This book is devoted to the special area of statistical mechanics that deals with the classical spin systems with quenched disorder. It is assumed to be of a pedagogical character, and it aims to help the reader to get into the subject starting from fundamentals. The book is supposed to be selfcontained (the reader is not required to go through all the references to understand something), being understandable for any student having basic knowledge of theoretical physics and statistical mechanics. Nevertheless, because this is only an introduction to the wide scope of statistical mechanics of disordered systems, in some cases to get to know more details about a particular topic the reader is advised to refer to the existing literature. Although throughout the book I have tried to present all the unavoidable calculations such that they would look as transparent as possible and have given everywhere (where it is at all possible) physical interpretations of what is going on, in many cases certain personal efforts and/or use of imagination are still required.

The first part of the book is devoted to the physics of spin-glass systems, where the quenched disorder is the dominant factor. The emphasis is made on a general qualitative description of the physical phenomena, being mostly based on the results obtained in the framework of the mean-field theory of spin-glasses with long-range interactions. First, the general problems of the spin-glass state are discussed at the qualitative level.

Introduction

Along with learning about specific aspects of matter transport and its relationship to materials engineering, a course in this subject area is an excellent place to learn about the role of modelling in materials science and engineering. Models are used to develop a better understanding of the processes we are trying to control in order to produce better, more reliable and consistent materials. We usually start with a problem (which is often ill-defined, at least initially) and some data. A model is an attempt to draw general conclusions about the behaviour of the system of interest in a quantitative way which allows us to make predictions about the behaviour of the system. Models invariably require certain assumptions and approximations to be made. Thus the model will be limited both in its precision and in its scope or range of applicability. However, a model can be a powerful tool if used carefully, and all materials engineers should have expertise in how to model materials behaviour.

Consider the problem of predicting weather behaviour. A set of observations of many days might lead to the hypothesis that the weather on any given day in a particular location is correlated to the weather on the previous day. Thus if you predict that the weather tomorrow is going to be the same as the weather today you will be correct say 50% of the time.

The field of matter transport is central to understanding the processing and subsequent behaviour of materials. While thermodynamics tells us the state in which a material system would like to exist, the kinetics of matter transport tell us how fast it can get there. The materials which we use in service are often not at equilibrium. This is particularly so in solid materials. Here kinetics are sufficiently slow that materials are almost never in complete thermodynamic equilibrium but attain a stable state which is kinetically limited. However, kinetics are equally important in fluids since the transport of matter in a liquid or gas often limits the rate at which many processes for materials fabrication can proceed. The aim of this book is to give students a solid grounding in the principles of matter transport and their application to the solution of engineering problems.

It is this emphasis on engineering application that distinguishes this text from many other books in this field. In the field of solid-state diffusion, for example, much has been written about the mechanisms of diffusion at the atomic scale. This is a fascinating topic and of great interest to materials scientists. However, only an elementary understanding is required in order to treat many practical diffusion problems. Thus, the emphasis throughout this book is on developing methods for solving problems involving the transport of matter in materials.

Up to now we have assumed that only concentration gradients provide a driving force for diffusion. We now generalize this, by including all forms of free-energy gradient. This will lead us fairly easily to a general theory of diffusion that includes Fick's First Law as a special case. We will apply our new understanding to problems such as diffusion due to an electrical field or a mechanical stress. We will finish the chapter by seeing how this method can help to understand diffusion in multicomponent systems.

Theory

Throughout this book we have considered that diffusional processes are solely driven by concentration gradients. This is based on the concept that a material strives towards equilibrium, and that equilibrium is defined as a state of uniform concentration. In materials science, however, we recognize that this is a rather naive definition of equilibrium. To be more precise, we should state that a body is at equilibrium when the free energy is uniform throughout. Using this definition, we can broaden considerably the range of situations in which diffusion occurs. Thus, we expect diffusion not only in response to concentration gradients, but also due to electrical and thermal fields, and mechanical

In this chapter we will consider classical experiments that have been performed on real spin glass materials, aiming to check to what extent the qualitative picture of the spin-glass state described in previous chapters does take place in the real world. The main problem of the experimental observations is that the concepts and quantities that are very convenient in theoretical considerations are rather far from the experimental realities, and it is a matter of the experimental art to invent convincing experimental procedures that would be able to confirm (or reject) the theoretical predictions.

A series of such brilliant experiments has been performed by M. Ocio, J. Hammann, F. Lefloch and E. Vincent (Saclay), and M. Lederman and R. Orbach (UCLA) [9]. Most of these experiments have been done on the crystals CdCr1.7In0.3S4. The magnetic disorder there is present due to the competition of the ferromagnetic nearest neighbor interactions and the antiferromagnetic higher-order neighbor interactions. This magnet has already been systematically studied some time ago [26], and its spin-glass phase transition point T = 16.7 K is well established. Some of the measurements have been also performed on the metallic spin glasses AgMn [27] and the results obtained were qualitatively quite similar. It indicates that presumably the qualitative physical phenomena observed do not depend very much on the concrete realization of the spin-glass system.

In this chapter we present a new method for studying statistical systems with quenched disorder in the low-temperature limit. The use of the replica method has turned out to be very efficient in some disordered systems. It allows for a detailed characterization of the low-temperature phase at least at the mean-field level. In all the mean-field spin-glass-like problems where one can expect the mean-field theory to be exact, the Parisi scheme of replica symmetry breaking is successful, and at the moment there is no counterexample showing that it does not work. On the other hand, the low-temperature phase of these systems is complicated enough, even at the mean-field level. One might hope that the very low-temperature limit could be easier to analyse, while its physical content should be basically the same. This very low-temperature limit is also an extreme case where one might hope to get a better understanding of the finite-dimensional problems. At first sight the low-temperature limit is indeed simpler because the partition function could be analysed at the level of a saddle-point approximation. However, it is easy to see that generically this limit does not commute with the limit of the number of replicas going to zero. There is a very basic origin to this non-commutation, namely the fact that there still exist, even at zero temperature, sample-to-sample fluctuations.

In this chapter we consider problems involving diffusion from a gas phase into the solid. The boundary conditions at the gas/solid interface are determined, assuming that local equilibrium exists, from a knowledge of thermodynamics. We will also consider an example in which the rate of reaction at the interface is the controlling step.

Introduction

We start by considering the interaction between gases and solid surfaces. This is a useful starting point for several reasons. First, this area includes a number of important technological problems in materials engineering. Second, these problems generally involve only a single diffusing species. This greatly simplifies matters. Examples of gas species which diffuse readily into solids include carbon and nitrogen in steel, hydrogen in many metals and ceramics, and water (H+ or H3O+ ions) into glass. One common feature of these gases is that they involve rather small ions, which diffuse interstitially into crystalline solids. They therefore exhibit fairly high diffusion coefficients.

There is a related set of problems, involving diffusion from liquids into solids, which can be handled in the same way. For example, glasses can be strengthened by allowing K+ or Ag+ to diffuse into the surface, through ion exchange from a liquid. Similarly, ions can be leached out of a glass when in contact with a liquid.

In this chapter the physical interpretation of the formal RSB solution will be proposed, and some new concepts and quantities will be introduced. The crucial concept that is needed to understand physics behind the RSB structures is that of the pure states.

The pure states

Consider again a simple example of the ferromagnetic system. Here, spontaneous symmetry breaking takes place below the critical temperature Tc, and at each site the non-zero spin magnetizations 〈σi〉 = ±m appear. As we have already discussed in Section 2.2, in the thermodynamic limit the two ground states with the global magnetizations 〈σi〉 = +m and 〈σi〉 = –m are separated by an infinite energy barrier. Therefore, once the system has happened to be in one of these states, it will never be able (during any finite time) to jump into the other one. In this sense, the observable state is not the Gibbs one (which is obtained by summing over all the states), but one of these two states with non-zero global magnetizations. To distinguish them from the Gibbs state they could be called the ‘pure states’. More formally, the pure states could also be defined by the property that all the connected correlation functions in these states, such as 〈σiσj〉c ≡ 〈σiσj〉 – 〈σi〉〈σj〉, tend towards zero at large distances.

In the previous chapter we obtained a special type of spin-glass ground-state solution.

Fick's First Law is still valid even when steady-state conditions do not exist. However, it is not very convenient to use it in this form for solving transient diffusion problems, for which the concentration at any position depends on both time and position. It is therefore more convenient to derive a second version of this equation, Fick's Second Law, which contains an explicit dependence on time. In this chapter we will first derive the equation; a second-order partial differential equation. We will then learn how solutions to this equation can be obtained for a large variety of situations depending on the problem geometry, the initial and boundary conditions and the time-frame over which we need a valid solution.

Fick's Second Law

We consider the situation illustrated in Fig. 3.1. At two positions along the y-axis at y1 and y2, a distance Δy apart, the concentration of the solute is C1 and C2 respectively. We can use Fick's First Law to determine the amount of solute per unit time which enters this element at y1 and which leaves it at y2. Because matter must be conserved, the difference between these must be equal to the rate at which solute accumulates within the element.