Defects in Liquid Crystals: Computer Simulations, Theory and Experiments

Transcription

1 Defects in Liquid Crystals: Computer Simulations, Theory and Experiments

2 NATO Science Series A Series presenting the results ofscientific meetings supported under the NATO Science Programme. The Series is published by los Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I. Life and Behavioural Sciences II. Mathematics, Physicsand Chemistry III. Computerand Systems Science IV. Earth and Environmental Sciences V. Science and Technology Policy los Press Kluwer Academic Publishers los Press Kluwer Academic Publishers los Press The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are "Advanced Study Institutes" and "Advanced Research Workshops, although other types of meeting are supported from time to time. The NATO Science Series collects together the results of these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from NATO's Partner countries - countries of the CIS and Central and Eastern Europe. Advanced StUdy Institutes are high-level tutorial courses offering in-depth study of latest advances ina field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above. Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series. hltp;// I -~ ~ I Series II: Mathematics, Physics and Chemistry - Vol. 43

3 Defects in Liquid Crystals: Computer Simulations, Theory and Experiments edited by Oleg D. Lavrentovich Liquid Crystallnstitute and Chemical Physics Interdisciplinary Program, Kent State University, Kent, Ohio, U.S.A. Paolo Pasini Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy Claudio Zannoni Dipartimento di Chimica Fisica ed Inorganica, Universita di Bologna, Italy and Siobodan LUmer Physics Department, University of Ljubljana, Slovenia " ~. Springer-Science+Business Media, B.V.

4 Proceedings of the NATO Advanced Research Workshop on Computer Simulations of Defects in Liquid Crystals Including their Relation to Theory and Experiment Erice, Sicily, ltaly September 2000 A C.1. P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper AII Rights Reserved 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1 st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized any form or by any means, electronic or mechanical, including photocopying, recognized or by any information storage and retrieval system, without written permission fram copyright owner.

5 CONTENTS Preface xiii 1 Classification of defects in liquid crystals H.-R. Trebin 1 Introduction Order and defects in a prototype model: the planar ferromagnet or the XY-model Perfect order The distorted ferromagnet Singularities Classification of defects Testloops and winding numbers Defect equivalence and stability Defect processes and group structure. 7 4 Further examples of reduced order parameter spaces and fundamental groups Two-dimensional nematics Two-dimensional smectics without rotations Two-dimensional periodic crystals Two-dimensional smectics with rotations. 9 5 Singularities in three dimensions Line singularities The three-dimensional ferromagnet (Heisenberg-model) and point singularities Defects of three-dimensional nematics Biaxial nematics and cholesteric liquid crystals Homotopy classification of defects in three dimensions 13 6 Topological solitons Planar and linear topological solitons Periodic boundary conditions 15 7 Summary and conclusions

6 vi 2 Alignment tensor versus director description in nematic liquid crystals A.M. Sonnet and S. Hess 1 Introduction... 2 Description of the Alignment Distribution Function and Averages 2.2 Alignment Tensor. 3 Dynamic equation for the alignment tensor 3.1 Director description 3.2 Tensor description. 4 Examples. 4.1 Visualization and s=1/2 disclination 4.2 Capillary 4.3 Droplet Liquid crystal colloidal dispersions 37 H. Stark, A. Borstnik and S. turner 1 Introduction Colloidal Dispersions in a Nematic Solvent What is the Director Field? One-Particle Properties Two-Particle Interactions in a Nematic Solvent Nematic Colloidal Dispersions in Complex Geometries 56 3 Colloidal Dispersions above the Nematic-Isotropic Phase Transition Stability criteria for colloidal dispersions Noncharged liquidcrystal colloidal dispersionsat temperatures above TNl Charged spherical particles immersed in an isotropic liquid crystal 74 4 Conclusions Computer simulations and defects in confined liquid crystal lattice models 87 C. Chiccoli, P. Pasini, 1. Feruli and C. Zannoni 1 Introduction... 2 Liquid Crystal Lattice Models The Lebwohl-Lasher model 2.2 Monte Carlo simulations

7 3 4 5 Polymer Dispersed Liquid Crystals Radial Boundary Conditions (RBC) Nematic Films. 4.1 Hybrid aligned cell. 4.2 Schlieren textures in planar aligned cells. Conclusions vii Molecular simulations and theory of planar interfaces and defects in nematic liquid crystals 113 M.P. Allen Computer simulations and theoretical approaches. 1.1 Simulation Models Simulation Methods. 1.3 Coarse-grained descriptions.. Liquid crystals between parallel walls. 2.1 Surface anchoring coefficient 2.2 Elastic boundary condition.. Nematic-isotropic interface. 3.1 Orientational order and density profiles 3.2 Results at high elongation. 3.3 Surface Tension Capillary waves Disclination in cylindrical pores. Defects near a colloid particle 5.1 Saturn ring defect 5.2 Satellite defect Off-center ring.. Acknowledgements Topological defect behavior in a quenched nematic liquid crystal 141 R.A. Pelcovits, J.L. Billeter, A.M. Smondyrev and G.B. Loriot 1 Introduction Numerical simulations Defect-finding algorithms Results Coarsening sequence Real-space correlation function Structure factor Conclusions

8 viii 6 Acknowledgements Restoring forces on nematic disclinations 167 R. Rosso and E. G. Vi1:ga 1 Introduction Restoring force and torque Core's influence Elliptic core Director's moulding Boundary's influence Challenges in the dynamics of point defects A.M. Sonnet and E. G. Virga Introduction. Elastic force... Defect drift... Discretization of planar director fields Artefacts in numerical defect dynamics Numerical simulation of elastic anisotropy in nematic liquid crystalline polymers 201 H. Tu, G. Goldbeck- Wood and A.H. Windle 1 Introduction Model The calibration of the model Reproducing the Freedericksz transitions Topological defects in thin films Coarsening of the defects Disclinations in bulk samples Twist type or wedge type Escaped integer disclinations Texture evolution under shear flow Out-of-plane Interaction of wedge disclination pairs under shear flow220 6 Summary

9 ix 10 Computer Simulations and Fluorescence Confocal Polarizing Microscopy of Structures in Cholesteric Liquid Crystals 229 S. V. Shiyanovskii, I.I. Smalyukh and D.D. Lavrentovich 1 Introduction Mathematical basis for computer simulations of equilibrium LC structures LC structure and elastic properties Electric field Surface anchoring Domain and scheme of simulations Confocal Microscopy and Fluorescence Confocal Microscopy Confocal Microscopy Fluorescence Confocal Microscopy Fluorescence Confocal Polarizing Microscopy Theoretical foundations ofconfocal microscopy oforientation patterns Experimental set up and materials Experimental set up Liquid crystal materials and fluorescent probes Results and Discussion FCPM of Fredericks transition FCPM of twisted nematic cell The FCPM imaging of basic cholesteric textures Field-induced cholesteric stripes in thin cells with planar boundary conditions Cholesteric stripesin thin cells with homeotropic boundary conditions Conclusions Acknowledgements Defects and Undulation in Layered Liquid Crystals 271 T. Ishikawa and D. D. Lavrentovich 1 Introduction Bulk Elastic and Surface Properties of Lamellar Systems Weakly distorted Smectic A phase in two dimensions Lubensky-de Gennes coarse-grained model ofthe cholesteric phase Surface Anchoring Edge Dislocations Linear Theory.. 279

10 x 3.2 Experimental Dislocation Profile and Non-linear Theory Layers undulations (Helfrich-Hurault instability) Helfrich-Hurault model Undulations Profile near the Threshold: Experiment Undulations in a Cell with a Finite Anchoring Undulations profile well above the Threshold Conclusion Liquid crystals under shear: role of defects M. Kleman and C. Meyer Introduction Disclinations Generalities Instabilities and defects in the LE regime Instabilities and defects in the Doi regime (De» 0.1) Isolated disclination Dislocations vs disclinations Core structure and physical properties Smectics General relationships Movement of isolated dislocations in a SmA phase Collective behavior of dislocations: climb of edges Collective behavior of dislocations: glide of screws A particular case ofreorganization in smectic phases: the onion texture Numerical simulation of defects in quasicrystals 323 H.-R. Trebin 1 Crystals and quasicrystals Characteristic features of crystals and quasicrystals Structure models for quasicrystals Model quasicrystals The cut formalism and phason-degree of freedom The cut formalism Phonon and phason degrees of freedom Dislocations in crystals and quasicrystals Geometric basis Experimental observations Atomic interaction and numerical algorithm. 331

11 3.4 Dislocation motion in the two-dimensional model Dislocation motion in the three-dimensional model Crack propagation Summary and conclusion Acknowledgement xi Index 337

12 A boojum in Erice "Returning from cool Florida to frigid Ithaca, I set to work preparing a set of lecture notes that I delivered that June at a summer school in Erice, a mountain-top Sicilian town three thousand feet straight above the sea, whose streets and alleys are paved in geometrical patterns of massive stones, polished smooth by feet and wheels. The views in all directions were spectacular and the weather was neither too hot nor too cold. Occasionally a cloud would settle over the mountain top for a day or two sending cool mists swirling through the steep alleyways. The perfect place to meet a boojum, though nobody ever did." N.D. Mermin, "Boojums all the way through", Cambridge U.P., PREFACE Liquid crystals remain in the focus of intensive interdisciplinary studies as an example of a well- defined soft condensed system with a rich variety of supra-molecular structures. The emphasis of research is however shifting more and more towards systems and complex geometries in which the liquid crystalline order is spatially non-uniform. This spatial non-uniformity can be caused by the heterogeneous character of the system, in which the liquid crystal is only one of few components, as in polymer-dispersed liquid crystals, in silica filled nematics or in liquid crystalline emulsions. All these systems are often populated by topological defects. Topological defects correspond to configurations of the order parameter that cannot be transformed continuously into a uniform state. They can occur during the symmetry breaking phase transitions, under an exxiii

13 XIV ternal field, or simply be a necessary element of the equilibrium state. For example, a sufficiently large spherical nematic droplet with perpendicular boundary conditions contains a radial point defect in equilibrium. Topological defects are of great interest not only in condensed matter physics, but also in particle physics and cosmology. A relative ease of experimenting with liquid crystals makes them well suited for the studies of defects. Furthermore, the rich variety ofmolecular order makes the liquid crystals a "drosophila fly" of the modern theoretical classification of defects based on homotopy groups. The microscopic organizations in systems with defects are often too complex to be treated by standard experimental and analytical tools. Computer simulations have thus become an indispensable part of defect studies, providinga set ofuseful and powerful tools for investigations starting from microscopic molecular interactions. They have matured over the last few years to capture successfully such features as order parameter gradients in the core of defects and far from the core and to incorporate the role of bulk elasticity, surface interactions, external fields. The continuous increase of computing power and the development of efficient simulation techniques now allow a deeper insight into the properties of systems with defects. The purposeofthe NATO Advanced Research Workshop on "Computer Simulations of Defects in Liquid Crystals Including their Relation to Theory and Experiment" of the International School of Liquid Crystals was to bring together specialists in the computer simulations, mathematics and physics of defects, both theorists and experimentalists. The Workshop, held at the Ettore Majorana Centre of Scientific Culture (EMFCSC) in Erice, Sicily from September 18th to September 23rd, 2000, was fortunate to have an excellent panel of international Lecturers and has attracted researchers from 15 different countries. The unusual combination of experts of different techniques brought a fruitful comparison of approaches and a very lively discussion throughout the meeting. The present book of the NATO Science Series contains a large portion of the lectures given at the NATO ARW. It starts with a general tutorial on the classification of defects in liquid crystals (Trebin), and an analysis of different approaches in simulating complex order parameter configurations (Sonnet and Hess). The next chapter (Stark et al.) discusses liquid crystal colloidal dispersions. Dispersions with liquid crystalline components are intimately related to the study of defects as in many cases the balance of surface and bulk energies leads to equilibrium states with topological defects. The influence of spatial confinement on the behavior of defects is further discussed in the chapters by Chiccoli et al. and by Allen. Pelcovits et al. present computer simulations of phase ordering dynamics during an isotropic-to-nematic phase transition in which the behavior of topological

14 defects determines theequilibrationprocess. Rosso and Virgaconsiderforce and torques acting on a linear nematic defect, a disclination. Sonnet and Virga then discuss the advantages of using the so-called alignment tensor description in computer simulations of point defects in nematics. The second half of the book discusses media that are somewhat more complex than low-molecular weight nematic liquid crystals. Tu et al. present the results of computer simulations of director configurations in nematic polymers. Shiyanovskii et al. describe a recently developed fluorescent polarizing confocal microscopy technique used in conjunction with computer simulations to image 3D director patterns in cholesteric liquid crystals. Cholesteric liquid crystals are used by Ishikawa and Lavrentovich as a model medium to study properties of defects (edge dislocations) and elastic instabilities (layers undulations) in lamellar systems. Kleman and Meyer present an overview of the role of defects such as dislocations and disclinations in the behavior of nematic and smectic liquid crystals under shear. The last chapter (Trebin), devoted to dynamics of dislocations accompanied by phasonic fields in quasicrystals, goes beyond the field of computer simulations in liquid crystals. The successful organizationofthe NATO ARW was only possible thanks to the help of many people and we would like to mention in particular Drs. R. Berardi and S. Orlandi. Moreover, it is a pleasure to thank the Director of EMFCSC, Prof. A. Zichichi, and the competent and helpful staff of the Erice Center. Last but not least we wish to thank NATO and its Science Committee, in particular Dr. F. Pedrazzini, for the essential support which made the organization of the Workshop and the production of this book possible. We also acknowledge financial contributions from Societ Italiana Cristalli Liquidi (SICL), Ente per Ie Nuove tecnologie, l'energia e l'ambiente (ENEA), and the Italian Consortium for Science and Technology of Materials (INSTM ). xv Oleg D. Lavrentovich Kent State University Paolo Pasini INFN, Bologna Claudio Zannoni Universita di Bologna Slobodan Zumer University of Ljubljana