Table Of ContentThe IMA Volumes
in Mathematics
and Its Applications
Volume 5
Series Editors
Geroge R. Sell Hans Weinberger
Institute for Mathematics and Its Applications
IMA
The Institute for Mathematics and Its Applications was established by a grant from the
National Science Foundation to the University of Minnesota in 1982. The IMA seeks to en
courage the developent and study of fresh mathematical concepts and questions of concern to
the other sciences by bringing together mathematicians and scientists from diverse fields in
an atmosphere that will stimulate discussion and collaboration.
The IMA Volues are intended to involve the broader scientific community in this process.
Hans Weinberger, Director
George R. Sell, Associate Director
IMA Programs
1982-1983 Statistical and Continuum Approaches to Phase Transition
1983-1984 Mathematical Models for the Economics of Decentralized Resource Allocation
1984-1985 Continuum Physics and Partial Differential Equations
1985-1986 Stochastic Differential Equations and Their Applications
1986-1987 Scientific Computation
1987-1988 Applied Combinatories
1988-1989 Nonlinear Waves
Springer Lecture Notes from the IMA
The Mathematics and Physics of Disordered Media
Editors: Barry Hughes and Barry Ninham
(Lecture Notes in Mathematics, Volume 1035, 1983)
Orienting Polymers
Editor: J. L. Ericksen
(Lecture Notes in Mathematics, Volume 1063, 1984)
New Perspectives in Thermodynamics
Editor: James Serrin
(Springer-Verlag, 1986)
Models of Economic Dynamics
Editor: Hugo Sonncnschein
(Lecture Notes in Economics, Volume 264, 1986)
J .L . Ericksen D. Kinderlehrer
Editors
Theory and Applications
of Liquid Crystals
With 60 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
1.L. Ericksen
Institute for Mathematics and Its Applications
Minneapolis, Minnesota 55455 USA
David Kinderlehrer
Institute for Mathematics and Its Applications
Minneapolis, Minnesota 55455 USA
AMS Classification: 82 XX
Library of Congress Cataloging in Publication Data
Theory and applications of liquid crystals.
(The IMA volumes in mathematics and its applications; v. 5)
Includes bibliographies.
I. Liquid crystals. I. Ericksen, J. L.
(Jerald L.)' 1924- II. Kinderlehrer, David.
III. Series.
QD923.T46 1987 548'.9 87-9858
© 1987 by Springer-Verlag New York, Inc.
Softcover reprint of the hardcover I st edition 1987
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987654321
ISBN-13: 978-1-4613-8745-9 e-ISBN-13: 978-1-4613-8743-5
001: 10.1007/978-1-4613-8743-5
The IMA Volumes in Mathematics and
Its Applications
Current Volumes:
Volume 1: Homogenization and Effective Moduli of Materials and Media
Editors: Jerry Ericksen, David Kinderlehrer, Robert Kohn, and J .-L. Lions
Volume 2: Oscillation Theory, Computation, and Methods of Compensated
Compactness
Editors: Constantine Dafermos, Jerry Ericksen, David Kinderlehrer, and
Marshall Slemrod
Volume 3: Metastability and Incompletely Posed Problems
Editors: Stuart Antman, Jerry Ericksen, David Kinderlehrer, and Ingo MUller
Volume 4: Dynamical Problems in Continuum Physics
Editors: Jerry Bona, Constantine Dafermos, Jerry Ericksen, and
David Kinderlehrer
Volume 5: Theory and Applications of Liquid Crystals
Editors: Jerry Ericksen and David Kinderlehrer
Volume 6: Amorphous Polymers and Non-Newtonian Fluids
Editors: Constantine Dafermos, Jerry Ericksen, and David Kinderlehrer
Volume 7: Random Media
Editor: George Papanicolaou
Volume 8: Percolation Theory and Ergodic Theory of Infinite Particle
Systems
Editor: Harry Kesten
Forthcoming Volumes:
1985-1986: Stochastic Differential Equations and Their Applications
Hydrodynamic Behavior and Interacting Particle Systems
Stochastic Differential Systems, Stochastic Control Theory and Applications
1986-1987: Scientific Computation
Computational Fluid Dynamics and Reacting Gas Flows
Numerical Algorithms for Modem Parallel Computer Architectures
Numerical Simulation in Oil Recovery
Atomic and Molecular Structure and Dynamics
Contents
Foreword................................................................. ix
Preface .................................................................. xi
Rheological and Rheo-Optical Studies with Nematogenic Solutions of a
Rod I ike Polymer: A Review of Data on Poly (Phenylene Benzobisthiazole)
G. Berry
Liquid Crystals and Energy Estimates for S2-Valued Maps 31
H. Brez i s
On Virtual Inertia Effects During Diffusion of a Dispersed Medium in
a Suspens ion. ............ .................. ......... .................. ... 53
G. Capriz and P. Giovine
Degenerate Harmonic Maps and Liquid Crystals ............................. 63
H. I. Choi
A Review of Cholesteric Blue Phases...................................... 73
P. Cladi s
Minimum Energy Configurations for Liquid Crystals: Computational
Results .................................................................. 99
R. Cohen, R. Hardt, D. Kinderlehrer, S.-Y. Lin, and M. Luskin
The Flow of Two Immiscible Liquids Through a Porous Medium:
Regularity of the Saturation ............................................. 123
E. Di Benede t to
Molecular Theory for the Nonl inear Viscoelasticity of Polymeric
Liquid Crystals .......................................................... 143
M. Doi
Mathematical Questions of Liquid Crystal Theory .......................... 151
R. Hardt and D. Kinderlehrer
The Effect of the Magnitude of the Disordered Phase Temperature Range
on the Given Phase Transition in Liquid Crystals ......................... 185
C.C. Huang
Some Topics in Equil ibrium Theory of Liquid Crystals ..................... 211
F. Lesl ie
Theory of Flow Phenomena in Nematic Liquid Crystals ...................... 235
F. Les lie
A Model for Disclinations in Nematic Liquid Crystals .........•........... 255
J. Maddocks
Some Remarks About a Free Boundary Type Problem .......................... 271
M. Miranda
Computer Simulation of Flow of Liquid Crystal Polymers ................... 281
G. Ryskin
Theory of the Blue Phases of Chiral Nematic Liquid Crystals .............. 305
J. Sethna
On the Global Structure of Solutions to Some Semilinear Elliptic
Problems ................................................................. 325
J. Spruck
Information About Other Volumes in this Program .......................... 355
Foreword
This IMA Volume in Mathematics and its Applications
AMORPHOUS POLYMERS AND NON-NEWTONIAN FLUIDS
is in part the proceedings of a workshop which was an integral part of the 1984-85 IMA
program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS
We are grateful to the Scientific Committee:
Haim Brezis
Constantine Dafermos
Jerry Eri cksen
David Kinderlehrer
for planning and implementing an exciting and stimulating year-long program. We espe
cially thank the Program Organizers, Jerry Ericksen, David Kinderlehrer, Stephen Prager
and Matthew Tirrell for organizing a workshop which brought together scientists and
mathematicians in a variety of areas for a fruitful exchange of ideas.
George R. Sell
Hans Weinberger
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
