a mass which reduces their effect on the N-A transition. The existence of this mass (discussed in different contexts in sections 10.7 and 13.4) in fact makes the mean field treatment and the associated type-I assumption essentially exact.

QUENCHED LAYER DISORDER AND THE N-A PHASE TRANSITION

377

3. However, in the special case where the nematic elastomer is ideally soft, elastic strain fluctuations [the phonons u (rr )] restore the HLM effect and the concomitant first order discontinuous phase transformation. The ψ -dependent part of the effective free energy density in this case is   2 3 4 1 1 1 (soft nematic), (13.60) kB T Fψ ≈ 2 ao |ψ | − βo |ψ | + 4 co |ψ | where the additional cubic coefficient is precisely the same as in the HLM case,

β0 = −

1 12π



q2o g⊥ K

3/2 .

4. If the nematic state is field-induced rather than spontaneous (i.e. the network has been formed to record the broken orientational symmetry and is, therefore, substantially semi-soft), the corresponding network phonon modes are conventional and the type-I transition N to A should be continuous. The free energy density has a non-analytic correction   2 4 1 1 1 (semi − soft), (13.61) kB T Fψ ≈ 2 as |ψ | + βs (ψ ) + 4 cs |ψ | where the additional term has a more complicated dependence on |ψ |:  3/2 4C5R D21 1 2 2 g⊥ qo |ψ | + R βs (ψ ) = − 4C5 D1 + (D1 + D2 )2 12π K 3/2 and C5R = C5 − D21 /8D2 is the familiar measure of nematic elastomer softness. In the limiting case of uncrosslinked material the second term in brackets is zero and we recover the HLM form of the free energy density (13.60). The basic effect of quenched disorder on the N-A transition in an elastomer network is a simple renormalization of the temperature and, depending on the nature of the nematic elastomer, a change in the order of the transition. However, on further cooling below the transition temperature TNA , the strength of layer coupling to the network crosslinks increases and the ‘classical’ smectic state changes to a glassy structure characterised by replica symmetry breaking. This crossover occurs at the same point where the mean-field replica-symmetric treatment becomes unstable (Olmsted and Terentjev, 1996). The estimate of crossover temperature, or more relevantly, of the magnitude of smectic order parameter ψ ∗ is ∗

|ψ | 



µ2 2 grf ns q4o L



1 2+η 

,

(13.62)

where µ ∼ n s kB T is a combination of rubber elastic constants, L the system size and η  = q2o kB T D1 /K/(8π µ ) is the exponent analogous to the Caille expression (Caille, 1972). Since ψ ∗ is inversely proportional to the system dimension L, the window within

378

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

which the replica-symmetric mean field solution holds should be very small. Thus, for understanding the properties of realistic systems, one must examine the properties of the disordered low temperature state within the framework of replica symmetry breaking or renormalisation group methods. 13.7

Smectic C and ferroelectric C∗ elastomers

The detailed theoretical description of smectic C∗ elastomers in continuum mechanics is rather tedious, due to the low symmetry of the phase and the resulting large number of independent anisotropic invariants in the free energy (Terentjev and Warner, 1994b). For instance, there are eight independent components of the ‘standard’ piezoelectric contribution to the free energy density, coupling the induced polarisation with the components of symmetric strain ε (coordinates from Fig. 2.21(b), taking c as the x and P s as y-axis): 4 γ14 Px εxy + γ16 Px εyz + γ34 Pz εxy + γ36 Pz εyz γi jk Pi ε jk ⇒ (13.63) +γ21 Py εxx + γ22 Py εyy + γ23 Py εzz + γ25 Py εxz However, in spite of the complexity13 , many terms have similar physical effects and the essence of the new effects can be explained very easily. The main coupling in smectic C∗ liquid crystal, eqn (2.58), is between the director orientation, represented by the twocomponent smectic-C order parameter ξ = (−nz ny , nz nx ) = 12 sin 2θ (− sin φ , cos φ ), and the dielectric polarisation P . In an elastomer, we have in addition the coupling between the director and the network strains (which has been the subject of the major portion of this whole book). Contributions to the free energy density take the form

+

1 2 αo sin 2θ (sin φ εyz − cos φ εxz )  2  2 2 2 1 1 4 α11 sin 2θ εzz + 4 α22 sin 2θ sin φ εxx − sin 2φ εxy + cos φ εyy ,

(13.64)

the second line showing the effects ∼ θ 2 ε , relevant further away from the A-C transition where the tilt angle becomes non-infinitesimal. The first term (αo ) has a similar action to the electroclinic effect of Garoff and Meyer (1977) and can thus be called mechanoclinic by analogy. At small tilt angle, the anisotropy in eqn (13.63) becomes small and the relevant piezoelectric terms in the free energy density reduce to

γ˜o (Px εyz − Py εxz ) (13.65) 1˜ 1˜ + 2 γ13 sin 2θ (Px sin φ − Py cos φ ) εzz + 2 γ33 sin 2θ Pz (εxz sin φ − εyz cos φ ) + 12 γ˜12 sin 2θ (Px [εxx sin φ − εxy cos φ ] + Py [εxy sin φ − εyy cos φ ]) The tilde above constants γ (the same as the one above mp in the electroclinic coupling) denotes the chiral nature of these terms; neither group would be present in an ordinary smectic C material – unlike the mechanoclinic effect(13.64), which is non-chiral. 13 The labelling convention here is standard in the formal description of piezoelectricity (and classical elasticity), combining pairs of stress-strain indices into one label running from 1 (for xx) to 6 (for yz).

SMECTIC C AND FERROELECTRIC C∗ ELASTOMERS

379

The three groups of terms describing the coupling between the director orientation (θ ), polarisation and strain, are always accompanied by the classical susceptibility penalty contributions to the free energy density, as seen before eqn (2.59). Qualitatively, ignoring the anisotropy of constants and assuming a uniform tilt, with φ = 0 (cc||x), the description includes F  12 Aθ 2 + 14 Bθ 4 − m˜ p Py θ − αo εxz θ − γ˜o Py εxz +

1 2 1 2 P + Cε , 2χ y 2 xz

(13.66)

where C is a generic shear rubber modulus. Taking into account the terms higher-order in θ would also involve the diagonal components of strain in this expression, such as εxx θ 2 , but their effect is smaller and we ignore them for this simple illustration. Minimising with respect to P and ε , we obtain that the spontaneous polarisation and spontaneous shear strain arise in the (uniform) smectic C∗ phase on the increase of director tilt: Py =

m˜ pC + αo γ˜o χθ ∼ m˜ p χ θ ; C − γ˜o2 χ

εxz =

αo + m˜ p γ˜o χ αo θ ∼ θ, 2 C − γ˜o χ C

(13.67)

assuming in all cases the rubber-elastic modulus is much larger than the SmC*-specific product of coupling constants γ˜o2 χ . However, since SmC elastomers are capable of soft deformations, one may want to be careful in making such an assumption. Diagonal strains εxx etc. will also arise, due to higher order coupling terms, proportional to θ 2 and higher (Terentjev and Warner, 1994b). Evidently, from the two parts of eqn (13.67), there is a direct piezoelectric effect: the induction or change in P by appropriate mechanical strains ε . Such an effect is called ‘improper piezoelectricity’ because the polar vector of polarisation is not the order parameter of the transition (the two-component vector ξ is). The linear piezoelectric coefficient, relating the polarisation and the stress, P  d σ (with σ  Cε ), can therefore be estimated as the ratio d ∼ (m˜ p χ /αo ) and is usually measured in units of (pC/N). In experiments (Lehmann et al., 1999) the smectic C∗ elastomer sample in bookshelf geometry was subjected to an oscillating external voltage, Fig. 2.22(b), and the resulting thickness change was measured by accurate interferometry. This is the inverse piezoelectric effect where strains are induced by electric fields, ε  (χ /dC)E). In Fig. 2.22(b) Ey is applied and εyy measured. Alternatively, by applying a periodic compression along the y-axis in Fig. 2.22(b) and measuring the depolarisation current associated with induced polarisation, the direct piezoelectric response has been detected. The values reported for SmC* elastomers are:  PZT ceramics 300 − 700 pC/N    85 pC/N BaTiO3 ferroelectric d ∼ 300 − 500 pC/N compare with : PVDF composite 30 pC/N    quartz 3 pC/N A suggested explanation (Lehmann et al., 1999) is that the azimuthal orientation of the director on the smectic C cone (or equivalently, the director re-alignment in the layer

380

CONTINUUM DESCRIPTION OF SMECTIC ELASTOMERS

plane) is fixed by crosslinking and is not likely to change under external field. Instead, the required polarisation switching is achieved by altering the tilt angle θ → π − θ (electroclinic effect). Accordingly, it was not surprising that the highest value of the response coefficient has been found at temperatures near the A∗ -C∗ phase transition. Again, we would like to refer the reader to Sect.12.6.1 where much more delicate aspects of SmC soft elasticity and microstructure were discussed. Their implications for the piezoelectric polarisation are a subject of future investigations. Consider again the ‘standard’ bookshelf geometry, Fig. 2.22(b) with φ = 0 and the external electric field Ey across the cell thickness. This geometry leads to a classical electrooptic effect in liquid crystal displays, when the optical axis switches, either around the cone (Goldstone mode) or by going over the state θ = 0 (soft mode) (de Gennes and Prost, 1994; Chaikin and Lubensky, 1995). Assume that at low temperatures, far below the A-C transition, the tilt angle θ of the elastomer is fixed and only the azimuthal director motion is possible. Observing the sample film between crossed polars would generate intensity change proportional to the change in the optical axis projection on the sample plane. The relevant part of the free energy density is, in this geometry, F  Ps E cos φ − 12 Cεs2 cos2 φ ,

(13.68)

where εs and Ps are the spontaneous values of shear strain εxz and polarisation Py defined in eqn (13.67). To switch its polarisation, the crosslinked state with φ = 0 now requires a threshold electric field, determined by the competition of electric and rubber-elastic effects, Ec = Cεs2 /Ps ∼ (αo2 /m˜ p χ C)θ . Since this threshold may not be very high (due to the proportionality to the tilt angle θ ), this electrooptical response could be observed on experiment. Indeed, Fig. 13.6(a) shows such a phenomenon, where one clearly detects the role of crosslinking the texture in one of the polarisation states thus shifting the central point of the hysteresis. The analysis of piezoelectric response was based on the linear coupling terms in the free energy density, schematically expressed by eqn (13.66). It is straightforward to bring into consideration the higher order terms in eqns (13.64) and (13.65) and obtain, after minimisation, contributions of the form

εyy 

1 (α22 − γ˜22 m˜ p χ )θ 2 . 2C

This would lead to the additional second-harmonic response of the mechanical shape to the oscillating external field, the electrostriction ε  aE 2 , Fig. 13.6(b). In the above discussion we have only focused on the case of uniformly aligned smectic C∗ , with its helix unwound. If, in equilibrium the in-plane director rotates helically, φ = q z, then a number of additional effects arise (Terentjev and Warner, 1994b), such as the helical mechanoclinic contribution − 12 α˜1 sin 2θ (cos φ εxz + sin φ εyz ) [∂ φ /∂ z].

(13.69)

Distortions of smectic layers v(rr ) would also generate additional responses proportional to ∇v, both in orientation and in polarisation. Clearly, although there are some good

SMECTIC C AND FERROELECTRIC C∗ ELASTOMERS

381

F IG . 13.6. (a) The electrooptic response of a smectic C∗ system in the bookshelf geometry, after (Brehmer et al., 1994). The thin line shows the response of the SmC∗ polymer melt before crosslinking; the data points are for the elastomer crosslinked in the ‘field-up’ state, indicated by an arrow. (b) The mechanical response of the thin free-standing film with layers in the film plane, after (Lehmann et al., 2001). Two data sets are for the fundamental (1ω , piezoelectricity) and first harmonic (2ω , electrostriction) of the response. In both graphs, the film thickness h and the smectic layer thickness do are indicated, for reference. ideas and observations in this field, much remains in a developing stage – wide open for theoretical and experimental studies.

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INDEX dopant (solvent), 221 separation, of racemic mixtures, 244 cholesteric coarse graining, 236 conical states, 35 photonics, 246 symmetry of, 36 twist and bend deformations, 35 cholesteric elastomers, 221 compressional strains, 221 imprinted, 221, 242 mechanical deformations, 227 monodomain helical, 222 stereo-selective swelling, 244 twist retention on imprinting, 224 uniaxial extension, 228, 233 cholesteric liquid crystals, 33 cis-trans isomerisation, 108, 132 clamp constraints to deformation, 187, 194, 202 elastic penalty for, 419 Clark-Meyer-Helfrich-Hurault effect in elastomers, 371, 373 CMHH instability geometrical basis of in SmA elastomers, 340 coarse-graining of cholecteric texture, 239 coarsening of cholesteric helix, 231 of stripe domains, 199, 422 common-tangent construction, 420 compatibility barriers in polydomain systems, 217 constraints on strain, 190, 203 geometric constraints, 78 compressibility of nematic networks, 257, 434, 436 of rubbery networks, 81, 91 compression, spontaneous on crosslinking, 92 Considere instability, 182 continuum theory of nematic elastomers, 256 of piezoelectricity, 237 of smectic elastomers, 350 convexity, of free energy, 172, 181 copolymers, random, in semi-soft elasticity, 184 Cosserat elasticity, 426 coupling D1 and D2 , relative rotation constants, 436 chiral mechanical, in cholesterics, 241 nematic-elastic network, 4, 140, 410

acoustic low impedance, 116 polarised (birefringent), 118 actuators, 118 light-driven, 109, 116, 131 temperature-controlled, 107, 116, 124 affine deformation, 64 anisotropy average, in random copolymers, 176 dielectric, 57 magnetic, 57 of average polymer shape, 57 of freely jointed chain, 60 of linear elastic moduli, 412 residual, at high temperature, 173 anticrossings, in photonic bands, 250 applications of liquid crystalline elastomers, 115 axial vectors, see pseudovectors azobenzene, see photochromic moiety band gap (stop), photonic, 34 photonic, reduced scheme, 247 structure for light, photonic, 119 biaxial nematic elastomer, 414 nematic elastomers, induced, 144, 408 nematic elastomers, induced by strain, 148 nematic liquid crystals, 14 bifocal lenses, 116 birefringence divergence approaching T ∗ , 142 optical, in nematic systems, 14, 18, 127 Bloch’s theorem, 250 Bragg glass, disordered textures, 376 X-ray scattering intensity, 370 Brillouin zone, photonic, 247 Burgers vector, 42 Caley-Hamilton theorem, 81, 434 cavities, mirrorless, in photonics, 253 characteristic length in Freedericks effect, ζ (E), 28 in nematic elastomer, ξ , 193, 268 characteristic time director relaxation, τ1 , 286, 301 lifetime of cis isomers τc , 133 Rouse, in polymer dynamics, τR , 291, 292 chiral, 33

393

394 relative rotation, 155 relative translation, 353 smectic layer-elastic network, 350 crosslinking bi-functional and tri-functional, 97 density, 66 effect on nematic texture, 214 effect on nematic transition, 137 history, of a network, 131 radiation induced, 97 two-step and multi-step, 105 crosslinks as sources of quenched disorder, 211 coupling to smectic layers, 318 flexible, 98 rigid rod-like, 98, 173, 186 damping of director fluctuations, 283 of mechanical energy, 118, 307 de Vries phases effect on SmA moduli, 343 in SmA elastomers, 343 Debye-Waller factor, 365 defects, hairpin, see hairpin defects deformation affine, 121 consistent with clamp constraints, 190 in stripe microstructure, 204 multiple steps, 121 spontaneous extension λm , 107, 123 deformation gradient full λik , 75, 257 small deformations uik , 258, 432 deformation tensor antisymmetric part of, 83 right and left Cauchy–Green, 76 density of states, in photonics, 253 diadic, form of tensors, 175 diamagnetic susceptibility, anisotropic, 57 dielectric anisotropy, 26 dimensionless numbers (dynamical) Ericksen, Er, 31, 300 nematic elastomer, Ne, 300 Reynolds, Re, 31 director anchoring to elastic network, 155 discontinuous strain-induced rotation of, 145 electrooptical response, 111 field-induced rotation, 25, 191 non-uniform profile of, 199, 422 rotation, coupling to shear, 158 rotation, in cholesteric elastomers, 231 strain-induced rotation, 111, 158, 171, 265 strain-induced, discontinuous, 146, 232 director alignment

INDEX at network formation, 121 by magnetic field, 101 by strain, in polydomain elastomers, 217 director deformation geometry half-space, 27 splay-bend, 26 twist, 26 director fluctuations damping of, 283 in nematic elastomers, 276 in smectic elastomers, 355 light scattering, 282 discotic elastomers, 98 dislocations, in smectic systems, 42 dispersion relation, in photonics, 247 dissipation, in nematic liquids, 29, 299 dynamic light scattering, 284 dynamic mechanical response, 293, 309 effective mass of nematic fluctuations, 281 smectic fluctuations, 376 elastic anisotropy high, 323 elasticity classical theory, 75 couple-stress and Cosserat, 426 effective, in nematic elastomers, 273 effective, in smectic elastomers, 360 linear and non-linear, 78 linear, for nematic elastomers, 258 semi-soft, see semi-soft elasticity elastomer networks chiral, intrinsic or imprinted, 221, 222 nematic phase transitions, 139 nematic, main and side-chain, 107 photochromic, 69 random copolymer, 174 volume conservation, 66 electroclinic coefficient in SmA* elastomers, 327 electroclinic effect in SmA* elastomers, 326 electroclininc effect, 380 electrostriction due to electroclinic effect, 322 second-harmonic response, 380 emission, stimulated, in cholesterics, 253 energy elastic, quasi-convexified, 204 interfacial, in stripe domains, 421 entanglements, 51 in elastomers, 70, 293, 294 in nematic elastomers, 150, 153 topological, 51, 71 entropy of polymer chain, 50

INDEX epoxy main-chain nematic elastomers, 97 Ericksen number, Er, 31 Euler-Lagrange problem, 26 extension applied perpendicular to director, 145 soft, perpendicular to director, 169 Fermi’s Golden rule, 253 ferroelectricity in smectic C∗ elastomers, 119, 378 in smectic C∗ liquid crystals, 44 proper and improper, 45 field contribution to free energy, 139 electric or magnetic, in cholesterics, 34 electric threshold, in Freedericks effect, 189 mechanical, in cholesteric elastomers, 227 orientational, randomly quenched, 139, 212 finite extensibility in isotropic networks, 70 in nematic elastomers, 150 flexoelectricity, 26 in cholesteric systems, 35, 239 Flory prefactor, generalised, 131 fluctuation-dissipation theorem, 291 fluctuations compositional, in semi-softness, 174 of director orientation, 284 of network junction points, 71 of smectic layers, 369 force, local ∇k σik , 79 forces and torques, balance of, 301 Fourier modes collective (Rouse) modes, 292 nematic distortions, 277–279, 356 network phonons, 280, 281, 367 smectic layer distortions, 361 Frank elasticity competition with rubber elasticity, 193 in liquid crystals, 22 in nematic elastomers, 62 free energy biaxial nematic, 16 effective rubber-elastic, 273, 360 entropic, of single chain, 50, 64 Frank elastic, 22 neo-classical, Trace formula, 121 quartic, in Freedericks geometry, 191 semi-soft corrections, 179, 185 small deformation expansion, 432 Freedericks effect, 25 characteristic length, 28 in nematic elastomers, 188 relaxation times, 32 freely jointed chain model, 128, 410

395 Gaussian distribution anisotropic, 55 deviations from, 150 for isotropic polymers, 49 generalised anisotropic, 58 normalisation factor, 121 geometry bookshelf, of smectic C∗ , 46 of deformations and rotations, 83 uniaxial stretching, 112 glass transition, 97 dynamic, 309 Golubovic-Lubensky, theorem, 183 hairpin defects, 54, 56 effect on splay Frank constant, 62 Halperin-Lubensky-Ma effect, 377 Hamiltonian effective, in nematic elastomers, 277 effective, in smectic elastomers, 366 Helfrich-Hurault effect in elastomers, 371, 373 helix, in cholesteric elastomers affine contraction with strain, 230 coarsening under strain, 231 history thermal, dependence on, 129 thermomechanical, for semi-softness, 183 imprinted elastomers, conical states, 236 imprinting of chiral texture, 223 of nematic order, 139 strong and weak limits, 243 Imry-Ma, random field scaling, 214 indicatrix, optical, 14 instability Considere, 182 layer buckling, 371 mechanical and nematic, 145 stripe domains, 194 integrating out phonons in nematic elastomers, 279 phonons in smectic elastomers, 366 interface, separating stripe domains, 196 internal field, electric, 26 invariance rotational of free energy, 166 Kramers-Kronig relations, 290 Lam´e coefficients, 80 Landau-de Gennes, expansion of free energy, 15 Landau-Peierls effect, 365 loss of long range order, 350 Langevin equation, 291

396 lasing in cholesteric systems, 253 lattice, one-dimensional smectic, 365 layer elasticity, see smectic layers layer rotation threshold elongation in SmA, 335 layers deforming affinely, 330 Legendre polynomial, 12 Leslie coefficients, 30, 299 Leslie-Ericksen theory, 30, 299 Levi-Civita tensor, 84 light circularly polarised, 247 guiding, 117 scattering, dynamic, 284 scattering, from director fluctuations, 282 loss tangent, in linear response, 307 lyotropic lamellar phase, Lα , 38, 318 lyotropic nematic liquid crystals, 10 Maxwell equal area construction, 232 Maxwell wave equation, 249 mean field, nematic, 21 mechanoclinic effect helical, 380 in smectic C∗ elastomers, 378 memory of chain shape anisotropy, 124 of formation conditions, 64 of nematic genesis, 125 micro-manipulators, micro-pumps, 118 microphase separation, 38 microstructure, 150 effect on SmA elastomer Poisson ratios, 344 of SmA elastomers, 344 strain induced, 194 stripes domains, 177 X-ray scattering from, 208, 344 modulus compressional, 66, 81 effects of nematic order, 408 extensional Young, 94 of smectic elastomers, 357, 359, 362 renormalised C5R , 274, 303, 359 shear, 81 smectic layer compression, 368 storage, in linear response, 308 mono-morph, bi-rubber elastic beam, 117 monodomain SmA elastomer, 320 monodomain nematic elastomer, 105 Mooney-Rivlin description, 74, 77 multiscale analysis of microstructure, 202 muscle, artificial, 117 nematic director, in smectic elastomers, 354

INDEX nematic director, integrating out, 277, 355 nematic elastomers classical analogy, 130 dynamical number Ne, 300 main and side-chain, 107 nematic-isotropic transition, 139 neo-classical theory, 120 nematic interactions, in polymers, 71 nematic liquid crystals, 9 nematic order induced in polymer backbone, 127 Maier-Saupe theory, 20 oblate and prolate, 17, 100, 108 orientational hAP2 B, 11 residual paranematic, 142 stabilised by crosslinking, 139 under strain, 97, 100, 408 nematic-smectic coupling, 354 nematohydrodynamics, 29, 298 neo-classical theory, Trace formula, 120 neutron scattering, of polymers, 55 non-ideality, 179 nuclear magnetic resonance, NMR, 12, 59, 100 opaque, polydomain nematic networks, 105 optical guiding, see light, guiding optical path length, 194 optical rotation of chiral network, 224 order parameter biaxial nematic, 15, 35, 411 nematic, 11, 14 paranematic state, 19 Parodi relation, 31 penetration depth in nematic elastomers, 188 phase transition first order, 15 in nematic elastomers, 136, 141 in nematic elastomers, under stress, 100, 139 nematic, in external field, 19 nematic, Landau-de Gennes description, 16 nematic-isotropic, with random disorder, 101 phonons, in rubbery network, 279, 366 photo-ferroelectricity in smectic C∗ elastomers, 119 photoactuation, see actuators, light-driven photochromic molecular moiety, 69, 131 photoisomerisation, 69 photomechanical effect, 108 dynamics of, 135 in nematic elastomers, 131 steady state, 132 photon correlation spectroscopy, 284 photonics

INDEX experimental observations, 251 of cholesteric elastomers, 119, 249 of liquid cholesterics, 246 piezo-electric responses in SmC*, 328 piezoelectricity experiments in elastomers, 241 applications of, 119 direct and inverse, 379 of cholesteric elastomers, 236 of smectic C∗ elastomers, 378 point group D∞h , 11, 39 point group D∞ , 40 Poisson ratio, 81 anisotropic, 411 deviations from ν = 12 , 94 zero, 160 Poisson ratios (1, 0), 337 for SmA, 333 crossover, 333 extreme, 329 for imposed λzz , 342 polar decomposition theorem, 89 polarisation dielectric, 26 flexoelectric, 35, 240 optical, 134 spontaneous in smectic C∗ elastomers, 379 polydomain texture alignment by stretching, 103 in nematic elastomers, 216 opaque, 106 polydomain-monodomain transition, 218 polymers anisotropy of backbone shape, 54 arc length, 48 beads and springs model, 291 configurations of, 48 finite extensibility, 70 freely jointed rods, 48 ideal Gaussian description, 51 nematic interactions, 71 nematic, Frank elasticity of, 61 nematic, freely jointed rods, 59 nematic, main- and side-chain, 53 radius of gyration, 55 step length , 48 population dynamics, on UV-isomerisation, 135 primitive path, 151 principal axes of strain, 85, 89, 165 projector tensor, perpendicular, 175 pseudoscalars, -vectors and -tensors, 36 quasi-convexification, 202 quasi-long range order, 365

397 quenched disorder N-A transition in smectic elastomers, 374 polydomains in nematic elastomers, 211 racemic mixture, separation of, 119 random disorder in nematic networks, 210 in smectic elastomers, 374 quenched sources of, 211 random field, 139, 212, 375 characteristic domain size, 213 random walk Gaussian, 49 self-avoiding, 51 reaction rate, of trans-cis isomerisation, 110 reflections, in cholesterics affine shift on deformation, 231 Bragg-like, 34 refractive index, optical, 12 relative rotations, 155, 259, 358 relative translations, 351 relaxation after UV radiation off, 136 of stress, 296, 313 times, in nematic viscoelasticity, 300 renormalisation group, functional, 219 replica method, symmetry breaking, 219 replica trick, 376 Reynolds number, Re, 31 rotation associated pseudovector, 237 in general vector form, 433 in matrix form, 84 local, elastic medium, 84 local, in fluids, 30 rotational invariance in general elasticity, 77 in soft elasticity, 272 Rouse dynamics, 291 Rouse time, 292 rubber, elastic modulus, 62 semi-soft elasticity, 173 strains and associated director rotation, 176 semi-softness continuum representation of, 267 dependence on nematic order, 275 effect of residual nematic order, 173 sensors opto-mechanical, 116 piezoelectric, 238, 381 shape change spontaneous, in nematic elastomers, 106 without energy cost, 114 shape memory alloys, 187 shear

398 compensating, in stripe domains, 196 coupling to director rotation, 159 pure, at constant volume, 84 simple, decomposition of, 87 spontaneous in smectic C∗ elastomers, 379 suppression at clamps, 195 smectic A elastomers, 350 smectic A, liquid crystals, 39 smectic C elastomers, 321, 378 smectic C, elastomers, 321, 378 smectic C, liquid crystals, 44 smectic layers compression strain, 40 buckling, see Helfrich-Hurault effect coupling to elastic network, 358 deformations, displacement field, 40, 41 fluctuations, 352, 369 relative displacement, 350 two-dimensional elasticity, 361 smectic liquid crystals, 38 smectic order, quasi-long range, 365 soft deformations continuum mechanism of, 271 geometry of strains, 164 nematic order unchanged, 162 rotation, shear and contraction, 171 soft elasticity, 5, 159 allowed forms of, 166 biaxial, 414, 416 continuum representation of, 268 necessary and sufficient condition for, 186 soft SmC elasticity biaxial, 437 space inversion, symmetry of, 36 spacer, odd-even effect, 99 spin-glass, 218 spontaneous distortion, simple model, 126 square roots of tensors, 90 steepest-descent integration, 277 step length anisotropy of, 126 of biaxial nematic polymers, 414 of isotropic polymers, 50 of nematic polymers, 56, 121 stereo-selectivity, 244 steric interactions, 10 Stokes theorem, 42 strain gradients of, 196 in soft deformations, 164 integrating out, 280, 364 oscillating response to, 304 photo-induced, 136 rate, velocity gradient, 29, 219, 299 reference state, 129 semi-soft threshold, 173

INDEX strain tensor asymmetric part of, 265 Cauchy–Green, 75 right and left, 79 symmetric part of, 79, 259 stress antisymmetric component of, 427 as aligning field, 103 microscopic definition, 293 semi-soft residual, 180, 181 steady relaxation, 296, 313 threshold, polydomain-monodomain, 218 true and engineering (nominal), 82, 181 stress tensor σik , 79 stress-optical effects, 99, 140 stripe domains, 113, 177, 194, 421 coarsening, 199, 422, 423, 425 energy of, 196 microstructure, 195, 419 period of, 200, 204, 423 threshold strain, 265, 424 supercritical phase transformation, 19 if crosslinked in nematic state, 139 in elastomers under strain, 100 surface tension, of stripe interfaces, 198 susceptibility diamagnetic, 57 dielectric, 26, 46 of nematic order, 16, 408 of nematic order to stress, 141 symmetry hidden, ‘soft elasticity’, 78 of cholesterics, 36 of smectics C-C∗ , 45 quadrupolar, nematic, 20 tensors second-rank, transformations of, 157 square roots and polar decomposition, 90 textures measurements of size, 215 polydomain nematic, 101 thermo-mechanical history, 105 thermotropic, nematic liquid crystals, 10 threshold electric field, in Freedericks effect, 103 electric field, in smectic C∗ , 380 for chiral imprinting, 244 in Clark-Meyer-Helfrich-Hurault effect, 43 in Helfrich-Hurault effect, 43, 373 strain, for director rotation, 176, 267 stress for polydomain-monodomain, 218 tilt mechanical SmA to SmC, 326 molecular, 326 time-temperature superposition, 309

INDEX topological imprinting, in chiral networks, 223 topological constraints (entanglements), 71 torques, local, balance of, 30, 427 Trace formula, 122 trans-cis isomerisation, 110 transistor, mechanical, 68 transition polydomain-monodomain, 104, 216 SmA to SmC, 326 strain-induced, anti-Freedericks, 146 transparent, aligned nematic networks, 105, 284 tube model in dynamical response, 293 in nematic elastomers, 151 in rubbery networks, 72 twist, in cholesterics reversal, on stretching, 253 topological defects, 231 UV light photoisomerisation, 69 rate of molecular absorption, 133 viscosity anisotropic, see Leslie coefficients isotropic, 29 rotational, in nematic elastomers, 287, 300 rotational, in nematic liquids, 30 white noise, 291 WLF, see time-temperature superposition Young moduli anisotropy of in smectic A elastomer, 363 parallel and perpendicular to director, 144

399

400

AUTHOR INDEX

Aβ falg, N., see Stein, P. 288 Abramchuk, S.S. 71, 124 Achard, M.F., see Lecommandoux, S. 55, 56 Adams, J.M. 328, 329, 335, 344, 345, 353 Ahir, S.V. 107 Aksenov, V., see Stannarius, R. 326 Als-Nielsen, J. 365, 370 Alvarez, E. 254 Amendola, E., see Carfagna, C. 97 Andereck, B.S. 375 Andersson, H. 318 Anisimov, M.A. 375 Anwer, A. viii, 102 Arfken, G. 37 Arrighi, V. 51 Ashby, M.F., see Fleck, N.A. 426 Atkins, R.J. 75, 89 Baekeland, L.H. 47 Ball, J.M. 150, 187 Ball, R.C. 51, 295 Ball, R.C., see Higgs, P.G. 72, 73, 295 Ball, R.C., see Mao, Y. 150, 151 Barclay, G.G. 96, 97 Barnes, N.R. 103, 178, 189 Bartolino, R. 414 Bastiaansen, C.W.M., see Harris, K.D. 111 Bastiaansen, C.W.M., see Mol, G.N. 111 Bates, F.S. 38, 39 Beattie, H.N., see Lacey, D. 104 Bellini, T. 211, 376 Belloq, A.M., see Safinya, C.R. 365 Belyakov, V.A. 34, 246–248 Ben-Shaul, A., see Gelbart, W.M. 24 Benn´e, I. 321 Benoit, H., see Higgins, J.S. 55 Bermel, P.A. 247, 248 Berreman, D.W. 34, 248 Beyer, P. 328 Bhargava, N., see Ortiz, C. 219 Bhattacharya, K. 201, 203, 205 Birgeneau, R.J., see Als-Nielsen, J. 365, 370 Birgeneau, R.J., see Garland, C.W. 375 Bladon, P. 14, 71, 231 Bladon, P., see Terentjev, E.M. 178, 188, 280 Bladon, P., see Warner, M. 154 Bl¨asing, J., see Stannarius, R. 326 Blumstein, A., see d’Allest, J.F. 53, 55 Blumstein, R.B., see d’Allest, J.F. 53, 55 Boiko, N.I., see Freidzon, Y.S. 221 Boue, F., see Fourmaux-Demange, V. 312 Bouligand, Y. 96 Bouwman, W.G. 365, 375 Brand, H.R 373 Brand, H.R., see Kaufhold, W. 141, 142 Brand, H.R., see Nishikawa, E. 8, 324, 326, 360

Brand, H.R., see Pleiner, H. 35 Brand, H.R., see Weilepp, J. 338, 350, 358 Bray, A. 211 Brehmer, M. 97, 321, 381 Brehmer, M., see Zentel, R. 322, 351 Broderix, K. 63 Brodowsky, H.M. 326 Broer, D.J. 249, 253 Broer, D.J., see Harris, K.D. 111 Broer, D.J., see Hikmet, R.A.M. 96 Broer, D.J., see Mol, G.N. 111 Brulet, A., see Fourmaux-Demange, V. 312 Brulet, A., see Li, M.H. 53, 54, 56 Bualek, S. 221 Caille, A. 365, 369, 370, 377 Callaghan, P.T., see Komlosh, M.E. 295 Camacho-Lopez, M. 110, 111 Carfagna, C. 97 Carlsson, T. 303 Carothers, W.H. 47 Chaikin, P.M. 43, 278, 355, 380 Chandrasekhar, S. 9, 14, 25, 29 Chang, C.-C. 103, 189, 194, 241, 242 Chang, C.H., see Saipa, A. 328 Chasset, R. 296, 314 Cheung, L., see Chou, S.G. 248 Chiaranza, T., see Bartolino, R. 414 Chien, L.-C., see Chang, C.-C. 103, 189, 194, 241, 242 Chien, L.-C., see Yang, D.-K. 223 Chou, S.G. 248 Chuang, I. 211 Chudnovsky, E.M. 216 Cicuta, P. 231, 251–253 Cimecioglu, A.L., see Arrighi, V. 51 Cladis, P.E., see Anisimov, M.A. 375 Cladis, P.E., see Bouligand, Y. 96 Clark, N.A. 43, 46, 372, 373 Clark, N.A., see Bellini, T. 211, 376 Clark, N.A., see Safinya, C.R. 365 Clarke, S.M. 105, 107, 127, 128, 139, 182, 215, 218, 219, 283, 288, 298, 310, 314, 376 Clarke, S.M., see Elias, F. 211, 212, 215, 313 Colby, R.H. 312 Compagnoni, R., see Bartolino, R. 414 Conti, S. viii, 188, 205, 207, 208, 210 Corbett, D. 111 Cosserat, E. 426 Cosserat, F., see Cosserat, E. 426 Cotton, J.P. 56 Cotton, J.P., see Fourmaux-Demange, V. 312 Cotton, J.P., see Li, M.H. 53, 54, 56 Cotton, J.P., see Noirez, L. 343 Coulter, M., see Mitchell, G.R. 57, 100 Courty, S. 224, 245

AUTHOR INDEX Cullity, B.D. 365 Curro, J.G. 297 Cuypers, R., see Harris, K.D. 111 Cviklinski, J. 109, 131–133 Cywinski, R., see Mitchell, G.R. 56, 126 d’Allest, J.F. 53, 55 Daniels, D.R. 343 Davidson, P., see Li, M.H. 53, 54, 56 Davis, F.J., see Barnes, N.R. 103, 178, 189 Davis, F.J., see Guo, W. 57 Davis, F.J., see Hasson, C.D. 223, 224, 242, 245 Davis, F.J., see Legge, C.H. 97 Davis, F.J., see Mitchell, G.R. 56, 57, 100, 112, 113, 126, 231 Davis, F.J., see Roberts, P.M. 107, 148, 149 Davis, R., see Deindorfer, P. 328 de Gennes, P.-G. 2, 9, 12, 22, 25–27, 29, 30, 33, 34, 36, 39, 41, 43, 44, 47, 51, 54, 56, 57, 62, 72, 100, 124, 140, 159, 223, 226, 227, 231, 237, 243, 282, 285, 294, 297, 299, 304, 313, 355, 365, 372, 380 de Gennes, P.-G., see Herbert, M. 117 de Jeu, W.H. 30 de Jeu, W.H., see Bouwman, W.G. 365, 375 de Jeu, W.H., see Gramsbergen, E.F. 17 de Jeu, W.H., see Wong, G.C.L. 370, 371, 374 de Vries, H. 246 Deam, R.T. 51 Deindorfer, P. 328 Dekker, A.J., see van der Meer, B.W. 33 Deloche, B. 71, 140 DeSimone, A. 150, 187, 205 DeSimone, A., see Conti, S. viii, 188, 205, 207, 208, 210 Dickie, R.A. 309 Diele, S., see Lehmann, W. 379 Dietrich, U., see Stannarius, R. 326, 328, 333 DiMarzio, E.A., see Gaylord, R.J. 296 Dimon, P., see Safinya, C.R. 365 Dingemans, T.J., see Madsen, L.A 414 Disch, S. 98, 100, 101 Dmitrienko, V.E., see Belyakov, V.A. 34, 246–248 Doane, J.W., see Yang, D.-K. 223 Doi, M. 47, 72, 289, 291, 293, 300 Doi, M., see Ball, R.C. 51, 295 Doi, M., see Kuzuu, N. 30 Dolzmann, G., see Conti, S. viii, 188, 205, 207, 208, 210 Dolzmann, G., see DeSimone, A. 205 Donald, A.M. 53 Donald, A.M., see Daniels, D.R. 343 Dotsenko, V.S. 211 Doublet, F., see Gallani, J.L. 312 Doussal, P. Le, see Giamarchi, T. 376 Dreher, R. 248

401

Dusek, ˇ K. 68 Dusek, ˇ K., see Matejka, ˇ L. 69, 70 Durrer, R., see Chuang, I. 211 Edwards, S.F. 51, 71, 72, 151, 212, 294–296 Edwards, S.F., see Ball, R.C. 51, 295 Edwards, S.F., see Deam, R.T. 51 Edwards, S.F., see Doi, M. 47, 72, 289, 291, 293, 300 Eisenbach, C.D. 69, 70, 132 Elias, F. 211, 212, 215, 313 Emig, T. 216 Eremin, A., see Deindorfer, P. 328 Ericksen, J.L. 29 Fan, B., see Kopp, V.I. 254 Ferry, J.D. 289, 308–310 Ferry, J.D., see Dickie, R.A. 309 Finkelmann, H. 53, 97, 99, 100, 107, 109, 110, 127, 128, 130, 131, 134, 136, 144, 177, 198, 199, 201, 221, 254, 273, 317, 321, 408, 410, 413, 425 Finkelmann, H., see Benn´e, I. 321 Finkelmann, H., see Camacho-Lopez, M. 110, 111 Finkelmann, H., see Clarke, S.M. 105, 215, 283, 376 Finkelmann, H., see Disch, S. 98, 100, 101 Finkelmann, H., see Fischer, P. 39, 319, 320 Finkelmann, H., see Gleim, W. 108 Finkelmann, H., see Greve, A. 108 Finkelmann, H., see Hammerschmidt, K. 58, 99, 108 Finkelmann, H., see Hiraoka, K. 8, 321, 326, 327, 347 Finkelmann, H., see Kaufhold, W. 141, 142 Finkelmann, H., see Kim, S.T. 222 Finkelmann, H., see Kundler, I. 97, 113, 177, 199 Finkelmann, H., see K¨upfer, J. 97, 102, 105–107, 182 Finkelmann, H., see Lebar, A. 101, 213 Finkelmann, H., see Lehmann, W. 379 Finkelmann, H., see Leube, H.F. 14, 414 Finkelmann, H., see Meier, W. 221, 225, 241 Finkelmann, H., see Nishikawa, E. 8, 319–326, 333, 335, 336, 342, 360, 374 Finkelmann, H., see Sch¨atzle, J. 107 Finkelmann, H., see Schmidtke, J. 253–255 Finkelmann, H., see Semmler, K. 319 Finkelmann, H., see Shibaev, V.P. 53 Finkelmann, H., see Stein, P. 288 Finkelmann, H., see Zubarev, E.R. 177, 199, 208–210, 351 Fischer, E.W., see Vallerien, S.U. 241 Fischer, P. 39, 319, 320 Fleck, N.A. 426 Flory, P.J. 47, 50, 68, 71

402

AUTHOR INDEX

Fourmaux-Demange, V. 312 Fox, N. 75 Fox, N., see Atkins, R.J. 89 Freidzon, Y.S. 221 Fridrikh, S.V. 212, 216, 218, 274 Galerne, Y. 414 Gallani, J.L. 312 Galli, G., see Colby, R.H. 312 Garland, C.W. 375 Garland, C.W., see Bellini, T. 211, 376 Garoff, S. 46, 327 Gaylord, R.J. 296 Gaylord, R.J., see Gottlieb, M. 74 Gaylord, R.J., see Higgs, P.G. 74 Gebhard, E. 322 Gebhard, E., see Lehmann, W. 322, 381 Gebhard, E., see Zentel, R. 322, 351 Gedde, U.W., see Andersson, H. 318 Geissler, E. 284, 285 Gelbart, W.M. 24 Gelling, K.P., see Warner, M. 124 Genack, A.Z., see Kopp, V.I. 247, 254 Giamarchi, T. 376 Giamberini, M., see Carfagna, C. 97 Giesselmann, F., see Langhoff, A. 328 Giesselmann, F., see R¨ossle, M. 343 Giesselmann, F., see Saipa, A. 328 Gillmor, J.R., see Colby, R.H. 312 Gleim, W. 108 Gleiss, W., see Finkelmann, H. 99, 100 Goldbart, P.M., see Broderix, K. 63 Goldberg, L.S. 253 Golubovic, L. 268 Gorodetskii, E.E., see Anisimov, M.A. 375 Gottlieb, M. 74 Gramsbergen, E.F. 17 Green, M.S. 293 Greve, A. 108 Greve, A., see Finkelmann, H. 107, 127, 128, 130, 144, 408, 410, 413 Grinsten, G. 366 Grosberg, A.Y. 62 Grun, F., see Kuhn, W. 70 Gubina, T.I., see Talroze, R.V. 53 Guo, W. 57 Guo, W., see Mitchell, G.R. 56, 57, 100, 112, 113, 126, 231 Hall, E., see Colby, R.H. 312 Halperin, A. 141 Halperin, B.I. 39, 374, 375 Hammerschmidt, K. 58, 99, 108 Hardouin, F., see Cotton, J.P. 56 Hardouin, F., see Lecommandoux, S. 55, 56 Harris, K.D. 111

Harris, K.D., see Mol, G.N. 111 Hartmann, L., see Lehmann, W. 379 Hasegawa, H., see Hashimoto, T. 282 Hashimoto, T. 282 Hasson, C.D. 223, 224, 242, 245 Haudin, J.M. 282 He, M., see Alvarez, E. 254 Heiney, P.A., see Spector, M.S. 343 Helfand, E., see Curro, J.G. 297 Helfrich, W. 41 Herbert, M. 117 Higgins, J.S. 55 Higgins, J.S., see Arrighi, V. 51 Higgs, P.G. 72–74, 295 Hikmet, R.A.M. 96 Hilliou, L., see Fourmaux-Demange, V. 312 Hilliou, L., see Gallani, J.L. 312 Hiraoka, K. 8, 321, 326, 327, 347 Hofmann, F. 47 Hogan, P.M. 109, 131 Holt, D.B., see Spector, M.S. 343 Honda, S., see Urayama, K. 178, 189, 280 Horn, R.A. 89 Hornreich, R.M. 17, 19, 248 Hotta, A. 219, 315 Hotta, A., see Clarke, S.M. 127, 128, 139, 182, 219, 310 Hult, A., see Andersson, H. 318 Huse, D.A., see Anisimov, M.A. 375 Hutchinson, J.W., see Fleck, N.A. 426 Ikeda, T., see Yu, Y. 111 Ilavsky, ´ M., see Matejka, ˇ L. 69, 70 Imry, Y. 211, 214 Imura, H. 30, 303 Jacobsen, B., see Saunders, K. 376 James, R.D., see Ball, J.M. 150, 187 Jarry, J.P. 71 Jeon, H., see Thomsen, D.L. 107 Johnson, C.R., see Horn, R.A. 89 Kant, R., see Herbert, M. 117 Kapitza, H. 321 Kapitza, H., see Bualek, S. 221 Kapitza, H., see Vallerien, S.U. 241 Kaplan, M., see Als-Nielsen, J. 365, 370 Kaufhold, W. 141, 142 Kawasumi, M., see Percec, V. 53 Keller, P., see Fourmaux-Demange, V. 312 Keller, P., see Li, M.H. 53, 54, 56 Keller, P., see Meyer, R.B. 44 Keller, P., see Noirez, L. 343 Keller, P., see Thomsen, D.L. 107 Kharitonov, A.V., see Shibaev, V.P. 53 Khokhlov, A.R., see Abramchuk, S.S. 71, 124

AUTHOR INDEX Kholodenko, A.L. 51, 71 Kim, R., see Ortiz, C. 218 Kim, S. T., see Finkelmann, H. 254 Kim, S. T., see Schmidtke, J. 254 Kim, S.T. 222 Kirste, R.G., see Ohm, H.G. 56 Kleman, M. 430 Kock, H.J., see Finkelmann, H. 53, 97, 99, 100, 221, 317, 321 K¨ohler, R. 327 K¨ohler, R., see Stannarius, R. 326, 328, 329, 333 Kohn, R.V. 201, 342 Komlosh, M.E. 295 Kopp, V.I. 247, 254 Korobeinikova, I.A., see Shibaev, V.P. 221 Kostromin, S.G., see Shibaev, V.P. 38, 53, 54 Kovalchuk, A.V., see Yuranova, T.I. 97, 98 Kramer, E.J., see Ortiz, C. 218, 219 Kremer, F., see Brodowsky, H.M. 326 Kremer, F., see Lehmann, W. 322, 379, 381 Kremer, F., see Vallerien, S.U. 241 Krost, A., see Stannarius, R. 326 Kr¨uger, P., see Lehmann, W. 322, 381 Kruth, H., see Lehmann, W. 379 Kuhn, W. 70 Kundler, I. 97, 113, 177, 199 Kundler, I., see Clarke, S.M. 105, 215, 283 Kundler, I., see Finkelmann, H. 177, 198, 199, 201, 425 Kunin, I.A. 426, 430 K¨upfer, J. 97, 102, 105–107, 182 Kuptsov, S.A., see Zubarev, E.R. 199, 208–210 Kutnjak, Z., see Lebar, A. 101, 213 Kutter, S. 72, 73, 151, 253, 295 Kutter, S., see Warner, M. 414 Kuzuu, N. 30 Lacey, D. 104 Lagerwall, J.P.F., see R¨ossle, M. 343 Lagerwall, S.T., see Clark, N.A. 46 Landau, L.D. 16, 32, 79, 82, 94, 232, 364, 427 Langhoff, A. 328 Lanham, K.W., see Saipa, A. 328 Larkin, A.I. 211 Laus, M., see Colby, R.H. 312 Lebar, A. 101, 213 Lecommandoux, S. 55, 56 Legge, C.H. 97 Lehmann, W. 322, 379, 381 Leslie, F.M. 29 Leube, H.F. 14, 414 Li, M.H. 53, 54, 56 Liang, K.S., see Wong, G.C.L. 370, 371, 374 Liebert, L., see Bouligand, Y. 96 Liebert, L., see Meyer, R.B. 44 Liebert, L., see Strzelecki, L. 96

403

Lifshitz, E.M., see Landau, L.D. 16, 32, 79, 82, 94, 232, 364, 427 Lindegaard-Andersen, A., see Als-Nielsen, J. 365, 370 Litster, J.D., see Als-Nielsen, J. 365, 370 Liu, H., see Bladon, P. 14 Lodge, A.S. 293 Long, D. 298 Longa, L., see Gramsbergen, E.F. 17 L¨osche, M., see Lehmann, W. 322, 381 L¨oscher, M., see Stannarius, R. 326, 328, 333 Lub, J, see Broer, D.J. 249, 253 Lub, J., see Harris, K.D. 111 Lubensky, T.C 350, 353, 355, 375 Lubensky, T.C., see Chaikin, P.M. 43, 278, 355, 380 Lubensky, T.C., see Halperin, B.I. 39, 374, 375 Lubensky, T.C., see Stenull, O. 164, 344, 345 Lubensky, T.C., see Terentjev, E.M. 366, 369 Ma, S.-K., see Halperin, B.I. 39, 374, 375 Ma, S.-K., see Imry, Y. 211, 214 Madsen, L.A 414 Mahadevan, L., see Warner, M. 110 Maier, W. 21 Maissa, P., see d’Allest, J.F. 53, 55 Mao, Y. 70, 150, 151, 229, 230, 232, 233, 243, 244 Marcerou, J.P., see Galerne, Y. 414 Martinoty, P., see Fourmaux-Demange, V. 312 Martinoty, P., see Gallani, J.L. 312 Martinoty, P., see Stein, P. 288 Matejka, ˇ L. 69, 70 Mathiesen, S., see Als-Nielsen, J. 365, 370 Mauzac, M., see Gallani, J.L. 312 Maxein, G. 225, 227 Mayer, S., see Maxein, G. 225, 227 Mazenko, G.F. 364, 366 Mazo, M. 291 McArdle, C.B. 53, 317 Meeten, G.H. 283 Meier, G., see Dreher, R. 248 Meier, W. 221, 225, 241, 318 Meuti, M., see Bartolino, R. 414 Meyer, R.B. 26, 34, 35, 44, 62, 227, 231, 243, 313 Meyer, R.B., see Chang, C.-C. 103, 189, 194, 241, 242 Meyer, R.B., see Chou, S.G. 248 Meyer, R.B., see Clark, N.A. 43, 372, 373 Meyer, R.B., see Garoff, S. 46, 327 Meyer, R.B., see Patel, J.S. 35 Meyer, R.B., see Pelcovits, R.A. 236, 238–241 Meyer, R.B., see Terentjev, E.M. 191, 192 Mezard, M. 211 Mitchell, G.R. 56, 57, 100, 112, 113, 126, 146, 231 Mitchell, G.R., see Barnes, N.R. 103, 178, 189

404

AUTHOR INDEX

Mitchell, G.R., see Guo, W. 57 Mitchell, G.R., see Hasson, C.D. 223, 224, 242, 245 Mitchell, G.R., see Lacey, D. 104 Mitchell, G.R., see Legge, C.H. 97 Mitchell, G.R., see Roberts, P.M. 107, 148, 149 Mol, G.N. 111 Mol, G.N., see Broer, D.J. 249, 253 Mollon, J.D. 117 Monnerie, M., see Jarry, J.P. 71 Morse, D., see Long, D. 298 Mukhopadhyay, R., see Lubensky, T.C. 77, 81, 164, 265, 269 Muller, G.M., see Fleck, N.A. 426 M¨uller, S., see Kohn, R.V. 201, 342 Mu˜noz, A.F., see Alvarez, E. 254 Mu˜noz, A.F., see Finkelmann, H. 254 Murnaghan, F.D. 75, 80, 81 Mutukumar, M., see Edwards, S.F. 212 Muzny, C.D., see Bellini, T. 211, 376 Naciri, J., see Spector, M.S. 343 Naciri, J., see Thomsen, D.L. 107 Nakai, A., see Hashimoto, T. 282 Nakano, M., see Yu, Y. 111 Nakata, M., see Madsen, L.A 414 Nattermann, T., see Emig, T. 216 Nishikawa, E. 8, 319–326, 333, 335, 336, 342, 360, 374 Nishikawa, E., see Clarke, S.M. 376 Nishikawa, E., see Finkelmann, H. 109, 110, 131, 134, 136 Noirez, L. 343 Noirez, L., see d’Allest, J.F. 53, 55 Noirez, L., see Lecommandoux, S. 55, 56 Nose, T., see Hiraoka, K. 8, 321, 326, 327, 347 Nounesis, G., see Garland, C.W. 375 Ober, C.K., see Barclay, G.G. 96, 97 Ober, C.K., see Colby, R.H. 312 Ober, C.K., see Ortiz, C. 218, 219 Obertur, R.C., see Ohm, H.G. 56 Ohm, H.G. 56 Okano, K., see Imura, H. 30, 303 Olivier, B.J., see Bellini, T. 211, 376 Olmsted, P.D. 154, 159, 271–273, 329, 375–377 Orlov, V.P., see Belyakov, V.A. 34, 246–248 Orsay Group on Liquid Crystals 42 Ortiz, C. 218, 219 Osborne, M.J. 356, 366 Osipov, M.A. 30 Osipov, M.A., see Saipa, A. 328 Ostrovskii, B.I. 45, 46 Otmakhova, O.A., see Yuranova, T.I. 97, 98 Pakula, T. 318

Palffy-Muhoray, P., see Alvarez, E. 254 Palffy-Muhoray, P., see Camacho-Lopez, M. 110, 111 Palffy-Muhoray, P., see Finkelmann, H. 254 Parisi, G., see Mezard, M. 211 Pasini, P. 213 Patel, J.S. 35 Patton, B.R., see Andereck, B.S. 375 Pearson, D.S., see Curro, J.G. 297 Peck, R., see Elias, F. 211, 212, 215, 313 Pelcovits, R.A. 236, 238–241 Pelcovits, R.A., see Grinsten, G. 366 Percec, V. 53 Pereira, G.G. 413 Pereira, G.G., see Finkelmann, H. 109, 110, 131, 134, 136 Petridis, L. 213, 216 Petschek, R.G. 62 Pikin, S.A. 26, 39, 44, 45 Pincus, P., see Curro, J.G. 297 Pink, R., see Thomsen, D.L. 107 Plate, N.A. 53 Plate, N.A., see Freidzon, Y.S. 221 Plate, N.A., see Shibaev, V.P. 38, 53, 54, 221 Plate, N.A., see Talroze, R.V. 53 Plate, N.A., see Zubarev, E.R. 97, 138, 177, 351 Pleiner, H. 35 Pleiner, H., see Brand, H.R. 298 Podneks, V.E., see Anisimov, M.A. 375 Pond, T.J. 297 Pople, J.A., see Lacey, D. 104 Popov, Y.O. 139, 173 Portugal, M., see Shibaev, V.P. 53 Prins, W., see Dusek, ˇ K. 68 Proceedings of IUTAM Symposium 426 Prost, J.P., see de Gennes, P.-G. 9, 12, 22, 25–27, 29, 30, 33, 36, 39, 41, 43, 44, 57, 100, 237, 282, 285, 299, 304, 355, 372, 380 Rabinovich, A.Z., see Ostrovskii, B.I. 45, 46 Radzihovsky, L. 375, 376 Radzihovsky, L., see Bellini, T. 376 Radzihovsky, L., see Saunders, K. 376 Radzikhovsky, L., see Lubensky, T.C. 77, 81, 164, 265, 269 Raja, V.N., see Chandrasekhar, S. 14 Ramaswamy, S., see Mazenko, G.F. 364, 366 Ratna, B.R., see Chandrasekhar, S. 14 Ratna, B.R., see Thomsen, D.L. 107 Reck, B., see Zentel, R. 58, 108, 221 Reckert, G., see Bualek, S. 221 Reckert, G., see Zentel, R. 58, 97, 108, 221 Rehage, G., see Finkelmann, H. 53, 97, 99, 100, 221, 317, 321 Rhodes, M.B. 282 Ringsdorf, H., see Disch, S. 98

AUTHOR INDEX Ringsdorf, H., see Shibaev, V.P. 53 Roberts, A.D. 289, 297 Roberts, P.M. 107, 148, 149 Rodigheiro, E., see Ortiz, C. 218 Rojstaczer, S., see Hashimoto, T. 282 R¨ossle, M. 343 R¨ossler, M., see Stannarius, R. 326, 329 Rouse, P.E. 291 Roux, D., see Safinya, C.R. 365 Rusakov, V.V. 70, 141 Safinya, C.R. 365 Safinya, C.R., see Als-Nielsen, J. 365, 370 Sagano, W., see Hiraoka, K. 8, 321, 326, 327, 347 Sahlen, F., see Andersson, H. 318 Saipa, A. 328 Samulski, E.T., see Deloche, B. 71, 140 Samulski, E.T., see Madsen, L.A 414 Sanchez-Ferrer, A., see Lebar, A. 101, 213 Saslow, W.M., see Chudnovsky, E.M. 216 Saunders, K. 376 Saupe, A., see Maier, W. 21 Savenkov, G.N., see Yuranova, T.I. 97, 98 Sch¨atzle, J. 107 Scheffer, T.J., see Berreman, D.W. 34, 248 Scheibe, P., see Harris, K.D. 111 Schmidt, C., see Disch, S. 100, 101 Schmidt, C., see Fischer, P. 39, 319, 320 Schmidtke, J. 193, 253–255, 286–288 Schnur, J.M., see Goldberg, L.S. 253 Sch¨onstein, M. 285–287, 300 Schumacher, P., see Disch, S. 98 Schuring, H. 321 Seddon, J.M. 38 Semenov, A.N., see Popov, Y.O. 139, 173 Semmler, K. 319 Semmler, K., see Benn´e, I. 321 Serak, S. V., see Alvarez, E. 254 Serota, R.A., see Chudnovsky, E.M. 216 Shadashiva, B.K., see Chandrasekhar, S. 14 Shaeffer, D.W, see Bellini, T. 211, 376 Shao, H., see Wong, G.C.L. 370, 371, 374 Shashidhar, R., see Spector, M.S. 343 Shelley, M., see Camacho-Lopez, M. 110, 111 Shenoy, D., see Thomsen, D.L. 107 Shibaev, V.P. 38, 53, 54, 221 Shibaev, V.P., see Freidzon, Y.S. 221 Shibaev, V.P., see Plate, N.A. 53 Shibaev, V.P., see Talroze, R.V. 53 Shiwaku, T., see Hashimoto, T. 282 Shliomis, M.I., see Rusakov, V.V. 70, 141 Shtrikman, S. 248 Shtrikman, S., see Hornreich, R.M. 248 Siemensmeyer, K., see Brehmer, M. 97, 321, 381 Singer, S.J. 44, 372, 373 Sinha, S.K., see Safinya, C.R. 365

405

Sixou, P., see d’Allest, J.F. 53, 55 Skacej, ˇ G. 190 Skacej, ˇ G., see Pasini, P. 213 Skupin, H., see Lehmann, W. 322, 381 Smith, G.S., see Safinya, C.R. 365 Sommers, C, see Hornreich, R.M. 248 Sonin, A.S., see Ostrovskii, B.I. 45, 46 Spector, M.S. 343 Stannarius, R. 326, 328, 329, 333 Stannarius, R., see Deindorfer, P. 328 Stannarius, R., see K¨ohler, R. 327 Stannarius, R., see Schuring, H. 321 Staudinger, H. 47 Stauffer, D. 63 Stein, P. 288 Stein, P., see Hiraoka, K. 326 Stein, P., see Lehmann, W. 379 Stein, R.S., see Hashimoto, T. 282 Stein, R.S., see Rhodes, M.B. 282 Stenull, O. 164, 344, 345 Stephen, M.J. 14, 31 Stille, W., see Schmidtke, J. 193, 253–255, 286–288 Stille, W., see Sch¨onstein, M. 285–287, 300 Straley, J.P. 24, 33 Straley, J.P., see Stephen, M.J. 14, 31 Strezelecki, L., see Bouligand, Y. 96 Strobl, G., see Schmidtke, J. 193, 286–288 Strobl, G., see Sch¨onstein, M. 285–287, 300 Struik, L.C.E. 309 Strukov, B.A., see Ostrovskii, B.I. 45, 46 Strzelecki, L. 96 Strzelecki, L., see Meyer, R.B. 44

Taheri, B., see Alvarez, E. 254 Taheri, B., see Finkelmann, H. 254 Tajbakhsh, A.R. 107, 108, 128 Tajbakhsh, A.R., see Ahir, S.V. 107 Tajbakhsh, A.R., see Cicuta, P. 231, 251–253 Tajbakhsh, A.R., see Clarke, S.M. 107, 127, 128, 139, 182, 219, 288, 298, 310 Tajbakhsh, A.R., see Courty, S. 224, 245 Tajbakhsh, A.R., see Cviklinski, J. 109, 131–133 Tajbakhsh, A.R., see Hogan, P.M. 109, 131 Takano, H., see Edwards, S.F. 151, 294–296 Takigawa, T., see Urayama, K. 178, 189, 280 Talroze, R.V. 53, 218 Talroze, R.V., see Shibaev, V.P. 221 Talroze, R.V., see Yuranova, T.I. 97, 98 Talroze, R.V., see Zubarev, E.R. 97, 138, 177, 199, 208–210, 351 Tanaka, R., see Tsutsui, T. 223 Taratuta, V.G., see Anisimov, M.A. 375 Teixeira, J., see d’Allest, J.F. 53, 55 ten Bosch, A., see d’Allest, J.F. 53, 55

406

AUTHOR INDEX

Terentjev, E.M. 28, 62, 178, 188, 191, 192, 211, 238, 280, 326, 327, 350, 366, 369, 378–380 Terentjev, E.M., see Ahir, S.V. 107 Terentjev, E.M., see Bladon, P. 231 Terentjev, E.M., see Brodowsky, H.M. 326 Terentjev, E.M., see Cicuta, P. 231, 251–253 Terentjev, E.M., see Clarke, S.M. 105, 107, 127, 128, 139, 182, 215, 219, 283, 288, 298, 310, 314, 376 Terentjev, E.M., see Courty, S. 224, 245 Terentjev, E.M., see Cviklinski, J. 109, 131–133 Terentjev, E.M., see Edwards, S.F. 151, 294–296 Terentjev, E.M., see Elias, F. 211, 212, 215, 313 Terentjev, E.M., see Finkelmann, H. 177, 198, 199, 201, 425 Terentjev, E.M., see Fridrikh, S.V. 212, 216, 218, 274 Terentjev, E.M., see Hogan, P.M. 109, 131 Terentjev, E.M., see Hotta, A. 219, 315 Terentjev, E.M., see Kutter, S. 72, 73, 151, 295 Terentjev, E.M., see Lubensky, T.C 350, 353 Terentjev, E.M., see Mao, Y. 150, 151, 229, 230, 232, 233 Terentjev, E.M., see Olmsted, P.D. 329, 375–377 Terentjev, E.M., see Osborne, M.J. 356, 366 Terentjev, E.M., see Osipov, M.A. 30 Terentjev, E.M., see Petridis, L. 213, 216 Terentjev, E.M., see Petschek, R.G. 62 Terentjev, E.M., see Tajbakhsh, A.R. 107, 108, 128 Terentjev, E.M., see Verwey, G.C. 170, 175, 187, 203, 204, 275 Terentjev, E.M., see Warner, M. 154 Thirion, P., see Chasset, R. 296, 314 Thomsen, D.L. 107 Tobolsky, A.V., see Green, M.S. 293 Tolksdorf, C., see K¨ohler, R. 327 Tolksdorf, C., see Lehmann, W. 322, 381 Tolksdorf, C., see Schuring, H. 321 Tolksdorf, C., see Stannarius, R. 326, 328, 333 Toner, J. 375 Toner, J., see Bellini, T. 376 Toner, J., see Mazenko, G.F. 364, 366 Toner, J., see Radzihovsky, L. 376 Toner, J., see Saunders, K. 376 Treloar, L.R.G. 70, 74, 75, 150 Tsutsui, T. 223 Tur, M., see Shtrikman, S. 248 Turok, N., see Chuang, I. 211 Twieg, R., see Alvarez, E. 254 Uchida, N. 212 Uematsu, Y., see Hiraoka, K. 326 Urayama, K. 178, 189, 280 Vallerien, S.U. 241

van der Hulst, H.C. 283 van der Meer, B.W. 33 van Haaren, J.A.M.M., see Broer, D.J. 249, 253 van Oosten, C.L., see Harris, K.D. 111 van Saarloos, W., see Anisimov, M.A. 375 Vasilets, V.N., see Yuranova, T.I. 97, 98 Vasilets, V.N., see Zubarev, E.R. 97 Vertogen, G., see van der Meer, B.W. 33 Verwey, G.C. 131, 170, 173, 175, 184–187, 203, 204, 275 Verwey, G.C., see Terentjev, E.M. 28, 62 Vilgis, T.A., see Edwards, S.F. 51, 71, 294 Vilgis, T.A., see Kholodenko, A.L. 51, 71 Vilgis, T.A., see Warner, M. 124 Virasoro, M.A., see Mezard, M. 211 Vithana, H.K.M., see Kopp, V.I. 254 Volovik, G.E. 211, 303 Voronov, V.P., see Anisimov, M.A. 375

Wagenblast, G., see Brehmer, M. 97, 321, 381 Wagner, M., see Ortiz, C. 219 Walba, D.M., see Saipa, A. 328 Wang, X.-J. 56 Warner, M. 110, 124, 154, 183, 236, 269, 271, 414 Warner, M., see Adams, J.M. 328, 329, 335, 344, 345, 353 Warner, M., see Ball, R.C. 51, 295 Warner, M., see Bermel, P.A. 247, 248 Warner, M., see Bladon, P. 14, 71, 231 Warner, M., see Clarke, S.M. 107, 288, 298 Warner, M., see Corbett, D. 111 Warner, M., see Finkelmann, H. 107, 109, 110, 127, 128, 130, 131, 134, 136, 144, 177, 198, 199, 201, 408, 410, 413, 425 Warner, M., see Kutter, S. 253 Warner, M., see Lubensky, T.C 350, 353 Warner, M., see Mao, Y. 150, 151, 229, 230, 232, 233, 243, 244 Warner, M., see Pereira, G.G. 413 Warner, M., see Terentjev, E.M. 28, 62, 178, 188, 191, 192, 280, 326, 327, 350, 366, 369, 378–380 Warner, M., see Verwey, G.C. 131, 170, 173, 175, 184–187, 203, 204, 275 Warner, M., see Wang, X.-J. 56 Weil, R. 47 Weilepp, J. 338, 350, 358, 373 Weiss, R.A., see Arrighi, V. 51 Wermter, H., see Finkelmann, H. 107 Weslowski, B.T., see Spector, M.S. 343 Wichterle, O., see Matejka, ˇ L. 69, 70 Windle, A.H., see Anwer, A. viii, 102 Windle, A.H., see Donald, A.M. 53 Wong, G.C.L. 370, 371, 374 Wu, L., see Bellini, T. 211, 376

AUTHOR INDEX Xing, X., see Lubensky, T.C. 77, 81, 164, 265, 269 Yablonovitch, E. 253 Yamamoto, J., see Nishikawa, E. 319, 320 Yamamoto, J., see Terentjev, E.M. 191, 192 Yamamoto, M. 293 Yang, D.-K. 223 Yokoyama, H., see Nishikawa, E. 319, 320 Young, M.J., see Garland, C.W. 375 Ypma, J.P.J., see van der Meer, B.W. 33 Yu, Y. 111 Yuranova, T.I. 97, 98 Yuranova, T.I, see Zubarev, E.R. 199, 208–210 Yurke, B., see Chuang, I. 211 Zalar, B., see Lebar, A. 101, 213 Zannoni, C., see Pasini, P. 213 Zannoni, C., see Skacej, ˇ G. 190 Zentel, R. 58, 97, 103, 108, 178, 189, 221, 317, 322, 351 Zentel, R., see Beyer, P. 328 Zentel, R., see Brehmer, M. 97, 321, 381 Zentel, R., see Brodowsky, H.M. 326 Zentel, R., see Bualek, S. 221 Zentel, R., see Deindorfer, P. 328 Zentel, R., see Gebhard, E. 322 Zentel, R., see Kapitza, H. 321 Zentel., R., see K¨ohler, R. 327 Zentel, R., see Lehmann, W. 322, 381 Zentel, R., see Maxein, G. 225, 227 Zentel, R., see Pakula, T. 318 Zentel, R., see R¨ossle, M. 343 Zentel, R., see Schuring, H. 321 Zentel, R., see Stannarius, R. 326, 328, 329, 333 Zentel, R., see Vallerien, S.U. 241 Zentel, R., see Wong, G.C.L. 370, 371, 374 Zhestkov, A.Z., see Grosberg, A.Y. 62 Zimm, B.H. 293 Zippelius, A., see Broderix, K. 63 Zubarev, E.R. 97, 138, 177, 199, 208–210, 218, 351 Zubarev, E.R., see Yuranova, T.I. 97, 98 Zumer, S., see Lebar, A. 101, 213

407