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Theory and applications of liquid crystals. (English) Zbl 0713.76006

The IMA Volumes in Mathematics and its Applications, Vol. 5. Institute for Mathematics and its Applications (IMA), University of Minnesota, Minneapolis. Nex York etc.: Springer-Verlag. XII, 353 p.; DM 75.00 (1987).
[The articles of this volume will not be indexed individually.]
(From the preface.) The diversity of experimental phenomena and the range of applications of liquid crystals present timely and challenging questions for experimentalists, mechanists, and mathematicians. The scope of this workshop was to bring together research workers and practitioners in these areas from laboratories, industry, and universities to explore common issues. The contents of this volume vary from descriptions of experimental phenomena, of which our understanding is insufficient, to questions of a mathematical nature and of efficient computation.
Interest in this area is stimulated by problems relating to the many familiar devices as well as by questions which arise in the processing of high strength polymer fibers such as Kevlar. From the standpoint of pure science, our concern is with mesomorphic phases of matter. These had received little or no serious mathematical treatment although the equations governing macroscopic behavior of small molecule liquid crystals are well established.
Among the workshops of the program, this was the most adventurous. In addition to describing recent activity in liquid crystal theory and experiment, our objective was to stimulate mathematical research connected to the discipline. Our thesis was that better mathematical understanding would lead to improved theory and more effective computational methods. Unlike most of the workshop topics, almost no mathematicians were engaged in liquid crystal research in January 1985. The contents of this volume are witness to the fruit of this effort. For example, the papers of Brezis, Cohen et. al., Hardt et. al., and Maddocks all report on investigations undertaken after the workshop took place. The paper of Maddocks attempts to place configurations with line singularities within a framework acceptable from the viewpoint of energetics. Those of Brezis, Cohen et. al., and Hardt et. al. establish, among other things, the notion of a stable point defect. This surprising phenomenon was discovered by a combination of analysis and computation and then precisely classified for harmonic mappings into spheres.
A brief introduction to the theory of small molecule liquid crystals is provided in the articles of Leslie. Various aspects of the study of liquid crystal polymers are presented by Berry, Doi, and Ryskin. The reader is introduced to blue phases in the papers of Cladis and Sethna. Phase transitions, especially connected to smectic states, are explored by Huang. The contributions of Capriz et. al., Choi, Di Benedetto, Miranda, and Spruck discuss mechanical or mathematical issues closely related to those encountered in the study of liquid crystals.
Contents:
G. Berry: Rheological and rheo-optical studies with nematogenic solutions of a rodlike polymer: A review of data on poly (phenylene benzobistiazole). H. Brezis: Liquid crystals and energy estimates for \(S^ 2\)-valued maps. G. Capriz and P. Giovine: On virtual inertia effects during diffusion of a dispersed medium in a suspension. H. I. Choi: Degenerate harmonic maps and liquid crystals. P. Cladis: A review of cholesteric blue phases. R. Cohen, R. Hardt, D. Kinderlehrer, S.-Y. Lin, and M. Luskin: Minimum energy configurations for liquid crystals: Computational results. E. Di Benedetto: The flow of two immiscible liquids through a porous medium: Regularity of the saturation. M. Doi: Molecular theory for the nonlinear viscoelasticity of polymeric liquid crystals. R. Hardt and D. Kinderlehrer: Mathematical questions of liquid crystal theory. C. C. Huang: The effect of the magnitude of the disordered phase temperature range on the given phase transition in liquid crystals. F. Leslie: Some topics in equilibrium theory of liquid crystals. F. Leslie: Theory of flow phenomena in nematic liquid crystals. J. Maddocks: A model of disclinations in nematic liquid crystals. M. Miranda: Some remarks about a free boundary type problem. G. Ryskin: Computer simulation of flow of liquid crystal polymers. J. Sethna: Theory of the blue phases of chiral nematic liquid crystals. J. Spruck: On the global structure of solutions to some semilinear elliptic problems.

MSC:

76-06 Proceedings, conferences, collections, etc. pertaining to fluid mechanics
00B25 Proceedings of conferences of miscellaneous specific interest
76A15 Liquid crystals