This volume (>Ie) NEMATICS Mathematical and Physical aspects constitutes the proceedings of a workshop which was held at l'Universite de Paris Sud (Orsay) in May 1990. This meeting was an Advanced Research Workshop sponsored by NATO. We gratefully acknowledge the help and support of the NATO Science Committee. Additional support has been provided by the Ministere des affaires etrangeres (Paris) and by the Direction des Recherches et Etudes Techniques (Paris). Also logistic support has been provided by the Association des Numericiens d'Orsay. (*) These proceedings are published in the framework of the "Contrat DRET W 90/316/ AOOO." v Contents (*) FOREWORD v INTRODUCTION 1. M. CORON, 1. M. GHIDAGLIA, F. HELEIN xi AN ENERGY-DECREASING ALGORITHM FOR HARMONIC MAPS F. ALOUGES 1 A COHOMOLOGICAL CRITERION FOR DENSITY OF SMOOTH MAPS IN SOBOLEV SPACES BETWEEN TWO MANIFOLDS F. BETHUEL, 1. M. CORON, F. DEMENGEL, F. HELEIN 15 ON THE MATHEMATICAL MODELING OF TEXTURES IN POLYMERIC LIQUID CRYSTALS M. C. CAmERER 25 A RESULT ON THE GLOBAL EXISTENCE FOR HEAT FLOWS OF HARMONIC MAPS FROM D2 INTO S2 K. C. CHANG, W. Y. DING 37 BLOW-UP ANALYSIS FOR HEAT FLOW OF HARMONIC MAPS Y. CHEN 49 T AYLOR-COUETTE INSTABILITY IN NEMATIC LIQUID CRYSTALS P. E. ClADIS 65 ON A CLASS OF SOLUTIONS IN THE THEORY OF NEMATIC PHASES B. D. COLEMAN, 1. T. JENKINS 93 RHEOLOGY OF THERMOTROPIC NEMATIC LIQUID CRYSTALLINE POLYMERS M. M. DENN, 1. A.

We present here the lectures and a selection of the seminars given at the Ninth International Workshop on Instabilities and Nonequilibrium Structures which took place in Vifiadel Mar, Chile, in December 2001. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Instituto de Fisica of Universidad Cat6lica de Valparaiso, Centro de Fisica No Lineal y Sistemas Complejos de Santiago and Facultad de Ingenieria, Universidad de los Andes, which starting from this year joins the other institutions in the coorganization ofthe Workshop. The organizers would like to express their gratitude to the following sponsors: Facultad de Ciencias Fisicas y Matematicas de la Universidad de Chile, Instituto de Fisica de la Universidad Cat6lica de Valparaiso, Facultad de Ingenieria de la Universidad de los Andes, Centro de Fisica No Lineal y Sistemas Complejos de Santiago, Academia Chilena de Ciencias, Ministere Francais des Affaires Etrangeres, CONICYT (Comisi6n Nacional de Investigaci6n Cientifica y Tecno16gicade Chile) and Departamento Tecnico de Investigaci6n y de Relaciones Internacionales de la Universidad de Chile. Enrique Tirapegui PREFACE This book consists of two parts, the first one has three lectures written by Professors H. R. Brand, M. Moreau and L. S. Tuckerman. H. R. Brand gives an overview about reorientation and undulation instabilities in liquid crystals, M. Moreau presents recent results on biased tracer diffusion in lattice gases, finally, L. S. Tuckerman summarizes some numerical methods used in bifurcation problems.

Micro/nanotribology as a field is concerned with experimental and theoretical investigations of processes ranging from atomic and molecular scales to the microscale, occurring during adhesion, friction, wear, and thin-film lubrication at sliding surfaces. As a field it is truly interdisciplinary, but this confronts the would-be entrant with the difficulty of becoming familiar with the basic theories and applications: the area is not covered in any undergraduate or graduate scientific curriculum. The present work commences with a history of tribology and micro/nanotribology, followed by discussions of instrumentation, basic theories of friction, wear and lubrication on nano- to microscales, and their industrial applications. A variety of research instruments are covered, including a variety of scanning probe microscopes and surface force apparatus. Experimental research and modelling are expertly dealt with, the emphasis throughout being applied aspects.

The importance of safety in any scientific endeavor is never in question. However, when cryogenic temperatures are involved, safety is especially important. In addition to observing the normal precautions, one must also take into account the variations of physical properties that occur at low temperatures. At these tempera tures, some properties not only exhibit large differences from their normal values but also can vary widely over a small temperature range. Before any cryogenic project is started, a thorough knowledge of the possible hazards is necessary. Only in this way can the safest operation be attained. Over the hundred-year history of cryogenic research, this has been shown to be the case. Keeping this requirement in mind is an essential ingredient in the quest for accident-free work. The past four or five decades have seen a great expansion of cryogenic technology. Cryogenic liquids, such as oxygen, nitrogen, hydrogen, and helium, have become commonly used in a number of different applications and are easily available in any part of the United States and, indeed, almost anywhere in the world. Not only are these liquids available, they have become less expensive and also available in ever larger quantities. As quantities increase, so also do the conse quences of mishaps. The future seems to hold promise of ever larger and more widespread use of the common cryogens. Thus, the importance of safety also increases as time progresses.

The study of defects and disorder in solids remains a central topic in solid state science. Developments in the field continue to be promoted by new experimental and theoretical techniques, while further impetus for the study of disorder in solids is provided by the growing range of applications of solid state materials in which disorder at the atomic level plays a crucial rOle. In this book we attempt to present a survey of fundamental and applied aspects of the field. We consider the basic aspects of defective crystalline and amorphous solids. We discuss recent studies of structural, electronic, transport, thermodynamic and spectroscopic properties of such materials. Experimental and theoretical methodologies are reviewed, and detailed consideration is given to materials such as fast ion conductors and amorphous semiconductors that are of importance in an applied context. Any survey of this large field is necessarily selective. We have chosen to emphasise insulating (especially oxidic) and semi-conducting materials. But many of the approaches and techniques we describe apply generally across the entire field of solid state science. This volume is based on a NATO ASI held at the Residencia Santa Teresa de Jesus, Madrid in September 1991. The Editor is grateful to the NATO Scientific Affairs Division for their sponsorship of this School. Thanks are also due to all who participated in and lectured at the school, but especially to the organising committee of A. V. Chadwick, G. N. Greaves, M. Grigorkiewicz, J. H. Harding and S. Kalbitzer. C. R. A.

Modern Synthetic and Application Aspects of Polysilanes: An Underestimated Class of Materials?, by A. Feigl, A. Bockholt, J. Weis, and B. Rieger;
*
Conjugated Organosilicon Materials for Organic Electronics and Photonics, by Sergei A. Ponomarenko and Stephan Kirchmeyer;
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Polycarbosilanes Based on Silicon-Carbon Cyclic Monomers, by E.Sh. Finkelshtein, N.V. Ushakov, and M.L. Gringolts;
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New Synthetic Strategies for Structured Silicones Using B(C6F5)3, by Michael A. Brook, John B. Grande, and Francois Ganachaud;
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Polyhedral Oligomeric Silsesquioxanes with Controlled Structure: Formation and Application in New Si-Based Polymer Systems, by Yusuke Kawakami, Yuriko Kakihana, Akio Miyazato, Seiji Tateyama, and Md. Asadul Hoque;"

Since the discovery that polymer single crystals are composed of chain folded macromolecules in 1957, the crystallization of polymers has attracted considerable interest and still provides fascinating and fruitful areas of research. Only a few books have been fully devoted to the crystallization of polymers in the past. This book contains the proceedings of the NATO ARW devoted to the `Crystallization of Polymers' which took place in September 1992 at the University of Mons-Hainaut (Belgium). In view of the variety of papers devoted to the crystallization of polymers, this book will be used in the next few years as a reference book for scientists concerned in the field of polymer physical chemistry. Crystallization of Polymers is mainly devoted to the experimental and theoretical study of the crystallization of synthetic polymers. As a kinetic study of the growth of polymer crystals should always be preceded by a morphological or a structural investigation, the structure, the morphology of polymer crystals and more particularly the lamellar and supralamellar organizations, as well as the nature of the crystal amorphous interface are reviewed and discussed.

Kratschmer and Huffman's revolutionary discovery of a new solid phase of carbon, solid C60, in 1990 opened the way to an entire new class of materials with physical properties so diverse that their richness has not yet been fully exploited. Moreover, as a by-product of fullerene research, carbon nanotubes were later identified, from which novel nanostructures originated that are currently fascinating materials scientists worldwide. Rivers of words have been written on both fullerenes and nanotubes, in the form of journal articles, conference proceedings and books. The present book offers, in a concise and self-contained manner, the basics of the science of these materials as well as detailed information on those aspects that have so far been better explored. Structural, electronic and dynamical properties are described as obtained from various measurements and state-of-the-art calculations. Their interrelation emerges as well as their possible dependence on, for example, preparation conditions or methods of investigation. By presenting and comparing data from different sources, experiment and theory, this book helps the reader to rapidly master the basic knowledge, to grasp important issues and critically discuss them. Ultimately, it aims to inspire him or her to find novel ways to approach still open questions. As such, this book is addressed to new researchers in the field as well as experts.

This vohune contains the papers presented at the Adriatico Research Conference on Structural and Phase Stability of Alloys held in Trieste, Italy, in May 1991, under the auspices of the International Centre for Theoretical Physics. The conference brought together participants with a variety of interests in theoretical and experimental aspects of alloys from Argentina, Belgium, Bulgaria, Czechslovakia, France, Germany, Italy, Japan, Mexico, People's Republic of Congo, Portugal, Switzerland, United Kingdom, United States, U. S. S. R., and Venezuela. The conference was purposely designed to succinctly cover experimental and the oretical aspects of magnetic and non-magnetic alloys, surfaces, thin films and nanos tructures. The Conference opened with an overview of a select class of advanced structural materials, with a potential in engineering applications, for which the con ventional "physics" approach, both theoretical and experimental, should have a sig nificant impact. A number of papers were dedicated to the use of phenomenological approaches for the description of thermodynamic bulk and surface properties. It was clear from these presentations that the phenomenological models and simulations in alloy theory have reached a high degree of sophistication. Although with somewhat limited predictive powers, the phenomenological models provide a valuable tool for the understanding of a variety of subtle phenomena such as short-range order, phase stability, kinetics and the thermodynamics of surfaces and antiphase boundaries, to name a few."

Addresses the use of MEMS (micro-electro-mechanical systems) and micromachined devices for the investigation of nanoscience and technology, as well as biotechnology. Such micromachined tools for nanotechnology can enhance the sensitivity, spatial resolution, dexterity, selectivity, and parallel processing capability in measuring and manipulating nano-objects. The book covers state-of-the-art MEMS and NEMS devices for DNA molecular handling and analysis, cell handling and culture on a chip, chemical lab-on-a-chip, multi-probes for vacuum tunneling microscopy and AFM, and characterization of quantum semiconductor structures. Readers will gain deep insight into such developments and students will learn about the emerging field of MEMS and nanotechnology

Toachieve design, implementation,and servicing ofcomplex systems and struc tures in an efficient and cost-effective way,a deeper knowledge and understanding of the subtle cast and detailed evolution of materials is needed. The analysis in demand borders with the molecular and atomic one, spanning all the way down from classical continua. The study of the behavior of complex materials in sophisticated devices also opens intricate questions about the applicability of primary axioms ofcontinuum mechanics such as the ultimate nature of the material element itselfand the possibility ofidentifying itperfectly. So it is necessary to develop tools that allow usto formulate both theoretical models and methods of numerical approximation for the analysis of material substructures. Multifield theories in continuum mechanics, which bridge classical materials science and modern continuum mechanics, provide precisely these tools. Multifield theories not only address problems of material substructures, but also encompass well-recognized approaches to the study of soft condensed matter and allow one to model disparate conditions in various states ofmatter. However, research inmultifield theories is vast, and there is little in the way of a comprehensive distillation of the subject from an engineer's perspective. Therefore, the papers in the present volume, 1 which grew out of our experience as editors for an engineeringjournal, tackle some fundamental questions,suggest solutions of concrete problems, and strive to interpret a host of experimental evidence. In this spirit, each of the authors has contributed original results having in mind their wider applicability.

Summary of the recent progress in ceramics research. Several novel concepts for materials selection and microstructural design are presented, as are experimental results that substantiate the ideas.

With this proceedings volume a new series of publications is started which will present the results of interdisciplinary research activities in the fields of materials science, coupling of biological and electronic systems and commu nication ergonomy. It will contain the contributions of the participants of the caesarium, a conference caesar will organize annually. The 1 st caesarium was held in Bonn on November 17-19, 1999 concentrating on Smart Materials. With the caesarium the recently founded research center caesar (center of advanced european studies and research) creates a forum for discussion of new developments in its fields of activities. caesar is an international research center, focusing on applied, interdisciplinary research projects in the areas of science and engineering. It was established as an independent foundation under private law as part of the compensatory actions under the Berlin/Bonn law of April 26, 1994 to support the structural change in the region of Bonn, when the German Government moved from Bonn to Berlin. The main donors of caesar are the Federal Republic of Germany and the State of North Rhine-Westphalia. A Board consisting of state and federal leg islators, members from the research community and industry and a Scientific Advisory Council assist caesar in all decisions concerning administration and research.

First-generation semiconductors could not be properly termed "doped- they were simply very impure. Uncontrolled impurities hindered the discovery of physical laws, baffling researchers and evoking pessimism and derision in advocates of the burgeoning "pure" physical disciplines. The eventual banish ment of the "dirt" heralded a new era in semiconductor physics, an era that had "purity" as its motto. It was this era that yielded the successes of the 1950s and brought about a new technology of "semiconductor electronics." Experiments with pure crystals provided a powerful stimulus to the develop ment of semiconductor theory. New methods and theories were developed and tested: the effective-mass method for complex bands, the theory of impurity states, and the theory of kinetic phenomena. These developments constitute what is now known as semiconductor phys ics. In the last fifteen years, however, there has been a noticeable shift towards impure semiconductors - a shift which came about because it is precisely the impurities that are essential to a number of major semiconductor devices. Technology needs impure semiconductors, which unlike the first-generation items, are termed "doped" rather than "impure" to indicate that the impurity levels can now be controlled to a certain extent."

Layered crystals, characterized by a quasi-two-dimensional character of certain physical properties, play an interesting role in surface science. First of all they provide excellent inert substrates for epitaxial deposition and physisorption studies. The surfaces of layered crystals, however, are interesting in their own right because they make a relevant class of low-dimensional phenomena accessible to surface probes. Change density waves, incommensurate structures, phonon anomalies and high Tc superconductivity are well known examples. This book collects a series of review articles written by outstanding specialists on the structural assessment and spectroscopy of layered structures with surface-sensitive probes such as scanning microscopy and helium atom scattering, the theoretical analysis of their electronic and vibrational surface states, and the investigation of physisorbed overlayers.

This volume focuses on modeling processes for which transport is one of the most complicated components, requiring different transport models in each region. The authors apply questions to a wide variety of application areas, such as semiconductors, plasmas, fluids, chemically reactive gases, etc.

The workshop on "Optical Properties of Low Dimensional Silicon sL Structures" was held in Meylan, France on March, I yd, 1993. The workshop took place inside the facilities of France Telecom- CNET. Around 45 leading scientists working on this rapidly moving field were in attendance. Principal support was provided by the Advanced Research Workshop Program of the North Atlantic Treaty Organisation (NATO). French Delegation a l'Armement and CNET gave also a small financial grant, the organisational part being undertaken by the SEE and CNET. There is currently intense research activity worldwide devoted to the optical properties of low dimensional silicon structures. This follow the recent discovery of efficient visible photoluminescence (PL) from highly porous silicon. This workshop was intended to bring together all the leading European scientists and laboratories in order to reveal the state of the art and to open new research fields on this subject. A large number of invited talks took place (12) together with regular contribution (20). The speakers were asked to leave nearly 1/3 of the time to the discussion with the audience, and that promoted both formal and informal discussions between the participants.

The idea of writing this book orIgmates from a suggestion of Bernard Sapoval: "Why don't you write it?" he asked. "Coulomb screening is a problem that everybody encounters in many different contexts, and there is no textbook that gathers the various aspects ofthe subject. " The content ofthe book, in a shorter form, was first taught for four years as a course in Dipl6me d'Etudes Approfondies Sciences des Materiaux, headed by Prof. J. -F. Petroff, at Paris VI University. The present extended version was written after discussions with Alia Margolina-Litvin. An essential feature of screening is its role in many different scientific areas. For that reason, the book is intended for use by a multidisciplinary readership. Reading it requires only a basic knowledge ofelectromagnetism, elementary quantum mechanics, and thermal physics. The spirit of the pre sentation is "simplicity first": new concepts (e. g. , dielectric function) are first introduced in their most elementary form and are progressively extended to more generality. The book stays at a basic level, and additional abstract developments that might have been included have been either omitted, rele gated to an appendix, or summarized in a qualitative manner. Apart from these restrictions, care has been taken to keep the presentation as rigorous as possible: the topics addressed are dealt with quantitatively, the results are given in mathematical form, and the interested reader should be able to fol low the algebra all the way through.

IMA Volumes 135: Transport in Transition Regimes and 136: Dispersive Transport Equations and Multiscale Models focus on the modeling of processes for which transport is one of the most complicated components. This includes processes that involve a wdie range of length scales over different spatio-temporal regions of the problem, ranging from the order of mean-free paths to many times this scale. Consequently, effective modeling techniques require different transport models in each region. The first issue is that of finding efficient simulations techniques, since a fully resolved kinetic simulation is often impractical. One therefore develops homogenization, stochastic, or moment based subgrid models. Another issue is to quantify the discrepancy between macroscopic models and the underlying kinetic description, especially when dispersive effects become macroscopic, for example due to quantum effects in semiconductors and superfluids. These two volumes address these questions in relation to a wide variety of application areas, such as semiconductors, plasmas, fluids, chemically reactive gases, etc.

The unexpected and therefore really amazing discovery of J.G. Bednorz and K.A. Muller, that certain oxide compounds enter a superconductivity state at temperatures above 30 K, pushed research on superconductivity into the limelight of science in general in a way that seemed reserved only for high energy or particle physics. It may therefore be expected that this entire review would solely deal with superconductivity at high temperatures, i.e. above the boiling point of hydrogen. Any unexpected occurrence of superconductivity is, however, a challenge to scientists interested either in the physics of this phenomenon or in its materials science aspects. In this respect, the eighties have been quite revolutionary in the sense that, on various occasions, superconductivity was discovered in materials whose physical properties were not obviously favourable for adopting this ground state. Therefore, apart from emphasizing the topic of oxide superconductors, this collection of reprints also contains a selection of papers that deal with other subjects, such as coexistence of magnetic order and superconductivity, heavy electron and organic superconductors. This is all the more justified when we consider the fact that various aspects of superconductivity in high Tc oxide compounds are, or might be, connected with features that are also observed in these other materials. For nonspecialists who might be interested in this collection of reprints the Editor briefly reviews the possibilities for identifying superconductivity and discusses some special features of the superconducting state. "

In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused works. Specific applications presented include plasma kinetic models, traffic flow models, granular media models, and coagulation-fragmentation problems. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.

Remarkable advances in semiconductor growth and processing technologies continue to have a profound impact on condensed-matter physics and to stimulate the invention of novel optoelectronic effects. Intensive research on the behaviors of free carriers has been carried out in the two-dimensional systems of semiconductor heterostructures and in the one and zero-dimensional systems of nanostructures created by the state-of-the-art fabrication methods. These studies have uncovered unexpected quantum mechanical correlations that arise because of the combined effects of strong electron-electron interactions and wave function confinement associated with reduced dimensionality. The investigations of these phenomena are currently at the frontiers of condensed-matter physics. They include areas like the fractional quantum Hall effect, the dynamics of electrons on an ultra short (femtosecond) time scale, electron behavior in quantum wires and dots, and studies of electron tunneling phenomena in ultra small semiconductor structures. Optical techniques have made important contributions to these fields in recent years, but there has been no coherent review of this work until now. The book provides an overview of these recent developments that will be of interest to semiconductor materials scientists in university, government and industrial laboratories.

Supramolecular stereochemistry is a topic with enormous breadth, and this book brings together experts in polymer chemistry, bioorganic chemistry, crystallography, materials science, dendrimer science, nanochemistry, conformational analysis, molecular recognition chemistry, and topological stereochemistry. Contains 19 plenary and 12 poster contributions.

Soils are complex materials: they have a particulate structure and fluids can seep through pores, mechanically interacting with the solid skeleton. Moreover, at a microscopic level, the behaviour of the solid skeleton is highly unstable. External loadings are in fact taken by grain chains which are continuously destroyed and rebuilt. Many issues of modeling, even of the physical details of the phenomena, remain open, even obscure; de Gennes listed them not long ago in a critical review. However, despite physical complexities, soil mechanics has developed on the assumption that a soil can be seen as a continuum, or better yet as a medium obtained by the superposition of two and sometimes three con and the other fluids, which occupy the same portion of tinua, one solid space. Furthermore, relatively simple and robust constitutive laws were adopted to describe the stress-strain behaviour and the interaction between the solid and the fluid continua. The contrast between the intrinsic nature of soil and the simplistic engi neering approach is self-evident. When trying to describe more and more sophisticated phenomena (static liquefaction, strain localisation, cyclic mo bility, effects of diagenesis and weathering, ..... ), the nalve description of soil must be abandoned or, at least, improved. Higher order continua, incrementally non-linear laws, micromechanical considerations must be taken into account. A new world was opened, where basic mathematical questions (such as the choice of the best tools to model phenomena and the proof of the well-posedness of the consequent problems) could be addressed.

Techniques for the preparation of condensed matter systems have advanced considerably in the last decade, principally due to the developments in microfabrication technologies. The widespread availability of millikelvin temperature facilities also led to the discovery of a large number of new quantum phenomena. Simultaneously, the quantum theory of small condensed matter systems has matured, allowing quantitative predictions. The effects discussed in Quantum Dynamics of Submicron Structures include typical quantum interference phenomena, such as the Aharonov-Bohm-like oscillations of the magnetoresistance of thin metallic cylinders and rings, transport through chaotic billiards, and such quantization effects as the integer and fractional quantum Hall effect and the quantization of the conductance of point contacts in integer multiples of the `conductance quantum'. Transport properties and tunnelling processes in various types of normal metal and superconductor tunnelling systems are treated. The statistical properties of the quantum states of electrons in spatially inhomogeneous systems, such as a random, inhomogeneous magnetic field, are investigated. Interacting systems, like the Luttinger liquid or electrons in a quantum dot, are also considered. Reviews are given of quantum blockade mechanisms for electrons that tunnel through small junctions, like the Coulomb blockade and spin blockade, the influence of dissipative coupling of charge carriers to an environment, and Andreev scattering. Coulomb interactions and quantization effects in transport through quantum dots and in double-well potentials, as well as quantum effects in the motion of vortices, as in the Aharonov-Casher effect, are discussed. The status of the theory of the metal-insulator and superconductor-insulator phase transitions in ordered and disordered granular systems are reviewed as examples in which such quantum effects are of great importance.