Introduction

Materials science in general and metallurgy in particular are concerned with understanding both the structure of useful materials, and also the relationship between that structure and the properties of the material. On the basis of this understanding, together with a large element of empirical development, considerable improvements in useful properties have been achieved, mainly by changes in the microstructure of the material. The term microstructure as normally used covers structural features in the size range from atoms (0.3 nm) up to the external shape of the specimen at a size of millimetres or metres. These structural features include the composition, the crystal structure, the grain size, the size and distribution of additional phases and so on, all of which are controlled by the normal methods of alloying, fabrication and heat treatment.

The materials scientist, having achieved some sort of optimum microstructure for a particular property or application, has not completed the task. The important area of the stability of the microstructure remains to be considered. This concern arises since almost none of the useful structures in materials science are thermodynarnically stable: changes that will increase the total entropy or decrease the material's free energy are almost always possible. So if the original structure was an optimum one then such changes will degrade the material's structure and properties.

Introduction

The three main interfaces which are important in metallic systems are the solid–gas interface (the external surface), the interface between two crystals of the same phase which differ only in orientation (the grain boundary) and the interface between two different phases (the interphase boundary). The interphase boundary provides an almost infinite range of possibilities since in addition to the possible difference of orientation of the two crystals, the crystals can also differ in crystal structure, lattice parameter and in composition. In this almost infinite array of possible structures and thus properties two limiting conditions can be recognised. In one case where the interface is formed, as it often is in metallic systems, by precipitation of a second phase within a primary crystal structure then a particular orientation relationship develops between the phases. This produces an interface with a close atomic fit which minimises the interfacial energy (see §2.2.2 and Doherty (1982)). This, in turn, can introduce difficulties for interfacial mobility in that growth ledges may be needed, see §1.3. The opposite extreme occurs when the two phases have no orientation relationship with each other. As a result their interface will be a high energy, incoherent one that usually provides no particular crystallographic barrier to mobility. Examples of this type of incoherent interface arise when the two phases come in contact by growth processes rather than by nucleation.

Almost all metallurgical materials are metastable in one way or another. Manipulating the metastability in alloy microstructures has proved to be essential in order to obtain the wide range of properties needed for different kinds of manufactured component. The conventional metallurgical processing methods of casting, deformation and heat treatment are used to control microstructural features such as chemical homogeneity, grain size, extent of precipitation and dislocation substructure. These are associated with relatively slight deviations from equilibrium, and are discussed in the other chapters of this book. In recent years a variety of processing methods have been developed to manufacture alloys with highly metastable microstructures, that is, with greater deviations from equilibrium. These highly metastable alloys are the subject of the present chapter.

The different methods of manufacturing highly metastable alloys all depend upon manoeuvring the material into a condition far from equilibrium, and simultaneously removing its thermal energy to freeze it into a metastable state. The microstructures that can then develop depend upon both thermodynamic and kinetic factors. Thermodynamic conditions define a set of possible alloy microstructures with lower free energy than the starting state. Kinetic behaviour determines which of these microstructures actually develops during manufacture. The main kinds of metastable material that can be manufactured are microcrystalline and nanocrystalline alloys with ultra-fine grain sizes, segregation-free highly supersaturated solid solutions, new metastable crystalline alloy compounds, amorphous alloys with non-crystalline disordered atomic structures, and quasi-crystalline alloys with ordered but non-periodic atomic structures.

For the second edition, the objectives and the approach previously used have been maintained. In many areas the science base of the subject has shown little change since the first edition and here the text has only been modified by improved examples where available. In other areas the subject has advanced significantly and the text has been updated with the insights. Topics previously covered incompletely, notably the highly unstable microstructure produced initially by rapid solidification but subsequently by other processing routes, have been greatly expanded. Other significant developments that have taken place include the detailed experimental studies of homogeneous nucleation, the growth of Widmanstätten precipitates and precipitate coarsening, and the new insights into the nucleation of recrystallisation, and grain growth and its inhibition by second-phase particles. In other areas, despite the importance of the subject, progress has been disappointingly slow. As in the first edition, we have tried to indicate where there are unsolved problems. The first edition provided the authors with a rich supply of fruitful research topics and we hope that this was also true for our readers and will be equally true for the second edition. Microstructural stability of metallic (and other industrially important) materials remains a field of research with many scientific and potential engineering applications.

The authors are again grateful to Professor Robert Cahn FRS for his efforts to get this volume completed and his much-appreciated enthusiasm.

Instability due to non-uniform solute distribution

The simplest instability in a metallic microstructure is that produced by a nonuniform distribution of solute in an otherwise stable single phase. Such a distribution always raises the free energy of the alloy and so it will decay to a uniform distribution at a rate determined by the thermodynamics and kinetics of diffusion. The kinetics of diffusion and its relationship to the concentration and mobility of point defects is one of the best-established topics in materials science, see, for example, Shewmon (1989), and so these topics need not be repeated here. However, the thermodynamics of diffusion is less widely discussed and is described below. Some new ideas on diffusion in alloys showing different rates of atomic motion in binary alloys as indicated by the Kirkendall effect are also described.

Thermodynamics of diffusion

Fig. 2.1 shows the free energy-composition diagram for a binary alloy. In stable regions of the system, where the second differential of the free energy with composition, ∂2G/∂C2, is less than zero, the free energy of composition C3, is increased from G3 to G′3 if it exists as a mixture of and rather than as a single uniform composition. The rise in G caused by any non-uniform solute distribution in stable regions of any phase provides the driving ‘force’ for the diffusion that homogenises the distribution.

Introduction

The earlier chapters of this book have dealt with the major causes of instability in the microstructure of metals and alloys, but there remains a series of other influences which can also modify the structure. Plastic deformation and irradiation can totally alter the defect structure in metal crystals, and corrosion can completely destroy not only the metallic microstructure but the metal itself. These subjects are, however, too extensive in scope and too important to materials science to be dealt with in a brief chapter. In addition they do not fall within a reasonable definition of the stability of microstructure. However, there have been some interesting and important investigations of the stability of dispersed second-phase precipitates under conditions of plastic deformation, and to a smaller extent, of their stability under irradiation. These instabilities will be discussed here. Other types of external influences include: temperature gradients; gravitational, electrical and magnetic fields; and, finally, annealing under imposed elastic stresses. All of these influences have been found to cause changes in precipitate morphology, and will form the subject of this chapter.

Many of the most informative experiments in this area have been carried out on transparent non-metallic materials such as ice and potassium chloride. The results of these investigations appear to be of direct application to metals and so, for the purposes of this chapter, any material subjected to a relevant and interesting experiment will be considered metallic.

When the metal complex [Ru(phen)2(dppz)]2+ is bound to DNA it can luminesce. If the metal complex [Rh(phi)2(phen)]3+ is nearby on the strand, the luminescence is quenched by electron transfer. By varying concentrations and by varying the DNA it is possible to probe the distribution of complexes in this one-dimensional (1D) system, and to gather information about the electron transfer length and interparticle forces. Our model assumes random deposition with allowance for interactions among the complexes. Long strands of calf thymus (CT) DNA and short strands of a synthetic 28-mer were used in the experiments and, for fixed [Ru(phen)2(dppz)]2+ concentration, quenching was measured as a function of [Rh(phi)2(phen)]3+ concentration. In previous work, to be cited later, we reported an electron transfer length based on the CT-DNA data. However, the short-strand (28-mer) experiments show a remarkable difference from the previously analyzed data. In particular, the electron-transfer quenching upon irradiation is enhanced by a factor of approximately four. This requires the consideration of new physical effects on the short strands. Our proposal is to introduce complexcomplex repulsion as an additional feature. This allows a reasonable fit within the context of the random-deposition model, although it does not take into account changes in the structure of the 28-mer introduced by the metal complexes during the loading process.

Editor's note

Kinetic Ising models in 1D provide a gallery of exactly solvable systems with nontrivial dynamics. The emphasis has traditionally been on their exact solvability, although much attention has also been devoted to models with conservation laws that have to be treated by numerical and approximation methods.

Chapter 4 reviews these models with emphasis on steady states and the approach to steady-state behavior. Chapter 5 puts the simplest 1D kinetic Ising models into a wider framework of the evaluation of dynamical critical behavior, analytically, in 1D, and numerically, for general dimension. Finally, Ch. 6 describes low-temperature nonequilibrium properties such as domain growth and freezing.

For a general description of dynamical critical behavior, not limited to 1D, as well as an excellent review and classification of various types of dynamics, the reader is directed to the classical work [1]. Certain probabilistic cellular automata are equivalent to kinetic Ising models.

Introduction

In many experiments on the adhesion of colloidal particles and proteins on substrates, the relaxation time scales are much longer than the times for the formation of the deposit. Owing to its relevance for the theoretical study of such systems, much attention has been devoted to the problem of irreversible monolayer particle deposition, termed random sequential adsorption (RSA) or the car parking problem; for reviews see. In RSA studies the depositing particles (on randomly chosen sites) are represented by hard-core extended objects; they are not allowed to overlap.

In this chapter, numerical Monte Carlo studies and analytical considerations are reported for 1D and 2D models of multilayer adsorption processes. Deposition without screening is investigated; in certain models the density may actually increase away from the substrate. Analytical studies of the RSA late stage coverage behavior show the crossover from exponential time dependence for the lattice case to the power-law behavior in continuum deposition. In 2D, lattice and continuum simulations rule out some ‘exact’ conjectures for the jamming coverage. For the deposition of dimers on a 1D lattice with diffusional relaxation the limiting coverage (100%) is approached according to the power law; this is preceded, for fast diffusion, by the mean-field crossover regime with intermediate, ∼ 1/t, behavior. In the case of k-mer deposition (k>> 3) with diffusion the void fraction decreases according to the power law t-1/(k-1).

Editor's note

Much interest has been devoted recently to various systems described in the continuum limit by variants of nonlinear diffusion equations. These include versions of the KPZ equation, Burgers’ equation, etc. Chapter 13 surveys nonlinear effects associated with shock formation in hard-core particle systems. Exact solution methods and results for such systems are then presented in Ch. 14.

Selected nonlinear effects in surface growth are reviewed in Ch. 15. Their relation to kinetic Ising models and a survey of some results were also presented in Ch. 4 (Sec. 4.6). This is a vast field with many recent results; see (and Chs. 4, 15) for review-type literature. Some surface-growth effects were also reviewed in Ch. 11.

The nonequilibrium ID systems covered in this book are effectively (1 + 1)- dimensional, where the second ‘dimension’ is time. For stochastic dynamics, the latter is frequently viewed as ‘Euclidean time’ in the field-theory nomenclature. Certain directed-walk models of surface fluctuations associated with wetting transitions, etc., as well as related models of polymer adsorption at surfaces, are effectively (0 + l)-dimensional in this classification, where the spatial dimension along the surface is effectively the Euclidean-time dimension. This property is shared by 1D quantum mechanics, to which the solution of many surface models reduces in the continuum limit. These models share simplicity, the availability of exact solutions, and the importance of fluctuations with the (l + l)-dimensional systems.

It has been well established by theory and simulations that the reaction kinetics of diffusion-limited reactions can be affected by the spatial dimension in which they occur. The types of reactions A + B → C, A + A → A. and A + C → C have been shown, theoretically and/or by simulation, to exhibit nonclassical reaction kinetics in 1D. We present here experimental results that have been collected for effectively 1D systems.

An A + B → C type reaction has been experimentally investigated in a long, thin capillary tube in which the reactants, A and B, are initially segregated. This initial segregation of reactants means that the net diffusion is along the length of the capillary only, making the system effectively 1D and allowing some of the properties of the resulting reaction front to be studied. The reaction rates of molecular coagulation and excitonic fusion reactions, A + A → A, well as trapping reactions, A + C → C, were observed via the phosphorescence (P) and delayed fluorescence (F) of naphthalene within the channels of Nuclepore membranes and Vycor glass and in the isolated chains of dilute polymer blends. In these experiments, the nonclassical kinetics is measured in terms of the heterogeneity exponent, h, from the equation rate ∼ F = kt-hPn, which gives the time dependence of the rate coefficient. Classically h = 0, while h = 1/2 in ID for A + A → A as well as A + C → C type reactions.

A generalized aggregation model of charged particles that diffuse and coalesce randomly in discrete space-time is studied, numerically and analytically. A statistically invariant steady state is established when randomly charged particles are uniformly and continuously injected. The exact steadystate size distribution obeys a power law whose exponent depends on the type of injection. The stability of the power-law size distribution is proved. The spatial correlations of the system are analyzed by a powerful new method, the interval distribution of a level set, and a scaling relation is obtained.

Introduction

The study of far-from-equilibrium systems has attracted much attention in the last two decades. Though many macroscopic phenomena in nature, such as turbulence, lightning, earthquakes, fracture, erosion, the formation of clouds, aerosols, and interstellar dusts, are typical far-from-equilibrium problems, no unified view has yet been established. The substantial difficulties in studying such systems are the following. First, far-from-equilibrium systems satisfy neither detailed balance nor, at the macroscopic level, the equipartition principle. Second, the system is usually open to an outside source. A common method to describe such systems is by abstracting the macroscopic essential features of the observed system and constructing a model in macroscopic terms irrespective of the microscopic (molecular) dynamics. In other words, we make a far-from-equilibrium model by assuming appropriate irreversible rules for the macroscopic dynamics.

Two recent developments involving activation and transport processes in simple stochastic nonlinear systems are reviewed in this chapter. The first is the idea of ‘resonant activation’ in which the mean first-passage time for escape over a fluctuating barrier passes through a minimum as the characteristic time scale of the fluctuating barrier is varied. The other is the notion of active transport in a fluctuating environment by so-called ‘ratchet’ mechanisms. Here, nonequilibrium fluctuations combined with spatial anisotropy conspire to generate systematic motion. The fundamental principles of these phenomena are covered, and some motivations for their study are described.

Introduction

The study of the interplay of noise and nonlinear dynamics presents many challenges, and interesting phenomena and insights appear even in onedimensional (1D) systems. Examples include Kramers’ fundamental theory of the Arrhenius temperature dependence of activated rate processes, Landauer's further insights into the role of multiplicative noise, and the theory of noise-induced transitions. This chapter reviews more recent developments which go beyond those studies in that the characteristic time scale of the fluctuations plays a major role in the dynamics of the system, whereas the phenomena in are fundamentally white-noise effects. Specifically, the two effects to be described in this chapter are the phenomena of ‘resonant activation’ and transport in ‘stochastic ratchets’.

Resonant activation is a generalization of Kramers’ model of activation over a potential barrier to the situation where the barrier itself is fluctuating randomly.