~ Pergamon Solid Sate Communications, Vol. 102, No. 2-3, pp. 187-197, 1997 O 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00 PII: S0038-1098(96)00718-1 SOFT CONDENSED MATrER PHYSICS T.C. Lubensky Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. Soft condensed matter physics is the study of materials, such as fluids, liquid crystals, polymers, colloids and emulsions, that are "soft" to the touch. This article will review some properties, such as the dominance of entropy, that are unique to soft materials and some properties such as the interplay between broken-symmetry, dynamic mode structure and topolo- gical defects that are common to all condensed matter systems but which are most easily studied in soft systems. © 1997 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION In recent years soft condensed matter physics, or simply soft physics, has emerged as an identifiable subfield of the broader field of condensed matter physics. As its title implies, it is the study of matter that is "soft", i.e., of materials that will not hurt your hand if you hit them. This is in contrast to "hard" materials such as aluminum or sodium chloride that are generally associated with the field of solid state physics. Though the term soft physics has only recently gained acceptance, its purview is vast. It subsumes all of fluid physics, including both micro- scopic structure and macroscopic phenomena such as hydrodynamic flow and instabilities. It includes liquid crystals and related materials with their vast variety of broken-symmetry states. It includes colloids, emulsions, microemulsions, membranes and a large fraction of biomaterials. It is a field that presents fundamental scientific challenges and one that has substantial economic impact. In this article, I will give a brief overview of some fundamental problems of soft condensed matter physics [1]. The defining property of soft materials is the ease with which they respond to external forces. This means not only that they distort and flow in response to modest shears but also that thermal fluctuations play an important if not dominant role in determining their properties. They cannot be described simply in terms of harmonic excitations about a quantum ground state as most hard materials can. There are soft materials that possess virtually every possible symmetry group, including three-dimensional crystalline symmetries normally associated with hard materials and many others not found at all in hard materials. Ordered phases of soft materials can easily be distorted, making it possible to study and to control states far from equilibrium or riddled with defects. Thus, soft materials offer an ideal testing ground for fundamental concepts, involving the connection between symmetry, low-energy excitations and topological defects, that are at the very heart of physics. In this article, I will discuss four broad problems that reflect the richness of soft physics: entroplc forces and entropically induced order, broken symmetries, topological defects and membranes. This is by no means an exhaustive list; it does not, for example, include nonequilibrium and non-linear phenomena or the vast field of polymer physics; it is, however, a list that has general applicability to hard as well as soft physics. 2. THE TRIUMPH OF ENTROPY 2.1. Introduction Phases in thermodynamic equilibrium correspond to minima of the free energy F = E - TS, where E is the internal energy, Tis the temperature and S is the entropy. In hard materials, E tends to dominate over entropy: to a good approximation, internal energy determines the structure of equilibrium phases and thermal fluctuations can be treated as perturbations about a minimum-energy phase. In soft systems, quite the opposite may be true: internal energy may either be small compared to TS, or it may not depend at all on configurational changes of the system. In the latter case, equilibrium states are those that maximize the entropy rather than minimize the internal 187188 SOFT CONDENSED energy. In addition, deviations of the entropy from its equilibrium maximum value create forces whose effects are every bit as real as those arising from the gradient of a potential. Perhaps the most familiar of such entropic forces is that required to stretch a polymer [2]. A polymer can be modeled as a sequence of N freely joined segments of length I. The entropy of such a chain is a maximum if it is completely unconstrained. Constraining its ends to have a separation R leads to an entropy reduction of ~S = - 3R21(2Nl 2), a free energy increase of AF = 3TR2/(2Nl 2) and a force f = - OF~OR = - 3TR/(2NI2). In this section, we will explore some systems where entropy determines structure and interparticle forces. 2.2. Hard spheres The interaction potential between spherical atoms such as the noble gases consists of a long-range Van der Waals attractive part and a short-range repulsive part arising largely from the Pauli principle. Though the formation of crystals at low temperature depends criti- cally on the existence of the attractive part of the potential, the properties of the liquid phases are deter- mined to a large degree by the repulsive part, which is well modeled by a hard-sphere interaction that is infinite for interparticle separations less the particles' diameter and zero otherwise. The hard-sphere gas was originally introduced as a mathematically simple model to describe fluid phases. Colloidal dispersions [3] of polystyrene spheres with radii ranging from 0.07 #m to 4 #m now provide nearly perfect experimental realizations of hard- sphere models. They provide marvelous laboratories in which to test a variety of theoretical predictions. They have also provided unexpected results. In a hard-sphere system, the internal energy is zero in every allowed configuration. Interparticle forces and the free energy are determined entirely by entropy, which depends on the fraction q~ of the total volume occupied by the hard spheres. At low volume fraction, collisions between particles are rare and the system is an ideal gas. As volume fraction increases,
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