Partially Ordered Systems

Transcription

1 Partially Ordered Systems Editorial Board: Lui Lam Department of Physics San Jose State University One Washington Square San Jose, CA USA Dominique Langevin Laboratoire de Physique ENS 24 Rue Lhomond F Paris, Cedex 05 France Advisory Board: J. Charvolin W. Helfrich P.A. Lee J.D. Litster D.R. Nelson M. Schadt Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

2 Partially Ordered Systems Editorial Board: L. Lam D. Langevin Solitons in Liquid Crystals Lui Lam and Jacques Prost, Editors Bond-Orientational Order in Condensed Matter Systems Katherine J. Strandburg, Editor Diffraction Optics of Complex-Structured Periodic Media V.A. Belyakov Fluctuational Effects in the Dynamics of Liquid Crystals E.I. Kats and V.V. Lebedev Nuclear Magnetic Resonance of Liquid Crystals Ronald Y. Dong Electrooptic Effects in Liquid Crystal Materials L.M. Blinov and V.G. Chigrinov Liquid Crystalline and Mesomorphic Polymers Valery P. Shibaev and Lui Lam, Editors Micelles, Microemulsions and Monolayers W. Gelbart, A. Ben-Shaul, and D. Roux Pattern Formation in Liquid Crystals A. Buka and L. Kramer, Editors

3 Agnes Buka Editors Lorenz Kramer Pattern Formation in Liquid Crystals With 89 Illustrations, Springer

4 Agnes Buka Research Institute for Solid-State Physics Hungarian Academy of Sciences Budapest M.u Hungary Lorenz Kramer Institut fur Physik Universitat Bayreuth Bayreuth D-9S440 Germany Editorial Board: Lui Lam Dominique Langevin Advisory Board: 1. Charvolin Directeur Adjoint Institut Laue-Langevin F Grenoble Cedex 9 FRANCE John D. Litster Francis Bitter National Magnet Laboratory Massachusetts Institute of Technology Cambridge, MA USA W. Helfrich Freie Universitat Berlin: Fachbereich Physik Institut fiir Theorie der Kondensierten Materie 1000 Berlin 33 GERMANY David R. Nelson Department of Physics Harvard University Cambridge, MA USA Patrick A. Lee Department of Physics Massachusetts Institute of Technology Cambridge, MA Martin Schadt Department ZFE/RLC F. Hoffman-La Roche & Co. CH-4002 Basel Switzerland Library of Congress Cataloging-in-Publication Data Pattern formation in liquid crystals I [edited by] Agnes Buka, Lorenz Kramer. p. em. - (Partially ordered systems) Includes bibliographical references and index. ISBN-13: e-isbn-13: : / Liquid crystals. 2. Pattern perception. I. Buka, Agnes. II. Kramer, L. III. Series. QD923.P '29-dc Printed on acid-free paper Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1 st edition 1996 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Robert Wexler; manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared by The Bartlett Press

5 Preface In the last 20 years the study of nonlinear nonequilibrium phenomena in spatially extended systems, with particular emphasis on pattern-forming phenomena, has been one of the very active areas in physics, exhibiting interesting ramifications into other sciences. During this time the study of the "classic" systems, like Rayleigh-Benard convection and Taylor vortex flow in simple fluids, has also been supplemented by the study of more complex systems. Here liquid crystals have played, and are still playing, a major role. One might say that liquid crystals provide just the right amount and right kind of complexity. They are full of nonlinearities and give rise to new symmetry classes, which are sometimes actually simpler to deal with qualitatively, but they still allow a quantitative description of experiments in many cases. In fact one of the attractions of the field is the close contact between experimentalists and theorists. Hydrodynamic instabilities in liquid crystals had already experienced a period of intense study in the late 1960s and early 1970s, but at that time neither the experimental and theoretical tools nor the concepts had been developed sufficiently far to address the questions that have since been found to be of particular interest. The renewed interest is also evidenced by the fact that a new series of workshops has evolved. The first one took place in 1989 in Bayreuth and united participants from almost all groups working in pattern formation in liquid crystals. It was taken up in Kitakyushu, Japan (1991), Santa Fe, New Mexico, USA (1993) and Copenhagen, Denmark (1995) under the broader topic of pattern formation in nonequilibrium complex systems. There exist excellent proceedings for the workshops in Kitakyushu and Santa Fe, which are frequently referred to in this book. Also, in the last International Liquid Crystal Conferences, which take place in even years, sessions on pattern formation were organized. The idea for this book, with contributions comprising reviews and introductory chapters, was conceived some years ago, but it was a letter from Hiap L. Ong (now at Prime View International Co., Ltd., Taiwan) that gave the critical impetus for its realization. We are grateful to the authors who have taken the task of writing a survey over their speciality-of course with particular emphasis on their own work and preferences-so seriously and have gone to great pains in the careful preparation of the manuscripts. The continuing state of activity and evolution of the field is evidenced by the fact that in several contributions material can be found that is yet unpublished at

6 vi Preface this time. Unfortunately not all major activities could be included in this volume. In the Introduction we have given very brief surveys over two such topics. We hope these (and other) topics Can be included more extensively in a second edition of this book. Financial support by the Volkswagen Foundation through a cooperation grant is gratefully acknowledged. Finally it is our pleasure to thank the staff of Springer Verlag, New York, for their kind support during the preparation and processing of the manuscript. Bayreuth and Budapest, July 1995 Agnes Buka and Lorenz Kramer

7 Contents Preface List of Contributors 1 Introduction to Pattern Formation in Nonequilibrium Systems Lorenz Kramer and Agnes Buka 1.1 General Remarks A Simple Model Pattern Formation in Liquid Crystals Transient Patterns in the Freedericksz Transition Patterns in Rotating Magnetic and Electric Fields References 2 Hydrodynamics and Electrohydrodynamics of Liquid Crystals Harald Pleiner and Helmut R. Brand 2.1 Introduction Symmetries and Broken Symmetries Conservation Laws Broken Symmetries Slowly Relaxing Variables. 2.3 Statics Thermodynamics Energy and Thermodynamic Forces 2.4 Dynamics Reversible Currents Irreversible Currents 2.5 Electrohydrodynamics External Fields Statics and Dynamics 2.6 Additions to Nematodynamics Fluctuating Forces Biaxial Nematics Order Parameter Variable v xi

8 viii Contents Side-Chain Polymers Nonlinearities and Higher-Order Gradient Terms 2.7 Director-Type Degrees of Freedom Smectic A Liquid Crystals Cholesteric Liquid Crystals Smectic C, C*, CM, and C Liquid ~ Crystals Smectic F, I, and L Liquid Crystals Appendix. 51 References 57 3 General Mathematical Description of Pattern-Forming Instabilities 69 Werner Pesch and Lorenz Kramer 3.1 Introductory Remarks Linear Analysis The Landau Equation The Ginzburg-Landau Equations Derivation Application of the Ginzburg-Landau Equations Extended Weakly Nonlinear Analysis Derivation of Order Parameter Equations From Order Parameter to Amplitude Equations Derivation of Coupled Amplitude Equations Concluding Remarks Swift-Hohenberg Equation Phase Equations 86 References 87 4 Flow Instabilities in Nematics 91 E. Dubois-Violette and P. Manneville 4.1 Introduction Continuous Description of Nematics and Viscometry Nematohydrodynamics Viscometry Apparent Non-Newtonian Behavior and Flow Alignment Anisotropy of Viscous Forces Viscous Relaxation of the Orientation, Flow, and the Ericksen Number Stability Analysis and Basic Mechanisms Stability Analysis The Pieranski-Guyon Mechanism Shear Flow Instabilities with the Director Perpendicular to the Shear Plane Simple Shear Flow Alternating Shear Flows Poiseuille Flow

9 Contents IX 4.5 Flow Instabilities with the Director Initially Parallel to the Shear Plane Elliptical Shear Instability in Homeotropic Configuration Experimental Results Theoretical Account Further Developments Appendix A: Linear stability problem when the director is perpendicular to the shear plane Appendix B: Elliptical Shear Equations 156 References Experiments on Thermally Driven Convection Guenter Ahlers 5.1 Introduction Instability Mechanisms Stability Analysis Pattern Formation Materials Planar Alignment and a Horizontal Magnetic Field Introductory Remarks Theoretical Predictions Experimental Results Homeotropic Alignment and a Vertical Magnetic Field General Remarks Heating from Below Heating from Above Two-Phase Convection Theoretical Predictions Experimental results Appendix A: Experimental Methods. 206 Appendix B: Physical Properties of 5CB. 211 References Electrohydrodynamic Instabilities in Nematic Liquid Crystals 221 Lorenz Kramer and Werner Pesch 6.1 Introduction General Considerations Theoretical preliminaries 6.2 Planar alignment: linear theory Conduction regime Dielectric regime 6.3 Planar alignment: nonlinear theory Results of Ginzburg-Landau Equation (GLE) Beyond the GLE. 6.4 Homeotropic alignment Case C

10 x Contents Case F Concluding remarks 247 References Mesophase Growth 257 John Bechhoefer 7.1 Introduction The Mullins-Sekerka Instability Undercooled Pure Material Thin Layer of a Binary Alloy in a Temperature Gradient Directional Growth Experiments The Initial Instability Secondary Instabilities Tertiary and Higher Instabilities Other Experimental Systems Free-Growth Experiments Microscopic-Solvability Theory Dendritic Growth at the Discotic-Isotropic Interface Dendritic Growth in Other Mesophase Systems Prospects. 283 References Viscous Fingering 291 Agnes Buka 8.1 Introduction Theoretical Background Experiments Isotropic Systems Anisotropic Systems Concluding Remarks. 303 References Thermal Fluctuations in Pattern Forming Instabilities 307 Martin Treiber 9.1 Introduction Macroscopic Stochastic Equations for Thermal Noise Stochastic Amplitude Equations Theoretical Results Rayleigh-Benard Convection Tay1or-Couette Flow Planar Electrohydrodynamic Convection Experimental Results Discussion 327 References 329 Index 333

11 List of Contributors Guenter Ahlers Department of Physics, University of California, Santa Barbara, California USA John Bechhoefer Department of Physics, Simon Fraser University, Burnaby, B.c. V5A IS6, Canada Helmut R. Brand Theoretische Physik III, University of Bayreuth, D Bayreuth, Germany Agnes Buka Research Insitute for Solid State Physics, H-1525 Budapest, POB. 49., Hungary Elisabeth Dubois-Violette Laboratoire de Physique des Solides, Batiment 510, Universite de Paris Sud, F Orsay Cedex, France Lorenz Kramer Institute of Physics, University of Bayreuth, D Bayreuth, Germany Paul Manneville Laboratoire d'hydrodynamique, Ecole Poly technique, F Palaiseau Cedex, France Werner Pesch Institute of Physics, University of Bayreuth, D Bayreuth, Germany Harald Pleiner Max-Planck-Institute for Polymer Research, D5502l, Mainz, Germany Martin Treiber Institute of Physics, University of Bayreuth, D Bayreuth, Germany